Geometry and duality in supersymmetric σ-models

-~ EI2EVIER

NUCLEAR PHYSICS

B

Nuclear Physics B 469 (1996) 488-512

Geometry and duality in supersymmetric o--models T h o m a s Curtright a'l, T s u n e o Uematsu

b,2,

C o s m a s Zachos ~'~

a DeparOnent of Physics. University of Miami. Box 248046, Coral Gables, FL 33124, USA b Department of Fundamental Sciences. kTHS, Kyom University, Kyoto 606-01. Japan c High Ener~,,y Physics Division. Argonne National Laboratory,. Argonne. IL 60439-4815, USA

Received 19 January 1996; accepted 14 March 1996

Abstract

The Supersymmetric Dual Sigma Model (SDSM) is a local field theory introduced to be nonlocally equivalent to the Supersymmetric Chiral nonlinear o'-Model (SCM), this dual equivalence being proven by explicit canonical transformation in tangent space. This model is here reconstructed in superspace and identified as a chiral-entwined supersymmetrization of the Dual Sigma Modcl (DSM). This analysis sheds light on the boson-fermion sympl,ysis of the dual transition, and on the new geometry of the DSM. PACS: I I.l().Ef; 11.30.Pb; I 1.30.Rd: 11.40.Ex: I 1.4().Dw Keyword~: Duality; Canonical Transformations; Sigma models; Supersymmetry; Symphysis; Bosonization

1. Introduction and conclusions A long-standing broad question in field theory involves the equivalence of tield theorics which may appear very different. Historically, physicists have been comfortable

with local changes of field variables (as in the Higgs mechanism) which correspond to point transformations in classical mechanics: these automatically preserve the canonical structure of the theory, i.e. the Poisson brackets, and the canonical commutators upon quantization - all of which may also be addressed equivalently in the conventional functional integral formalism. A subtler issue arises, however, when nonlocal translbrmations are considered, which link two local field theories. As a rule, nonabelian ( t - ) d u a l i t y transformations in two i E-mail: [email protected] -' E-mail: [email protected] E-mail: [email protected] 0550-3213/96/SI 5.00 Copyright @ 1996 Elsevier Science B.V. All rights reserved PIIS0550-3213(96)00138-1

T. Curtright et al./Nuclear Physics B 469 (1996) 488-512

489

dimensions (popular in string culture [ I - 3 ] ) , which broadly map gradients to curls, hence waves to solitons, are of this type: equivalence results when these are c a n o n i c a l 4 of the nontrivial type that mixes canonical momenta with field gradients, which results in nonlocal maps. Conversely, failure of such transtormations to preserve the canonical structure leads to striking inequivalences in such theories (cf. the PCM, a double limit contraction of the W Z W N o'-modcl, in [8,9] ). Unfortunately, so far, there is no systematic theory of such transformations in field theory', and most discussions merely abstract and change notations on a handful of examples. The only examples available are in two dimensions: Sine-Gordon/Thirring 1101" C M / D S M [8,9]; and, tinally, S C M / S D S M [1 1]. The second system was first introduced via tirst-order functional integral manipulations [12] and the equivalence was shown to be canonical [8] at the classical level, and argued 18] to extend to the quantizcd theory 5 ; moreover, a credible case has been made for complete equivalence of the respective S-matrices of the two theories [ 15]. The third system was introduced in [11] and illustrates some intricacies of the nonlocal canonical equivalence best, by dint of symphysis, which is the conjoining in the transition maps of fermion bilinears with bosons to yield transformed bosons, and of t'ermions with bosons to yield transformed fermions. It was first constructed via a generating functional ansatz [ 11 ] and first-order Lagrangian quadrature of the type 1121, and not as the supersymmetrization of the bosonic DSM (which, unlike the CM, contains torsion). However, we demonstrate here that the s a m e model may also be constructed by judiciously supersymmetrizing the bosonic DSM; in the past [ 16,17] we have provided a general procedure of supersymmetrizing any given torsionful manifold, such as that of the DSM. Thus, a superfield construction of the same model from a different starting point sheds new light on the supersymmetry realization at work. Here, we discuss more explicitly detailed aspects of the SDSM, with special emphasis on the special realization of supersymmetry and the superfields controlling it, to shed light on the intriguing chiral entwining structure at work. We review the bosonic case in Section 2. We then review the supersymmetric theories including tangent-space supersymmetric actions in Section 3 and then we then summarize Ref. [ 11 ] by way of introduction of the SDSM in Section 4. In Section 5 we proceed to rederive the SDSM via supcrficld extension of the DSM and Fridling-Jevicki-type quadrature in superspace, and explain and support special features observed before which appeared accidental. As a guide to insight of the field theory results, a brief review of canonical transformations of classical mechanics from the more modern, Poisson-bracket-based point of view is provided in Appendix A. Subtle relations between curved manifold and tangent space in the dual thcory, specifying a n e w inhomogeneous gca~metry are illustrated in Appendix B. Every eflbrt is made to stay consistent to the conventions of Ref. [ 1 1 ] ; we ~ I.e.. preserve canonical commutation relations. Canonical transformations in quantum mechanics underlie l)hac's path integral formulation 14 I. and have been discussed extensively in field theory, e.g. [ 5-7 ]. 5 Rcf. [ 13 [ has introduced objections to this identification, based on observation of the respective effective actions at two loops, supported by ]14]. We care to suppose thai, with proper appreciation of the new underlying geometry, some form of identitication may eventually go through.

77 Curtright et al./Nuclear Ph)'sic.v B 469 (1996) 488-512

.190

correct a typographical error in the part of the action involving a factor of 3/8 in the quartic fermion interaction.

2. Review of the bosonic theories Recall the standard bosonic nonlinear Chiral Model (CM) on 0 ( 4 ) ~_ 0 ( 3 ) × O ( 3 ) ~ _ S U ( 2 ) × S U ( 2 ) . In geometrical language, I

~

a~#

h

£cM = 5g,,ha~,q~ a ~p ,

(2.1)

where g,,/, is the metric on the field manifold (three-sphere). Explicitly, with group elements parameterized as U = q~0 + irJ~oi ( j = 1,2,3), where (~p0)2 + q~2 = 1, and ~p2 =_ ~ i ( ~ i ) 2 , we may resolve q~0 = + X / I _ q~2 to obtain

g,,j, = 6 'a' + q~"~J' / ( 1 - ~2),

(2.2)

and hence

1 (

£cM = ~

~°i~J ~

"

~,JwJ

30 + 1 - ~p2J a,uq~'a'%Pa = ~"u ~

(2.3)

'

where we re-expressed the action in terms of tangent quantities through the use of either left- or right-invariant vielbeine (e.g. [ 17] ) - either choice yields this Sugawara current-current form. Choosing left-invariant dreibeine gives "V + A" currents which are vectors on the tangent space. In terms of the above explicit coordinates,

U-J auU = - i r J J~u,

(2.4)

J~ = -t:~,a u ~oa ,

( 2.5 )

c[, = VII - ¢2g, v + e'Oh ~ph.

(2.6)

where

Note that these currents are pure gauge, or curvature-free, such that

e'**~" (31aJ~, + - ~i.ikjj - # -jk~ v ] = O. •

(2.7)

i

This is canonically equivalent to the Dual o'-Model [12] ( D S M ) 6 with torsion and a new geometry,

EnsM - ~1 + 14@ (~ (t~ ij -~ 4qbiqbJ)cg~z~io#tl~ i -- glz"eijkqbioolxtlrlavtlbk )

.

