- Email: [email protected]
S1, and holomorphic geometry
Nuclear PhysicsB294 (1987) 556-572 North-Holland, Amsterdam
THE
SUPERSTRING, DIFF $ 1 / S l, AND HOLOMORPHIC GEOMETRY* Diego HARARI,DeogKi HONG, Pierre RAMOND and V.G.J. RODGERS Physics Department, University of Florida, Gainesville, FL 32611, USA Received 29 April 1987
We incorporate superstrings into the non-perturbative formulation of string field theories based on KS.hlergeometryrecentlyproposed by Bowickand Rajeev. The string field is conjectured to be the Kiihler potential of loop space, its equation of motion given by the vanishing of the curvature of a product bundle constructed over a graded Diff SI/s 1, as required for reparametrization invariance of the theory. We find that hosonic and fermionic loops in a Minkowski background solve the equation for the K~thlerpotential only in ten dimensions. We use geometric quantization techniques to calculate the curvature of the super-holomorphicvector bundle, since they emphasize the role of the complex geometry, and flag manifold techniques to calculate the curvature of the line bundle over super-Diff S1/S1.
I. Introduction In ref. [1] Bowick and Rajeev put forth a new approach to understanding string field theories on a purely geometric, nonperturbative basis. This new approach stems from the recognition that loops over Minkowski space form an infinite dimensional KLhler manifold. Kahler geometry [2] is a natural complex analogue of riemannian geometry, with both the metric and a complex structure preserved under parallel transport. The complex structure, however, changes under reparametrizations of the loops (diffeomorphisms of the circle) except for the rigid rotations. The space of all complex structures over a loop space is then the manifold M = Diff S1/S 1. A complex structure defined in the phase space of a classical system described in terms of symplectic geometry [3] plays an important role in the construction of the Hilbert space of the related quantum mechanical system according to the method of geometric quantization [4-6]. The complex structure provides a natural polarization: a division of the phase space into generalized "configuration" and " m o m e n tum" spaces. If the change in the quantum operators constructed using different complex structures define a nontrivial connection, its curvature represents an obstruction to an invariant definition of the vacuum state of the system. In the case of bosonic loops in a flat background, the curvature of the holomorphic vector * Work supported in part by the US Department of Energyunder grant no. FG05-86-ER40272. 0550-3213/87/$03.50©Elsevier Science Publishers B.V. (North-Holland Physics Pubhshing Division)
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bundle constructed over Diff s l / / s 1, that represents alternative Fock spaces for bosonic excitations, is just the central term of the Virasoro algebra. However, another bundle over Diff S1/S t needs to be considered simultaneously: the canonical line bundle, that can be interpreted as the vacuum for the ghosts of the theory. The condition necessary to define a reparametrization invariant vacuum for a bosonic string theory is then given by the vanishing of the Ricci tensor of the product bundle. This can be seen as an equation for the Kahler potential of the bosonic loop space, which is to be taken as the dynamical variable of the theory. In ref. [1] it was shown that the K~_ler potential for loops in a Minkowski background satisfies the equation only in 26 dimensions. Here we extend this proposal to include a fermionic loop space and a graded Diff S1/S 1. In sect. 2 we first review the salient features of geometric quantization, with particular emphasis on the case of KSJaler polarizations. Then we apply this method to open superstrings propagating in a flat background, since their phase space can be seen as a space of loops. We calculate the curvature of the holomorphic vector bundle constructed over super-Diff S1/S 1 with the Fock space of the open superstring as fiber, which turns out to be the usual anomaly in the super-Virasoro algebra. In sect. 3, using flag manifold techniques [7], we calculate the curvature of the complex line bundle over super-Diff S1/S 1 representing the ghosts excitations. No regularization is needed, using this technique, to calculate the Ricci tensor of this infinite dimensional manifold. The Ricci tensor of the product bundle representing fermionic and bosonic plus ghosts excitations is shown to cancel only in the critical dimension d = 10. The conclusions are drawn in sect. 4.
2. Geometric quantization of super-manifolds 2.1. REVIEW OF GEOMETRIC QUANTIZATION
What we will need is to generalize the methods of geometric quantization to graded Lie algebras. This has in fact been investigated in mathematics [8]. We shall use the salient features of the results but first let us review the scheme of geometric quantization [4-6]. Geometric quantization was developed by mathematicians to lay a rigorous ground work to the passage from classical to quantum mechanics. To begin with we assume the classical system under investigation can be represented as a phase space, F, of 2n dimensions endowed with a symplectic structure given by the 2-form ,0, which requires d,0 = 0 and det ,0 :# 0. The condition d~o = 0 implies the Jacobi identity for the Poisson bracket and det ~0 4= 0 is needed for ~0 to be invertible. Classical observables are represented as functions on the phase space, F. For any functions f and g on F, the Poisson bracket is given by {f, g} = - ~ o - l ( d f , d g ) = - , , ( x I, xg), where Xf is the hamiltonian vector field associated to the classical observable f and is given as Xf= -~0-1(df). The hamiltonian vector fields preserve the symplectic structure and form a Lie algebra. In formulas:
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.oqqxOa= 0 where *~x: is the Lie derivative in the direction of X/, and [Xf, Xg] = X{f, g}. If .Lax~0 = 0 globally, then the Lie algebra of the globally hamiltonian vector fields is the symmetry algebra of the classical system. Starting from the classical system one performs the following steps (1) Prequantization. (2) Polarization for quantum theory. One begins by defining a hermitian inner product using the 2-form, oa, on F to construct a prequantum Hilbert space, 171.One then seeks a correspondence between the classical observables, f and a set of operators, Of which act on 171. Geometric quantization provides such a correspondence by requiring the operators Of to satisfy =
(2.1)
Thus we have a mapping from Poisson brackets to commutators. The explicit operator relation to f is given by Of = - iVx: + f .
