Topological phase entanglements of membrane solitons in division algebra sigma models with a Hopf term

ANNALS

OF PHYSICS

193, 419471

(1989)

Topological Phase Entanglements of Membrane Solitons in Division Algebra Sigma Models with a Hopf Term* CHIA-HSIUNG

Virginia

Instituie Polytechnic

AND WJNKEON NAM

TZE

of High Institute

Energy Physics. Physics Department, and State Unit~ersiry, Blackshurg.

Received

January

I’irginia

24061

31, 1989

Exploiting the unique connection between the division algebras of the complex numbers (C), quaternions (H), octonions (0) and the essential Hopf maps S’” ’ -+ S” with n = 2,4.X, we study S” 2-membrane solitons in three D-dimensional KP( 1) u-models with a Hopf term. (D, K) = (3, C), (7, H), and (15, a). We present a comprehensive analysis of their topological phase entanglements. Extending Polyakov’s approach to Fermi-Bose transmutations to higher dimensions, we detail a geometric regularization of Gauss’ linking coefficient, its connections to the self-linking, twisting, writhing numbers of the Feynman paths of the solitons in their thin membrane limit. Alternative forms of the Hopf invariant show the latter as an Aharonov-Bohm-Berry phase of topologically massive, rank (n - 1) antisymmetric tensor U( 1) gauge fields coupled to the S”-2-membranes. Via a K-bundle formulation of the dynamics of electrically and magnetically charged extended objects these phases are shown to induce a dyon-like structure on these membranes. We briefly discuss the connections to harmonic mappings, higher dimensional monopoles and instantons. We point out the relevance of the Gauss-Bonnet-Chern theorem on the connection between spin and statistics. By way of the topology of the infinite groups of sphere mappings S” -+ S”, n = 2.4. 8, we also analyze the implications of the Hopf phases on the fractional spin and statistics of the membranes. (

1989 Academic

Press. Inc.

I. BACKGROUND

AND

SUMMARY

The possible existence of quantum excitations bearing fractional spins and intermediate statistics had attracted little attention [l]. While a few field theories supporting such objects in D < 3 spacetime had been known for some time [2], their potential testing grounds, actual physical systems, were only available recently in the laboratory. In large measure the current interest in these exotics states has been stimulated by attempts to understand the fractional quantum Hall effect [3] and high T, superconductivity [4-51. Against appearance, physics in spatial dimensions less than three is in some * Work

supported

by the U.S. DOE

under

Grant

DE-AS05-80ERl0713.

419 0003-4916/89

$7.50

Copyright C‘ 1989 by Academic Press, Inc All rights of reproductmn ,n any form reserved

420

TZE

AND

NAM

respect richer than that in higher dimensions. In recent years D < 3 field systems endowed with a manifest or hidden gauge symmetry have been widely studied as they were found to admit states of fractional spins and exotic statistics [6-151. This special feature of a “planiverse” [16] is already clear at the level of its rotation group SO(2) x S’, the unit circle, which is infinitely connected, i.e., nI(SO(2) % S’) = 2, where 2 is the additive group of the integers. Thus, let us consider the unitary irreducible representations (UIR) of the universal covering group B = R3 x SO(2, 1) of the D = 3 Poincare group P [ 17, 151. Topologically, we have SO(2, 1)~s’ x R2 hence lr,(SO(2, l))%Z. Since we have SO(2, 1)x Rx R2, the little group is isomorphic to R, the additive group of the real line, and it follows that the spin s, which along with the mass m labels the UZR’s of P, assumes a continuum of real values. Kinematically, the arbitrary value of the spin is unique to three spacetime dimensions. In contrast to SO(2), SO(n) (n # 2) is doubly connected; ni(SO(n)) z Z2, the cyclic group of order 2. This latter property is the familiar kinematical reason for the existence in spatial dimensions d> 3 of only half-integral spin fermions and integral spin bosons. Moreover, the Wigner-Bargman analysis of the D > 3 Poincare group [17], rigorous studies in local quantum field theories, algebraic [ 181, and otherwise [ 191, as well as all the particles observed thus far, have upheld Pauli’s spin and statistics connection [2]. So if deviations from this norm are possible in higher odd spacetime dimensions, they may conceivably arise from dynamically induced effects among extended, topological objects, ones which cannot be created by local field operators. Even in three spacetime dimensions, though kinematically allowed, solitons with intermediate spin and statistics are only known to occur through the long range “magnetic” interactions of a hidden gauge field, induced by an added Hopf-Chern-Simons action. This greater flexibility in spin and statistics reflects the multi-connectedness of the configuration space f of the solitons [2]. Specifically it is embodied in the relative phases of the amplitudes for homotopically inequivalent paths in r, phases which are determined by the added Chern-Simons action. We shall refer generally to these effects as “Hopf phase entanglements,” a terminology appropriately borrowed from Wheeler’s global characterization of spinors [20] and from the statistical physics of polymer chains

WIAs common sense suggests, specific candidate theories are to be found among the of the o-model with a Hopf term [7], the model par excellence admitting exotic spin and statistics. In this comprehensive paper we endeavor to analyze two such D > 3 field systems with nontrivial Hopf phase entanglements. Their uniqueness and overall structure were only sketched in Ref. [22] in connection with the ribbon approach [22,23] to Polyakov regularization [24]. By exploiting the salient features of Hopf fibrations, we set up the proper framework to address two key questions: (a) the higher dimensional counterparts of fractional spin and statistics of solitonic excitations, (b) the physical significance of such Hopf entanglements. For notational definitions and future comparison, we recall the main aspects of D > 3 counterparts

TOPOLOGICAL

the CP(l)( ~2:s’) o-model action reads

PHASE

421

ENTANGLEMENTS

with a Hopf term [7-151.

In its most telling form, the

A= d3X ,D,z,2+ & EPYiA,,Fyi + A,P 1 [

1

with 0 < 8 d 7~.The chiral field ZT = (Z,, Z,), subject to the constraint IZI’ = 1, is a two component complex spinor; it parametrizes a 3-sphere S3. D,, = ii,+ iA,, denotes the covariant derivative, with as composite U( 1) gauge field A,, = iZ+ii,Z. F,,, = 8, A,, - d,.A,‘ is the associated field strength, and J,, = -(i/2x) Ed,.,.D”ZD’Z is the conserved topological current. That (1.1) admits exact static multi-S2-solitons, the Skyrmions, is well known [25]. Following the terminology of Goldhaber et al. [13], we call the third term in (1.1) the Aharonov-Bohm term. Linear in time derivative in A,, the second (Hopf) term, written as a Chern-Simons action, has been recognized as a WessZumino-Novikov-Witten (WZNW) action [lo] with an unquantized coefficient 0. As it may be converted into the form S,=

-$f=s2ds.B.

(1.2)

where B =V x A, this Hopf term acts as a magnetic flux, emanating from a configuration space Dirac monopole. Like in the D = 4 case of the Skyrmion [26], exp(iSi,) is akin to an Aharonov-Bohm-Berry phase, dictating the spin and statistics of the solitons. One of the main objectives of this work is to see exactly how this Berry phase connection generalizes in D > 3 hypercomplex projective a-models augmented with a Hopf term. While the action density in Eq. (1.1) could be expressed in terms of the standard unit vector n = ZtaZe S2, the Hopf density would be a nonlocal functional in the field n and its derivatives [lo]. This nonlocality underscores an essential global property of the system (1.1) i.e., its singularity free action is actually not valued in the field space S2 but in its covering space, the complex Hopf bundle space S3 with base space S2 and fiber S ‘, which gives the hidden U( 1) gauge group of the field A,. More details on this bundle structure will be presented later. In the context of (1.1) in a path integral approach with the choice of H = n, Polyakov [24] showed that the Z-quanta, in their pointlike limiz, obey a Dirac equation. Namely the spin zero bosons at large momenta turn into spin + fermions at small momenta. For any 8 (0 < 8 < 7c), one is dealing with the more general quantum field theory of anyons [6], carrying fractional spin and intermediate statistics. While only confirming previous results [7-153, Ref. [24] nevertheless casts the Hopf phase interactions of the solitons in a geometrical framework which is physically revealing and mathematically tractable. Readily generalized to higher dimensions [22], this approach points the way to at least two exciting areas of 595’193,‘?-I2

422

TZE

AND

NAM

investigations. They are: (1) the construction of new, purely bosonic path integrals for (interacting) spinning point-like [27] and extended objects [28] and (2) the study of D > 3 Hopf phase entanglements, of higher dimensional exotic spin and statistics connections. Two notable features of Ref. [24] are: (a) Upon regularization, Sn for a Feynman path P of a point soliton is given by the total geometric torsion of the space curve P in S3. (b) It also explicitly assumes the identity of a Berry phase [29]. Specifically it is given by the action integral for the interaction of a charged particle and a Dirac monopole. However, as remarked recently by several authors [13], there is a subtle difference with the usual electrodynamics as one is working with a topologically massive Chern-Simons QED. Consequently there is an important physical effect due to the charge screening coming from the massive nature of the gauge quanta. Though the latter give rise to a short-range magnetic field, the magnetic potential is long range. In computing from (l.l), which corresponds to the e -+ co (e = electric charge) limit of a fully gauged CP( 1)-model with a ChernSimons term [S, 131, the statistical phase factor arising from the interchange of two solitons, both the Aharonov-Bohm term and the Chern-Simons term contribute, the latter giving exactly minus one-half of the phase due to the first, i.e., (1.2). Namely due the magnetic field strength-charge density coupling [7] arising from the Hopf-Chern-Simons action, the statistical phase is half of the more familiar Aharonov-Bohm phase coming from the interchange of two dyons or cyons. It follows that, instead of the usual eg = n/2 (n E 2) quantization rule of the MaxwellDirac electrodynamics of dyons, we encounter the peculiar relation eg=n/4. Since all these features will have their proper generalizations in higher dimensions, we shall review them in some detail next. A consequence of the point soliton approximation, the entity requiring regularization in Ref. l-241 is the double integral (1.3)

in the limit where the two smooth closed 3-space curves C, and C, coincide, namely C, = C, = P, the worldline of the point soliton. Were the above curves disjoint, ZG would just be the linking coefficient Lk(a, 8) for C, and C, [30], a topological invariant. The magnitude of Lk measures the number of times C, is linked through the loop of C,. Its sign depends on the orientation assigned to the curves, the components of an oriented link. Lk can be defined [31] by projecting the two oriented curves on a plane (Fig. la). To the projected image, we then assign an index E= +l if the priority of the overpass is from the right and an index E = -1 otherwise. Then Lk(a,P)=l

1 PpanB

E(P),

(1.4)

TOPOLOGICAL

PHASE

423

ENTANGLEMENTS

b Lk(cx,P) = 1

a

CD

Lk(a,@=-1

FIG. 1. (a) We can assign crossing with given signs of + 1 as shown here. Given two components r and /3. let n n p denote the set of crossings of the components a with the component fi. (Thus it does not include self crossings of a or ,8.) Then the linking number is defined to be Lk( a, b) = f I,,. lci B c( p ). (b) Simple examples of links. (c) The Whitehead link is a famous example of a nontrivial link (named after J. H. C. Whitehead) which has zero linking number: Lk(a, fl) = l/2( 1 + I - 1 - I ) = 0.

where o! n /I denotes the set of crossings of c1with fi. Namely, Lk is one-half the sum of all the indices of the intersections of C, with C,. A few simple illustrations are given in Fig. lb. Clearly the example of the nontrivial Whitehead link (Fig. lc) with Lk = 0 shows Lk to be only a first-order topological invariant. Higher order invariants such as the Alexander, Jones [32], and other polynomials [33] are needed to distinguish and classify knots and links [31] in 3-space. There is an alternative, physicist’s derivation [34] of the integer value of ( 1.3). By Stokes’ theorem, (1.3) also reads

for disjoint C, and C,. We obtain ZG(C1, C,) = Lk(a, /3) = an integer n after a combined use of the relation V x V = V(V. ) - V2, Green’s lemma

s

V

-= dy y’lx-Yl

-

s

dy.(x-y) lx-YIP

= -

5

dS.VxV-

1 Jx-y,=O

(1.6)

424

TZE AND

NAM

along with v2

(1IX-Y1 >=

-47&53(x

- y)

(1.7)

and

s P

dS, . dy d3(x - y) = n.

(1.8)

n is the algebraic number of times the curve y, C,, intersects the surface S, with C, as its perimeter. Stokes’ theorem also yields yet another revealing form [30] of (1.3), Z, = (l/471) J dQ(M,), where Q(y) is the solid angle subtended by C, at the point y of C,. With 47rZ, being the variation D as y runs along C,, ZG measures the algebraic number of loops of one curve around the other. It is an invariant under (isotopic) deformations of C, and C,, which entail no intersections of these curves. Widely applied in mathematics [30], Zc, and its relative, the Gauss map, along with their higher dimensional generalizations [36], are key elements in our work. We recall that the linking coefficient Zo, which marks the birth of knot theory, was discovered by Gauss [37] during his investigations of electromagnetism. To exhibit this abelian gauge field connection we rewrite (1.3) as

10= 4 A&) cm

dx:,

P = 192, 3,

(1.9)

where (1.10) So A,(x) is identical to the potential due to a closed magnetic vortex line C,. By the Biot-Savart law A,(x) can also be interpreted as the total magnetic field produced at x by a steady unit electric current flowing along C,. (1.9) is then the work done by this field on a unit Dirac monopole making one circuit around C,

C381.

Finally, ZG embodies a local gauge symmetry [39], for under an isotopic deformation C, --) C,., ZG + Z& = ZG , there arises jointly the transformation A&,)

+ A;(&)

= A,(&)

+ a,Q(x,)

(1.11)

a”A, = 0,

where Sz is the solid angle subtended by the curve (C, - C,,) at x. We will subsequently show that the above picture admits higher dimensional antisymmetric tensor gaugefield (ATGF) generalizations.

TOPOLOGICAL

PHASE

ENTANGLEMENTS

425

As noted and solved by Polyakov [24], the problem with formula (1.3) lies in its region of integration: if we set C, = C, = P, (1.3) is at first sight an undetermined quantity. This very problem had been encountered and left unresolved by Regge and Rasetti [40] in their quantum knot formulation of superfluid helium. In Section II, we will recall Polyakov’s regularization of ( 1.3). He found the regularized k+, say ZRegy to be given by the total torsion T(P) of the curve P. As with anomalies, this regularization has a deeper topological and geometrical basis [22]; exactly such a procedure had been applied to the Gauss linking coefficient by mathematicians [35-361 not long ago. It led to an alternative formulation of the Gauss-Bonnet-Chern theorem for Riemanian manifolds and to interesting theorems in the geometry of submanifolds. Some of these theorems have found applications in the statistical mechanics of polymer chains [41], in the molecular biology of supercoiled DNA molecules [42-441, and more recently in quantum held theory [22]. Next we recall the relevant aspects of this mathematical development. Seeking higher order knot invariants, Calugareanu [35] considered the limiting form of the Gauss integral (1.3), as one space curve C, approaches another, C,. He discovered in this process a new topological invariant, the self-linking number SL = Lk(a -+ 0) for a simple closed space ribbon. Specifically, SL is the linking number of the curve C, with a twin curve C, moved an infinitesimally small distance along the principle normal vector field to C,. Notably there exists the “conservation law”: SL = T+ W, i.e., SL, a topological invariant, is given as the algebraic sum of two differential geometric characteristics of a closed ribbon, its total torsion or twisting number T, and its writhing number W [42]. While both T and W, which are metrical properties [43], can take a continuum of values, their sum SL must be an integer. Now, in the differential geometry of surfaces, there is a star theorem, the Gauss-Bonnet theorem [30], which also forms a bridge between entities defined solely in terms of topology such as the Euler characteristics x of a closed surface and metrical entities defined purely in terms of distances and angles such as total Gaussian curvature K for that surface. Its reads K= 27~~. It turns out [36,42] that the relation SL = T+ W is just a special case of the Gauss Bonnet formula! To give an intuitive feeling for the above notions [42-44] while deferring analytic details till Section II, we define a ribbon P to be a smooth embedding in 3-space off: S’ x ( - 1, 1) -+ R3 and as its axis f(S’ x (0 )). the line running along its center. Then the total torsion T(P) measures the amount P twists about its axis while the writhing number W(P) measures the nonplanarity of this axis. Figures 2a,b illustrate in a specific case of a ribbon with Sl. = 1 the topological equivalence (or isotopy) of two configurations, one (Fig. 2a) with pure twisting T= 0, W= 1, the other (Fig. 2b) with pure writhing W = 1, T = 0. Figure 2c shows the generic situation of a more complex link where SL = T+ W. In everyday life, this relation between the twisting and writhing of a ribbon is most apparent in the case of a coiled phone cord. When the cord is unstressed with its axis curling as a helix in space, its writhing is large while its twist is small, However, in its stretched

426

TZE AND NAM

3-

a)Writhhg

-a

b) Twisting

FIG. 2. These figures illustrate the familiar phenomenon that two contigurations, one with writhing and the other with twisting are topologically the same, thus giving rise to same linking numbers. Note that when we evaluate the linking number from the writhing case we do not count the self-crossings. A general ribbon will consist of writhings and twists and counting them gives the linking number immediately. However, for certain cases it is not clear whether it is really twisting or writhing. Then we get noninteger contributions for the writhing number and the twist.