(2.8)

This new geometry is explained in Appendix B, and, in particular, how the tangent- and curved-space indices of the field/coordinates q~ in this geometry are indistinguishable. a This m o d e l was also c o n s i d e r e d in the last reference o f Ref. J I ] and the c o m b i n a t i o n o f [ 15 I.

T. Curtright et al./Nuclear Physics B 469 (1996) 488-512

491

The generator for a canonical transformation relating ¢> and 40 at any fixed time is the tangent-space functional [ 8 ] F[ 40, ¢] = f-~oo dx 40ij) [ ~]. Specifically,

F[40,¢1 =

-e~cfdx40 ! i (1vi_~_~2~.~ x q3i+~ijk~j~_~ - -

.

(2.9)

IeX3

Although we originally constructed F in the Hamiltonian framework by the indirect reasoning reviewed below, its structure is also evident within the Lagrangian framework as follows. Treating J as independent variables in £CM = ~J~uJ uj, impose, with Ref. [ 12], the pure gauge condition on the currents by adding a Lagrange multiplier term, ~-~a = 40i8#r('~ IJ + ~_jkllktl ,-,~,, J~,J,,). Then, complete a square for the J's and eliminate them from the dynamics (integrate them out) in favor of the DSM. But to do this, t The total divergence one first writes £a = cT~, ( ~lVeU~ J)~ ) eu" JJ P,~,q~i + .t:..kt~zu~JkJ . . . u - ~" term. intcgrated over a world-sheet with a constant time boundary, gives our generating functional relating the CM to the DSM 7. The conjugate momentum of 40i specified by the generating functional is -

(

¢+>'

'

•

)

L,,J

(2.10)

Thc conjugate of ¢" is

~i -

~F[40,¢] 6¢ i ~ _ "}- siJk qgk) 'oA gO=. q~' = ( ~/1-- q92~ ij "}- V/I --¢ ' ---g92 ~- (

2

(qgiclfl -- qgiq~) -- 2t3ijk40k)

(2.11)

To solve for the fields themselves, e.g. 40[¢], their canonical momenta may be eliminated through substitution for Hi and "nri, in terms of a~oi/ot and a40J/~gt, as follows from £1 and £3:

Hi - ~1 + '440 ( ( 6 0 + 4 4 0 i 4 0 a ) ~ iOt wi =

~ii

-F 1 - ¢ 2 ]

~go .

+ 2eiJkq~l ff--~ q~ ) ' (2.12)

7 Roughly speaking, the Lagrange multiplier dual field q~ characterizes the ratio of normal magnitudes of the Sugawara Lagrangian and the zero-curvature constraint, respectively, in the function space of currents, upon cxtremization. Recall constrained extremization of a surface f(x, y) - ag(x, y) via the Lagrange multiplier ~.. Vanishing variation specifies that the constraint surface g ( x , y ) = 0 touches the surface z = f ( x , y ) at the contact point (.to, Yo, zo = .f(xo, Yo)). The section plane z = za~ intersects the respective surfaces at two curves which are tangent to each other at the point of contact; the normals to these two curves on this plane arc parallel, the ratio of their magnitudes being A.

7~ Curtright et

492

al./Nuclear Physics B 469 (1996) 488-512

However, the resulting transformation laws are complicated and nonlocal, as illustrated at the end of this section. Instead, it is relatively more instructive to simply identify the conserved, curvature-free current in the two theories, consistently to the above, an identitication which will turn out to be local. It is then straighttbrward to exploit the current-current form of the respective Hamiltonian densities which will thus likewise identify. Now, then, in the DSM, what is the conserved, curvature-free current? In contrast to the PCM, where it was essentially to be a topological current, here a topological current by itself will not suffice; neither will a conserved Noether current. (Under isospin transformations, 6 0 ' = the Noether current of £3 is = ~/~3/6(c)#O9 i) = 8,iikqsJHk, but it is not curvature-free.) Instead, the conserved, curvature-free current f f / u [ O , H ] = Jff[~o,w'] (identified with Jff of the CM) is a of the Noether isocurrent and a topological current, fffl = 21ff - em'0,,O i, so that ~ l = Both conservation and curvature-freedom now hold on-shell, for the on-shell identified

forced

eOkOJwk,

SO li°

lff

mixture

I1,.

current g ,

-I "7'u - ~1 -+~ 4 0

((60 + 4@ioi)eu,,a,,q~a + 2eUkCaauOk).

(2.13)

Neverthclcss, the following canonical identifications of currents can be shown [8] to hold off-shell:

( Z ~ ~/6

Z°

-

=

,,/I -

~2au +

~oi~oI sijk~k) Lq~J~J], v/1 _ ¢,~

a 0'-2v.u~¢allk = - v / I - ¢ , 2 ~ i -

.... a x

(2.14)

ax

~iikCJmk = ~ .

(2.15)

These two relations combine to integrate to the explicit nonlocal map [9], O( x) • r = U-J ( x)U(O)q~(O)

• r U -1

(0)U(x)

~-U-'(x) (bfdyiU(y)aoU-'(y)) U(x ).

(2.16)

The dual character of this nonlocal transition is manifest in the weak tield limit. It bears repeating that canonical commutators of such complicated nonlocal expressions are simple and conventional, precisely because the transformation is canonical: e.g. equal time commutators of two expressions of this type at different space points x and z vanish, as these expressions are essentially local, despite formidable appearances! Substituted into the Sugawara-current-current Hamiltonians, these locally identified currents (2.14), (2.15) further lead to mutual local identification of the respective Hamiltonian densities, 7-~CM =

JoJo + JlJl

= JoJo

4- Jl,,~l = 7~I)SM.

(2.17)

The equations of motion are necessary, as Hamihon's equations have already been utilized in the elimination of the canonical momenta from the subsequent expressions.

T. Curtright et al./Nuclear Physics B 469 (1996) 488-512

493

In terms of the dreibeine of the new dual geometry (discussed in Appendix B), the currents of the DSM read ~

(2.18)

= - ( g ( j a ) e # v O v q ~a + VljalOlaqsa).

The reader may contrast these currents with those of a WZWN model on a group manifold. The currents for the latter are obtained by keeping the vielbein intact: J~ = -~,,{ (aaeJ" + neu,Y4)"). (For the CM, r / = 0.) Also note that the DSM dreibein already appeared in the above action, £DSM = /V~¢ (Olz CI)aO#q~l + 13#vOtxcl)ac~v~)J).

3. Supersymmetric theories In a direct application of the general construction for supersymmetric or-models with torsion 117,16] (whose conventions we use), the supersymmetric extensions of the two bosonic models above through the addition of Majorana fermions, the SCM [ 18] without torsion, and the s d s m with torsion, are readily read off. In the following sections we canonically transform between the two. We first review this in component formalism [ 11 ] but in the next section we will provide a complementary picture in superspace which illuminates and confirms our construction. The SCM is £SC~.1 = 7J ( g"t'bu q~''o#qJ' +~g,,b~ls~)Js " -~'~ l, +g~,,hcd~U to 7 , , .qJc-rb--d'~ qJ ~U )

(3.1)

= 51 ( gaha#qgacg#q~h + tgah~P • --a O~b/'+ ](galAb I --a ~Pb ) 2' ,

where rb ~

crd

a

D.~lfl' = c~u~pI' + tcdau~v ~ ,

,aja

d

F~,c = ¢"gh,: = t: ,,¢~,t.c),

R,z,,d = g,,cg~,d -- g,,dgl, c =Rcd,#,, g,,I, = ~,h _ ~p,tqJ, = vail,hi,

( 3.2 ) t,ai = V/1

_

~2~lj

+

13ajb~b

(3.3)

t',~ = V/I - q~2ga i + e"it'qJ '.