(2.2)
The covariant derivative, V, must have curvature [Vx, Wr] - Vtx, r I = - i o a ( X , Y )
(2.3)
to guarantee that the operator relation eq. (2.2) has the desired commutation relations (2.1). In a coordinate basis we may write eq. (2.2) as Of = --i( oJJ Oif )wj + f ,
(2.4)
where i, j = 1 . . . . . 2n; and define our curvature as iWtiWj] =- % j .
(2.5)
This construction provides a complex vector space of a U(1) line bundle over F. The sections, xO(~/), are elements of this Hilbert space, 171,where 3' is a point on F. To see that this is indeed a U(1) line bundle let us introduce a covariant derivative acting on sections '/' by
V,J' = ( 3 i - iA,)g'.
(2.6)
Then the symplectic 2-form, ~0= dA, is not affected by a U(1) gauge transformation A ~ A + dX under which xr, --, exp(iX)'/'.
(2.7)
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The inner product defined by (g', q,) = f 2 ~ , det oa
for ~/', q~~ 171
(2.8)
remains invariant under the gauge transformation. This completes the first step of prequantization. Clearly the above defined functions cannot describe quantum mechanics since they are functions of the whole phase space. In other words in order to be in accord with the uncertainty principle we are required to eliminate n degrees of freedom. In geometric quantization this is achieved by assigning to each point of the phase space an n-dimensional subspace of the 2n-dimensional tangent space. The vectors spanning such n-dimensional subspaces must be closed under commutation. Moreover, if Va, Vb (a, b = 1,..., n) are any of those vectors, then oa(Va, Vb) = 0. Such a choice of subspaces at each point in F is called a polarization, P. Abusing the notation, we shall also denote by P the n-dimensional subspaces at each point in F. Using the vectors spanning P we select those functions g' satisfying £°vg'('/) -= ViVrfl• = 0
for any V~ P.
(2.9)
This implies '/'(3') is only a function of n-variables. These functions span the prequantum Hilbert space H e and contain the quantum wave functions. To extract the quantum Hilbert space from these states one must provide an inner product and select those functions which are suitably normalized. If a classical observable f is such that its hamiltonian vector field Xf satisfies
£#vXf~ P
for any V~ P
(2.10)
then the prequantum operator O! given by eq. (2.2) is also appropriate at the quantum level, since it preserves the polarization condition. Indeed, if '/' satisfies X'v~ = 0 so does Ofg'. We shall introduce, however, the notation r ( f ) for the operators at the quantum level because when eq. (2.10) is not verified the quantum operators will not be given by O/. We shall see later how to deal with such observables. From now on we shall concentrate on the case where the phase space is a K~hler manifold, both because that is the case for the system under investigation in this paper and because Kiihler manifolds are particularly suitable for the geometric quantization techniques. Indeed, K~aler manifolds not only provide a natural choice of polarization, but also provide the correct quantum hermitian inner product, already in the prequantum Hilbert space. A K~ihler manifold admits both a symplectic structure and a complex structure, i.e. there exist a tensor field J satisfying ,I2_ - 1 .
(2.11)
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Its action on a vector field X maps the space of vector dissolves the space of vectors into two components (0,1) eigenvalue + i o r - i respectively of the operator J on fact to define what is called the Kahler polarization. Let
fields onto itself. One then and (1, 0) depending on the a vector. One exploits this V be any (1, 0) vector; then
.gave/' = 0
(2.12)
defines the Kahler polarization. In a K~ihler manifold the metric g( X, Y ) = ,.,( X , J Y )
(2.13)
is hermitian. The inner product is simply, as before
( 7 , , , ) = f ~ ¢ d e t ~0.
(2.14)
By performing a gauge transformation one can relate the functions satisfying the polarization condition to holomorphic or anti-holomorphic functions. Let us introduce complex coordinates {z% ~ } with a = 1,... n on the 2n-dimensional phase space. In a Kahler manifold (2.15)
OOaa= i O[aOa]K
and the covariant derivative may be written as V,, = Oa + Oa + ½( O, K + OaK ) .
(2.16)
We may gauge transform q" and ¢ by dP ~- e l / 2 K ~
and
~
= el/2K~/* .
(2.17)
This enables us to restate the polarization condition (2.12) as a a¢ = OaqZt~-"O.
(2.18)
(~, q ' ) = f ~ q ' e - K d e t oa.
(2.19)
The new inner product is
We wish now to extend these notions to the open superstring. 2.2. G E O M E T R I C Q U A N T I Z A T I O N OF T H E S U P E R S T R I N G
We wish to describe the closed superstring in terms of the Kahler geometry of superloop space, the space of maps from the circle S1 to the supermanifold ~d-1,1, with periodic boundary conditions in the bosonic coordinates and periodic or antiperiodic boundary conditions in the fermionic coordinates. In order to apply the
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geometric quantization techniques to this system it is convenient to think of superloop space as the phase space of the open superstring. To define the configuration space we consider the space of maps q: [0, q'/'] ~ ~ d - l , 1 ,
(2.20)
where ~d-1,1 is a supermanifold. These maps may be written as ~ = (qB, qF) where qB and qF refer to bosonic and fermionic coordinates. As in the bosonic case qB satisfies aoq B = 0 for o = 0, ~r and qF is written as a spinor distribution with two spinor components q+ and q_. The boundary conditions on qF are q + (0) = q_ (0),
(2.21a)
q + ( ~r ) = eq_ ( ~r),
(2.21b)
The parameter e specifies the Neveu-Schwarz (e = - 1 ) and Ramond (e = +1) sectors of the theory. As in the case of bosonic strings it is convenient to redefine the interval to [-~r, ~'] and map
q: In this way we are defining loops in R d-1,1. The coordinates in the classical phase space are ( qB, qF; PB, PF }, where PB and PF are the bosonic and fermionic conjugate momenta. On the above extended interval we may define (PB + q~)(o),
x(o)=
(PB--q~)(--°),
(eSPv +
- ~r ~< o ~<0,
qF)(O),
0~
(pFwegqF)(_o) '
(2.22a)
(2.22b)
/
T h e ' denotes derivatives with respect to o and the matrix S = |0 x and ~b form \ 1 +1. 0] components of ~ which maps S1 to loops in a supermanifold. Hence superloop space is the phase space of open superstrings. To proceed with the geometric quantization we need to display the symplectic form ~o. This ~o is given by 1
dofdo
u
Dv "~,~,
(2.23)
where u = u B + Ouv is a tangent vector of superloop space and 0 is a Grassmann
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variable. D = - i d / d 0 + 0 d / d o is the supersymmetric covariant derivative. It is straightforward to show that d~ = 0 and 0J is nondegenerate. We can expand the vectors u and v in their normal modes by writing oo
u~(°)=
E tl m
u~ , e '"° ,
(2.24a)
on~ein°.