state where the cord axis is almost straight, it is the twist which is large and the writhing small. In a higher dimensional world, by generalizing the results of Polyakov and of Grunberg ef al. [23), we will argue that the equality W = -T modulo I 136,221 provides the needed relation between fractional statistics and spin for topological Nambu-Goto membranes. In his thesis, White [36] generalized Calugareanu’s formula to higher dimensional spaces. In Ref. [22], by introducing the notion of self-linkage of a spacetime ribbon [41], one of us (H.C.T.) adapted White’s results to the geometric extension of Polyakov’s regularization and to the analysis of Hopf phase entanglements in higher dimensions. Within the point soliton approximation, the regularized Hopf phase xW in model (1.1) was also identified [22] with the global action of Balachandran et al. [453 for the interaction between a charge e and a Dirac monopole g, albeit obeying the quantization condition eg = i. As alluded to previously, this modification of the Dirac quantization rule is a reflection of charge screening in topologically massive QED [13]. It came about as we corrected for a multiplicative factor of $ missing in the Polyakov expression of the Hopf phase in Refs. [24, 22, 281. The crucial role played in a Chern-Simons electrodynamics by the complex Hopf-Dirac monopole bundle [46] was underscored [22,28]. The latter identification combined with the spacetime ribbon picture shows [22] how the Hopf phase effectively dresses the D = 3 Skyrmion into a “dyonic” bound complex of a magnetic vortex line twisting and writhing about the charged pointlike Skyrmion, which accounts for the phenomenon of Bose-Fermi transmutations. Not yet seen in Minkowski spacetime, magnetic monopoles have shown up in the

TOPOLOGICAL

PHASE

ENTANGLEMENTS

427

configuration spaces of classical, quantum mechanical systems and in quantum field theory. This is so since the dynamics of these systems [47] are isomorphic to that of a charged particle in a Dirac monopole field; namely, their actions are multiplevalued functionals on the space of closed curves on the 2-sphere. Some early examples [48] are Kirchhoff’s equations for a rigid body in an incompressible fluid and Leggett’s equation for the spin dynamics in superfluid 3He in its A-phase [49]. Recent realizations are Berry’s non-integrable phases in diatomic physics [29], in nonlinear optics [50], in theories of the quantum Hall effect [3], and the WZNW term in effective chiral soliton models for QCD [Sl, 261. Most striking perhaps is the effect of such a phase on the spin and statistics structure of solitonic excitations. A case in point is provided by the semi-classical quantization of the D = 4 chiral SU(3) soliton; here the WZNW term simplifies to the action for the interaction of a charged particle with a U,( 1) monopole in chiral field space [26]. It supplies the Skyrmion wave functional with a projective Berry-phase enforcing its quantization as a fermion (if the number of color is odd). As we have witnessed in the CP( 1) model (1.1) in consequence of the added Hopf-Chern-Simons action, a more exotic situation [7-133 still obtains in odd dimensional spacetimes. The phenomena above are most easily understood in a unified way after they have been cast as Aharonov-Bohm effects of topologically massive Abelian [47] and non-Abelian gauge fields [52] in configuration spaces. As in the CP( 1) a-model [7-151, the primary task is to bring forth the hidden U( 1) gauge field structure. In higher dimensions, only a few studies [53, 22, 28 J have been undertaken in the search for nontrivial phase interactions due to Chern-Simons terms of Abelian tensor gauge fields coupled to extended geometric sources [54, 551. As part of this endeavor, our paper elaborates on the theme of two previous communications [22, 281, the Hopf phase entanglements of relativistic membranes in two specific models. By combining White’s theorems [36] on self-linking of manifolds with Adams’ theorem [56] on Hopf mappings with invariant one, Ref. [22] singled out, besides the D = 3 CP( 1) o-model, only two others, the D = 7 HP( 1) and the D = 15 s2P(l) a-models augmented with a corresponding Hopf term. The latter theories admitting topological S4 -+ S4 and S* + S8 solitonic membranes are uniquely tied to the existence of the quaternion and octonion algebras in 4 and 8 dimensions, respectively. In the present paper we implement the scenario laid down in Ref. [22]. TO render our highly nonlinear field theories tractable we are led to consider the London-Nielsen-Olesen strong coupling limit [57, 581, where the solitons are infinitesimally thin, geometrical membranes. Due to the geometric nonlinearity and regularization of these KP( 1) models, this long wave length approximation preserves the topological phase entanglements that we seek. In such a limit, the physics of the problem translate into deep theorems of geometric topology and differential geometry. Thus, by applying the work of White [36], we show how the connection between statistics and spin of the Nambu-Goto membranes is a particular realization of the Gauss-Bonnet-Chern theorem. We also make contact with the generalized Aharonov-Bohm effects of p-form electrodynamics [55, 281 augmented

428

TZE

AND

NAM

by corresponding Abelian Chern-Simons terms and their accompanying subtle topological effects [lo]. They are systems of electric and magnetic membrane singularities in D = 7 and 15 spacetimes interacting via their Abelian ATGFs. In particular they are electric-magnetically dual in D + 1 dimensions. We extend our generalization [28] of the global formulation of Balachandran et al. [45] to the Chern-Simons electrodynamics of the membranes. Then by exploiting the topology of the configuration space, and the associated complex, quaternionic and octonionic Klhlerian symplectic structures we cast the Hopf phases as Aharonov-Bohm-Berry phases of the division algebra ATGFs and discuss their implications for exotic spin and statistics. The rest of our paper is organized as follows: In Section II, the Polyakov’s regularization [24] of Gauss linking number is cast in a geometric framework and generalized in terms of the self-linking, the twisting, and the writhing number of higher dimensional spacetime ribbons. In Section III, a unified division algebra construction of Hopf mappings is presented. We give a compact construction of the Hopf invariant and its alternative and physically revealing forms. Applying Adams’ celebrated theorem relating the four division algebras to mappings of Hopf invariant one, we single out, then discuss the relevant solitonic aspects of the two D-dimensional KP(1) u-models with a Hopf invariant (D, K) = (7, H) and (15, a). After a brief analysis of the associated harmonic mappings and the soliton stability problem, we proceed to show that these models admit Y-membrane solitons (n = 2 and 6, respectively) as topological defects, in accordance with the Toulouse-Kleman classification [SS]. In Section IV, we exploit the formalism of Umezawa et al. [60] for extended objects with topological singularities. These topological singularities are the sources of effective Kalb-Ramond fields [61] obeying not a Maxwellian but a ChernSimons-Hopf invariant or, more generally, a mixed Maxwellian-Chern-Simons action [62]. We focus on their associated conserved currents in the limit where the solitons are Nambu-Goto membranes. Applying the formalisms of previous sections, the Hopf phases of the membranes are regularized into either the generalized torsions or writhing numbers of hyper-ribbons in spacetime or Aharonov-Bohm phases of composite U(1) topological gauge fields of rank 1, 3, and 7, respectively. Mainly we shall argue that the equivalence between torsion and writhing provides the suitable generalization of the connection between fractional spin and statistics for the higher dimensional Nambo-Goto membranes. By way of our K-Hopf bundle path space generalization [28] of the Balachandran et al’s [45] charge-monopole action involving electric and magnetic membranes arising in p-form electrodynamics with a Chern-Simons term, we show explicitly that the Hopf phase interactions induce a “dyonic” structure on the membranes. In Section V, we discuss the physical interpretation of the phase entanglements of the membranes. In connection with the question of spin and statistics of the membranes and with an eye on future developments, we initiate a discussion on the topology and current algebra representation theory of the infinite loop groups of sphere mappings. We also point out the relevance of our membranes to supermembrane compactitication, quantum gravity, and condensed matter physics.

TOPOLOGICALPHASEENTANGLEMENTS II. SELF-LINKING,

429

TWISTING, AND WRITHING OF FEYNMAN PATHS

A. Geometry and Physics of Polyakov’s Regularization Among theories of high T,. superconductivity, the resonating valence bond approach [4] has uncovered new nonperturbative aspects of the d = 2 Heisenberg antiferromagnet. One such feature [S] is that magnons may be fermions, i.e., they can undergo a Bose-Fermi transmutation. Seen in the context of the D = 3 CP( 1) sigma model with a Hopf term (1.1) this phenomenon is but a special case of the well-studied intermediate spin and statistics connection [615]. The dynamics of magnons can be modelled by the continuum action (1.1) with 0 = n. That model can have a twofold interpretation. In the first viewpoint, ( 1.1), is the Wilczek-Zee action [7-l 51. Its hidden U( 1) gauge invariance comes from the phase invariance Z -+ exp(ia(x))Z. Its Hopf term also reads as SH = -5 J d’x A,,J”, as an interaction between the gauge field A, = iZtd,LZ and the conserved topological current

where n” = Z +a”Z E S’. With J, so normalized, the integral topological charge Q = j d2.x J, labels the homotopy classes of rc2(S2) = Z, of the model’s S * -+ S2 solitons ~241. Alternatively [S], (1.1) could model the infrared truncation or strong coupling e + co limit [ 131 of a fully gauged CP( 1) model with a Hopf term, i.e., of a topologically massive electrodynamics of the Z-quanta. At small distances, Z and A, are independent spin 0 and spin 1 fields. At large distances they are related asymptotically by A, = iZ +8,Z. Such a correlation derives from the requirement of vanishing covariant derivative D,Z, i.e., of the first term in ( 1.1) as /x”J -+ a. This condition implies the S3-compactification of Euclidean spacetime R3, wherefore the relevance of the Hopf maps Z from S3 to S2, the field space. Quadratic in the field derivatives, the usual Maxwellian action, dominant at shorter distances, has been dropped at large distances leaving as sole kinetic term the Chern-Simons characteristic or Hopf invariant y(Z), up to a suitable normalization. Irrespective of the interpretation, the parametric 8 angle is a priori not fixed by gauge invariance. One typically chooses [ 5, 241, 0 = rr in ( 1.l ), a value which can be derived if the Hopf-Chern-Simons term is radiatively generated [8] from an underlying theory of Dirac fermions. As in the path integral approach of Ref. [24], we shall focus on the nontrivial infrared asymptotics induced by the Hopf term on Green’s functions and on the spectrum of states of the system (1.1). For sufficiently large mutual separations, the S2-Skyrmions can be considered as pointlike topological singularities. This geometric limit has commonly been used to describe defects in condensed matter, e.g., in Popov’s analysis [63] of effects of vortices on the infrared Green’s functions’ asymptotics of superfluid helium. Such an approximation has the practical merit of drastically reducing the number of degrees of freedom from an infinite dimensional configuration field space of a soliton down

430

TZE

AND

NAM

to the 3-dimensional physical spacetime of the quantum mechanics of a point particle (or a higher dimensional geometrical object) in a background gauge field [24,22,28]. For a point soliton [24] the partition function reads all closed

paths

~+“Lip)(expi(~~A”dr,)).

Z=

(2.2)

P stands for a closed Feynman path, a non-self-intersecting 3-spacetime curve in the configuration space S 3; L(P) denotes its length. The first exponential factor is the relativistic Schwinger-Nambu action of the worldline P of a free Z-particle of mass m. The other exponential factor provides a modulating dynamical phase induced by the low frequency part of A,; the functional averaging ( . ..) is done w.r.t. the Hopf action. Being Gaussian, this “infrared factor” is readily calculated. For arbitrary 8 and by way of a singular source qr we set j dx” A,(x) = j d3x q”(x) A,(x) [63]; then upon cancellation of this linear functional by a standard field shift A” + A’p + Afi we obtain [22] Q(P)=

( exp ( i$ p A”d %))=exp{

-iJJp~~(x)D.,(X-y)Il*(y)d3xdPj,

(2.3)

where D,, = i(n/28) E~,,~((x’ - $)/lx - y13) is the A, field propagator. Then for 8 = n, Q(P) = exp(irrZG) with Zc, given by (1.3). Patterning after Refs. [53, 133, an equivalent derivation of the phase @(P) proceeds by the direct integration of the equations of motion. In the point soliton limit Q(P) is given by exponentiating an effective action; i.e.,

where SO is the free relativistic action of a point particle of mass m, N a suitable normalization. The effective conserved current f,(x) is that of a point source of unit charge

4 (r)

J,(x) = J dr S3(x - y(z)) -+

The effective action in (2.4) thus linearizes the field problem (2.3) by partitioning the nonlinear field Z into a singular charged point source and its gauge field A,. From (2.4) the key equation for J, is J,(x)=

-&p”pa’AP

TOPOLOGICAL

which upon substitution

PHASE

ENTANGLEMENTS

431

in (2.3) gives (2.7)

So for a giuen current J,, namely (2.4) A, can be exactly solved through “Ampere equation” (2.6) in the Lorentz gauge 8”A, = 0,

the

(2.8)

Upon substitution of this A, and JG in (2.6), we recover the result (2.3) of Q(P) = exp(inZ,). While with the point soliton approximation we could solve for the statistical [ 131 phase factor Q(P), we did so at a price. Though the integrand in (2.2) is that of Gauss’ linking formula (1.3), the integration is not over two disjoinr closed curves but over one and the same curve. I(P), which represents the net action of the Aharonov-Bohm and the Chern-Simons terms in (1.1) on a pointlike charged soliton, is thus undetermined. Yet from analyses [7-151 of the field theory (1.1 ), the infrared asymptotics of the Z-field embodied in the Hopf action are nontrivial. To remedy this unsatisfactory situation, a proper regularization of (1.3) is therefore necessary. Besides, physics tells us that isotopically equivalent closed spacetime paths P of the soliton in interaction with the gauge field A, should generally give rise to different values of Z(P). So we expect IReg to both assume any value and still be very closely related to Gauss’ linking invariant. This turns out to be indeed the case. In this regularization, Polyakov [24] found I(P) to be given by the total torsion or twist T(P) [64] of the curve P in spacetime with (2.9)

s denotes the arc length and n the principal normal vector to P at the point x(s). We recall that while a space curve P does not generally lie in one plane, it is possible to associate to each of its point p a plane I7, the osculating plane, which lies closer to the curve at p than any other plane. Z7 is spanned by the tangent t and the principal normal n (with n . t = 0) to the curve at p. The vector field n plays the same role for a space curve as the ordinary unique normal for a plane curve. The torsion r(s), the integrand in (2.9), is the rate of change of direction of the osculating plane (see Fig. 3). Its sign depends on the side toward which 17 turns as it moves along P. One can visualize the osculating plane as a fan blade with two lines, t and n, drawn on it. At each instant t is turning in the direction of n at the rate determined by the curvature K(S), while ZZ rotates around t with a speed and direction determined by the torsion r(s).

432

TZE AND NAM

king

Plane

FIG. 3. A space curve in R’ and the osculating plane at a given point is defined by the tangent and normal vectors of the space curve at the point.

Polyakov’s regulating procedure [24] leading to Q(P) = exp( - ircT(P)) consists in replacing the delta function in the alternative expression for 47rZo of Jp dxp s.rp d’y,S(x - JJ) by the Gaussian 6,(x - v) = (2~s))~‘~ exp( - (x - y( */s). Since the dominant contribution of the surface integral now comes from an infinitesimal strip L”,, this procedure effectively turns a spacetime curve into a ribbon [22]. Indeed, if instead of only one curve P, we have a ribbon with its two edges, the first P and the other edge P,, a separate twin sister curve, which nowhere intersects P and is located a minute distance E< 1 away from P. Being disjoint these two curves can therefore be linked and unlinked, like the two strands of a circular supercoiled DNA molecule [43, 441. In fact, as we will see, behind this straightforward lifting of undeterminacy lies a mathematically and physically revealing topological structure. This “loop-splitting” regularization of the Gauss integral, Eq. (1.3), as C, --) C,, was recognized [22] as precisely that of Calugareanu [35]. To be more specific, consider a smooth, closed, simple (i.e., without self-intersections) space curve P in R3 (or S3), defined by its position vector x(t), 0 < t d L, x(0)=x(L), of total length L and with length parameter t. Next we construct a closed and simple ribbon (x(t), n) whose edges are two disjoint curves, P with x(t) and P, with x,(t) = x(t) + m(t). P, is moved a small distance E from P along II, a smoothly varying unit vector everywhere normal to P. For small enough separation E < .Q, which always exists iff the curvature is nowhere zero ((x”(t)1 < const with the superscript prime denoting t-differentiation), the linking number of the two curves no longer changes. To these two disjoint curves, we can apply Gauss’ linking formula, then let E + 0. This E independent limit is perfectly well defined [34], it is a topological invariant, the self-linking number SL(P, P,) of P, denoted henceforth by SL. A pedestrian derivation [41] of Calugareanu’s formula for SL consists in evaluating the limiting form of the double integral Z(C,+C2)=G

1 L L dt EpJXP’(S) + d’(s)][x”(t) ds s0 s0 Ix(t) -x(s)

-x”(s) - &II”(S)] x”‘(t) - &n(s)13 (2.10)

as E + 0. The interval of integration is subdivided into two regions: an infinitesimal problem neighborhood (s - 6 < t < s + 6) of the point s and the remaining segment

TOPOLOGICAL

PHASE

433

ENTANGLEMENTS

[0, L] where the integrand in (2.10) is regular. It is readily checked that only MO finite contributions survive as E+ 0. In the b-neighborhood one sets Y’(t) = x~(s) + (t-s) xP’(s) and x”‘(t) = x@(s) and ignores terms small compared to E. The t-integration in (2.10) then gives T=