The Cartan-Maurer relations of the dreibeine merit recall .

-

= ~

?yaq~l, _ 8it,~,,

car # = ej,a, +

(3.4)

V/1 - q)2

The conserved currents now consist of the previous bosonic terms augmented by spinor hilinears, Cui = Jui + K ~i ,

Jui = - c ,i,,c?u~ a '

1-;°ijk"j"k'~'a~' "l'b K~i = 2,~ ,',,'~t,w r u w •

(3.5)

The last term, explicitly, is given by the above results for the Cartan-Maurer relations, , Ku

=

i • iah7 a --h 5_te qs yuqs

i ~"~7"Yu~bi -

V/I

_~2

(3.6) "

494

"ECurtright el al./Nuclear

Physics B 469 (1996) 488-512

This is to bc expected: the tangent-space lefi-invariant spinor of (A.51) specified in Ref. II71, X j = c~',~//',

(3.7)

transforms as 6 X .~ = eJkt,~kXt under a full V + A transformation. The right rotation in tangent space transforms linearly, and the spinor's contribution to the corresponding current is that of a conventional isorotation. Recall (Ref. [17], (A.52), (A.29)) that now the Lagrangian simplifies significantly to a mere function of tangent-space spinors and currents, £scxl = ~I( J)uJ u' + i ~ X

' " t + ¼()(X) 2). l + ie'ktXJ~Ikx

(3.8)

This tangent-space formulation is at the heart of the canonical transformation, as will become apparent in the following section. The supersymmetry transformation laws are 6 X ) = i]J e - ½eJkt(yt, e2kyt, y t + y u e f ( k y t z X t ) , 3J~ = -@( OuX -i + eikt J ~ x t ) .

(3.9)

The bosonic generating functional can then be re-presented as DO

FIq~, ~] =

/ •-

dx@t.',,Oiq~ .

(3.10)

.'2v.~

The above supersymmetry transformations follow from the general case, ,3~ = @¢,,

6¢; = (~ - i 4 ¢ ) e ,

6~ = - i g ~ ¢ ,

(3.1 l)

where wc use the conventions of Ref. [ 17], Eq. (5.7), 2 ~ " = l'~c~t'~b '" - g~,.~l'rt,¢",

(3.12)

where the torsion S vanishes here - but, of course, not for the £sdsm model, below. In this case, the SCM equations of motion, written covariantly, directly lead to conservation of the supercurrent of Ref. [ 18], s u = -iq~o"yug,#,~, I' = i~l i y u X j.

( 3.13 )

Correspondingly, we may express the Lagrangian in terms of conserved vector currents,

CscM =

I

(c/,c"' + r-xx x; +

3 (f(jxJ)2)

(3.14)

with supersymmetry transformations 8 X j = i6Se - ½ y p e e i k t ( f ( k y p X t ) ,

v k X 1 ). 6 C )u = - g ( O u X ) + e J'kl euC~,Yt,

(3.15)

Taking n = 3/2 in Ref. [ 17], the conventional supersymmetrization Z2.~smof the DSM (which, unlike the SCM, contains torsion) is readily seen to be

T. Curtright et al./Nuclear Ph)'sics B 469 (1996) 488-512

495

I ( G+~hO~+qJ a3 u ~ h + iG,,h-~qtb + Eah~uvOu~aO~b £sd~m = 5_ I --a *g~,,bc,ltt " ( 1 + yp)g-'c~t'( 1 + y p ) ~ a ) ,

(3.16)

with a ttew geometo' elaborated in Appendix B. The curly-covariant derivative on the

fcrmions is

"D~,'F" = 8utl]a + ~h,~. ou~ - Gc v" 8u,,a q, ,

(3.17)

in terms of the tensors of the new geometry specified in Appendix B.

4. Canonical equivalence of the supersymmetric models If these two supersymmetric theories are canonically equivalent like their bosonic limits, a generating functional in tangent space for such a canonical transformation is needed. Taking into consideration dimensional consistency, Lorentz-invariance, and a good free-field limit, we posit the Ansatz for the supersymmetric theory in tangent space rclating ¢ and X at any fixed time to q' and X (the bosons and fermions of the dual theory),

F[q,,X,q~,X] = f dx(~JJ'J[q~] - li-xJTIxJ =f dx(q)i(

)

lx/i-~-~2~-~x~Pi+e'Jt~dOq~k)-½i-XJy'XJ ) . (4.1)

Classically, the canonical conjugate to X, rr x = 6£SCM/6OoX = - i x t / 2 ,

(4.2)

is obtained from F as

.-6F/6 X = -iXtyp/2, where

(4.3)

"yp :_ yOyJ. So under the canonical transformation,

X; = Y;, XJ.

(4.4)

Likewise, the momentum conjugate to X is

6F/6X = -ix~y;,/2,

(4.5)

leading to

7r~ =_ --iXt /2, which specifies part of the dual Lagrangian.

(4.6)

496

Z

Curtright et al./Nuclear Physics B 469 (1996) 488-512

This chiral rotation of the fermions reflects the duality transition of their bosonic superpartners, whose gradients map to curls in the weak field limit, as already noted. ( F o r a mathematically subtle interpretation of tangent-space canonical transformation in terms of Cartan's equivalence problem see Ref. [ 19].) As a result of (4.4), the equal-time anticommutation relations for Majorana spinors in tangent space, { y ' ( x ) . g~'(y) } = {X~(x), Xk(y) } = 2~ik6(x - y ) ,

(4.7)

are preserved in the above transformation, thus confirming its identification as canonical. To handle the bosons, it may be advantageous to resort to the first order Lagrangian quadrature mentioned before. Adding the customary pure-gauge-enforcing Lagrange multiplier to the tangent-space SCM and integrating by parts leads to g'SCMa = ½ \{'Jj,, Mu~" )k ",, - 2eu" JJ Ou~i + i-xx~xJ + iUkt-xJ~lkxl + ¼(Y(X)2) , (4.8) where M ,UP ,I,

gUv~ah + 2e u" eat'c@ ',

:-=-

(4.9)

with an inverse which satisfies Mu#~,(NO)h,. = guaa,,c, H.l'

Nah = g

p.u

G,,h + eU"E,,#.

(4.10)

The crucial transition bridge to the SDSM relies on the bosonic current encountered in Section 2, j/u = -~*ik ~,,a,, "~

I + 4 q )2 (~t+4q~JqDt)eu"O"q~t+2~Jtk@tO~*q~k

'

(4.11)

which conjoins with the fermionic bilinear component into j~,i = -Nil~l,, (e,,ac)A q~h - ~te I. &d--c X V,,X dx)

=

j u J + N~k **vK,,, k

(4.12)

which follows from varying J in the first order Lagrangian £SCMa. (Note from (4.4), Kiu[ X ] = KJu[X].) This is the on-shell relation linking J to ~. Below, this is shown to be derivable from the generating functional F.~.. The Lagrangian resulting from Fridling-Jevicki-type [ 12] substitution for J, or, equivalently, completing the quadratic in £SCMa and integrating the shifted Js out, is [ 1 I ] = ½i-~,4X i + ½ ( f ( X ) 2 ~, (g,,aO a~ a ZZSDSM • × N ~ (F.vp(~Pq)b -- ½iehef-x'eYvXf ) =

+ ½i 'J

-

I• ~te

acd.--; X

Y**X

as )

'j'x

E To-E + ½ (Gi, at, q~iO~'cbk + Eikeu"Ou~i&,qsk ) .