(2.24b)
--OO
O0
Vt*(O) --- £ n ~
--o0
Then O0
~(u,o)
=
Y'
O0
- i ( - ) g~""~n t "U
" ' )~n' l ,
0 ~ ']"- n ~ p , , "1-
E
(
- , )"( u
F ) .Iz( o F ) -~. n . .
.
(2.25) The degree g(u B) appears because the "bosonic" part u a can be either ordinary or Grassmann valued.S Once the two-form o~ is given the Poisson bracket of any two functions f and g can be evaluated as ( f , g ) -- -o~ (X/, Xg). In the phase space of the superstring one can define functions ~, and % that provide a representation for the super-Virasoro Poisson algebra (h,,,,)k. } = - i ( m - n ) X m + . , {X,.,%} = - i ( ½ m - n ) % . + . , {~%, % ) = - i 2 ) L , + , ,
(2.26)
where m, n ~ Z. We shall work out explicitly the R sector of the theory and just quote the results for the NS sector. In the NS sector the indices of the Grassmann valued quantities (like %) run over the set of half integers. The functions satisfying (2.26) are given by 1 X . = - 4-7
f],~ d°ei"°[(x')2
+ iq~/] ,
(2.27a)
~7
i
q~.= ~-~ _ doe~"Ü+x '
(2.27b)
After a mode expansion of x and ~p x . ( e ino -
x(o) = /I ~
~(o) =
ein~)
(2.28a)
--0C
~
q,.(e ~"°-e~"~),
(2.28b)
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563
we can write a mode decomposition of )~n and % as oo ~rl ~
----
oo
21 ~,, p(n +p)xpX_n_ p 2 E p = - oe
t~n =
P~Pp~P-p-,
1
(2.29a)
p = - oo
f ( p q- ll ) @ p X _ n _ p . p = -oo
(2.29b)
The next step is the construction of the hamiltonian vector fields Xx, and X~o" and the corresponding operators Ox, and O~,. Afterwards, we need to introduce a polarization. Since we are dealing with a Kahler manifold, we will use its complex structure to define it. The complex structure J can be written as [ B. O v. J ( X ) = - i . . o E sgn(n) [X~[ --Ox~+ X~
O
]
- ~ . ].
(2.30)
Then J ( X ) = T-iX = X is of type (1, 0) and (0,1) (i.e. n < 0 or n > 0 respectively.) By construction j 2 = _ 1. Given the complex structure the natural (K~.hler) choice for the polarization is L~evg"= 0
for V ~ P = ( (1,0)-type vectors ).
(2.31)
In a coordinate system as the one used in (2.30) we can regard the polarization as spanned by all the vectors ( a/Ox,, O/O~pmwith n, m > 0). The wave function, q' can then be written as if' = e - r / 2 ~ with ~" a function of ft, a n d ~m only. The hamiltonian vector fields of functions t , , % preserve the polarization only if n >/0. In other words, ~vXx, and £PvX~. ~ P(y) only if n >/0. For all functions f which preserve the above polarization we may construct a quantum operator r(f) = Of on the Hilbert space, i.e.
r(f ). ~ = -i~Txi~ + f ~ .
(2.32)
It is straightforward to show that r(f)q" satisfies the polarization condition if does. Upon computation we find, choosing a gauge for the connection A such that Ap is zero for p > 0. 0
r(X~)=-
E
(P+n)xp+~O~ p
p>0
0 71 E ( 2 p + n ) f f p + ~ 3 f f p p>0
+½ f P(n-p)xpxn-p+½ ~ (½n-p)~n-I,~Pp p=l
p=l
(2.33a)
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564 and
0 r(q%) = - E ~ + , O--~p + Y', (n+p)~.+p p>0 p>0 +1
0
O~/p
i P~n-p'~p+1 i (n--P)~/pXn-p"
(2.33b)
p=l
p=l
Also for n = 0, we get 0 r(Xo) = - E n~, n>O O'¥n
~-~P~e
p>O
0
r(%) = -
0
~p
+ ia,
0
~P-~xp+ E pX--pO~p
(2.34a)
(2.34b)
p>O
The constant term (+ia) was added in eq. (2.34a) to account for possible normal ordering ambiguities, a is the ground state energy. Since the hamiltonian vector fields associated with the negatives n's do not satisfy (2.10), we need an indirect method to quantize those observables. By using a method similar to Bowick and Rajeev we can find the corresponding quantum operators for these observables. First note that since the polarization changes if we change the complex structure, so do the operators r(A,) and r(%). A change of the complex structure can be seen as a motion on super-Diff S1/S 1, since reparametrizations of the superloops, except for the rigid rotations, do not leave the complex structure invariant. The alternative Hilbert spaces for the superstring thus form a holomorphic vector bundle over super-Diff S1/S 1. Let L., F. be tangent vectors at the origin of super-Diff S1/S 1 that generate the superconforrnal algebra. If we require that r(X.) for a different complex structure be the left translation of r(X.) at the origin, we can define a connection by VL. =-oq°L~+ r(X.), and analogously VFn= ~ . + r(%). Now, since the superloop space is a KLlaler manifold, the metric is covariantly constant, i.e.