-&{oLd

[n(~)xx’(~)l-n~(s)(E2

j:I:

rcl_g;f:E2,1~2)=~b;

dx’ . [n x n’]. P

(2.11 ) If n(t) is the principal normal of the curve x(t), T(P) is the E-independent, normalized integrated twist of P [64]. In view of subsequent D > 3 generalizations we recall some specifics of the geometry of space curves: With x: P + R’ as our embedded curve in 3-space, we choose a Serret-Frenet moving frame (e, , e,, e3) at each point of s E P. The triad of orthonormal vectors are e, = ax/&, the unit tangent vector to P, s being the arc length parameter; e2 =n, the unit normal vector field along P and e, =e, x e,. If o-o,~~z $eiikrrjke, is the rotation rate of the Frenet frame w.r.t. s, de,/& = o x ei (i= 1, 2, 3) then the component wi = rrZ3= de, .e, is the twist of the ribbon at each point of the curve x. The total twist (or torsion) T of e2 along P is (l/271) jP w, ds; its sign depends on the orientation of the coordinate axes. If the curvature K = Ix”(s)1 and T are given as functions of s, then the curve P is uniquely determined up to motions. The remaining finite contribution to (2.10) is called the writhing number [42] or the writhe of the curve: W=&

jLdl jLds 0

0

(Cx(t)-x(s)l xx’tt)).x’(s) Ix(s) - x(t)13

(2.12)

To writhe means to coil or fold, like a snake or a phone cord. As in (2.11), the integral over the remaining part of the interval is singularity free, so the integration has been extended over the whole [0, L] interval. As 6 + 0 the value of the integral in the (s - 6 < t
434

TZE AND

NAM

27cT is the ratio of the integrated angle of twist along the length of the ribbon. It is also invariant under rigid motions and dilatations. Though for a simple plane ribbon, T= SL, in general T is not an integer. Its value is not invariant under arbitrary deformations and so varies with the exact spatial shape of the ribbon. We note that while the above given definition of the total twist T depends on the choice of the vector field e,, its value modulo 1 and hence exp( -inT) do not. To see this feature [65], let us take a new triad (el, n,, n3) such that n3 = e, x n, with n2 = cos Be, + sin ee,,

n3 = -sin 8e, + cos Be,

(2.13)

and jtZ3 = dn, . n3. Then since it,, = 7r23+ de, it follows that

1I - 1s

G pn23

--

2n p=23=G

1

i Pde,

(2.14)

where the RHS, which counts the number of turns n2 makes around e2, is clearly an integer. With the axis being the line running along the center of the closed ribbon, then W is a property of this axis since the vector n(t) is absent in (2.12). Like T, W is invariant under rigid motions and dilatations. Its value is independent of the direction of travel along the axis and is a continuous function of the exact spatial shape of the curve. Its sign is changed by reflection in a plane or a sphere if the ribbon avoids the center of the sphere. So W= 0 for any closed curve in a plane or on a sphere; T= SL for a closed ribbon with its axis lying on a plane or on a sphere. Thus W measures the left-right asymmetry, the chirality of a curve. Furthermore, if the ribbon intersects itself the value of W jumps by + 2. As emphasized by Crick [43], T and Ware not topological but rather metrical properties of the ribbon and of its axis, respectively. Coming back to our regularization, with the notion of a spacetime ribbon in mind, a mere comparison of the formal identity of Eqs. (1.3) and (2.12) identifies Polyakov’s Z,(P) as the writhing number W of the path P. Indeed if we are dealing with a ribbon as defined above, then the integral (1.3) with C, = C, is nothing but W(P). In other words the process of framing P or of ribbon creation from two paths P and P, by doubling P along its whole length points to the choice of the axis of the ribbon as the regularized spacetime curve. This identification is consistent with Ref. [24], since W= -T (mod 2). Also it is perhaps to be expected that it is in this chirality W(P) of the path P that the spin of the particle is encoded for a given Feynman path, On the other hand, the more intuitive connection between the torsion of the path of a point particle and its spin has recently been made clear by Polyakov in his 1988 Les Houches lectures [24]. The writhing number W, also known in mathematics [30,36] as the Gauss integral for the map 4: S1 x S ’ --* S 2, is also the element solid angle or the pullback volume 2-form dl2, of the 2-sphere S2 under 4 [36]. Exploiting the invariance of

TOPOLOGICAL

PHASE

ENTANGLEMENTS

435

W under dilatations and selecting for 4, the unit vector-valued map, e(s, u) (e’ = 1 ), a local Frenet frame vector attached to the curve, we can write W=-& a, h, c = (1,2, 3) and

joL ds j,’ du EuhceUa,,eb(?Uec,

a, = a/as, a, = a/au.

A conformally invariant action for the frame field e, W in (2.15) is manifestly a WZNW term. Up to a necessary factor of i, it is Polyakov’s double integral representation (modulo an integer) [24] for the torsion T. In our geometric language, this agreement is rooted in the equality T= - W (mod Z). Upon exponentiation, the solid angle character of W = ( 1/4n) {sl x .Y~dQ,, dQ, = the area element of S2, makes it a Berry phase [29]. To see this relation explicitly we invoke Fuller’s interpretation [42] of W(P), a direct consequence of the Gauss-Bonnet theorem. The latter states that for a closed curve on the unit 2-sphere, the sum of total rotation of a unit tangent to the sphere defined at each point of the curve and the area enclosed by the curve is identically zero modulo 27~. Thus consider as a curve a closed ribbon, the unit tangent t(s) to its axis sweeps out a closed curve on S*, n(s), the unit normal to t(s) is hence tangent to the sphere. So the total rotation of n(s) is 2xT. If Q denotes the solid angle in steradians, then by the stated Gauss-Bonnet theorem: T= - W = -(Q/271) mod 1. From this result and the fact that W jumps by +2 for each selfintersection, Fuller’s theorem follows: For a non self-intersecting ribbon W = Q/2 - 1 mod 2, namely W = Q/2 + n; n = odd integer. So the regularized Hopf phase factor exp( in W) = exp(i(R/2)) ex p( rnz ) is indeed, up a to inconsequential sign, Berry’s phase. This consequence of Fuller’s theorem has independently been noted in Ref. [23]. Solely from the above formulation can we infer that the statistical phase factor Q(P) = exp { in W} is a spin phase factor in disguise? The answer is affirmative. Already at the level of the partition function Z its spinorial character can be deduced merely from the topological content of our regularization. In naively applying the point soliton approximation to a field model with sophisticated topology, a price was extracted, namely the apparently indeterminate answer in (1.3). By integrating out the gauge field, in the process we lost the expected [7, lo] topological effects of the Hopf term, the signature of the “twists” in the field Z due to the long range interaction between the Hopf gauge field, and a Z-particle at low momenta. This lost structure must be recovered in our regularization and must be encoded in the ribbon picture, specifically in the writhing and twisting numbers W, T= - W+ SL of the particle world line. NOW let us consider the connection between spin and statistics. The statistical factor (i.e., the factor arising when we adiabatically interchange two identical objects) yields the phase involving the writhing number. Now by the Gauss-Bonnet theorem (disguised as the equation SL = T + W) the phase becomes that involving the twist, which has to do with the spin of the object [24]. So we see an interesting

436

TZE AND

NAM

role that the Gauss-Bonnet theorem plays in a fundamental physics principle-the relation between spin and statistics! The following physical picture, apparent from the proposed regularization [22], is simple, and consistent with that of Refs. [ 12, 13, 24 J. It has become a common folklore that exotic statistics of D = 3 particles can be achieved by attaching fictitious magnetic vortices and charges to them. This mechanism is precisely what is realized by the Polyakov regularization. Here due to the topological nature of the Hopf action and the geometry of the point soliton approximation, the world line or Feynman path of the bare pointlike Z-particle gets renormalized into a self-linking spacetime ribbon, and the charge gets screened. The net effect of the long-range interactions between the Z-particle with the field A, is to produce an anyon [6] for an arbitrary value of 0 by dressing the path of the charged Z particle, one edge of the ribbon with a fictitious magnetic vortex line, a Dirac monopole propagating along the other edge infinitely close by but never intersecting it by reason of Dirac’s veto. For a given ribbon, the constituent curves consisting of the interlocking of the world line of a charge particle and a magnetic flux line have SL links. The ribbon twists and writhes in spacetime such that SL(P) = T(P) + W(P). The emergence of the self-linking number in the point soliton limit is to be expected from any proper regularization; it mirrors a theorem in homotopy theory due to Hopf [66] equating his invariant to the linking number between two curves. Another way of understanding the physics behind the ribbon picture is the following [13]. As a constituent of the ribbon, the charged particle revolves around the flux line, and there is a phase due to the long range character of the magnetic potential. On the other hand, since the fields are screened due to the massiveness of the gauge quanta, there is no contribution coming from the reciprocal movement of the flux line around the charge. The above interpretation is further confirmed when rt W is recognized [22] as the nowhere singular, global action of Balachandran et al. [45] for the interaction of an electrically charged point particle of charge e* = e/2 with a Dirac monopole g such that the quantization condition e*g = i is satisfied. The reason [ 131 for this has already been given. This configuration space monopole is located at the origin of the 2-sphere. So exp(i7c W) is a Berry’s phase [29] with its associated monopole bundle structure. Recall that while the monopole gauge potential is singular on S2, due to the necessity of a Dirac string, it is perfectly regular on S3. Since the configuration space Q w R3 - (0) x S2 x R, the field of frame e in (2.14) is not globally defined on S2, we must lift the field e from S2 to S3 and thus link up with the complex Hopf bundle S3 + S2, where S ’ = U( 1) is the fiber and S2 is an, orbit. This principal bundle is defined as follows: Let e, be any chosen fixed reference point in Q. A point in PQ M S3 is a spacetime dependent path e(s, U) such e(s, u = 0) = e, and e(s, u = 1) = e(s). These boundary conditions are precisely those of Polyakov [24]. This globalization of the dynamics [28] was automatically achieved by our regularization scheme. Being nonsingular, 7~W is valued not on the path space Q but on its covering PQ x S3, the U( 1) 1-Dirac monopole bundle over S’. In this fashion the topological content of the field Hopf term is fully recovered within the

TOPOLOGICAL

PHASE

437

ENTANGLEMENTS

point soliton approximation. There we obtain the extreme situation where the abstract and the intuitive merge since the configuration space is just physical 3-spacetime with the configuration space monopole turning into a physical space monopole! B. Higher Dimensional

Self-Linking,

Twisting, and Writhing Numbers

For a Polyakovian approach to the D > 3 counterparts of the CP( 1) model ( 1.1). we shall require the higher dimensional generalization of the D = 3 self-linking formula for a ribbon P

SL( P) is the algebraic sum of the writhing number (2.11) (or the Gauss integral for the map of the unit vector frame field e) and the total torsion T= (1/2x) lp T ds. Happily such an extension has been known for some time [36,67]. The interested reader should consult these papers for proofs. We only gather below the relevant theorems and formulae. First we should define the linking coefficient for manifolds. Generalizing the construction in Section IIA, we consider two continuous maps/(M) and g(N) from two smooth, oriented, non-intersecting manifolds M and N, dim(M) = m and dim(N)=n, into R”‘+“+‘. Let S”+” be a unit (m + n)-sphere centered at the origin of R m+n+l and da,,,,, be the pull-back volume form of S”+” under the map e:MxN-+S”+“,

(2.17)

where we associate to each pair of points (m, n) EM x N the unit vector e in R”‘+“+*: e(m, n)= ((g(n)-f(m))/Jg(n)-f(m)\). The degree of this map [29, 361 is then the generalized Gauss linking number of M and N, namely, W-(W,

hWW&N)=~j-

n+M M x N

da,,,,,.

(2.18)

Here Q, (= 2n (n+1)‘2/ZJ(n + 1)/2)) is the volume of S”. We note down the important non-commutative property of L(M, N): L(A4, N) = (-l)“+

*lfn- *’ L(N, M),

(2.19)

which implies, for instance, a vanishing linking number for et,jen dimensional submanifolds M and N. Let the mapping f: M + R2” + ’ be a smooth embedding of M in R2”+ ‘. We then consider two neighboring submanifolds M and N such that N = M x I,, where I,, is a closed, minute real interval [0, EJ. Let v be a nowhere zero, differentiable normal unit vector field on M and E (
438

TZE AND

NAM

are tangent to M at f =f(m), where m E M, and an+ i is along vrc,,,, and a n + *, .... a2,,+ i span the remaining space at f = f(m). Define the entities nU = dai. aj on M and expand e, in the a-basis, i.e., e, = uljaj. In the above notations, the selflinking number for f(M) and f,(M), i.e., of the hyper-ribbon M, is given by (2.20) for n odd (i.e., D = 3, 7, 11, 15, etc.). t* dV, the torsion form of M w.r.t. the vector field u is given by t* dV= O,d

(2.21)

@k 1 .3...(n-2k)2(“+“/2k!

(2.22)

with 1 (n--1)/2 A=-----+ + 1l/2 k;, (-1)” @,EE

a,cq...qn*n+la,

A

...

A ‘Il,+10rn-2k

A

fb->~+,a”_Z+Z

A

..*

A

A.-,a.

(2.23)

and ,4,,=Cr=, nair\ rriB. The numerical tensor E,~.,.~~= +l (-1) if ai . ..a., is an even (odd) permutation of n + 1, .... 2n + 1 and is zero otherwise. The first term on the RHS of (2.20), the Gauss integral for the map f, is the writhing number W of the hyper-ribbon. The other term is the total twisting number or torsion T. In contrast to the 3-dimensional case, the fact that the simple relation SL = T+ W still holds for a higher dimensional ribbon is rather nonintuitive. Since we see from the work of Polyakov [24] (see also Ref. [23]) on spin factor of a point particle that the torsion of its path is intrinsically related to its spin, it is natural for us to infer that White’s formula T= - W (mod 2) provides the vehicle to link the statistical phase to the spin phase for higher dimensional extended objects. Of note also are the duals of the torsion forms (2.21). They first appeared in Chern’s intrinsic proof [68] of the generalized Gauss-Bonnet formula. Indeed White’s work [36] provides a new formulation of that formula. The fact that the conservation law T + W = SL generalizes to higher dimensions is merely a reflection of the generalization due to Chern of the Gauss-Bonnet formula for a Riemannian manifold. We quoted Fuller’s result (1978) [42] that the D = 3 writhing number W=Q/2n + k. This result follows directly from the Gauss-Bonnet theorem. It should generalize [36, 371 in the case of the higher dimensional writhing number in (2.20). Our argument goes as follows. From the Gauss-Bonnet-Chern formula [68, 36, 671, if we denote by 52, the solid angle of S2” (say n = 1, 3,7), we expect to have T+ 28/Q, =integer. The latter may be called the closed-hyperpath theorem [see Abelson and disessa, Ref. [68]], it holds for closed simple S” hypercurves on S2”. The fact that we need the normalization QZ,, is easy to understand when we consider two specific cases of closed hyper-ribbons, one involving T = 0 and the other ribbon from the same curve but with T= 1. The corresponding solid

TOPOLOGICALPHASEENTANGLEMENTS

439

angle should change in units of Q,,,. Thus we obtain W = 252/Q,, + k; k = odd integer. For R2”+ ’ with n even (D = 1, 5, 9, ...). the situation is uninteresting as far as applications to topological phase entanglements are concerned. The reason is that both W and T vanish identically and so the self-linking coefficient SL(f, f,) is zero. For n = 1, formula (2.20) gives back Calugareanu’s formula when v is the principal normal vector field. As applied to physics, (2.20) then defines and relates the twisting and writhing of odd dimensional closed S3-, S5-, S’-... hyper-ribbons in D = 7, 11, 15,... dimensional spacetime, respectively. It provides the needed formula for the D > 3 extension of our regularization and for a connection between spin and statistics, which brings us to the next topic of essential Hopf librations and KP( 1) o-models. III.

MEMBRANES SOLITONS IN THE HP( 1) AND RP( 1) CT-MODELS WITH HOPF TERM

A. Division Algebras and Hopf Fibrations To provide a natural field theory setting for the above generalized twisting and writhing numbers, we seek c-model realizations of the possible higher dimensional Hopf maps S’” ’ + s”. Rather, as we will explain, particular focus will be on two such classes of maps with n = 4 and 8, the quaternionic and the octonionic Hopf liberings. To give an unified treatment of these K-Hopf fibrations, we first exploit the one-to-one connection, first established by Hopf [69], between fibrations of s Z1lm~ ’ -+ S” by great (n - I)-sphere and n-dimensional division algebras K for n = 1, 2, 4 and 8. Consequently, along with some necessary notation, we sum up the defining properties of K [70]. A division algebra is a linear algebra K over the real numbers R. Denote two of its elements by X, .VE A, their multiplicative norm NE R is such that N(Q) = N(s) N(y),

N>O

N(x) = 0 -+ s = 0

(3.1)

with N to be defined shortly. By a theorem of Frobenius [71], every associative division algebra is isomorphic to the real (R), complex numbers (C) or quaternions (H). Moreover, a classical theorem of Hurwitz [72] states that every normed algebras must be isomorphic to R, C, H or the Cayley algebra Q of the octonions. The canonical basis vectors ei (i = 0, 1, ,.., (2” ’ - 1)) of these 2” ~~‘-dimensional algebras K for m = 1, 2, 3,4 satisfy the multiplication law: eOeO= e,,

eOe, = eieO = P,

e,e,= -c?,,+~~)!~ ‘*tikek, k=l i,,j, k= 1, 2, .... (2”- ’ - I),

(3.2)

440

TZE

AND

NAM

where qijk is a completely antisymmetric numerical tensor. The latter is zero when K = R and C, it is the Levi-Civita tensor .siik for K = H and for K = Q the seven nonzero triples can be chosen as +ijk = 1, zjk = 123, 145, 176, 246, 257, 347, 365. It follows from the multiplication table (3.2) that while R and C are associative and commutative, H is associative but noncommutative, D is neither associative nor commutative, the lack of associativity is measured by the assaciutor {ei, ej, ek > s (e,ej) ek - ei(eje,) =

(3.3)

-6,e,+6j~ei+(*,,~,,,-I(/jkqICIisr)e,.