J (4.13)

T. Curtright et al./Nuclear Physics B 469 (1996) 488-512

497

This result forces the variation 6/&l) a of the generating functional to be just the q~ bosonic current J, while the .t,i = yj, Xi's are regarded as independent variables. As a result, the generating functional automatically yields

l l = J j =,.71 + ( N . K )

I.

(4.14)

Wc now use the variation w.r.t. ~ to match the timelike components as well. The arguments of the bosonic model (Ref. [81, Eqs. (3.11), (3.12), which connect timelike components of currents) remain exactly as they were, _ a__~, _ 2 g ~ j k ~ r 6

,gx

= _,/1

- ~2~r~ - gikCY~k,

(4.15)

since both sides are equal to

c?X

~

but now 11 contains an additional fermionic piece beyond its bosonic component, as seen above. And likewise w-, nr' = bosonic + K~( v/l - ~p28iJ + (piqd/x/l - ~z + eiJtq~t).

(4.16)

Thcsc then introduce fermionic current pieces in the above equation (4.15),

~ i - 2eiitcl)J( N" K) kl = J'o - K6.

(4.17)

As a direct consequence, .Io = ,fr0 + ( N . K) °,

(4.18)

and the above pivotal identification of currents (4.12) holds. Note how duality is implemented on thcse manifolds: Bosons in one theory contain both bosons attd fermions in the dual theor): Further note that the conserved current is

k.k + K # j • '--7ui + Nu, i~, "',,

(4.19)

The supercurrent for the DSM is

0~, = iT t, (,~.i +1~. K j ) y u X ),

(4.20)

which holds by the symphysis identification of currents (4.12) above. Consequently, supercharges identify, and whence Hamiltonians (which are their squares), energymomentum tensors, and all such quantities conventionally constrained by supersymmetry. From the structure of this supercurrent, it then follows that this is a "chirally twisted" realization of supersymmetry,

~q)" = g( X a + 2yt, e"~k~hX k ),

(4.21)

which preserves the pariO" of the original X. and thus of the action. This is the combination entering into the spinors of the £.~d~m model (the fermion entering in the respective superfield of the next section) for comparison with the SDSM,

T. Curtright et al./Nuclear Physics B 469 (1996) 488-512

49;";

t[" a = X " 4- 2"YI,P.ahi(]hbg i = ( V (ai) 4- ~pV la'il ) X i, ( 1 + "yp)h/t''= ( I + y p ) v a i x

( 1 - Yt, )W"= ( 1

-

(4.22)

i,

(4.23)

"yp)ValX j ,

(4.24)

where ~"' is the transpose of V ''i. Consequently, its inverse ~,i = Gai - Eai is also the transpose of V,,i. As a result,

(4.25)

XI = (V(,u) + ypVta)l )~k".

This last relation strongly echoes Eq. (2.18), which finds its explanation in the superfield formulation below. It turns out that £SDSM = £~dxr,. The pure bosonic piece of £SDSM above is, naturally, £DSM- To further match with the pieces of Esdsm quadratic and quartic in fermion tields, respectively, we need to check the following two relations. For the function W in -I ~. -t-Xi ~ m Wik,,X k , one needs to show I.:'k / G im ~/p -4"-,~i,~'jEh,, = 2 ( - ~'iyt, -4- 28 ain@n ) Gab~t,k., + ( 6 ai - 2 y p e ain(l)" ) ( I'abm -- ypSabm ) ( ¢~bk _ 2yt, ehk.,.q), ),

(4.26) which, indeed, holds. Secondly, /'or the fermion quartic terms, use is made of the group-Fierz identities

(o.

a,)

*) = =

(4.27)

xx')(x%x'),

to prove

(XiXi)

(;

2

(-XiXJ)

(1 -F4qb 2)

'

(qr~j~j~kXk) + (1 q - 4 ~ 2) (dktq~t-~jyt, Xk )

I ,r) ,F, a t = iZ,'..,,,ca~" ~ I + .)q,)q.,,.g,h( 1 + ")q,)V.'J I "D f'zawcf?hwd~,i[ = G,v,,i,,.,tv i vj Vk */ "~ ~l +Yt' )xJf(k( 1 + y p ) X t.

) (4.28)

Consequently, supersymmetrization of the mutually dual bosonic models produces mutually dual theories, as demonstrated, susy CM , SCM

DSM

Xll£y

,sdsm

4=SDSM.

(4.29)

In these variables, the supercurrent for the DSM now reads

Ou = i~D"YuGat' ~t'

+

iyP~'"~/I~( N . K)£ ( g(aj) + ,)/pVlajl ) ~ a ,

which may now be compared to Eq. (3.13).

(4.30)

7~ Curtright et al./Nuclear Physics B 469 (1996) 488-512

499

5. Super,space formulation General supertield formulations tbr supersymmetric o--models with torsion are given in Rcf. [171. Recall

D=~

,9 c~0

-,9 D = - - - + i04, ,90

-i~O,

c)

Q=--=+i~O.

(5.1)

c)0

Thc scalar supertield for the supersymmetric chiral model, SCM, is 1-O ~"' = ~," + O~b" + ~00~

(5.2)

and has superderivatives

17~" = ~ ' + -0 (q~" + iq~") -- ~iO0~" ~.

(5.3)

From this, one may construct the bilinear

~q~,,yt, Oq91,= f,,yp~/,h + ~,,yp (~t, _ i~tch) 0 -- ~b'yt, (q~" - i~p a) 0 +`9~, ~tO ~',,0 " - ~"O,,q ~h] eu"-O0.

(5.4)

Thc corresponding supcrfield for the dual theory is q~,, = cp,,+ffq,,,,+ _~00 i - Y,,,

(5.5)

hence

O ~ " = q'" + (Y" - i0/q~") 0 + ½iq~q,'"-O0.

(5.6)

Thc dual transition between two field theories, however, ~s normally effected in tangent spacc. To address tangent space in superspace, start from the chiral element superfield C, = U ( x ) (g + iOx. ~ + ½-O0(iZ . r + ½-iX)),

(5.7)

a most general ansatz, such that, for unconstrained X and Z,

GiG = 11.

(5.8)

Rccalling U-IOuU = -i7-. Ju, obtain its superspacc analog [20], the spinorial current

supe .~eld, G--i D G = i x • 7"+ (iZ . 7" -~[. 7")0

-

½i?/#o~JklT"J-iky#XI

_

~ITI,0 ikt TJ--kx ~/t'.~l

+ ~-O0(2ieJktT"iZkxt - CX" ~" - 2UktT"J~[kXt + ix" r ~ . X),

(5.9)

notably valued in the SU(2) Lie algebra (unlike G). Consequently, the tangent-space SCM action is specified in superspace by

71 Curtright et al./Nuclear Physics B 469 (1996) 488-512

500

,/ d2OTrDGDG

£,lield = g

= '~ ( z 2 + J.. : + ~ . ~ x + ~:'-~J/'x' + ¼(-)i. x)~).

(5.10)

Thus, Z = 0 on-shell, matching the component supersymmetry transformations on the tangent-space objects of the SCM exhibited in Eq. (3.9). The variational result for the (right-) conservation law in superspace is thus

.[ d20 ~ ( G-i DG) = _2i<7u (ju . r - l ieSUr.i~yUxt

(5.11)

which again identilies with the conserved current. Now consider the superspace curvature identity [20]

I~T/,( G-I DG) + ( G-I-DG)Tt,( G-I DG) = 0.