v,o(,, ,> = + <,, ~z~o,>. But since
VLo<*, *> = ~eLn<*,*> = <~eZp, *> + <*, ~ p > , we have
r(X_n) = --r(XO*. And similarly we get r(rp_.)= + r(%)*. Therefore we have quantum operators for
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565
2~+. and ~ + n with n < 0. Explicitly 0
0
r()~_.) = E pX-p-Cg~p+. + ~1 E (2p q_ n )lpp O~p+n p>O
p>O
1
l
- ~ p=l 0~._~ 0 ~
o
e(
n_p
~ p=l 2
o{, ~ 0~._~ '
(2.35a)
o
l~p0--Xn+p p>0
r ( ep_ . ) = - E p x-'p-pV/aS'n + -'k E p>0
n
o
0
o
0
(2.35b)
p=l
The action of super-Diff S1/S 1 in the superloop space is tantamount to a change of the complex structure used to polarize the phase space, and thus to a change in the quantum operators r(~,n), r(cp.). As said before, such a change defines a connection through VL. =L#L. + r()%), VF. =LPF. + r(cp.). If the connection has curvature, the quantum theory has an anomaly. The curvature is given, in the R sector, by
F(L., Lm) = [VL, VLm] - VI/.., t-ml = [r(~n),r(Xm) ]-r((x.,~m})
= ( ~ d n 3 - 2 a n ) 8 . _m,
(2.36a)
F(Fn, Fro) = [r(e&),r(%,)]+-r({~p~.epm}) =(½dn2-2a)~n,_m,
(2.36b)
where d is the number of space-time dimensions. Recall that a is an arbitrary additive constant term in r(~,o). In the NS sector the result is
F( L,,Lm) = [~d(n 3 - n ) - 2an]8,,_m,
(2.36c)
F(Fr, Fs)= [ ½ d ( r 2 - ¼) - 2a] 8r,_s,
(2.36d)
where r, s are half integers. From eqs. (2.36) we see that the hermitian operators do not form a representation to super-Diff Sk The r.h.s, indeed represents the anomaly in the super-Virasoro
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algebra. However since this anomaly is a 2-cocycle these operators form a projective representation of super-Diff S1. Only by introducing ghosts are we able to construct a unitary representation of super-Diff S1. Throughout, the Kiihler potential K stands on special ground since it induces the symplectic structure. It is for this reason that K is identified with the closed string or superstring field. As it stands, however, we still do not have a gauge invariant description due to the anomaly. We must therefore find a geometric description of the ghost sector. In the next section we compute the curvature of the line bundle over super-Diff S1/S 1 This is interpreted as the ghost contribution to the representation. We then put the two bundles together to form the flat background solution to the super-Bowick-Rajeev equations of motion. A few remarks are in order. Since the superloop case also contains a bosonic sector, the constant maps (or rigid deformations) are lost. This renders ~0(u, v) to be degenerate for the zero modes. Bowick and Rajeev are able to remedy this by using the notion of a contact manifold [3]. Since the zero modes are not lost for the fermionic sector of the theory we can rely on this notion of a contact manifold just for the bosonic part of the theory, 3. Curvature of the line bundle
The physical processes of the string theory will of course be independent of the way the string is parametrized. As we have just seen, canonical transformations can induce a different parametrization and change the complex structure. Thus a particular complex structure carries with it a particular parametrization. By applying super-Diff S1 to J we can effectively generate all possible paramerizations. However super-Diff S1 (as Diff S1 for the bosonic case) contains generators which leave the complex structure invariant. Indeed the action of L 0 and F o on J leaves it invariant. By dividing out these rigid rotations (super S1) we may represent the space of all complex structures. This is the manifold super-Dill S1/S 1. We will consider super-Diff Sx/S 1 as an abstract manifold. By studing the properties of its curvature we can extract information on the anomaly structure of super-Diff S1/S 1. Then by combining the holomorphic vector bundle of the last section with the line bundle we are about to describe we will have a gauge, i.e. reparametrization invariant theory. The manifold M = super-Diff S1/S 1 is a homogeneous and KLlaler manifold since it is a coset space and has a KLlaler 2-form. The tangent vectors of M form the superconformal algebra, [Ln, L m ] = ( n - m ) L n + m,
[Fn, Fro]+= 2Ln+m,
(3.1)
D. Harari et al. / Superstring
567
and carry the complex structure J given by
JG., =
iGm -iG.,
if m < 0 if m > 0,
(3 .2)
where G., ~ (L,., F., for m ~ Z} for the R sector, and Gm ~ {Lm, Fro+l~2 for the m ~ Z) for the NS sector. As in the previous section, we write all intermediate steps for the R sector and quote the results for both sectors. The most general solution to the closure condition of the Kahler 2-form is given by [9,10] o~( Lm, L , ) = (am 3 + brn)Sm, _ , , 0~(Lm, r , ) = 0, o~( F,,, F,) = (4am z + b )8,,,, _ , ,