A general element of K, X= xOeO+ x,e,, x0, xi E R, its K-conjugate is the involution X= xOeo - xiei such that 8= X, xy= 7;. We define the scalar and vector parts of X as x0 = SCX = i(X+ 8) and xiei = Vet(X) = $(X- X). The norm N(X) is N(X)=XX=XX=(x~+xf)e,.

(3.4)

C, H, and Q can be obtained in a unified manner via the Cayley-Dickson process. It consists in carrying out three successive applications of the following doubling procedure on R: Let Z = (R, C, H), denote by C2 the system formed by pairs (X, Y) of numbers from Z. In Z2 addition is defined in the usual fashion. Multiplication is given by the rule

(Xl, Y,).(X,, Y,)=(X,-x*-Y*.

Yl, Y,X2+

(X, Y)= (F, - P).

Y,.X,) (3.5)

A unique connection exists between the four division algebras K = R, C, H, and 0 and the librations of S2”- ’ by a great P-r-sphere, n = 1, 2, 4, and 8, respectively. Hurwitz’s theorem is equivalent to the following [73]: the only dimensions n of R” with a multiplication R” x R” + R”, denoted by F(X, Y) = XP with XP= 0 t) X= 0 or Y = 0 are n = 1,2,4,8; namely these multiplications can be realized by K, i.e., R n z K. Then, by a linear identification of the product space K x K with R”‘, the product F(X, Y) with X, YE K defines a bilinear map, the Hopf map H: p+Sn+l

(3.6.1)

with H(X,

Y)=(JXJ’-jYyJ=,2F(X,

Y))=((X(2-jYy12,2X~).

(3.6.2)

So for (XJ2+)Y12=1, we have IH(X, Y)12=(jX(2-jY12)2+4JXYJ2=1. Hopf’s construction [69] consists in taking two spheres, S2n-1 as the space of pairs (X, Y) of K with (XI2 + ) Y12 = 1 and S” as the space of all pairs (s, k) of a real number s = IX12- 1Y12 and k =2XPoK. Thus H restricts to the map H: SZnhl + S” with S2” - ’ as the libre space, S 2n and S”- ’ as base space and libre, respectively. For our purposes the following form for the maps (3.6.2) is most illuminating.

TOPOLOGICAL

PHASE

441

ENTANGLEMENTS

Let N, N’ = 1, be a unit (n + 1)-vector parametrizing S” and KT = (K,, K,), K,, K, E K, KtK = 1, be a unit normed K-valued 2-spinor parametrizing S2” ‘. Then the Hopf map (3.6) reads (3.7)

N = Sc(K+yK)

with K’ = (I?, , Kz) and O yl’z ( < e,0’ 1

p = 0, 1) ...) m - 1)

1 0 Ym= ( o -1 >3

(3.8)

m = 1, 2, 4, and 8. This relation

between the y’s and the K-units e, reflects the fact [73] that every orthogonal multiplication Rk x R” -+ R” is a Clifford module Ck , In particular, we have the irreducible Clifford modules C,, C,, C3, and C, corresponding to R, C, H, and $2. The y’s are just the Dirac matrices of s/(2, K) with y,, being the analog of y5 of the standard D = 4 formalism. Alternatively if we take a “unit sphere” S 2n- ’ in K x K and an n-sphere s” = K u ( cc }, the Hopf projection map, n: S2”- ’ -+ s” also reads n(X Y) =

X/Y cc

provided if Y=O,

Y# 0

(3.9)

where IX\*+)Y)‘=l, Jr’, YEK. We observe that the pre-image (or inverse) of this Hopf map, 7c-.‘(X, Y), is geometrically the intersection of S 2n~ ’ with an n-dimensional subspace of K x K, hence a great (n - 1) sphere S” - ‘, i.e., an (n - l)-cycle. So the image of any point on S” is a S’-‘-sphere on S”‘--I. This is apparent since N (or the ratio X/Y) is invariant under the phase transformation K+ KU (X4 XU, Y -+ YU). U = 8, 1U12 = I, is a unit normed, pure imaginary K-number, i.e., U E So z Z2, S ’ x U( 1 ), s3z W(2), and S’, namely an (n - I)-cycle for n = 1, 2, 4, and 8, respectively. To make our general treatment more concrete, let us illustrate the n = 2 case of the complex Hopf libration. Take any map f: R3 -+ S* with the boundary condition f(x) --, ,li, _ T (0, 0, 1). It implies that R3 is compactilied into S3. Equivalently we consider the map J s+s, 7r3(Sz) % Z. Given 7, Hopf’s construction of N(J), a generator of the homotopy group n3(S2), proceeds as follows. Pick any point q5, = ZJZ, of S’, Zi, Z, E C, its image r-‘(d,) in the covering space S3, parametrized by the pair [Z,, Z,] with IZ,12+ jZ,I’= 1, is a loop C, or fibre U = exp(irp), since 4, is invariant under the transformation Z, + Zi exp(icp). If we pick any other point q& of S*, then?-‘(4,) is also a loop C2 in S3. If $i #d, then C, and C2, represented by the position vectors rl and r2, are disjoint curves. Then the Hopf invariant y(f) = r(f) is given by Gauss linking number of C, and C, in S3 or R3.

442

TZE AND NAM

9-l

S2n-l

s” (-zi!f)

/- -=.\\ I \ *t I .. -a-__ i /’ : I’ I //’ :,/ : /’ 5

FIG. 4. Two points on a great curve in S” can correspond to two linked hyperribbons in S2”-‘.

To explicitly see that the Hopf mapping (3.6) maps circles in S3 into single points in S2 and has Hopf invariant 1, we consider the following: Take the 4-position vector in R4, v=($~,$~, ti3,ti4), tiieR and so that S3 is given by v2 = 1 or IXJ2+(Y(2=1 with I=($,+$*), Y=($,+i$J. The complex Hopf map fi S3 + S2 is then H(X, Y)=(IXJ2-~Y~*,2X~)=(N3,N1-iN2), where (3.10)

N2=2(+2q3-(C/LtiJ

fv,=I):+&+II/:=

-1+2($:++;).

Hence N2 = (I# + I# + I# - (c/j)’ = 1, N = (N,, N2, N3) E S2. We next select any two points of S*, e.g., its North and South poles, N = (0, 0, 5 1). Their images, the loops C, and CZ, are described by the equations II/f + +i = 1 and +f + $3 = 0. By deforming S3 into R3 through a stereographic projection onto the e4 =0 plane with, as the center of projection, the North pole (ti4 = 0), we get Fig. 4. It shows the curves C, and C2 to be linking once, consequently the Hopf map (3.10) has invariant one. Parallel constructions go through for K = H and Sz. B. The Many Faces of the Hopf Invariant and Adams’ Theorem The Hopf invariant y(D) classifies the mappings @: S2”-’ + S”. Its presence in the chiral model action (1.1) where n = 2 is essential to a dynamical realization of exotic spin and statistics. Our work is in essence about the many faces of y(Q), its various mathematically equivalent, physically telling expressions [ 743. We begin with a compact derivation of the Whitehead form of y(Q). Let V’(M) be the space of p-forms on a manifold M, p d dim M. Let us define two objects. On S”, we select a normalized n-form area element w,, fsn o, = 1. On

TOPOLOGICAL

PHASE

ENTANGLEMENTS

443

S2”- ‘, by pulling back the Hopf map @, we define a second induced n-form p,, = @ * 0, E V”‘(P ’ ). That the latter is closed (dpn = 0) follows from d(@ * 0,) = @ * (do,) = 0 and dw, = 0, the pull-backs of closed forms are closed. By de Rham’s second theorem H”( S 2n~ ’ ) x 0, all closed n-forms on S’” ’ are exact, there is a such that aA,.-, = F,,. So the non-unique (n- l)-form A”, _’ E Ycfl-‘J(Sz”~-‘) integral v(G) of the exterior product A,- , A Fm over S’“- ’ is defined (3.11) It is easily proved [74] that (a) v(Q) is independent of the choice of either 2, , such that dA”, ~ ’ = Fn or of w,. (b) y(Q) = 0 for all maps @: S’“-’ + S” with n odd. (c) y(Q) is invariant for any two smooth and homotopic maps S’” ’ -+ s”. (3.11) is called the Whitehead form [75] of the Hopf invariant y(Q). For physics applications the expression (3.11) for the Hopf index is recognized as an abelian ChernSimons term for the Kalb-Ramond field A,- , = 27rd,-, and property (a) translates into the gauge invariance of this ATGF. Now (3.1) is just one of several equivalent definitions of the Hopf invariant. Next we derive these alternatives forms of H(Q). For that purpose we parametrize the map @: S2”-’ -+ S” by the (n + 1) component unit vector NE S”, (N* = 1). Let N,, be an arbitrary fixed point on s”, then as illustrated above N(x) = N, is the equation of a closed hypercurve Co % S”- ’ on S”‘- ‘. Equivalently stated, the pre-image of Co = @‘(N,,) of N, is an (n - l)-cycle in S2” ‘. Now if ,YO is some n-dimensional closed connected submanifold on S’” ’ with, as its boundary 2,X,,, C,, then N(x) maps JYO,called a Seifert surface, onto the whole n-sphere. Then the Hopf invariant y(N) can be defined as the number of times N maps ,YOonto s”: i.e., it is then the mapping degree of N(x) restricted to ,Y,, from C, to S”, N(x): ,X0 --) s” and is independent of the point N,, of S”. With n,(Y) 2 Z, the Hopf invariant is an integer as it should be. According to a theorem of Eilenberg and Niven [70], representative maps S -+ s” for n = 2, 4, and 8 with winding number m are given simply by P with XE C, H, and Q respectively. Consequently besides the form (3.1 l), we also [74] have the following generalized flux and loop integral representation of y(N) (3.12) where ?;;, = d2, ~, is the area element n-form of S” mapped by N into S’” ‘. Indeed, as it should be, these Fn and A,- ’ are the same ones occurring in the above construction of the Whitehead form (3.11) of y(N). In contrast to Eq. (3.11), the Hopf invariant as (3.12) manifestly gives rise upon exponentiation to a generalized Aharonov-Bohm phase factor associated with the corresponding antisymmetric U( 1) gauge fields 1551. As to the relation between the Hopf invariant and Gauss’ linking number, the answer cannot be simpler: according to Hopf [69], y(Q) was originally defined as

444

TZE

AND

NAM

a linking number! The map n represents an element a in the homotopy group JQ-,(S”). Pick two distinct regular values of n, namely two points n, and n, on S”, then their pre-image @(n,) = C, (a = 1,2) is a (n - l)-manifold in S2+l. UpOn assigning a natural orientation to these hypercurves we obtain two (n - 1 )-spheres in S2”- ’ or (n - l)-cycles C, and C2, which can be linked or unlinked. In fact y(a) is just the linking numbers Lk(a,, ~1~)of C, and CZ. It depends only on c( and is thus a homomorphism: H:Tc~,-,(S”)+Z,

(3.13)

with the expression for the Gauss linking coefficient given earlier in (2.18). Finally we cite some useful properties of the Hopf invariant [74, 761: (a) (b) (c) (d)

For n odd, His zero by the anticommutativity of linking numbers (2.19), For n even, Hopf proved that maps of an even H always exist, If the map J? SZn-’ -+ S2n-1 has degree p the y(@o r) = py(@), If the map !?? S” + S” has degree q then y( Yo @) = q2y(@),

where the degree of the map S” + S” is an element of rc,(S”). From White’s theorems on self-linkages (Section IIB), we can already deduce interesting restrictions on the nontrivial D > 3 counterparts of Polyakov’s regularization and on some general features of the sought for field models. These theorems point to the possibility of an infinite sequence of D > 3 field theories in specific higher odd dimensions. Specifically, due to the noncommutativity of the linking coefficient (2.19), nontrivial Calugareanu-White formulae only exist if the spacetime paths of embedded geometrical objects to be regularized are odd-dimensional submanifolds (n = 1, 3, 5, 7, ...) of D (= 3, 7, 11, 15, ...) spacetimes. What then are our expectations? In three spacetime dimensions [24], the closed Feynman paths of pointlike solitons are topological l-spheres S’. So the regularization of Gauss’ integral [24, 221 consists in the splitting of space curves into self-linking, twisting, and writhing S’-ribbons. The exotic spin and statistics are induced by magnetic effects of a topological U( 1) gauge field coupled to charged point particles. In higher dimensions, analogous nontrivial phase entanglements effects should arise from the twistings and writhings of d= 3, 5, 7, ... dimensional hyperribbons obtained via a manifold-splitting regularization [22]. The latter are the world volumes of 2-, 4-, 6-, ... membranes embedded in odd D = 7, 11, 15, ... spacetimes, respectively. These effects should similarly be encoded as Hopf phase factors, the counterparts of (2.3). They arise from the long-range magnetic interactions of the membranes through their higher rank topological U(1) ATGFs. The latter could be the ATGF of compactilied superstring [77], supergravity [78], and particularly of supermembrane theories [79]. The importance of the topological effects of the ATGF was amply demonstrated in the compactification of D = 11 supergravity with its Chern-Simons term [SO] and in the anomaly cancellation [77] in superstrings. To cut down the plethora of possibilities indicated by White’s theorems alone,

TOPOLOGICAL

PHASE

445

ENTANGLEMENTS

further physical restrictions are necessary. In our search among higher dimensional extended objects for analogs of intermediate spin and statistics, it is clear that at least three essential features of the CP( 1) model ( 1.l) should be required of their D > 3 counterparts. These are [7]: (1) The existence of topological solitons; (2) the presence in the action of an Abelian Chern-Simons term, the Hopf invariant; (3) the associated Hopf mappings S2” ~ ’ -+ S” include ones with Hopf invariant 1. The first two requirements are embodied in the time component of the key equation (2.6). Upon integration over all of space on both sides of this equation, one obtains the topological charge-magnetic flux coupling which is at the very basis of the fractional statistics phenomenon in (2 + 1) dimensions. With regards to the third requirement, an essential element in the proof of the fractional spin and statistics for one soliton [7], the following striking feature holds true for the Hopf mappings in higher dimensions. While for any n euen there always exists a map fi S’+’ + S” with only even integer Hopf invariant y(f‘), possibilities for Hopf maps of invariant 1 are severely limited. This existence question of Hopf maps of invariant 1 has a long and fascinating history [81], the final answer is provided by a celebrated theorem of Adams [56]: If there exists a Hopf map f: S” -+ S’D+l’iz of invariant y(f) = 1, in fact with y(f) = any integer, then D must equal 1, 3, 7, and 15 or m = (D + 1)/2 = 1, 2, 4, and 8. So there are altogether four and only four classes of Hopf maps with y(f) = 1. Thus its has been proved [82] that there exists no maps: S3’ -+ S” of Hopf invariant 1 and hence there is no real division algebra of dimension 16. These unique four families of Hopf maps and their associated hidden or holonomic gauge field structures are best displayed by the following diagram of sphere bundles over spheres: U(l)=SO(2) II Z2 = O(1) = So + S’ + S1/Z2 = RP(1)

x SO(2)/Z2

II

SU(2)

=

Sp( I ) = .S3 + S’ -. S4 = HP(l)

SpiMVSpin(7)

r

Sp(Z)/Sp( I) x Sp( 1)

II = S’ + S”(-Spin(9)/Spin(7))

+ S* = QP(1)

z Spin(9)/Spin(8).