(5.12)

This is an identity. But, if the pure gauge form of Ju is not noted/utilized in the current superfield (only 9 ), now dubbed for this purpose J, it turns to a constraint instead, with all components vanishing identically, except for the O0 term which consists of (only) the zero-curvature constraint for Ju,

~,y/,j + ~yt, J = i-~Orieu~ (flu j[. + e/tt jk jt ) = 0. #--P

(5.13)

:

As a consequence, enforcing this constraint through a Lagrange multiplier superlield in an appendage to the superspace action equation (5.10), ~sticld

=

5,1 .

f d2OTr (~-]J + iqJ . r(-Dy:,J + Jy,,J))

,

(5.14)

will only involve ¢I;~ but not q' or Y, as already observed in practice in the component calculation in the previous section. Thus. Fridling-Jevicki-type [12] quadrature in supcrspace will lead to a superlield formulation of the SDSM, see below. Evcn though the fermions ~ appeared to be projected out and thus superfluous "gauge freedom" components of the original • superlield, they emerge in the final answer below [20]. Supcrspace integration by parts yields the usual quadratic,

'

/ / d2x

d2OTr(i-]ypDq~.r+i~.r-]y/,J+iJJ)

,

+

(5.15)

]~MikJk/)

and a total divergence term discussed below, where

M i~ = ~Stk - 2yt, eit
(5.16)


501

T Curtright et al.INuclear Physics B 469 (1996) 488-512

This is the dreibein-twisted-chiral structure observed before, and it is inherited by its inverse

N i~ = Git(cl)) - yt, Eik(@),

(5.17)

also already encountered in the previous section. The customary elimination of

jk = _ iNkJ yp DcI~j

( 5. I 8 )

identifies, to O(0°), Y i = - ' y p ( G j k - - y p E j k ) ~ k encountered in Eq. (4.25) and the further terms are expected to yield the current symphysis formulas. The algebraic equations for the auxiliary fields result from substitution into the superspace action,

/

'fl =i i

'/~nfield = 7 , d2 x

-~

d2x

d2 0-Oel~J NJk D¢I)t

d20-D(I~(G jk _ yt, EJk)D~ k.

(5.19)

This action automatically coincides with /~sdsm, as it is the superfield formulation of a (r-model with torsion possessing a given geometry (cf., e.g., Eq. (5.3) of Ref. [17]), in this case this new geometry is already discussed. The generating functional F with spinors should emerge out of the total divergence term resulting from the above superspace integration by parts,

- ¼ . f d2x / d2OTrDY1, ( i q ) ' r J ) I::'J"l't'lrJ = / d 2 x S d20 ½-OOcg"e,,#(J#jtlp + ~'X t" ~r -- ½i,Y(JyueJkt@kxt),

(5.20)

i.e. £

= Jax (jue J +

~if(Jylg, J _

½if(JyleP,t@kxt).

(5.21)

However, superfield tormulations of general or-models with torsion always contain a total divergence of a fermion bilinear term in the reduction from superspace, cf. Eq. (5.8) in [17]. In this case, this extra term contributes --½i'~Jyle)kt@kq-'t/(l + 4@z) tO the above, to produce the generating functional

F'=

/ ( dx

. . . .

._

.

°,)

jU@, + ½iY(,yiCJ _ ½tX.,yiejktekxt_ ½i-~2yid~:t(l +4@ 2) g,'

.

(5.22) Variation with respect to .~" yields ¢rx, hence

yt, x j = ,l.,.i _ 2eikt@k xt '

(5.23)

so that y~,X-~= ( V~,,i) + yI, Vl,,jl ) q-'",

(5.24)

502

7~ Curtright et al./Nuclear Physics B 469 (1996) 488-512

the right-hand side fermion actually being X as introduced in (4.25). Note that we never really introduced tangent-space fermions for the dual theory in superspace - they emerge as an output. Moreover, variation with respect to 1/-' yields --h'q, for £~d~m, hence yt, Gjkqrk = X i + E i k ~ k,

(5.25)

and hence y~,X i = G i k ~ ~ - yt, Eikqtk = X j,

(5.26)

consistently to the above. Eliminating ~ s in favor of X's in F ~ yields F' = ~ d x (jUqsi + ½iS(iyIX.i _ ½i(5(J + f(J.y/,).yl~.jklqSk(xI

_

yPXI) ) .

(5.27)

!

This provides an alternate route to the same theory, and the final term vanishes given the above results of the transformation.

Acknowledgements This work was supported by the NSF grant PHY-95-07829 and the U.S. Department of Energy, Division of High Energy Physics, Contract W-31-109-ENG-38. T.U. and T.C. thank Argonne National Laboratory for its hospitality during important phases of this project. We are thankful to L. Palla for communicating to us a draft of [14] before release.

Appendix A. Overview of canonical transformations in mechanics We review some rarely emphasized aspects of canonical transformations in classical mechanics: we take as our starting point the invariance of Poisson brackets (PB), instead of the more conventional preservation of Hamilton's equations. Poisson brackets are suitable for eventual quantization, {u, t,.} ~ ( l / i h ) [u, v], as they turn into canonical commutators. Extension to field theory merely involves arraying a continuum of modes, and transcription of sums into integrals, as exemplified in the text. Poisson brackets can be defined as &' {u'c}q/' - &~ aq ap

&l at: ap aq "

(A. I )

PB's are antisymmetric, linear, they obey the Jacobi identity (from associativity of the underlying operators), and convert by the chain rule (A.2)

{u, c }~_)e= {u, t;}qt,{q,p}Qe. Now the transformations (q,p) H (Q(q,p),P(q,p)

)

(A.3)

T. Curtrightet al./Nuclear Physics B 469 (1996) 488-512

503

are called canonical when they yield trivial Jacobians,

{Q,P}qp = I,

hence

{q,p}op = 1,

(A.4)

so that they preserve the Poisson brackets ("canonical invariants') of their functions. Equivalently,

{q,p} = {Q,P}

(A.5)

in any basis. Following Poincar6, the measure of phase-space area/volume is seen to be preserved,

dQ dP = dqdp {Q, P}.

(A.6)

For instance, point transformations (which generalize to usual local field redefinitions in field theory) are

Q= J(q),

P=P/J'(q)

~

{ J ( q ) , P/J'(q)}up = 1 .

(A.7)

Now consider the transformation generated by the hybrid function F2(q, P ) , p -

cgF2(q , P ) , Oq

Q =

aF2( q, P ) , OP

(A.8)

where, in principle, the first equation can be inverted to produce P ( q , p ) , which is then substituted into the second to yield Q as a function of q,p. (The point transformation just illustrated is generated by F2 = PJ(q).) This transformation is canonical, seen explicitly as follows. Define p = OF2( q, P)/Oq It, = 'F and Q = OF2( q, P)/OPIq = F', contradistinguishing the arguments being differentiated. Now switch basis to q,p, and consider P as a function of q, p, partly specified via the partial differential equation Op/c)p = 1 = 'UP.t,. It is then straightforward to show in this q,p basis that I = {q,'F}

(A.9)

= {F',P},

since the middle expression equals 1 of the I.h.s. by the above differential equation, and, by the same token, the r.h.s, also equals ( F"Pq + tF')P,, - F'P4,P q = 1.

(A. 10)

Infinitesimally, this transformation is also easily seen to be canonical, as follows. Introduce a generating function infinitesimally expanded around the identity through an expansion parameter w,

F2(q, P) = qP - w G ( q , P ) .