(3.3)
o~ is not invertible for - b / a = m 2 for nonzero integers m. Let us consider M as a graded flag manifold G / H and exploit its complexification by splitting it into M+ and M . Then we can take the decomposition of the associated Lie algebra of G as
Gc=Hc+M++M_ where
H e = {G 0= Lo,F0}. Since M is a flag manifold, we can calculate the curvature of M by using the Toeplitz operator method [7]. For any tangent vector, X, we can define the Toeplitz operator ~(X) = Vx-~x, where V'x is a covariant derivative and -~x is the Lie derivative. Since cp(X)(fY) = f ~ p ( X ) Y , it preserves the tensoriality and is a linear operator. So the Toeplitz operator is an intrinsic quantity of the manifold. Since M is a homogeneous Kahler manifold, the Lie derivative preserves the Kahler structure of M (namely the complex and symplectic structure of M) and there is a natural connection W which preserves the KLlaler structure. The formulae for the Toeplitz operators in terms of the complexified quantities can be determined from the requirement that both ~7x and LPx preserve the Kahler structure [7]. They are ~(H) = -ad H
for H ~ He,
cp(X ) = - ~ r + . a d X
for X ~ M
¢p(X+) = _ ( _ ) g ( x + ) ~ ( ~ + ) t
for X + ~ M + ,
, (3.4)
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where g(X) is the grade of X (0 for L, and 1 for Fn) and we define ~(X+) as the adjoint of q0(X_) with respect to the metric, rt+ is the projection operator over M_. We need the phase factor for qo(X+), since Fn's anticommute with themselves. Since M is homogeneous, it is enough to calculate the curvature at one point. Then the curvature is, following the expression for a supermanifold Ill], for any vector Z on M and for any 1-form W on M,
w-R(x, Y)z=
Vrx,
}z
= (-)g(W)g(Z)w. ([¢p(X), ¢p(Y)] ± - ¢p([ X, Y]:L ) ) Z ,
(3.5)
where g(W) and g(Z) are the grades of W and Z respectively. Applied to (3.1) eq.
(3.4) gives, cP(Lo)L_ p = - [ L o , L p ] = - p L _ p , ¢P(Lo)F_ p = - [ L o , F_p] = - p F _ p , ¢P(ro)L_ p = - [ g o , L_p]-- - ½ P r p , ¢p( go)F_p = - { g o , F p } = - 2 L _ p , ¢P(Lm)L_ p = - O ( p - m)(p + m)Lm_p,
m > O,
¢p(L,,,)F_ p = - O ( p - m ) ( ½ m + p ) F m _ p,
m>0,
¢p( Fm)L_ p = - O ( p - m ) ( m + ½ p ) r m _ p,
m>0,
~(Fm)F_ p = --20(p-- m)Lm_p, =
m > O,
ap3 + bp ( p + 2m) a(p + m) 3 + b(p + m) L-m-P'
m>0,
(p + ~m)(4ap z + b) ep( L_m)F_ p =
¢p(F ,,,) L_ p = _
4a(p + rn)2 + b
F_p_m,
m>0,
2( ap3 + bp ) 4a( p + m)2 + b
g-(p+m),
m>0,
(3.6)
and cp(F_m)g p =
½(p+ 3m)(4ap 2 +b) a(p+m)3+b(p+m)L_(m+p),
re>O,
D. Harariet aL/ Superstring
569
where p > 0. From eq. (3.5) we get
R( L ,,,, L,,)L_p= - O ( p - m)(p + m) 2a(p
+
m) 3 +
b(p ap3 + bp
m)
(p + 2m)2(ap3 + bp) -2mp}8~,,8~L q, (3.7a) a(p+m)3+b(p+m)
t l m124a(p- m)2 + b R(L_m,L,,)F - e = ~t O(p-m)[p , + -~ ] -4-ap2--+
( p + 3m)2(4ap2 + b ) + 2rap} ~m,n~qF-q, 4a(p+m)Z+b R(F_m,F.)L_r=
(3.7b)
4a(p_m)2+b O(p-m)(½P+m) 2 -~p'~p
4(apa +bP)
+ 4a( p + m )2 + b
+ 2p)~m n~qZ_q,
(3.7C)
and R(F
-~'
F.)F- e
= f-40(p-
m) a ( p -
I
m) 3 +
b ( p - m)
4ap2 + b 1 (p+ 3m)E(4ap2 + b) -2p~Sm,SqF q, (3.7d) 4 a(p+m)3+b(p+m) ] ' -
where O is the step function. All other components vanish. To get the Ricci two-form we have to take trace over positive p of the Riemann curvature. Performing the trace one finds, in the R sector
RJc( L_m, L.) =
1°m36
Ric(F_,,,, F,,) = - 2t°m28 re,el
•
(3.8a)
In the NS sector the result is Ric(L_m, Ric(F_r,
L.) = ( - ~m 3+ ¼m)3.... 1 ~ V~) = ( - ~ r z + ~) r,~ ,
(3.8b)
D. Harariet al. / Superstring
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with r, s half integers. When calculating the Ricci form, we find that the infinite series converge remarkably. So again we do not need any regularization to calculate the trace. We combine the result of this section and the curvature of the holomorphic vector bundle, eq. (2.36), to write the curvature of the twisted bundle in the R sector as
G( Lm, L _ , ) = F( Lm, L _ , ) + Ric(L_m, L + , ) = [ 1 ( d - 10)m 3 - 2arn]8 . . . .
G( F,,, V_,) = F( Fm, V_,) + R i c ( F _ m, F,) = [ ½ ( d - 10)m 2 - 2a]/Jm, . .
(3.9a)
Thus for d = 10 and a = 0 we have a gauge invariant description of the theory. In passing we note that the vanishing of the ground state energy in the R sector signals the presence of space-time supersymmetry. In the NS sector the curvature of the twisted bundle is given by G ( L m, L _ , ) = [ l ( d _ 10)m 3 - ( I d - ¼ + 2a)m]/Jm,,, G(F~, F_s) = [ ½ ( d - 10)r 2 - ( I d - j + 2a )] 6r, s . Thus d = 10 and a = sector, as expected.