The four rows mirror the one-to-one correspondence between the four (and only four) division algebras over R and the real (R), complex (C), quaternionic (H), and octonionic (Q) Hopf bundles (tabulated in bold letters). The first three principal bundles are just the simplest examples of the three infinite sequences of the K = R, C, H universal Stiefel bundles over Grassmannian manifolds [83]. The latter are

446

TZE AND

NAM

instrumental in obtaining the general instanton solutions of Atiyah, Drinfeld, Hitchin, and Manin [84]. The last bundle stands alone, reflecting the nonassociativity of octonions. The spheres SP, p = 0, 1, 3, and 7 are the fibers, with the first three being Lie groups while S7 is a very special coset space, the space of the unit octonions. The latter has been the focus of much fascination and discoveries in mathematics [SS] and in the Kaluza-Klein compactification of D = 11 supergravity [80] and supermembrane theories [79]. An n-sphere S” is parallelizable if there is a continuous family of n orthonormal vectors at each its points. The fact [73] that S’, S3, and S’ are the only parallelizable spheres is yet another corollary to Adams’ theorem. s’, r = 1, 3, 7, 15 constitute the corresponding fibre spaces. Finally the sequence of base spaces S”, n = 1, 2, 4, 8 are equally interesting as K-projective lines, as is clear from their coset forms. With their holonomy groups Z,, SO(2), SO(4), and SO(8) being the norm groups of R, C, H, and Q they can be said to have a real, complex, quaternionic [86], and octonionic Kahler structures [28]. In the past decade Hopf maps f: S2n-’ -+ S”, n = 1, 2, 4, 8 with Hopf invariant one have found important physical applications in condensed matter physics [87] and in quantum field theory [SS, 891. Even the connection between Hopf maps and nonstandard spin and statistics had been lurking in the background. Thus, in the D = 2 4” field theory [90, 151 the n = 1 real Hopf map realizes the l-kink soliton, which carries intermediate spin and admits exotic statistics [ 151. The n = 2 complex Hopf map, besides being the Dirac 1-monopole bundle, underlies the 8 spin and statistics of D = 3 CP( 1) model. The n = 4 quaternionic Hopf map is the embedding map for the SO(4) invariant, D =4 SU(2) BPST 1-instanton [91 J or the SO(5) invariant, D = 5 SU(2) Yang monopole with eg= 4 [92]. The n = 8 octonionic Hopf map appears as a SO(9) invariant, D = 8 SO(8) 1-instanton [93]. The latter two maps have been shown [28] to admit yet further realizations in terms of U( 1) tensor gauge fields associated with extended Dirac monopoles with eg = 4 in p-form Maxwellian electrodynamics [SS], their role in determining the spin and statistics of membranes is the principal remaining theme of this work. Having observed that the field theory realizations of the real and complex Hopf fiberings both admit exotic spin and statistics, an obvious question arises. Does this pattern persist in theories built on the remaining two Hopf fibrations, SZn-’ + S” for n = 4,8? The answer should be sought within the quaternionic D = 7 HP( 1) ( wS4) and the octonionic D = 15 Q2p( 1) (Z S8) a-models augmented with their respective Hopf invariant term [22, 53, 281. Making use of our outlay of geometric and algebraic tools, these models are studied next. C. Quaternionic

and Octonionic a-Models

with a Hopf Term

In mathematics, the standard nonlinear cr-models [94] are better known as harmonic maps [95]. One associates with the map Y? M+ N between two Riemannian manifolds M and N an action (or “energy” for the mathematician) S,( !P) = JM a( Vu) d”x = i j

M

IdY’(x)l’

dmx,

(3.15)

TOPOLOGICAL

PHASE

447

ENTANGLEMENTS

dY(x) is the differential of Y at the point x E M and d”zc, the volume element of M. In a coordinate patch, l&PI* = gti(dYY” aYa/axi dxj) h,, is the pullback on M of the metric ds2 = h,, dYa dYYB on N. (3.15) is just a compact expression of the usual o-model action. Y is called harmonic if it leads to a vanishing Euler-Lagrange operator (or tension field) div(dY) = 0. In particular, it is well known [96] that the quadratic Hopf map Y(X, Y): S2”-’ -+ S” (n = 2,4, 8) is a harmonic polynomial map, with constant Lagrangian density a(Y) = 2n. As such they are the simplest harmonic representatives of the maps with Hopf invariant one. While the D = 3 CP( 1) o-model (1.1) admits exact finite energy static solitons [25], the corresponding D=7 HP(l) (z.S4) and the D=15 sLP(1) (z:S’) a-models, (3.15), do not. This conclusion derives from the Hobart-Derrick scaling argument [97]. As is usually done in practice, dynamical stability can be achieved in two ways. Either the needed repulsive interaction is provided by coupling the model to a gauge field or by adding to the standard KP( 1) action (3.15) suitable chiral invariant terms of higher order in the field derivatives, the Skyrme terms. Opting for the second alternative, the generic action with the added Hopf term is in the form S,,,,=j.

M

t?,,N-I?“Nd*“-!x+Oj

uhf

A dA ,I , + suitable Skyrme terms;

A,-,

(3.16)

n=4, 8,

M=S7,

SL5,

where the unit vector N with K = H and fi is given by (3.7). The composite U(, 1) ATGF A,, I, being nonlocal in N, is local in the 2-spinor K (3.7). Its expression in terms of K will be given later. We can write the e-term as 1 i

“=(n-l)! i.e., as an interaction

of the potential J n-,=

-

d*”

A,_,

(n-l)! 4n’

(3.17)

‘x Jp’ ““” ‘A,, ../,,, ,,

with the topological

0 *F n

current

(n=4, 8).

The latter’s conservation and expression in terms of N will be shortly deduced solely from the field topology. Since the sources of J,_ , are charged solitons, we shall first determine what types of solitons are allowed in our KP( 1) models. To a condensed matter physicist, our a-models are the well-known n-vector models. As field theories of a 5- and 9-unit-vector parameter N, they are the quaternionic and octonionic counterparts [98, 22, 53, 281 of the isotropic Heisenberg ferromagnet. Consequently the nature and dimensionality of their allowed topological defects should only depend on the dimensionalities of the order parameters and

448

TZE

AND

NAM

of the compactified spacetime. Specifically they should obey the defect formula of Toulouse and Kleman [59]. Consider a topological defect of spatial dimension d’ in D-space or D-Euclidean spacetime. To measure its homotopic charge, we need to completely “surround” this defect by a submanifold of dimension r such that d’ + r + 1 = D. The geometric meaning of the contribution 1 on the LHS of this last relation is evident for a vortex line. It corresponds to the distance in 3-space (D = 3) between the line defect (d’ = 1) and its surrounding loop (r = 1). The topological charge labels the equivalence classes of the group rcI(Sn) of mappings s’ + S”, of the spatial submanifold S into the space of the (n + 1) unit vector order parameter N. With r D and (n + 1) < 0, but for 0 < (n + 1) 4 such as in Kaluza-Klein-type theories and if r > m so that RJS”‘) is generally nontrivial, even a richer variety of defects are possible. Applying the Toulouse-Kleman formula to our cases of (D, r = n) = (3, Z), (7,4), and (15, 8) we find that the allowed topological defects in the CP( 1 ), HP( 1 ), and QP(l) o-models (3.16) to be 0-, 2-, and 6-membrane solitons, their topological charges being the generators of rr,JS”) x Z, n = 2,4, 8. Since our solitons are charged 2- and 6-membranes, we expect the associated a-models to possess a rank 3 and rank 7 topological conserved current Jppa and JppouBvA.Their conservation [99] follows solely from the constraint N2 = 1, hence N . a,N = 0, and the fact that n the dimension of the unit vector N is less than or equal to the dimension D of spacetime. Since here (D, n) = (7,4), (15, 8), the latter condition is satisfied. Indeed if n < D, then the (D x n) matrix [aN] must have rank less than n, so we have (3.18) Consequently,

where JW~+I-.~PD=~

P”.‘PD&,,....~(a,,N”‘.

. . apn-,w--L) A'-'.

(3.20)

As with the U(l) model, these 3- and 7-form conserved currents, suitably normalized, are just the D = 7 and 15 Hodge duals of the respective 4- and (I-forms antisymmetric gauge fields F,, = dA,- , appearing in the Hopf invariant action in (3.17): J,= -(n! O/4n2) *Fn+l, with the star operation denoting the Hodge dual *~Pl~..Pn--p = (l/n!) EPI __,I& Fpn-p+l’..pn.

TOPOLOGICAL

PHASE

449

ENTANGLEMENTS

Following the analysis of Teitelboim [SS], the conserved current (3.20) can be readily converted into a conservation law. Two equalities are used: (a) Stokes’ theorem J,u dw = JdM o, o is a p-form and M is an oriented compact manifold with boundary d&f, D = dim M = (p + 1) and dim(aM) = p; h) the relation between the up. With the divergence and the exterior derivative: a,oEj l,r = ( - 1 )D’ [I *d*w],, latter identity (3.i9) reads (3.22.1)

d*J=O.

The integration ary akf gives

of (3.22) over a (D - p + 1)-dimensional

manifold

M with bound-

P(?M) *J=O.

(3.22.2)

If aM consists of two spacelike hypersurfaces 2 (with dim(C) = D - p), connected by a remote timelike tube aT and since the topological current J in our o-models is localized in space [ 1001, the integral over dT vanishes and (3.22) yields the Lorentz invariant and conserved charge

Q = J: *J, its value being independent of JY:.Applied to our KP( 1) o-models, where the equations of motion (2.6) forces a e-dependent linear relation between topological charge and flux, (3.23) reduces for (D, p) = (3, 1) to the Skyrmion winding number, the generator x2( CP( 1)) z Z, Q=~js-d~‘nF~=~ji,d\‘Ai=C,

(for 0 = 7r).

(Note that we are using Roman indices for the spacelike components.) It is also the first Chern index C, of the U( 1) bundle which is the complex S3 + Sz Hopf fibration [loll. For (D, p) = (7, 3) and (l&7), the topological charges of the membrane S“- and Ss-solitons and the generators of n,(HP(l)) z 2 and r,(QP( I )) z Z are similarly given for 6 = 7~by (3.25)

450

TZE AND

NAM

and Q

=

js,

Jo’1

...k

dZi ,.,. i,~=~SssFi,...i,,dZ”“.i’* =-

-1

27c I s7

A i, i,4 d.Z” ‘. i’4,

(3.26)

respectively. One way to see that Q is equal to n, an integer, is through the mentioned gauge field connection between our problem and the D = 2 complex [90], D = 4 quaternionic [91, 1011, and D = 8 octonionic [93] instanton. We take for the U(l) field coordinate, the mapping K(x) = xn, where x is the space position K-number in Z, K= C, H, and Q. While these maps are not 0-, 2-, or 6-membrane solutions to the systems (3.16), they are the simplest harmonic representatives maps: S” + S” (m = 2, 4, and 8) with topological number Q = n [70]. As will be clear the charges (3.25) and (3.26) can be identified with the 2nd and 4th Chern index which reflects the relation between the U(1) ATGF and the hidden non-Abelian gauge structure of the o-models, namely the Sp(1) quaternionic and associated Spin(8) [93] octonionic Hopf fibrations, respectively. Though our subsequent analysis of the thin soliton limit deals primarily with the o-term in (3.16), the Hopf term, to make clear the hidden gauge connection we now consider the HP( 1) zzS4 model in greater detail. A parallel treatment of the QP( 1) model is left as an exercise for the reader. As the coset space Sp(2)/Sp(l) x Sp(l), the quaternionic projective line HP(l) can be parametrized [loll either by two real quaternions q1 and q2 with (q,12+ (q2)‘= 1, i.e., by a 2-component H-spinor QT=(ql,qZ), QtQ= 1, coordinatizing the sphere S’ or by one quaternionic inhomogeneous coordinate h = q2qc1. An alternative parametrization is by way of the unit 5-vector N defined by the Hopf projection map (3.7) from S’ to S4, 2h l+hh

-,

N5=-

To make manifest the local Sp( 1) z SU(2) gauge invariance a = 1,2;

U(x) E SP( 1)

(3.27)

of the HP( 1) model, we introduce the covariant derivative D,,Q = (8, + a,)Q. The holonomic Sp( 1) gauge field is a,=a;e=Q’d,Q=~~~d,q,=--

1 h8h 21+hh

TOPOLOGICAL

PHASE

451

ENTANGLEMENTS

is purely vectorial and takes the ADHM form [84] solution. So the first term in (3.16) reads S (410=

for the 1 - SU(2) instanton (3.28)

scuqLa+ (WI)1

and similarly for the Skyrme terms. As for the Hopf term, we can readily check after some algebra that the 3-form Ao, is the D = 4 Sp( 1) Chern-Simons form of L&her ef al. [ 1023:

=Tr

(

A AdA+fA’

1

(3.29.2 )

F,,, = dA,,,.

where d-xl‘ d..u’ -d-P

A dx’, etc. In component A!,,,, = Wqp.frp~ F pvp” = =xr;,.r,~

(329.1 I

form, we have [53, 1033

- $~,,a,a,,,

+&L

1

+ /;,,.rl,, 1

( 330.1 ) (3.30.2)

in terms of the quaternionic-valued Sp( 1) gauge potential a, and field strength .f,,,,- ?,,a,. - ?,.a,, + [a,,, a,,]. More compactly, in terms of the 2-spinor Q, we have

A,,,=Se[Q+dQdQdQ+f(Q+dQ)31

(3.31.1 i

F(,, = Sc[dQ+ dQ + (Q+ dQ)‘].

(X31.2)

These forms manifestly show the Hopf term (3. I1 ) to be local when expressed in terms of the bundle space S’-valued field Q. It is thus locally a total divergence as is already clear from (3.12). While a parallel derivation of A(,, and hence of Fo, = dA,,, can be similarly performed for the D = 15 !ZJP( 1) a-model, the connection to the D = 8 octonionic instanton problem readily identifies the 7-form ,4 ,:, to be that given by D = 8 Chern-Simons term of a Spin( 8) gauge field, namely [ 1041 A,,) =Tr[A(dA)3

+ $A3(dA)’

+ :A5 dA + +A’].

(3.32)

In what follows, the above specifics of the o-models are sufficient for our analysis. As highly nonlinear field theories, our models are analytically intractable in their details. Besides, there is much arbitrariness in the choice of Skyrme terms. Being higher order in the field derivatives, the latter control the shorter distance structure of the solitons. As in the lower dimensional case [24], to study the phase entanglements of membranes it is enough to analyze the effective theories obtained in the geometrical NambuGoto limit of widely separated membranes. In such a limit, the particulars of soliton structure are irrelevant, only the existence and not the details of the Skyrme terms matter.

452

TZE AND

NAM

IV. EXTENDED OBJECTS WITH TOPOLOGICAL

SINGULARITIES

A. Thin Membrane Limit Extended objects with topological singularities can be created by the condensation of Nambu-Goldstone bosons [60]. This fact accounts for their appearance in certain ordered states as crystal dislocations, grain boundaries, crystal point defects, solitons in chiral theories, vortices in superfluids, in superconductors, and in broken gauge theories. Generally, due to the nonlinearity, the field description of their motions has been technically difficult, if not impossible, except in the London limit where these “vortices” can be parametrized geometrically by points, lines, or surfaces, There are several techniques for extracting from a given field theory an effective, more tractable low energy theory of interacting geometrical extended objects. They allow for a systematic freezing out of the short distance structures. Examples are the method of collective coordinates Cl051 and Popov’s continual integral formalism [ 106,631 applied to vortices in superfluid helium. Indeed the analogy between our systems with the electromagnetic theory of currents, with vortices in superfluids or in superconductivity, suggests that the effective theories are Kalb-Ramond type theories of interacting charged Nambu-Goto membranes coupled to a nonMaxwellian Chern-Simons action of a U( 1) ATGF. For our analysis of phase entanglements, the crucial ingredient of this low frequency correspondence is the following identification of the topological current (2.1) with the singular DiracNambu membrane current. For the U(l) model (l.l), we have

where the summation is taken over n isolated I-solitons. That the correspondence (4.1) is realized for our a-models in the London limit can be seen from the following general analysis of the topological singularities [60]. First, let our unit (m + 1)-vector field order parameter N = (N,, .... N,, ,), N2 = 1 in D-dimensional spacetime be well defined, i.e., [a,, a,] N(x) = 0 except at the sites A of the singularities. To analyze the structure of the singular domains A, we consider maps of all possible closed d-dimensional surfaces C onto the sphere S”. Since some parameters or combinations thereof must stay invariant under the mapping of C on S” when d > m, we assume d < m. Take the case of d= m with D > m. When C does not cross any singularities, it can be uniquely mapped onto a closed surface on S”. When C intersects with any singularities in A, N is defined only along the particular path specified by C. Generally we consider C, + C,. Now consider a domain S of S”. The area occupied by S is given by js o where the volume form is given by 1

w=~Ei~,...i,,,NidN,

A .‘. A dNi,,,.

(4.2)

TOPOLOGICAL

PHASE

453

ENTANGLEMENTS

So the total surface area is a,=j

(4.3 1

0. .sm

Since S” is m-sphere, we adopt the following polar parametrization N,=sin%,sin%,~~.sin%,_,cos%, N2 = sin 9, sin 9, . . sin 9, _ , sin 9, (4.4)

N, = sin 9 I cos %I N m+l =cos %,, so that the volume form on S” is w = -t(sin %l)m -’ (sin%,)“-2...(sin%,_,)d%,...d%,,.

(4.5)

Let S = C,. The volume integral can be written in terms of the fraction of the volume that C, occupies in S”, v

s,vqcm0.

(4.6)

To convert this integration to the one over x-space we write the order parameters N’(x), i= 1, .... m + 1, in the form involving the angles 9,(x). ‘.9,,,(x). The surface integral on S” becomes

s2,v =s

gp,

~, dx”’ A dxp” A

. A d-Pm,

cm

(4.7)

where

=(-1)

(1~2~~m~‘)(m+2)[sin~,(x)J”~‘...~in~,,

~~(~~)“‘~;i:.l:.:~~~;“, . (4.8)

By Stokes’ theorem, (x) dxp’ FPI...P,*I

59%193:?-14

A d.? A ... A dxpm+‘,

(4.9)

454

TZE AND

NAM

where we define the field strength

Fcll...~‘m+L(X)=Ca~L1g~*...~(m+l.