(A.11)

The O(w °) piece is the identity, as

OG(q, P) 8q

p = P - w - -

aG(q, P ) Q = q - w - -

8P

To leading order in w, one can substitute G(q,p) for G ( q , P ) , so that

(A.12)

"F Curtright et al./Nuclear Physics B 469 (1996) 488-512

50.1.

OG(q,p)

Q

q = - w - P - p = w

OG(q,p) 0q

+ O(w 2) = w{G,q}qt , + O(w2),

(A.13)

+ O ( w 2) = w{G,p}ql, + O(w2).

(A.14)

Then it is easy to see, by the Jacobi identity, that this transformation is canonical to O(w:). { q + w { G , q } , p + w { G , p } } = 1 + w{G, {q,p}} = 1.

(a.15)

Actually, the full, exponcntiated transformation

O = e'"~C'q = q + w{G,q} 4-w2{G, { G , q } } / 2 ! + . . . , P = e"{Gp = p + w { G , p } + w2{G, { G , p } } / 2 ! + . . .

a.16)

,

A. I7)

is canonical to all-orders in w, Y(w) = { Q , P } = I,

A. I8)

for essentially the same reason. Pro(~ Note that, from the very definition of the Hadamard exponential operator above, for any w-independent a, such as q or p,

d(e~,{C,a) dw - {G, eW{Ga}.

(A.19)

Consequently, by the Jacobi identity,

dY(w)

- {{G,Q}, P} + {Q, {G,P}} = {G,

dl, i.'

Y(w)}.

(A.20)

Now Y(w) = 1 + ~,,~I w"y,, yields the recursion y,,.. = {G,y,,},

(A.21)

and all the Y,,>0 = 0 by induction based on Y0 = 1. Consequently Y(w) = 1.

[]

Note this is a standard integrated Lie evolution, but it is still hard to fully connect to the unfolded general F2. The other three types of canonical transformation can be obtained through the trivial symplectic reflection, which is is also canonical,

( q , p ) ~--* (Q = - p , P = q )

since

{ - p , q } = 1,

(A.22)

and thus may combine with other canonical transformations to yield more canonical transformations, by the chain rule of PBs. Applied to the variables Q, P of the above F2, it produces the canonical transformation, generated by the same functional form, now called Fi, p -

c)Vl (q, Q)

c~q

P -

c)Fl ( q, Q) , OQ

(A.23)

T Curtrightet al./Nuclear Physics B 469 (1996) 488-512

505

and likewise for the other two types. The Field Theoretical construction in this article on the tr-model is of the Fi (q, Q) type, but it could be trivially converted into the F2(q, P) type by this trivial symplectic reflection (readily generated by Fi (q, Q) = qQ). The classical mechanics analog of our transtormation is Fi (q, Q) = QJ(q), which then goes by the above symplectic retlection to

F2(q, P) = - P J ( q ) ,

(A.24)

s.t. 3k)( q , P ) -Q, c~P

OF2Cq, P) -p, c~q

(A.25)

Q = JCq),

(A.26)

p aJCq)/Oq

(A.27)

thus a point transformation

p=P--

c+J(q) aq

hence

P-

In other words, the canonical transformation used, in more conventional F2 hmguage, is simply a transition from the q's to the J(q)'s with the standard point determinant scaling for the momenta to preserve the PB's, 1

{J(q),pilj(q)/Oq}=l.

(1.28)

In field theory, it is a transition from ~p to JI (,P). But interchange of q:"s and H's in the Hamiltonian of the DSM, yields something unconventional, involving space derivatives of the IIs. Now', how is the identity transformation generated by Fl (q, Q), and why should one choose to base the discussion on F2, in the first place? Some awkward features of the Fi ( q , Q ) generating function have been pointed out by Schwinger [21J. The inverse canonical transformation is, evidently, F~ ( Q, q) = -F~ (q, Q).

(A.29)

The composition of two successive transformations (q,p) H (O,P) ~ (Q, P) is simply generated by the sum of the respective generating functions, WCq. Q ) = F~(q, gl) + F~ (0, Q),

(A.30)

whcre each term generates the respective piece of the total transformation, and the dependence on the intermediate point ("superfluous variable") vanishes between the two in the total transformation, i~W(q,Q) _ O. O0

(A.31)

506

T Curtright et al./Nuclear Phy,~'ics B 469 (1996) 488-512

But, by zhe above defined inverse, the identity should be then generated by W(q, Q) = O. This singularity of the generator can be made more palatable by retaining the superlluous variable <7 which serves as a Lagrange multiplier, W(q,Q) = cT(q - Q),

(A.32)

hence

81.V -- =0

~

3W

3W

a0 &l

-~ -

q = Q, ~

OQ

(A.33) p = P.

(A.34)

Motion is a canonical transformation, of the above infinitesimal F2 type. When G is chosen to be the Hamihonian H, and w = - d t , dq = dt {q, H}w,,

dp = dt {p, H}qp,

(A.35)

which comprise Hamilton's equations, gl = aH/3p,

p = -c?H/aq,

(A.36)

dictating incompressible flow of the phase fluid, 3il/3 q + Ol)/Op = 0. (As seen above, such a transformation readily exponentiates.) Consequently, for any function f ( q , p ) , {f(q,p)

'

H}qp

df =dt'

(A.37)

Now consider some arbitrary canonical translbrmation to Q, P, and take f to be Q and P, respectively; switching PB basis by virtue of the canonical nature of the transformation, one sees directly that Q = {Q, H}op = aH/aP,

P = {P, H}oe = -aH/OQ.

(A.38)

That is. any canonical transformation also preserves Hamilton's equations of motion, ft qt

,-J Q,

'

qT

)

Or.

l ,--

It

Motion is also generated by the Action Integral S (Hamilton's principal function), but on the classical path, via an FI transformation; this is the one utilized by Dirac in his celebrated quantum Hamilton-Jacobi functional integral [4], Pt -

6 f/. d'r L ~qr '

6 fT dr L PT --

~qr

(A.40)

The intermediate-times variables q(t) are not arbitrao,, but must be specified by the equations of motion. To appreciate this, recall that the action integral may be effec-

T. Curtright et al./Nuclear Physics B 469 (1996) 488-512

507

tively regarded as a sum of inlinitesimal transformation generators of type Fi, namely dt L( (q + Q ) / 2 , ( q - Q) / d t ) ; thus

f

I

T

T

t

t

dr L =

d r 8q

(6Ld6L) ~q ~ ~-O

SL' + M/, 8q,

~LT6qr ¢50r

(A.4I) .

Vanishing of the first term in parentheses (the Euler-Lagrange equations of motion) to yield the above result is dictated by the requirement of independence from the (intcrmcdiate timcs) superfluous variables. That is, the requirement of continuous canottical tran.@~rmation.for motion underlies the classical variational principle. Dirac 141 discovcrcd that in Quantum Mechanics thc generator must be exponentiated and the supcrlluous variables must be integrated over instead - the above classical path is then only the contribution to leading order in h.

A p p e n d i x B. Explicit g e o m e t r y of the dual m o d e l s

The dual sigma model is not a WZWN model on a group manifold. The geometry of the DSM is described in detail by the following, in the conventions of Ref. [ 17]" 1 (8,,,, + 40"Ot') , G,,,, = V,~V~[ - I + 402

(B.l)

d e t G = I / ( I + 402) 2,

(B.2)

- _ _1 1 + 402

(B.3)

Ed,

_28ahco c) 1

= G,d + E,,j - 1 + 402 (6,# + 4 0 " 0 a - 2s'VcOc) , det V,( = ~/det G = ( 1 + 402) - ~, Va l

(B.5)

= ~a/ - 2e""jcoc,

GJ =

V " i V ;')

(B.4)

(B.6)

= (I + 402) 8,,t, - 40"O;'.