(3.98)
½ gives a reparametrization invariant theory in the NS
4. Conclusions and remarks
As we remarked earlier by combining the holomorphic vector bundle with the line bundle corresponding to the ghost sector we are able to describe a gauge, i.e. reparametrization, invariant theory. We have shown that the twisted bundle with curvature G given in eq. (3.9) is precisely this combination for d = 10. Throughout we have focused only on the flat space case. It would be interesting to perform an adiabatic approximation for the super manifold to find the graded analog of the bosonic case. Bowick and Rajeev found that such an approximation for the bosonic case yields the vacuum Einstein equations. The more general setting realizes a manifold endowed with a KLlaler potential K. From here one can construct the covariant derivatives and all other geometric objects. The equation GK = 0 is to be interpreted as the equation of motion for the closed superstring field. Also we remark that in computing the curvature for super-Diff S1/S x we relied on techniques which go through without regularization. It has been shown that coset space techniques designed for finite-dimensional graded Lie algebras may be used if
D. Harariet al. / Superstring
571
one regulates with zeta functions [12]. This places the computation of the curvature in a more familiar setting. Since the superstring enjoys a global supersymmetry, we may exploit this fact to argue that M describes the ghost sector of the superstring just as in the bosonic case where here this sector also admits commuting ghost for the fermions. Let us make a further remark about supersymmetry in general. Consider a 2n-dimensional graded vector space with 2n bosonic and 2n fermionic vector fields. For the bosonic sector the complex structures are parametrized by O(2n)/U(n), since we consider only those orthogonal rotations which do not leave the complex structure invariant. For O(2n) its overlap with SP(2n) is U(n) therefore we arrive at O(2n)/U(n). However for the fermionic sector the complex structure is symmetric (given by O(2n)) and its vector transformations are SP(2n). Therefore the space of fermionic complex structure is parameterized by SP(2n)/U(n). One can see that dimensionally these spaces are different, i.e. Dim(bosonic complex structure) =
2n(2n - 1)
2 2n (2n + 1)
Dim(fermion complex structure) =
n2=n2-n,
n2=n2+n.
A global supersymmetry transformation maps the bosonic and fermionic structures into each other but this mapping is not 1-1. It is interesting to ask what further structure is needed to define the complex structure of a graded vector space which has supersymmetry. In the ~ dimensional case the mapping may be 1-1. Rajeev [13] has found a prescription for taking n to oo in the bosonic case (namely by considering finite energy requirements). Perhaps similar reasoning is needed for the graded case. Finally in ref. [14], Pilch and Warner have used still other methods to compute the curvature due to the holomorphic vector bundle and the contribution to the ghost. We would like to thank the high energy theory group at the University of Florida especially P. Oh. and E. Piard for very fruitful discussions. We would also like to thank M. Bowick and S.G. Rajeev for explaining their work. One of us (D.K.H.) is supported by the Division of Sponsored Research at the University of Florida.
References
[1] [2] [3] [4]
M. Bowickand S.G. Rajeev,MIT preprint CTP.~1450; Phys. Rev. Lett. 58 (1987) 535 S. Goldberg, Curvature and homology(Dover, NY, 1982) V.I. Arnold, Mathematicalmethodsof classicalmechanics(Springer, New York, 1978) NJ.M. Woodhouse,Geometricquantization(ClarendonPress, Oxford, UK, 1980)
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D. Harari et al. / Superstring
[5] J. Sniatycki, Geometric quantization and quantum mechanics (Springer, New York, 1980) [6] A. Ashtekar and M. StiUerman, Syracuse University preprint (1985) [7] D. Freed, in Infinite dimensional groups with applications, ed. V. Kac (Springer, Berlin, 1985); Ph.D. thesis, University of California at Berkeley (1985) [8] B. Kostant, Graded manifolds, graded Lie theory and prequantization, in Differential geometrical methods in mathematical physics, Lecture Notes in Mathematics, vol. 570 (Springer, Berlin, 1975) [9] P. van Alstine, Phys. Rev. D12 (1975) 1834 [10] G. Segal, Commun. Math. Phys. 80 (1981) 30 [11] B. DeWitt, Supermanifolds (Cambridge University Press, 1984) [12] P. Oh and P. Ramond, University of Florida preprint (April 1987) [13] S.G. Rajeev, private communication [14] K. Pilch and N.P. Warner, MIT preprint CTP#1457 (Feb. 1987)
THE
SUPERSTRING, DIFF $ 1 / S l, AND HOLOMORPHIC GEOMETRY* Diego HARARI,DeogKi HONG, Pierre RAMOND and V.G.J. RODGERS Physics Department, University of Florida, Gainesville, FL 32611, USA Received 29 April 1987
We incorporate superstrings into the non-perturbative formulation of string field theories based on KS.hlergeometryrecentlyproposed by Bowickand Rajeev. The string field is conjectured to be the Kiihler potential of loop space, its equation of motion given by the vanishing of the curvature of a product bundle constructed over a graded Diff SI/s 1, as required for reparametrization invariance of the theory. We find that hosonic and fermionic loops in a Minkowski background solve the equation for the K~thlerpotential only in ten dimensions. We use geometric quantization techniques to calculate the curvature of the super-holomorphicvector bundle, since they emphasize the role of the complex geometry, and flag manifold techniques to calculate the curvature of the line bundle over super-Diff S1/S1.