(4.10)

A

The index A means summing over all the terms given by antisymmetrizing the indices. Now by differentiating Eq. (4.8), summing over, and antisymmetrizing the indices we get a[Ni~(x)+“Nim+l(X)] ~‘~,g~*...~-+,=~(‘il...i~+l A

~~xp,~~~xpm+,l

aCNi*(x).“Njm+,(X)] +CEit...im+lNi,(x) A

a~Xp2~~~X~m+,l

.

(4.11)

>

The first term of (4.11) vanishes since N2 = 1 and so using Eq. (3.18) we get

aCNi,(x)“‘Ni,+,(x)l F~,..~/lm+~(X)=CEi~.~~im+~Ni~(X) a~xj42..expm+tl . A

(4.12)

Since it is made up solely of commutators of derivatives, F,, ...Pm+,(~) is nonzero only at the loci of the topological singularities of N(x), where the angle 0;s are not single valued. Such an object is an assembly of (m + I)-dimensional b-functions; it specifies a (D - m - 1 )-dimensional region in D-dimensional x-space. Call such a domain A,-, _ 1 and parametrize it by the functions y”(a,, .*., cDern- r ) depending on (D -m - 1) proper coordinates Ini}. We write

VQ, ,?,...1&,-1 F~,.-.~,+,(~)=(~_~_~)!&~,~~ s da,...do,-,,,-I

ab.,

. . . Yl&-J

Xa(O,...O,p,-,)

W[x-

y(a,

. ..o.-,-,)I.

(4.13)

As seen in the previous section, its dual is nothing but our conserved topological current.

By the above procedure we explicitly see that our 5’” 4 S” topological defects (n = 4,8) indeed behave as 2- (6-) Nambu-Goto membranes, which span respectively S3- (S7-) singular hypervolumes in D = 7 (15)~spacetimes. We are ready to analyze the phase entanglements of these thin membranes next.

TOPOLOGICAL

PHASE

ENTANGLEMENTS

455

B. The Hopf Phase in the Thin Membrane Limit Following Polyakov [24], Nepomechie and Zee [53], we proceed to calculate the Hopf phase in the above higher dimensional a-models. The D = 7 case will be treated explicitly. Its D = 15 counterpart will be subsumed under a general formula displayed at the end of this subsection. To derive the statistical phase involving extended objects, we first consider the propagation of two pairs of membranes-antimembranes and see the phase when we adiabatically exchange them. We then get

(4.15) As in D = 3 case [23], the resulting phase is the sum of three phases. The first phase is a well-defined double integral involving distinct disjoined hyper-paths P, and Pz, the other two terms are double integrals of the same Pi, hence requiring regularization. The first contribution yields the phase factor exp{ 2i(n2/0)L 1, where L is the Gauss’ linking coefficient (2.18) for two S3-loops. It shows that rr’/8 is the statistical phase. The other two phase factors @(Pi) are given by the expectation value of one loop: @p(P) =

where the functional average ( . . . ) is taken over the Hopf action in (3.16). In the London limit we use the effective action d ‘.y J,,\,j A’““..

S, is the free Nambu-Goto action for a relativistic 2-membrane, constant a is yet to be chosen. We are to evaluate

(4.17)

0 < 0 < n, and the

AI-,,,,,, d.9 A dx” A dx”

=exp(ild’lL)exp(~~A~,,~,.,hl’“d.~~’”dr.).

The equation of motion

derived from (4.17) gives

(4.18)

456

TZE AND

NAM

where the current J,,yI only has support on the geometric membrane. It reads

world volume of the

(4.20) If we substitute in Eq. (4.20) for the derivative of the tensor gauge field we get

)I

JpvlApVE. .

Let us next solve for the tensor gauge field AcIvl in terms of the current. Taking the curl of the equation of motion (4.19), we get (4.22.1) which leads to the generalized Poisson equation 28 upon the choice of the Lorentz gauge PA,,

(4.22.2) = 0. Since

we obtain (4.24) Therefore,

upon using (4.20) for the singular current distribution. Were the two hypercurves S3-disconnected in the two 3-volume integrals of (4.25), the integral would be (up to a multiplicative constant) just the Gauss’ linking coefficient (2.18) for two 3-spheres in S 7. However, in exact analogy with Eq. (1.3) [24], the double integration is over one and the same hypercurve S3, so the phase (4.16) is undetermined. Proper regularization is required. For that

TOPOLOGICAL

PHASE

457

ENTANGLEMENTS

purpose we only have to apply the corresponding formulae of Section IIB on the D > 3 geometric extension of Polyakov’s scheme. The regularized phase is given by @(P=S’)=exp(i$

(4.26.1)

W(P)),

where

is the writhing number (see Eq. (2.20)) of the closed Feynman path P of the Nambu-Goto membrane, a S3 hyper-ribbon in 7-spacetimes. Note that for the value of a = 4n2, the functionally averaged regularized statistical phase is Q(P) =exp(n2iW/B). By setting 8 = rc, we get the S3-counterpart of Polyakov’s phase factor @(P) = exp( - niT( P)) exp( nin). T is the generalized torsion for an S3-ribbon P, its explicit integral expression was obtained from (2.21)-(2.23) by White in Ref. [36]. Intuitively speaking, as in the CP( 1) model [24], we expect this torsion term to embody the thin membrane’s spin in a functional integral formalism. Provided that this rather reasonable conjecture is supported by a construction a la Polyakov of the spin factor for membranes, we thus obtain a higher dimensional analog of the D = 3 Fermi-Bose transmutation. With the value of 8 not fixed by the U( 1) gauge invariance of the ATGF, we thus have the possibility of fractional statistics and spin via W= -T (mod Z) for the membranes for arbitrary values of 0. This and other issues will be further discussed from a connected viewpoint in the next section. Instead of (4.26) or its octonionic 15-dimensional counterpart, we can derive a more general formula for the phase G(P), one which covers all three cases of the 0-, 2-, and 6-membranes. From the Lagrangian density L=L,+

8 A~I’~~~rn~~‘m+LA~~+2~~~I~2rn+~ 47c2(m!) 2 E!4~~..2m+,

with m = 1, 3, and 7, a computation phase O(P)=exp(

+iJp,.

paralleling

-ig

(4.17))(4.26)

W),

~p,Ap~~“ILm

(4.27)

yields for the same

(4.28)

W being the generalized writhing number. Note that, in the thin membrane limit, the parameter 0 appears here in the denominator [lo73 rather than in the numerator in the field theory case [7]. Having gotten the general regularized phase (4.28), let us comment on a more explicit form of the Berry’s phase rr W for extended objects, i.e., for the system of a

458

TZE

AND

NAM

spin i (0 = a) membrane in a magnetic ATGF. Since the Aharonov-Bohm term contributes minus twice that of the Chern-Simons term to the statistical phase, we expect [23] that the dynamical phase will be twice the Berry phase. This together with the relation W= 252/52,, + k discussed previously (Section IIB) determines the Berry phase [29] to be @(P) = exp(ircW) = exp( - i252/$2,,).

V. PHYSICAL MEANING

OF HOPF PHASES AND RELATED QUESTIONS

In this closing section we advance further topological arguments in support of our physical interpretation of the Hopf phases as higher dimensional statistical or spin factors. Two perspectives are available; that of the nontrivial field topology of our a-models and/or that of the global dynamics of the effective p-forms ChernSimons electrodynamics of the thin membranes. Fortunately, without probing the analytic complexity of the soliton structure and the detailed canonical quantization [ 14, 1071 of our U(l)-models, much can be deduced solely from the topology of the configuration space of fields. In the Schrijdinger picture, the configuration space r of finite energy static solutions of our U(1) a-model is the mapping space of all based preserving smooth soliton maps: N(x): x E S” --) N(x) E S”,

n = 2,4, 8.

(5.1)

The space r is an infinite Lie group [lOS]. The connectivity of r is given by n,,(I’=

{N: S”+S”})xn,(S”)~z.

(5.2)

So r is divided into an infinite collection of pathwise-connected components r,, CIE Z. The rol’s correspond to the various soliton sectors, each labelled by the winding number or topological charge Q (3.24)-(3.26). Hence the nontriviality of (5.2) merely reflects the existence of topological membrane solitons. Moreover, as is well known from studies of Skyrmions and Yang-Mills instantons [109], each connected component r, can have further internal topological obstructions of physical significance. Such is the case with our membrane solitons. As noted by G. W. Whitehead [ 1 lo] all components of r have the same homotopy type, i.e., ni(&) % rci(rs). Of relevance to the question of exotic spin and statistics, we have in particular for the 1-soliton sector 71i(rl)~:n,(ro)~ni+n(~")~z

for

(i, n)= (1,2), (3,4), (7, 8).

These important relations [ 1111, which reflect the multi-ualuedness of ri, consequences of the Whitehead [llO] and Hurewicz [ 1121 isomorphisms, latter stating p(rM)=2:i+.(Sn).

(5.3) are the

TOPOLOGICAL

PHASE

ENTANGLEMENTS

459

In fact the relations (5.3) imply the possibility of adding to the KP( 1) a-model action a Hopf invariant y(N), the generator of the torsion free part of rr,+,,( s”), specifically rc3(S2)%Z, .rr,(S4)%Z@Z1, and rr,5(S7)xZOZ,,,. They therefore signal the existence of a higher dimensional analog of an exotic spin and statistics connection for the membrane solitons. Generalizing the case of the CP( 1) model where {(i n) = (1, 2)), the nontriviality of these homotopy groups rc,(Z-,) implies the possibilities of Aharonov-Bohm type quantum effects of a multiply connected configuration space r. In the case of the CP( 1) model [7-153, the exotic spin and statistics connection arises from the Hopf invariant action Sn. Being a local homotopic invariant linear in the field time derivative, this action induces, upon an adiabatic 27r rotation P of the Skyrmion or an interchange of two Skyrmions, a projective spin phase factor a(P) = exp{ i0) = exp(i2rr.s). s = 8/27c is the spin of the soliton. This equality Q = 271s for this process of rotation is a physical realization of the following homomorphism:

~,(so(2))~:7c,(s~)~.,(f,)~z.

(5.4)

It establishes the equality of the kinematically allowed exotic spin to the dynamically induced O-spin by way of the Hopf term. However, (5.4) is but a special case of the Hopf-Whitehead J-homomorphism rr,(SO(n)) % x/, +n(Sn) [ 1131. In light of (5.3) and

713(s0(4))z71'(s4)~z

and

~7(~0(~))~:,,(s*)~z,

(5.5)

we have the following chain of homomorphisms: n;(r,)D71,(~0)~4:i+~(Sfl)~x;(so(n))~z

(5.6)

with (i, n) = (1,2), (3,4), (7, 8). When applied to our KP( 1) a-models, the following physical interpretation may be inferred. Without any loss of relevant information, our solitons can be taken to be Nambu-Goto membranes in the limit of large separation. This limit affords better visualization and easier interpretation. Consider now a continuous dynamical deformation which begins from a particular l-soliton configuration, a point y in f, , and which subsequently returns to the same configuration. Specifically let us perform a 2n rotation of our I-soliton membrane about some axis. While a point CP( 1) soliton traces out a closed Jordan curve, a l-sphere within I-, = R' - {O) z S' x R of the configuration space r, a 2-, &membrane traces out respectively a closed hypercurve, a S’- and S7-sphere or cycle in their respective f, The charged membranes always interact via the Hopf term through the background U( 1) ATGFs. So any rotation of a membrane or interchange of two membranes necessarily involve the corresponding dragging of their associated gauge fields, which allows the possibility of dynamically induced exotic spin and statistics. Since these rotational or exchange deformations must respect the topological

460 interlocking

TZE AND

NAM

between physical space and the field space via the nontrivial

membrane

mappings s4 + s4 z SO(5)/SO(4),

S8 + S= w SO(9)/SO(8)

(5.7)

and

s’+ s4, slS+ sa,

(5.8)

the associated rotation groups, ones which are dynamically allowed by the Hopf interaction, are not SO(6) and SO(14) as expected on intuitive and kinematical grounds. Thus even for free membranes devoid of Hopf interactions, simple stable rotations are quite non-intuitive: an example is a rotation made up a product of two mutually commuting rotations around the same angle [ 1143. Moreover, as discussed in Ref. [114], the spacetime dimension D must be at least (2M+ 3) for free pulsating and/or rigidly rotating M-membranes minimally embedded in D-spheres to exist. This condition is indeed satisfted by the C, H, and Q Hopf vibrations giving respectively 0-, 2-, and 6-membranes in 3, 7 and 15 dimensions. While our 3-dimensional intuition can go astray in higher dimensions, the underlying mathematics suggest that it is the topology of the o-models which singles out the holonomy groups of HP(l) and sZP( l), O(4), and O(8) as the relevant rotation groups. As in theories of interfaces [115], the emergence of these groups examplifies nonlinearly realized spatial symmetries via the soliton maps. Namely the membranes, being extended solutions to the a-model field equations, have spontaneously broken the rotational symmetry groups O(6) and O(14) down to the subgroups O(4) and O(8), respectively.

4(t)

Fro. 5. Membrane-antimembrane the two solitons results in paths

which

creation followed are linked.

by an adiabatic

rotation

or the interchange

of

TOPOLOGICAL

PHASE

ENTANGLEMENTS

461

The rotation of the membrane soliton and its connection to the exchange of two membranes can be described by a K-Hopf bundle generalization of the geometrically transparent analysis of Wilczek and Zee [7]. The central mathematical ingredient is the already mentioned linking theorem of Hopf [69]. In Section IIIB, we stated that the Hopf invariant, y(Q), the generator of the maps @: S’“-r -+ s” (n = 1,2,4,8) can also be defined as the generalized Gauss linking number between two hypercurves S”-’ embedded in S*+’ (or R*” ’ ). From this identification, the map @ with Hopf invariant 1 can be realized physically in our o-models by the following sequence of soliton events in (2n - 1 )-spacetime: Imagine a process of pair creation of a (n - 2)-membrane-antimembrane followed

I .-‘\ \ \, III’ I1 I1 \\ ,I I \ I ‘.-.I 0

FIG.

6.

Creation

of two pairs

of membrane-antimembranes

462

TZE AND

NAM

by a 27c rotation of the membrane about some axis before the pair is allowed to annihilate. Figure 5 is the Euclidean space pictorial representation, albeit in 3-space, of this process. There the two hypercurves or (n - 1)-cycle, traced out by two distinct regular values of N, say the North and South poles of S”, clearly are linked once. To determine the statistics of the membranes, one considers a process in S’“-’ where two membrane-antimembrane pairs are now created, then allowed to annihilate only after two of their membranes have been exchanged. Figure 6 shows the net result to be again a Hopf map @ of invariant 1 by reason of the above linking number theorem. So the spin and statistics connection of our (n - 2) soliton membranes is established through the Hopf invariant seen as a generalized linking coefficient. With rcZn- r(Y) z 2, it follows that the (n - 2) membranes obey exotic spin and statistics. A most explicit, heuristic demonstration of the fractional statistics of our solitons is by way of path integral approach to systems of indistinguishable membranes. The path integral formalism has provided an ideal global setting for the study of quantum effects of the multiply connected configuration spaces [ 1163. Let us first consider the problem in its field theoretic setting. We know from (5.2) that the configuration space r = {N: S” + S”}, n = 4, 8 is a disjoint union of path connected pieces, ro, Q being the soliton winding number. From the analysis in Section IIIC, closed classical paths of the solitons are continuous hypercurves S”-‘. They can be grouped into distinct homotopy classes according to zn- ,(r,) x rcnzn-i(Y) z 2, Eq. (5.3). Let the &&N(x), t; N’(x), t’) be the single membrane propagator obtained by summing over all paths connecting the point N = {N(x) at t} and N’= (N’(x) at t’} that are homotopic to a particular path P. The full propagator is

K,(N, N’) = 1 x(CPl) K&N

W

(5.9)

[PI

where, for each [P], x( [P] ) is a complex phase factor. Now select any fixed basepoint N, in r, and join it to the points N and N’ by a pair of paths L and L’, respectively. The composite path LPL’ -’ defines an element of n,- i(Z-r, NO) and so (5.9) reads K,(N,

N’) = c x(a) KAY

OL

N’),

~ET-I(C

No).