(B.7)

Nole that. in this remarkable geometry, base- and target-space indices are innocuously interchangcablcfor our choice of coordinates (O's), since Oa = GabO I' = 0", 4)h = 0" V~'. Connections are obtained in the usual way, i l',a,,. = 5 ( c)hG,,. + c)L.G,,t, - OaGbc )

4 (O"&,. + O"G~,,. - ObG,,. - O"G,,b) I + 402 4 1"ii' - I + 402 (O"8hc + O"Gl,,. - Oh8,,c - O"8,#,) _

= 16

q~'Ohclf 80,,61,,. ( I + 202 ) + (1 + 4 0 2 ) 2 (1 + 4 0 2 ) 2

S,a,, = ~I ( O,,Eh,. + OI,Ec, + OcE,d,)

40h8 " + 4qY6 "t' (! + 4 0 2 )

(B.8)

(B.9)

508

Z Curtright et al./Nuclear Physics B 469 (1996) 488-512

_ ( 3 + 4* 2 ) (_ e,,i,,. )

(B.10)

(1 +4'1'2) 2 Note that the DSM dreibein does not satisfy Cartan-Maurer equations. Rather, ,,

~,k/,,k,,t

(1

4,/' 2)

-

8,, Vj: - 81, V,i, + .-,~ h, % = -4 (l + 4--~ 2 ( .a(~l,j _ *" (~,,tj)

(1

16

(.bgajc.c

__ ( 1 9 a s h j c . c

(B.ll)

2

where

~;k;Vk.,; ~"%

-

1 +

--24 *

2

1

(*"&'J

eahj"

--*b6"i) " + -1 +- 442

(B.12)

Alternatively.

O.V/ -..el, V/-

-4

1 +442

e,,i,c G . _ 4e,,i,.S, jd.a"

(B.13)

'1

Contrast these to the corresponding relations for the CM.

a,,,.,/,

•

a,,,,,{ = 'Ag" ' k ' baV '~ "h = 2~"" + ~ - 2

(~,,a~,_¢,6,,3 .

"

(B.14)

However, note that, in contrast to group manifolds, Sah¢ --

3 + 442 eikl V.i Vfl ill I +4¢b ---5 " J'""

(B.15)

(~hG,,i = 2 S,,I,,. V I'J I.

(B.16)

So, e.g., 8. Gh i

Hence, (B.17)

]'1 o h l c = Sahd V Idcl. The geometry follows from direct, albeit lengthy computahon hup://daisy.waterloo.edu / ), 16

4((3+12*2+16*4)(3ac6hd-aad61,c)) --

I,c.,,

Maple,

(B.18)

l),1S,,,,. - ( 1 -- 4 * 2 )3 .,1 e,,h,, R"

(using

+ (16* 2)

(1 + 4 4 2 ) 3

(*c

+ ( 1 2 + 16"2)*"(*d6

(3 + 4*2) 2 Sf"dS/h -- SfacSdfh- (1 + 442) 4

( (

__ . d

~w)

,

(B.19)

t'c --*cr3h'l)

-

) )

__4.b ( . c ~ a d __ . d s a c ) _ 4 . a (.d6t, c_ .c6,,,t )

S,, ,..t _ S" S f - (3 + 442) 2 ( (~'"8 t'd - ~'d6h") ) ,#°,4, ,J m, ( 1 + 442 ) 3 _4@ h (."8 "1 _ 4'16 '':) "

,

(B.20)

(B.21)

7:

Curtrightet al./Nuclear PhysicsB 469 (1996) 488-512

509

Thus, the torsionful Riemann tensor amounts to (B.22)

R'l,,<' ,l -=- R "#,,.j - S<~ssS~:#, ' I + ~!tS
(,S<,<<¢,1_ a,,,~#,, )

3

(1 + 4c1~2) 4- 4 (9 + 4 @ 2))2rb" ((l,"lg '<` - q # 6 <'<) + 16(_3 4-4@ 2) @" (q5,61,,. _q;,:aJ,,l) ( 1 + 4(/,2) 3 ( l + 4~'/J2 " ( 1 +164q52) 2

(@cgabd -- qf)dgalw) +

64 ( l + 4 ~ 2 ) 3 ,/,,,qss (@<%,cS

-
(B.23) Actually, curly-R without a raised index is a little easier to look at, since it has the usual antisymmetries under interchange of pairs of indices, but not the symmetry under interchange of pairs of pairs that the torsionless curvatures have, "]~ah( d --

3

(a<,
a,,aa,,<)

( ] + 4q02) 2 4 ( 9 + 4q~2 ) (¢ba (cbd 6h c _ q~cf/,d) _ Cbl~ (qscl ~/,c _ q y ~,d) ) " (1 + 4 ~ 2 ) :~ +

16 (4'%,#,d - qYle,,bc) ( 1 + 402) 3

(B.24)

There is an alternate route to this torsionfu] curvature. The spin connection [ 17] is worked out to be xd - v"~,,~

- V"'(D,,Vi/ + S,,~,
1

- ( I + 4q02 )2 ( ( 14 + 24q )2 ) (6"J~b ~ - ~ " ~ q # ) + ( - 5 - 4@ 2 ) (e. ''u + 4qb% iJm qbm ) ) - (14+24qJ2)(~'i(pi(I +4c/~2)2 _~,,iq#)

4O2 )2)2 (8aij + 4qpeij,,,O,,,). (1(5 ++4 @

(B.25)

This is not the spin connection on any group manifold. On the one hand, first differentiate the spin connection and then combine it with its square, antisymmetrically, to yield the curly curvature,

(1 + 4 ,2)3 3 x (l+4~2)2(6bJq5 ai- 6bi6a)) x

)

- 1 6 x (4@ 2 + 5) (q:,a(fihjq~, _ <$hicpi ) _ @t,(aai@i _ Saiqbt)) - 2 x ( 16(I~4 + 24q~ 2 - 11 )((PaebiJ - (]~b~aij

On the other hand, for our dreibein ( R J i ),,,, = V"iVdi~c
.

(B.26)

511)

T. Curtright et al./Nuclear Physics B 469 (1996) 488-512 + ( ec'lir])j

ecdiqsi)"]~cdah,

- -

(B.27)

which gives explicit agreement with the RHS of the previous expression (providing some algebraic checks), so ('Ri'),,,, = G I Z / - 8 M ( { + ~2~ d2ki - $~k ~2~",

(B.28)

as it should be. Recall that, for three-dimensional manitolds, the Weyl tensor vanishes (even when torsion is present, as in the model at hand), permitting us to re-express the Riemann tensor in terms of Ricci and scalar curvatures, T4,#,,.d = ~I ( G,,aGhc - G,,cGhd ) 74 + G,,cRhd -- G,,dRh,: + GhdT4,,c - Gh,.T4,,d = (r~,,,.-

'G,,,.r~)

G~,, -

(r~,,,,

-

¼G,,,tr~) S~,.