I. Introduction In ref. [1] Bowick and Rajeev put forth a new approach to understanding string field theories on a purely geometric, nonperturbative basis. This new approach stems from the recognition that loops over Minkowski space form an infinite dimensional KLhler manifold. Kahler geometry [2] is a natural complex analogue of riemannian geometry, with both the metric and a complex structure preserved under parallel transport. The complex structure, however, changes under reparametrizations of the loops (diffeomorphisms of the circle) except for the rigid rotations. The space of all complex structures over a loop space is then the manifold M = Diff S1/S 1. A complex structure defined in the phase space of a classical system described in terms of symplectic geometry [3] plays an important role in the construction of the Hilbert space of the related quantum mechanical system according to the method of geometric quantization [4-6]. The complex structure provides a natural polarization: a division of the phase space into generalized "configuration" and " m o m e n tum" spaces. If the change in the quantum operators constructed using different complex structures define a nontrivial connection, its curvature represents an obstruction to an invariant definition of the vacuum state of the system. In the case of bosonic loops in a flat background, the curvature of the holomorphic vector * Work supported in part by the US Department of Energyunder grant no. FG05-86-ER40272. 0550-3213/87/$03.50©Elsevier Science Publishers B.V. (North-Holland Physics Pubhshing Division)
D. Harari et al. / Superstring
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bundle constructed over Diff s l / / s 1, that represents alternative Fock spaces for bosonic excitations, is just the central term of the Virasoro algebra. However, another bundle over Diff S1/S t needs to be considered simultaneously: the canonical line bundle, that can be interpreted as the vacuum for the ghosts of the theory. The condition necessary to define a reparametrization invariant vacuum for a bosonic string theory is then given by the vanishing of the Ricci tensor of the product bundle. This can be seen as an equation for the Kahler potential of the bosonic loop space, which is to be taken as the dynamical variable of the theory. In ref. [1] it was shown that the K~_ler potential for loops in a Minkowski background satisfies the equation only in 26 dimensions. Here we extend this proposal to include a fermionic loop space and a graded Diff S1/S 1. In sect. 2 we first review the salient features of geometric quantization, with particular emphasis on the case of KSJaler polarizations. Then we apply this method to open superstrings propagating in a flat background, since their phase space can be seen as a space of loops. We calculate the curvature of the holomorphic vector bundle constructed over super-Diff S1/S 1 with the Fock space of the open superstring as fiber, which turns out to be the usual anomaly in the super-Virasoro algebra. In sect. 3, using flag manifold techniques [7], we calculate the curvature of the complex line bundle over super-Diff S1/S 1 representing the ghosts excitations. No regularization is needed, using this technique, to calculate the Ricci tensor of this infinite dimensional manifold. The Ricci tensor of the product bundle representing fermionic and bosonic plus ghosts excitations is shown to cancel only in the critical dimension d = 10. The conclusions are drawn in sect. 4.
2. Geometric quantization of super-manifolds 2.1. REVIEW OF GEOMETRIC QUANTIZATION
What we will need is to generalize the methods of geometric quantization to graded Lie algebras. This has in fact been investigated in mathematics [8]. We shall use the salient features of the results but first let us review the scheme of geometric quantization [4-6]. Geometric quantization was developed by mathematicians to lay a rigorous ground work to the passage from classical to quantum mechanics. To begin with we assume the classical system under investigation can be represented as a phase space, F, of 2n dimensions endowed with a symplectic structure given by the 2-form ,0, which requires d,0 = 0 and det ,0 :# 0. The condition d~o = 0 implies the Jacobi identity for the Poisson bracket and det ~0 4= 0 is needed for ~0 to be invertible. Classical observables are represented as functions on the phase space, F. For any functions f and g on F, the Poisson bracket is given by {f, g} = - ~ o - l ( d f , d g ) = - , , ( x I, xg), where Xf is the hamiltonian vector field associated to the classical observable f and is given as Xf= -~0-1(df). The hamiltonian vector fields preserve the symplectic structure and form a Lie algebra. In formulas:
558
D. Harari et aL / Superstring
.oqqxOa= 0 where *~x: is the Lie derivative in the direction of X/, and [Xf, Xg] = X{f, g}. If .Lax~0 = 0 globally, then the Lie algebra of the globally hamiltonian vector fields is the symmetry algebra of the classical system. Starting from the classical system one performs the following steps (1) Prequantization. (2) Polarization for quantum theory. One begins by defining a hermitian inner product using the 2-form, oa, on F to construct a prequantum Hilbert space, 171.One then seeks a correspondence between the classical observables, f and a set of operators, Of which act on 171. Geometric quantization provides such a correspondence by requiring the operators Of to satisfy =
(2.1)
Thus we have a mapping from Poisson brackets to commutators. The explicit operator relation to f is given by Of = - iVx: + f .
(2.2)
The covariant derivative, V, must have curvature [Vx, Wr] - Vtx, r I = - i o a ( X , Y )
(2.3)
to guarantee that the operator relation eq. (2.2) has the desired commutation relations (2.1). In a coordinate basis we may write eq. (2.2) as Of = --i( oJJ Oif )wj + f ,
(2.4)
where i, j = 1 . . . . . 2n; and define our curvature as iWtiWj] =- % j .
(2.5)
This construction provides a complex vector space of a U(1) line bundle over F. The sections, xO(~/), are elements of this Hilbert space, 171,where 3' is a point on F. To see that this is indeed a U(1) line bundle let us introduce a covariant derivative acting on sections '/' by
V,J' = ( 3 i - iA,)g'.
(2.6)
Then the symplectic 2-form, ~0= dA, is not affected by a U(1) gauge transformation A ~ A + dX under which xr, --, exp(iX)'/'.