(5.10)

Though the element c1 of z,- i(r, ) depends on the choice of L and L’, another choice of the latter only multiplies K, by an inconsequential multiplicative phase ry x is a character of n,- l(f,). More specifically for our membranes we have K,(N,

N’) = I ~9N(x)

ew{~S,,,JN(x))~

(5.11)

TOPOLOGICAL

PHASE

ENTANGLEMENTS

463

or after using (5.10) K,(N

N’)

=I

x(a)

jNtx,,,

gN(x)

exp~~~,,,dN(x)))~

(5.12)

a

where Scm,O stands for the total action S,m, without the Hopf term. The complex weight x(a) is given by (5.13) which depends solely on the homotopy class ~1, in accordance with (5.3). So the possible two point functions of the 1-soliton are labelled by the character group Hom(n,_,(r,, U(1))) of rcn-,(r,). Generalizing the CP(l) case [ll], (5.12) shows that there are as many distinct propagators as there are scalar unitary representations of rr,, .~,(r,) = Z, the associated (n - 1)-dimensional analogues of the usual [ 1173 fundamental Poincare group. Quantization is therefore not unique; there are e-vacua, or in our case, the soliton obeys intermediate statistics. From the viewpoint of a Hilbert space approach, the connection of the above result to the hidden gauge field can be obtained as follows. Usually the state vector ‘Y(N) is seen as a wave functional on r and carries a representation of the canonical commutation relations. Since rc,,- ,(f ,) x Z, Y’ can “twist itself around’ as N varies over r, i.e., it should actually be a cross section of a nontrivial V( 1) bundle of a ATGF A,. 1 over lY However, to our knowledge, a Chern-Weil tibre bundle theory for ATGFs has not been systematically formulated in the field theory framework. We shall work instead in the more familiar London limit of Nambu Goto membranes [ 1 IS] and analyze the corresponding global aspects of the quantum mechanics of the geometric Sm-membranes (m = 2 and 6) interacting with their ATGFs. If d is the spatial dimension and X is the coordinate space of the one-membrane system, the configuration space M, of r distinguishable Nambu-Goto membranes is Y” = Xd x . x Xd (r factors). What is it for r indistinguishable membranes? If they are indistinguishable, then there should be no physical distinction between m-spheres (m = 2 or 6) in Xd’ which mutually differ solely by the ordering of their labelling indices. So the symmetric m-spheres in Xdr under the action of the symmetric group S, of n-objects should be identified. The Sm-membranes are impenetrable or non-intersecting, so the defect set D of all diagonal Y-domains where two membranes overlap must also be excluded from M,. Consequently we find that M, = (X” - D)/S,. Due to the exclusion of D leading to the multiplyconnectedness of M,, it is clear that there are Y-typed obstructions in M,. These “holes” can be detected by lassoing the space M, from any basepoint with S” + Iloops, namely rcm+ ,(M,) #O in agreement with the previous field theory analysis. If we now construct the propagator for the Nambu-Goto membrane corresponding to (5.9) the nontriviality of 71m+ ,(M,) will be reflected by the presence of a phase

464

TZE AND

NAM

factor x(a), a representation of rc,, ,(M,) with a summation over all c1labelling the homotopy classes of paths of this group. As detailed in Section IV, the above phase I has provided the effective gauge interaction such as (4.22). Our membranes are extended objects with the topology of m-spheres (m = 2 and 6). In a “first quantized” theory of such membranes, we define a wave functional Y(u(x(a)) over a configuration space in which each point is a coordinatized m-manifold x”(a) (p = 1, .... D = (7, 15)) [55]. Every configuration of the ATGFs A,,,+ 1 coupled to these membranes defines locally a generalized electromagnetic field on the functional space P’ of basepoint preserving maps from S” to the spacetime manifold S2” + 3. Since by (3.25,3.26) and [28] (1/47c) fZ F,,,,, = an integer, for any (m +2)-dimensional submanifold without boundary, A,, i defines a smooth connection in a nontrivial U( 1 )-bundle over the “loop space” P’ with yl(x(a)) seen as a cross section. We refer the reader to Ref. [28] for a discussion of the cohomology of these ATGF bundles and their connections to the Hopf bundles. For completeness we should also mention that an alternative and less intuitive analysis of membrane rotations can be performed [ 1131, without appealing to the linking theorem. Instead of the usual single angle parametrization, the allowed shape dependent rotations R(t) of the (n-2)-membranes are effectively parametrized by t E S”- ‘, n = 4, 8. With the set of 27r rotations, i.e., a map S”-’ -P SO(n) and the soliton map N(x) describing an element of rc,- ,(SO(n)) and Z,,(P), respectively, N(R(t)x) then defines a map S”-’ x S” --) S”. We thus have the map of nc,- ,(SO(n)) x xJS”) + [S”-’ x S”, Sn]. It is then possible to define a map into n2n- i(P) by the following J-homomorphisms: 7c,- ,(SO(n)) x n,(S”) -5

n2n- 1(Sn) x 7rn(Sn) -5

~T~~-~(S~).

(5.14)

C defines the map S2”- ’ + S” obtained by composing the two maps @: S2+i + S” and !P S” + S”. As stated in Section IIIB, if the soliton map Y has winding number Q, then the Hopf invariant for the product map !Po @, y( !Po @) = Q2y(@). Consequently, from a 1-soliton map Y acted on by rotations, we can obtain via the J-homomorphism a Hopf map of invariant 1 and we recover the result obtained before. The above discussions are mainly topological. A more involved analytic approach, yet to be implemented, is the explicit determination of the relation between the membrane spin, its topological charge Q = 1, and the coefficient of the Hopf term. It calls for the computation of the angular momentum of the soliton upon a canonical quantization of our U(1) o-model. In particular, as for the CP( 1) model [ 14, 1071 this could be accomplished through a semi-classical collective coordinate quantization of the 1-soliton membrane sector. To underscore the technical difficulty of a fully quantized Hopf membrane theory the following must be noted. Due to the nonlinearity of the constraints from reparametrization invariance, even the canonical quantization of neutral free 2-dimensional NambuGoto membranes still remains an open problem [119]. Of special concern and

TOPOLOGICAL

PHASE

ENTANGLEMENTS

465

interest to us will be the dimensionality of spacetimes in which a Chern-Simons quantum electrodynamics of charged Nambu-Goto membranes will be consistent. In fact even the simpler functional approach to the higher dimensional FermiBose transmutation of the charged Nambu-Goto membrane appears at the moment technically difficult to implement as fully as Polyakov [24] did for the point particle case. The following discussion is therefore incomplete. Patterning after Ref. [24], we may attempt to compute the partition function Z,, for the thin membrane soliton. It reads

-cl=

cflt...icn7 dx,, A . . . A dxpn

c P = closed

(5.15)

paths

The first exponential is the Schwinger action for a free Nambu-Goto membrane which spans a closed spacetime hypercurve S”- ’ in D = (2n - 1)-spacetime, n = 2,4, 8. The sum is over all closed spacetime paths s”- ’ and functional average ( ... ) is performed over the Hopf invariant action. In the last section, we have shown that, after regularization, this functionally averaged phase is given in terms of the twisting T(S”- ‘) or the writhing W( S” ~ ’ ) number, namely,

z,=

c all closed

exp(iS,(P)

- 7liW(P)).

(5.16)

paths

While it remains a technical challenge to evaluate this path integral, the spacetime arguments via the linking theorem for fractional spin and statistics and the physical interpretation of the regularization process giving rise to T or W are clear. Anomalous spin and statistics are induced by the Hopf action via the Aharonov-Bohm effects of their nontrivial topological U( 1) gauge fields in the configuration field space. In the Nambu-Goto limit, the configuration space simplifies enormously and so does the field problem which reduces to the quantum mechanics of a relativistic charged membrane in a antisymmetric U( 1) gauge field background. The Aharonov-Bohm effect becomes effectively the one in compactilied spacetime s 2n-- ‘. The picture is then one of bare charged (n - 2) membranes tracing out s” ’ hypercurve in M. For their currents, the effective action (3.16) gives the linking number of the membrane trajectories. For a single membrane, the regularized infrared effect of the ATGF is to dress the world volume into a S”- ’ ribbon. Indeed it has little choice since by Dirac’s veto magnetic fluxes and charged membranes cannot overlap [55, 281. The latter is a membrane “dyon,” a twisting and writhing composite of a membrane with a nonintersecting sister magnetic membrane. Due to the mentioned dual effect of an Aharonov-Bohm phase and a Chern-Simons-Hopf phase, we have the peculiar quantization condition eg = i. The relevant Hopf bundle formulation of the quantum electrodynamics of such electric and magnetic membranes is still relevant in the presence of the Chern-Simons term; it has been detailed in Ref. [28]. We will not repeat it here. To proceed further, we need to follow Polyakov’s footsteps [24] and explicitly compute the corresponding membrane propagator, a difficult task to be performed.

466

TZE AND

NAM

In conclusion we have attempted a comprehensive analysis of the phase entanglement structures of two remarkable higher dimensional field theories with membrane solitons and added Hopf invariant action. We have exploited the joint implications and constraints from the geometric topology of linkages, the Gauss-Bonnet-Chern theorem, division algebras, variants of the Hopf index, the hidden gauge field structures of the K-Hopf bundles, and the topology of the configuration spaces. A case is made for the existence of higher dimensional analog of exotic spin and statistics. Several challenging issues remain to be studied: First, it is important to perform a semi-classical quantization of the single soliton membrane. As was done for the C&l) case, such an analysis will establish the explicit analytic connection between spin, statistics and the coefficient of the Hopf term. A related problem is the completion for 0 = rc of the counterpart of Polyakov’s proof [24] of Bose-Fermi transmutation for membrane solitons in the London limit. Second, it would be interesting to carry our analysis over to the supersymmetric Hopf bundles, specifically to the supersymmetric extensions of the KP( 1) a-models. In this connection and in the London limit of geometric membranes, we observe that (1) among the possible supersymmetric p-membranes lying along four sequences connected to the four division algebras K [79], a 2-supermembrane quantizable by parastatistics exists in critical dimension 7. (2) Moreover, our effective action (3.16) for the HP( l)-model coincides precisely with the D = 7 bosonic action of the 11 -+ 7 + 4 compactification [79] of the D = 11 supermembrane theory. The Hopf term corresponds to the compactified bosonic part of the Wess-Zumino term whose &coefficient is quantized as dictated by local supersymmetry. Presumably as an embedding [I201 of supersymmetric manifolds is not possible beyond D = 11, D = 15 octonionic supersymmetric S6-membranes cannot exist. These questions merit a detailed study. Finally, even more intriguing questions regarding our Hopf phases come to mind. They are (a) the statistical mechanics of classical and quantum of membranes [ 123; (b) a Chern-Weil theory for the ATGF leading to a generalized Berry phase for extended geometrical objects; (c) the formulation and representation theory of the Kadanoff-Ceva-‘t Hooft [ 121) dual algebras for the membranes; (d) the theory of D ( = 3, 7, 15)-dimensional m-links and m-knots (m = 1, 3, 7) [ 1221, of the latter’s Seifert matrices of which the generalized Gauss linking number is but a first order invariant, its connection to higher dimensional braids, and to the diffeomorphism group of S”, n = 2,4, 8; (e) the related representation theory of the current algebras with Abelian extension underlying our KP(1) o-models [14]; (f) regarding nonAbelian generalizations, besides the D > 3 counterparts to Witten’s work Cl231 connecting Jones polynomials and the D = 3 non-Abelian Chern-Simons field system, there is the phase entanglement problem of D = 3 solitons whose spacetime trajectories are the non-Abelian Z,-vortices of Nielsen and Olesen [SS, 124, 125 J. Do these solitons admit higher dimensional counterparts? What novel knot and link invariant structures do they imply? These questions are surely enticing joint areas of investigation for physicists and mathematicians.

TOPOLOGICAL

PHASE

ENTANGLEMENTS

467

ACKNOWLEDGMENTS We take pleasure in thanking Lay Nam Chang, Bob Marshak, Peter Orland, and Royce Zia for their encouragement and for helpful discussions. We are indebted to Peter Orland for a careful reading of the manuscript. C-H.T. is grateful to Dr. T. Jacobson for informing him in May 1988 of the missing numerical factor of f in the spin factor Ref. [24] and also in Ref. [22] in connection with the ribbon picture. He also wishes to thank Howard Garland for a clarifying discussion on Freudenthal suspensions and on the homotopy groups of mapping spaces. REFERENCES k. A. B. GOVORKOV. Sot. P&Y. JETP 27 (1968). 960; .4nn. Ph.vs. (N. Y.) 53 (1969), 349: A. J. BRACKEN ANU H. S. GREEN, J. Math. Phys. 14 ( 1973), 1784; .I M. LEINAAS AND J. MYRHEIM. Nuovo Cimento B 31 (1977). 1. “PCT, Spin and Statistics and All That.” 2. See. for instance, R. F. STREATER AND A. S. WIGHTMAN, Benjamin, New York, 1964; D. FINKELSTEIN, .I Math. Phys. 7 (1966). 1218; D. FINKELSTEIN ANI) J. RUBINSTEIN, J. Malh. Phys. 9 (1968), 1762; R. F. STREATEK AND 1. F. WILDE, Nucl. Phxs. B 24 (1970). 561 and references therein; J. A. SWIECA, in “Proceedings, Karpacz Winter School, 1980” (L. Turko and A. Pekalski, Eds.), Harwood Academic, Chur, 1981; Y. S. WV. Fractional quantum statistics in two-dimensional systems, in “Proceedings, 2nd Int. Symp. Found. of Quantum Mechanics, Tokyo 1986” (M. Namiki et al.. Eds. ). p. 171 and references therein. Phys. Sot. of Japan. Tokyo, 1987. 3. B. 1. HALPERIN, Phys. Rev. Lett. 52 (1984). 1583; R. B. LAUGHLIN. Phw Rec. Letr. 50 (1983). 1395: R. B. LAUGHLIN. in “The Quantum Hall Effect” (R. E. Prange and S. M. Girvin, Eds.), p. 233. Springer-Verlag, New York, 1987; G. W. SEMENOFF AND P. SODANO, Phys. Rev. Letr. 57 (1986). 119s; A. J. NIEMI AND G. W. SEMENOFF, Phys. Rec. Lett. 56 (1986). 1019. 4. S. K. KIVELSON, D. S. RORHSAR. ANI) J. P. SETHNA. Phys. Rec. B 35 (1987), 8865; S. KIVELSON. Phys. Rer:. B 36 (1987). 7237; V. KALMEYER AND R. B. LAUGHLIN. PhJ:v. Rev. Leaf. 59 (1987), 2095: R. B. LAUGHLIN, Php. Rec. Left. 60 (1988), 2677. 5. P. B. WIEGMAN. Phys. Ret!. Let/. 60 (1988), 821; 1. E. DLIALOSHINXY. A. M. POLYAKOV, i\\v P. B. WIEC;MAN, Phys. Ret. Letr. 127 (1988). 112. 6. F. WILCZEK. Phys. Ret>. Lett. 48 (1982), 1144; 49 (1982). 957; J. FR~~HLICH AND P.-A. MAR~HETTI. “Quantum Field Theory of Anyons,” ETH-HGnggerberg preprint, 1988. 7. F. WILCZEK AND A. ZEE. Phys. Ret]. Lert. 51 (1983). 2250. 8. A. J. NIEMI AND G. W. SEMENOFF, Phys. Reo. LetI. 51 (1983), 2077. 9. Y-S. WU. Phys. Rev. Left. 52 (1984). 2103; Phys. Rec. Left. 53 (1984). 111. 10. A. M. DIN ANY W. J. ZAKRZEWSKI, Phys. Left. B 146 (1984), 341; Y-S. Wu ANL) A. ZEE. Ph1.t Lc,rt. B 147 (1984), 325; T. JAROSZEWICZ. Phys. Letf. B 159 (1985). 299. 12. D. AROVAS, R. SCHRIEFFER, F. WILCZEK. AND A. ZEE, Nuc~. Phx B 25 ( 1985). 117. 13. T. H. HANSSON, M. ROCEK. 1. ZAHEV, AND S. C. ZHANC. Phy,s. Letr. B 214 (1988). 475; A. S. GOLUHABER. R. MACKENZIE, AND F. WILCZEI(. “Field Corrections to Induced Statistics.” Stony Brook-Harvard Preprint S/88. 14. M. J. BOWICK, D. KARABAU AND L. C. R. WIJEWARDHANA. Nucl. Phy,t B 271 (1986), 417. 15. see also J. FRBHLICH, Statistics of fields, the Yang-Baxter equation, and the theory of knots and links. in “Nonperturbative Quantum Field Theory. 1987 Cargese” (G. ‘t Hooft et al., Eds.). Plenum, New York, 1988; Statistics and monodromy in two and three dimensional quantum field theory. in “Proceedings, 1987 Como Conference” (K. Bleuler et al.. Eds.). 16. A. K. DEWDNEY, “The Planiverse,” Poseidon Press, New York, 1984. 17. E. P. WIC;NER, Ann. qf Math. 40 (1939). 149; V. BARGMANN. Ann. of’ Math. 48 (1947). 568. 18. S. DOPLI~HER, R. HAAG, AND J. E. ROBERTS. Commun. Marh. Phys. 35 (1974). 49. 19. D. BWHHOLZ AND K. FREDENHAGEN, Nucl. Phys. B 154 (1979), 226: Commun. Math. Phl,s. 84 (1982), 1.