- G,d (T4hc +Gac (~r~hd - ~GhaT-4) I

(B.29)

( n . b . a . b . c = 1,2,3, only). On the other hand, for three-dimensional manifolds, the issue of contormal flatness is decided not by the Weyl tensor, but rather by the Cotton tensor [22], obtained by taking derivatives of the Ricci and scalar curvature combinations exhibited in the last equality. Without torsion, the Cotton tensor is defined as c,,,,,

= D,. (R,,,,-

¼C,,,,,R)

- D,, (R,,,.--

¼S,,,.R).

(B.30)

The manifold is conformally equivalent to a flat space if C,,b,. = 0. It is straightforward to check that C,l,,. = 0 for the dual sigma model. (This is also true, rather obviously, for the usual chiral model.) Along these lines, it is interesting to note the same linear combination of Ricci and scalar curvature appears in the quartic spinor terms for a general supersymmetric model detined on a three-dimensional manitold. Taking into account the Majorana property of the spinors, and making Fierz rearrangements, gives T~al,,d~a ( 1 + yt, ) qt"-~t' ( 1 + yl, ) q td I = 4G,,;,(qT"q t;') (7"4~.,1 -- sG,.d~'~) ~ ' ( l + yl,)~ "/d

= 4G,,t,(~-"¢") (?-£(,:d) -- ¼G,.dT£) ~'~,p.d + 4G,b(~"~b)7~l,.al~'yp~d

(B.31)

( n . b . a . b , c = 1,2,3, only). Another way to say this, valid for higher-dimensional manifolds, is to write the quartic term in general cases using the Weyl tensor, (?,,h,,t = 7g,,I,,d + ½ ( G,,cGbd - G,,dGt,c) 7~ -- G.cT~t,d + G,,dT-4t,,: - Gl,,tT£,,c + Gl,,:TP..,,d , (B.32) T4,1,,.dqT" ( I + yt,)qt':q7J' ( 1 + y t , ) ~ d

= C,,~,,.d~"( 1 + 7,)~"q71'( I + yt,)q .''t •+4G,,~,(~"q "t') ('R.,.d .-- ¼GcaT£) 97"(I + yp)7 "d.

(B.33)

511

T Curtright et a l./Nuclear Physics B 469 (1996) 488-512

Specification of the curly curvature now allows evaluation of the curly-Ricci tensor, crucial to the computation of the B-function to one loop [ 17], "~I;,,/) - ( I + 24~2) 3 ((3 + 16~4)~/~d - 4 ( 3 + 3 2 ~ 2 + 1 6 ~ 4 ) ~ / ~

d) .

(B.34)

Thcre is also the antisymmetric part of the curly-Ricci tensor, 16 )ael'dfdP y = --8 (det G) E,,h. ~ : I,,JI = D,, ~l,t - ( I + 4 ~ 2

(B.35)

(We disagree here with the corresponding expressions of Ref. [ 13] and agrce with [14]; we are grateful to L. Palla for communicating to us this result betbre release.) Finally, there is thc scalar curly curvature, 2_

(1 + 4 q 3 2 ) 2 ( 3

_ 4q)2) 2 --2

4

1 +4~ 2

!

(B.36)

with small and large field limits, 18 and 2, respectively. I.e., the curvature starts as a sphere of 18/r 2, goes to zero at ~2 = 3/4, and stabilizes asymptotically to 2/r 2. It is evident that 7~ is invariant not only under isorotations of the qSis, but also under the discrctc "isometry" q~2 .__, (r/,2 + ~)/(4q~2 _ 1); however, this discrete transformation does not leave the torsionless curvature R invariant. It goes without saying that full appreciation of this new geometry should hold the key to renormalization of the dual models beyond one loop. References Ill E. Alvarez, L. ,~lvarez-Gaume and Y. Lozano, Phys. Lett. B 336 (1994) 183: Nucl. Phys. B (Proc. Suppl.) 41 (1995) 1; Y. l,ozano, Phys. Lctt. B 355 (1995) 165: E. Alvarcz, L. ,~lvarez-Gaum6, J. Barb6n and Y. Lozano, Nucl. Phys. B 415 (1994) 71. 121 W. Siegel. Phys. Rev. D 48 (1993) 2826; A. I)as and J. Maharana, Mod. Phys. Lctt. A 9 (1994) 1361; C. Klimd~ and I~ ,~evera, Phys. Lett. B 351 (1995) 455; S. Ilassan, Nucl. Phys. B 460 (1996) 362; E ~lvarez, L. Alvarez-Gaum6 and 1. Bakas, Nucl. Phys. B 457 (1995) 3; I. Bakas and K. Sfetsos, Phys. Lett. B 349 (1995) 448; K. Sfctsos. Nucl. Phys. B 463 (1996) 33; hep-th/9510103. [ 3 I A. Givcon. E. Rabinovici and G. Vcneziano, Nucl. Phys. B 322 (1989) 167; A. Givcon. M. Porrati and E. Rabinovici. Phys. Rep. 244 (1994) 77. 141 P. Dirac, Phys. Z. Sowjetunion 3 (1933) (,,4. 151 G.I. Ghandour, Phys. Rev. D 35 (1987) 1289; T.I,. Curtright, in: Differential Geometric Methods in Theoretical Physics: Physics and Geometry, eds. L.-L Chau and W. Nahm (Plenum Press. New York, 1990) pp. 279-289, 16 ] T. Curtright and G. Ghandour, in: Quantum Field Theory. Statistical Mechanics, Quantum Groups and Topology, eds. T. Curtright, L. Mezincescu and R. Nepomechie (World Scientific. Singapore, 1992) pp. 333-345. 171 H. Kastrup, Phys. Rep. 101 (1983) I. [ 81 T. Cul'tright and C. Zachos, Phys. Re','. D 49 (1994) 5408. 191 T. Curtright and C. Zachos, in: PASCOS '94, ed. K.C. Wali (World Scientific, Singapore, 1995) pp. 381--390.

512

7~ Curtrik, ht et aL/Nuclear Physics B 469 (1996) 488-512

]10[ S. Mandelstam, Phys. Rev. D II 3027 (1975). I I I I I'. Cunright and C. Zachos, Phys. Re','. 1) 52 (1995) R573. 112[ FL Fridling and A. Jevicki, Phys. l.ett. 13 134 (1984) 70: E. Fradkin and A. Tscytlin, Ann. Phys. 162 (1985) 31; First-order formulations of chiral models go back to, e.g., S. Deser, Phys. Rev. 187 (1969) 1931 il3[ A. Subbolin and 1. Tyutin, Int. J. Mod. Phys. A II (1996) 1315. [ 14] J. Balog, E Forg,'ics, Z. tTorv,'lth and L. Palla, hep-th/9601091. 115[ J. Balog, P Forg,'ics, Z. Horvfith and L. Palla, Phys. Lctt. B 324 (1994) 403; J. Evans and T. Hollowood, Nucl. Phys. B 438 (FS) C1995) 469. [ 16] T. Curtright anct C. Zachos, Phys. Rev. l,ett. 53 (1984) 1799. [17[ E Braaten, T. Curtright and C. Zachos, Nucl. Phys. B 260 (1985) 630. [18[ E Witten, Phys. Rev. I) 16 (1977) 2991. J 19l O. Alvarez and C.-II. Liu, hcp-th/9503226; O. Ah,arez. hep-lh/9511024. 1201 J. Schouteld, Nucl. Phys. B 169 (1980) 49. I 21 I J. Schwinger, Quantum Kinematics and Dynamics ( Benjamin, New York, 1970) pp. 175-177. [22] E. Cotton. C.R Acad. Sci. Paris 127 (1898) 447; Ik)n-a brief review, see S. Goldberg, Curvature and Homology (Dover, New York, 1982) ch. 3.9.