468

TZE AND NAM

20. C. W. MISNER, K. S. THORNE, AND J. A. WHEELER,“Gravitation,” Freeman, San Francisco, 1973. 21. F. W. WIEGEL, “Introduction to Path Integral Methods in Physics and Polymer Sciences,” World Scientific, Singapore, 1986. 22. C-H. TZE, Inf. Jour. Mod. Phys. A 3 (1988), 1959. Erralwn: Inconsistency in the normalization of the Hopf action led to a net missing factor of 1 in the Hopf phase factor G(P). In this connection see Ref. [27] and our acknowledgments, and see Polyakov’s 1988 Les Houches Lecture Notes, Refs. [24,23]. 23. J. GRUNDBERG,T. H. HANSSON,A. KARLHEDE, AND U. LINDSTR~~M,Spin, statistics and linked loops, Stockholm preprint, 1988. 24. A. M. POLYAKOV, Mod. Phys. Left. A 3 (1988), 325; see, in particular, his 1988 Les Houches Lectures. There the missing factor of f in his spin factor was restored. 25. A. BELAVIN AND A. M. POLYAKOV, JETP Left. (Engl. Transl.) 22 (1975), 245. 26. E. WI~TEN, Nucl. Phys. B 223 (1983), 422,433; A. P. BALACHANDRAN,Skyrmions, in “High Energy Physics 1985%Proceedings, of the Yale TASI” (M. J. Bowick and F. Giirsey, Eds.), World Scientific, Singapore, 1985; Wess Zumino terms and quantum symmetries, in “Proceedings, Workshop on Skyrmions and Anomalies, Cracow, 1987” (M. Jezahek and M. Praszatowicz, Eds.), p. 375, World Scientific, Singapore, 1987. 27. T. JACOBSON,Boston University Preprint BUHEP-88-12; P. ORLAND, VP1 Preprint VPI-IHEP88/6; K. JOHNSON,MIT Preprint CTP 1656 (1988) and references therein. 28. C-H. TZE AND S. NAM, Phys. Left. B 210 (1988) 76. We note that here as in Ref. [22] a missing numerical factor of 4 in the Hopf phase Q(P) should be corrected. 29. M. V. BERRY,Proc. Roy. Sot. London A 392 (1984), 45; B. SIMON, Phys. Rev. L&r. 51 (1983) 2167; H. KURATSLJJIAND S. IIDA, Prog. Theor. Phys. 74 (1985), 439. 30. B. A. DUBROVIN, A. T. FOMENKO, AND S. P. NOVIKOV, “Modern Geometry-Methods and Applications,” Part II, Chap. 3, Springer-Verlag, New York, 1985. 31. See, for example, L. H. KAUFFMAN, “On Knots,” Princeton Univ. Press, Princeton, 1987. 32. V. F. R. JONES,Bull. Amer. Math. Sot. 12 (1985), 103. 33. P. FREYD, D. YETTER, J. HOSTE, W. B. R. LICKORISH,K. MILLETT, AND A. OCNEANU, Bull. Amer. Mud Sot. 12 (1985), 239. 34. S. F. EDWARDS, J. Phys. A, Ser. 2, 1 (1968), 15; see also M. DELBRUCH,Proc. Symp. Appl. Math. 14 (1962), 55. 35. G. CALLJGAREANU,Rev. Math. Pures Appl. 4 (1959), 5; Czech. Math. J. 11 (1961), 588. 36. J. H. WHITE, Amer. J. Math. 91 (1969), 693. See also W. F. POHL, J. Math. Mech. 17 (1968), 975. 37. K. F. GAUSS, “Werke,” Vol. 5, p. 605, Konig. Gesellschaft der Wissen., Gottingen, 1877; G. W. DUNNINGTON, “Carl Friedrich Gauss, Titan of Science,” p. 222, Hafner, New York, 1955. 38. H. FLANDERS,“Differential Forms,” Chap. 6, Academic Press, New York, 1963. 39. M. G. BRERETONAND S. SHAH, J. Phys. A 13 (1980), 2751. 40. M. RASETTI AND T. REGGE, Physica A 80 (1975), 217. 41. M. D. FRANK-KAMENETSKII AND A. V. VOLCMXDSKII, Sov. Phys. Usp. (Engl. Transl.) 24, No. 8 (1981), 679. 42. F. BROCK-FULLER, Proc. Nat. Acad. Sci. U.S.A. 68 (1971), 815; Proc. Natl. Acad. Sci. U.S.A. 75 (1978), 3557. 43. F. H. C. CRICK, Proc. Nat. Acad. Sci. U.S.A. 73 (1976), 2639. 44. W. R. BAUER, F. H. C. CRICK, AND J. H. WHITE, Sci. Amer. 243 (1980), 118. 45. F. ZACCARIA et al., Phys. Rev. D 27 (1983), 2327; A. P. BALACHANDRAN, H. GOMM, AND R. D. SORKIN, Nucl. Phys. B 281 (1987) 612. 46. M. MINAMI, Prog. Theor. Phys. 62 (1979), 1128; L. H. RYDER,J. Phys. 13 (1980) 437. 47. I. J. R. ATCHISON, Acta Phys. Pal. B 18 (1987), 207 and references therein. 43. S. P. NOVIKOV, Russ. Marh. Surv. 37, No. 5 (1981 j. 1 49. W. F. BRINKMAN AND M. C. CROSS,in “Progress in Low Temperature Physics” (D. F. Brewer, Ed.), pp. 105-198, North-Holland, Amsterdam, 1978. 50. J. C. GARRISONAND R. Y. CHIAO, Phys. Rev. tiff. 60 (1988), 165 and references therein.

TOPOLOGICAL

PHASE

ENTANGLEMENTS

469

51. J. WESS AND B. ZUMINO, Phys. Left. B 37 (1971), 95; S. P. NOVIKOV, Usp. Mat. Nauk 37 (1982). 3. 52. F. WILCZEK AND A. ZEE, Phys. Rev. Left. 52 (1984). 2111. 53. Y. S. WV AND A. ZEE, Phys. Left. B 207 (1988). 39; R. I. NEPOMECHIEAND A. ZEE. in “Quantum Field Theory and Quantum Statistics (Fradkin Festschrift)” (I. A. Batalin rf al., Eds.). Vol. 2. Hilger, Bristol, 1986. 54. P. G. 0. FREUND,Chicago preprint (1981). EFI 81/07 (unpublished): R. I. NEPOMECHIE,Phys. Rrr. D 32 (1985). 1921. 55. C. TEITELBOIM, Phvs. Left. B 167 (1986), 69; M. HENNEAUX AND C. TEITELBOIM, Found. Phw. 16 (1986), 593; see also P. ORLAND, Nucl. Phys. B 205 (1982). 107. 56. J. F. ADAMS, Ann. q” Math. 72 (1960), 20; J. F. ADAMS ANI) M. F. ATIYAH. Q. J. Math. 17, No. 2 (1966). 31. 57. F. LONDON AND H. LONDON, Proc. R. Sot. London A 149 (1935). 71. 58. H. B. NIELSEN AND P. OLESEN, Nucl. Phys. B 61 (1973), 45. 59. G. TOULOUSEAND M. J. KLEMAN, J. Phys. Left. 37 (1976), L-149. 60. H. UWEZAWA. H. MATSUMOTO, AND M. TACHIKI, “Therm0 Field Dynamics and Condensed States,” North-Holland, Amsterdam, 1982. 61. M. KALE AND P. RAMOND, Phys. Rev. D 8 (1974) 2273. 62. S. DESER, R. JACXIW, AND S. TEMPLETON, Phw. Rev. Lert. 48 (1982), 975; .4nn. Phys. f N. Y.j 140 (1982). 372; J. SCHONFELD,Nucl. Phys. B 185 (1981), 157; S. S. CHERN AND J. SIMONS. Ann. Murh. 99 (1974), 48. 63. V. N. POPOV, “Functional Integrals in Quantum Field Theory and Statistical Mechanics.” Reidel. Dordrecht, 1983. 64. See, for example, H. W. GUGGENHEIMER.“Differential Geometry,” Dover. New York. 1977: M. SPIVAK. “Differential Geometry,” Vol. II, Chap. 1, Publish or Perish. Berkeley, 1979. 65. T. F. BANCHOFFAND J. H. WHITE, Math. Stand. P 36 (1975), 254. 66. P. G. HILTON. “An Introduction to Homotopy Theory,” Chap. 6, Cambridge Univ. Press, Cambridge. 1961. 67. J. H. WHITE. J. Diff: Geom. 4 (1970), 207. 68. S. S. CHERN. Ann. Math. 43 (1942), 178; 46 (1945), 674; “Topics in Differential Geometry,” IAS lecture notes. Princeton, 1951; for an explanation for pedestrians see H. ABELSON AW A. A. DISESSA,“Turtle Geometry,” MIT Press, Cambridge, 1979. 69. H. HOPF. Math. Ann. 104 (1931), 637; Fund. Marh. 25 (1935). 427. 70. See, for instance, 0. K. KALASNIKOV, S. E. KONSHTIXN, AND E. S. FRADKIN, SOC. J. NW+. P/~J,s. 29, No. 6 (1979). 852; S. EILENBERGAND I. NIVEN. Bull. Amer. Math. Sot. 50 (1944), 246; F. G~TRSEY AND C-H. TZE, Phps. Rep. C, in preparation, 71. F. G. FROBI?NIUS. “Gesammelte Abhandlungen,” Springer-Verlag, Berlin, 1968. 72. A. HURWITZ, Nachr. Ges. Wiss. Giiltingen (1898). 309; Math. Werke, Bd.II, 565. 73. D. HUSEMOLLER.“Fibre Bundles,” Springer-Verlag, Berlin, 1975. 74. C. GODBILLON. “Elkments de Topologie Algkbrique,” Hermann, Paris, 1971; J. HERTEL. DESY Report, 1976 (unpublished); H. J. DE VEGA, Phys. Reo. D 8 (1978), 2945. 75. J. H. C. WHITEHEAD. “The Mathematical Works” (I. M. James, Ed.). Vol. 1, p, 317, Pergamon. Oxford, 1962. 76. M. W. HIRSCH.“Differential Topology,” Springer-Verlag, Berlin, 1976. 77. M. B. GREEN. J. H. SCHWARZ.AND E. WITTEN. “Superstring Theory.” Vols. 1, 11. Cambridge Univ. Press. Cambridge, 1987. 78. M. J. DUFF. B. E. W. NILSSON, AND C. N. POPE, Ph~.~. Rep. 130 ( 1986 ). I. 79. M. J. DUFF. Class. Quanf. Grau. 5 (1987), 189. 80. F. ENGLFRT, Phw. Lelt. B 119 (1982). 339; F. G~;‘RSEYAW C-H. TZE. Ph?s. Lerr. B 127 (1983). 191. 81. N. STEENROD. “Topology of Fibre Bundles,” Princeton Math. Series. Princeton, 1974; P. HILTON. “General Cohomology Theory and K-Theory.” Cambridge Univ. Press, Cambridge, 1971;

470

TZE AND

NAM

F. VAN DER BLIJ, History of the Octaves, in “Wis- en Natuurkundig Tijdschrift” (S. Stevin, Ed.), 34” Jaargang, Aflevering III (1961), 106. 82. H. TODA, C.R. Acad. Sci. Paris 241 (1955), 849. 83. A. TRAUTMAN, Int. J. Theor. Phys. 16, No. 8 (1977), 561; J. NOWAKOWSKI AND A. TRAUTMAN, J. Math. Phys. 19, No. 5 (1978), 1100. 84. M. F. ATIYAH, N. J. HITCHIN, V. G. DRINFELD, AND Yu. 1. MANIN, Phys. Lelt. A 65 (1978), 185; V. G. DRINFELD AND Yu. I. MANIN, Commun. Math. Phys. 63 (1978), 177. 85. J. W. MILNOR,

86. 87.

88. 89. 90. 91. 92. 93. 94.

in “Lectures

on Modern

Mathematics”

(T. L. Saaty,

Ed.),

pp. 165-183,

Wiley,

New York, 1963; J. EELLS AND N. KUPIER, Ann. Math. Pure Appl. 60 (1963), 63; “Differential and Combinatorial Topology,” Princeton Univ. Press, Princeton, NJ, 1965. E. MARTINELLI, Ann. Mat. Pura. Appl. 49 (1960), 73; 50 (1960), 135. G. E. VOLOVIK AND V. P. MINEEV, Zh. Eksp. Teor. Fiz. 72 (1977), 2256; 73 (1977), 767; Y. BOULIGAND, B. DERRIDA, V. POENARU, Y. POMEAU, AND G. TOULOUSE, J. Phys. 39 (1978), 863. D. J. FINKELSTEIN,J. Math. Phys. 7 (1966), 1218. L. D. FADDEEV, in “Relativity, Quanta and Cosmology” (M. Pantaleo and F. de Finis, Eds.), Vol. 1, p. 247, Johnson Reprint, New York, 1979. L. J. BOYA AND J. MATEOS, J. Phys. A 13 (1980), L285. Y. S. Wu AND H.-Y. Kuo, Sci. Sinica 20 (1977), 186; M. MINAMI, Prog. Theor. Phys. 63 (1980), 303. C. N. YANG, J. Math. Phys. 19 (1978), 320. B. GROSSMAN, T. W. KEPHART, AND J. D. STASHEFF, Commun. Math. Phys. 96 (1984), 431; F. G~~RSEY AND C-H. TZE, Phys. Z&t. B 127 (1983), 191; R. DONDARER, F. G~~RSEY,AND C-H. TZE, Nucl. Phys. B 266 (1986), 440. K. NISHIJIMA, Nuovo Cimento 11 (1959), 698; F. G~RSEY, Nuovo Cimento 16 (1960), 230; Ann. Phys.

(N.Y.)

12 (1961),

91; M. GELL-MANN

AND M. LEVY, Nuovo

Cimento

16 (1958),

705.

95. J. EELLS AND L. LEMAIRE, Bull. London Math. Sot. 10 (1978), 1. 96. G. T~TH, “Harmonic and Minimal Maps with Applications to Geometry and Physics,” Ellis Horwood, Chichester, 1984. 97. R. HOBART, Proc. Phys. Sot. London 82 (1963), 210; G. H. DERRICK, J. Math. Phys. 5 (1964), 1253. 98. Higher dimensional o-models built on spheres and K-Hopf maps, but without an added Hopf invariant action, have been studied by K. Fujii, Left. Math. Phys. 10 (1985), 49; H. DATE, K. FUJII, AND H. SATO, Lett.

Mad

Phys. 13 (1987),

195; K. FUJII,

H. So, AND M. SUWA, Lelr.

Math.

Phys.

15 (1988), 151 and references therein. M. SCHLINDWEIN, AND Q. SHAFI, J. Phys. A 9 (1976), 1513. L. D. FADDEEV, JETP Lett. 21 (1974), 64. See, for instance, F. G~~RSEY AND C-H. TZE, Ann. Phys. (N. Y.) 128 (1980), 29. M. MSCHER, Phys. Lett. B 78 (1978), 465; A. AURILIA et al., Phys. Lett. B 95 (1980), 265. An independent derivation of these formulae was obtained by Y-S. Wu, private communication from Y-S. Wu to C-H. TZE. 104. P. H. FRAMPTON AND T. W. KEPHART, Phys. Rev. D 28 (1983), 1010. 105. See, for example, D. FORSTER,Nucl. Phys. B 81 (1974), 84. 106. V. N. POPOV, Sov. Phys. JETP 37 (1973), 341. 107. G. SEMENOFF, Phys. Rev. LetI. 61 (1988), 517. 108. J. W. MILNOR, in “Relativity Groups and Topology II” (B.S. de Witt and R. Stora, Eds.), Les Houches Session XL, 1983, North-Holland, Amsterdam, 1984. 109. J. WILLIAMS, J. Mafh. Phys. 11 (1970), 2611; M. F. ATIYAH AND J. D. S. JONES,Commun. Math. Phys. 61 (1978), 97. 110. G. W. WHITEHEAD, Ann. Math. 47 (1946), 460; J. H. C. WHITEHEAD, Ann. Math. 58 (1953), 418. 111. V. L. HANSEN, Q. J. Math. 25 (1974). 313. 112. W. HUREWICZ, Proc. Akad. Amsterdam 38 (1935). 112; 39 (1936), 117, 215. 113. D. RAVENEL AND A. ZEE, Commun. Math. Phys. 98 (1985), 239. 114. J. HOPPE AND H. NICOLAI, Phys. Letf. B 1% (1987), 451. 99. A. PATANI,

100. 101. 102. 103.

TOPOLOGICAL

PHASE

ENTANGLEMENTS

471

115. M. LOWE, J. Phys. A 15 (1982), 1741 and references therein. 116. L. SCHLILMAN. Phys. Rev. 176 (1968), 1558; “Techniques and Applications of Path Integration.” Wiley, New York, 1981. 117. M. G. G. LAIDLAW AND C. M. DEWITT, Phys. Rev. D 3 (1971). 1375; J. S. DOWKER, J. Phys. .1 5 (1972), 936; C. J. ISHAM. Phys. Left. B 106 (1981), 188. 118. P. G. 0. FREUND AND R. I. NEPOMECHIE, Nucl. Phys. B 199 (1982). 482. 119. See, for example, M. DUFF et al., Nucl. Phys. B 297 (1988), 515. 120. T. CURTRIGHT. CERN Preprint CERN-TH-4924/87, 1987. 121. L. KADAN~FF ANL) H. CEVA. Phys. Rev. B 11 (1971). 3918; G. ‘T HOOFT. Nucl. Phw. B 138 (1978). 122. M. KERVAIRE, Bull. Sot. Math. France 93 (1965). 225; J. LEVINE. Comment. Math. Heit!. 44 ( 1969). 229; D. ROLFSEN, “Knots and Links,” Publish or Perish, Berkeley, 1976. 123. E. WITTEN, “Quantum Field Theory and The Jones Polynomial.” IASSNS-HEP-88133. 19X8; in “Proceedings, IAMP Congress, Swansea. 1988.” 124. S. MANDELSTAM, Phys. Letl. B 53 (1975). 476; C-H. TZE ANI) 2. F. EZAWA. Phys. Rev. B 14 (1976). 1006. 125. Y-H. CHEN AND F. WILCZEK, HUTP-88/A016.