(2.7)
D. Harari et al. / Superstring
559
The inner product defined by (g', q,) = f 2 ~ , det oa
for ~/', q~~ 171
(2.8)
remains invariant under the gauge transformation. This completes the first step of prequantization. Clearly the above defined functions cannot describe quantum mechanics since they are functions of the whole phase space. In other words in order to be in accord with the uncertainty principle we are required to eliminate n degrees of freedom. In geometric quantization this is achieved by assigning to each point of the phase space an n-dimensional subspace of the 2n-dimensional tangent space. The vectors spanning such n-dimensional subspaces must be closed under commutation. Moreover, if Va, Vb (a, b = 1,..., n) are any of those vectors, then oa(Va, Vb) = 0. Such a choice of subspaces at each point in F is called a polarization, P. Abusing the notation, we shall also denote by P the n-dimensional subspaces at each point in F. Using the vectors spanning P we select those functions g' satisfying £°vg'('/) -= ViVrfl• = 0
for any V~ P.
(2.9)
This implies '/'(3') is only a function of n-variables. These functions span the prequantum Hilbert space H e and contain the quantum wave functions. To extract the quantum Hilbert space from these states one must provide an inner product and select those functions which are suitably normalized. If a classical observable f is such that its hamiltonian vector field Xf satisfies
£#vXf~ P
for any V~ P
(2.10)
then the prequantum operator O! given by eq. (2.2) is also appropriate at the quantum level, since it preserves the polarization condition. Indeed, if '/' satisfies X'v~ = 0 so does Ofg'. We shall introduce, however, the notation r ( f ) for the operators at the quantum level because when eq. (2.10) is not verified the quantum operators will not be given by O/. We shall see later how to deal with such observables. From now on we shall concentrate on the case where the phase space is a K~hler manifold, both because that is the case for the system under investigation in this paper and because Kiihler manifolds are particularly suitable for the geometric quantization techniques. Indeed, K~aler manifolds not only provide a natural choice of polarization, but also provide the correct quantum hermitian inner product, already in the prequantum Hilbert space. A K~ihler manifold admits both a symplectic structure and a complex structure, i.e. there exist a tensor field J satisfying ,I2_ - 1 .
(2.11)
D. Harari et al. / Superstring
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Its action on a vector field X maps the space of vector dissolves the space of vectors into two components (0,1) eigenvalue + i o r - i respectively of the operator J on fact to define what is called the Kahler polarization. Let
fields onto itself. One then and (1, 0) depending on the a vector. One exploits this V be any (1, 0) vector; then
.gave/' = 0
(2.12)
defines the Kahler polarization. In a K~ihler manifold the metric g( X, Y ) = ,.,( X , J Y )
(2.13)
is hermitian. The inner product is simply, as before
( 7 , , , ) = f ~ ¢ d e t ~0.
(2.14)
By performing a gauge transformation one can relate the functions satisfying the polarization condition to holomorphic or anti-holomorphic functions. Let us introduce complex coordinates {z% ~ } with a = 1,... n on the 2n-dimensional phase space. In a Kahler manifold (2.15)
OOaa= i O[aOa]K
and the covariant derivative may be written as V,, = Oa + Oa + ½( O, K + OaK ) .
(2.16)
We may gauge transform q" and ¢ by dP ~- e l / 2 K ~
and
~
= el/2K~/* .
(2.17)
This enables us to restate the polarization condition (2.12) as a a¢ = OaqZt~-"O.
(2.18)
(~, q ' ) = f ~ q ' e - K d e t oa.
(2.19)
The new inner product is
We wish now to extend these notions to the open superstring. 2.2. G E O M E T R I C Q U A N T I Z A T I O N OF T H E S U P E R S T R I N G
We wish to describe the closed superstring in terms of the Kahler geometry of superloop space, the space of maps from the circle S1 to the supermanifold ~d-1,1, with periodic boundary conditions in the bosonic coordinates and periodic or antiperiodic boundary conditions in the fermionic coordinates. In order to apply the
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D. Harari et al. / Superstring
geometric quantization techniques to this system it is convenient to think of superloop space as the phase space of the open superstring. To define the configuration space we consider the space of maps q: [0, q'/'] ~ ~ d - l , 1 ,
(2.20)
where ~d-1,1 is a supermanifold. These maps may be written as ~ = (qB, qF) where qB and qF refer to bosonic and fermionic coordinates. As in the bosonic case qB satisfies aoq B = 0 for o = 0, ~r and qF is written as a spinor distribution with two spinor components q+ and q_. The boundary conditions on qF are q + (0) = q_ (0),
(2.21a)
q + ( ~r ) = eq_ ( ~r),
(2.21b)
The parameter e specifies the Neveu-Schwarz (e = - 1 ) and Ramond (e = +1) sectors of the theory. As in the case of bosonic strings it is convenient to redefine the interval to [-~r, ~'] and map
q: In this way we are defining loops in R d-1,1. The coordinates in the classical phase space are ( qB, qF; PB, PF }, where PB and PF are the bosonic and fermionic conjugate momenta. On the above extended interval we may define (PB + q~)(o),
x(o)=
(PB--q~)(--°),
(eSPv +
- ~r ~< o ~<0,
qF)(O),
0~
(pFwegqF)(_o) '
(2.22a)
(2.22b)
/
T h e ' denotes derivatives with respect to o and the matrix S = |0 x and ~b form \ 1 +1. 0] components of ~ which maps S1 to loops in a supermanifold. Hence superloop space is the phase space of open superstrings. To proceed with the geometric quantization we need to display the symplectic form ~o. This ~o is given by 1
dofdo
u
Dv "~,~,
(2.23)
where u = u B + Ouv is a tangent vector of superloop space and 0 is a Grassmann
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D. Harari et al. / Superstring
variable. D = - i d / d 0 + 0 d / d o is the supersymmetric covariant derivative. It is straightforward to show that d~ = 0 and 0J is nondegenerate. We can expand the vectors u and v in their normal modes by writing oo
u~(°)=
E tl m
u~ , e '"° ,
(2.24a)
on~ein°.
(2.24b)
--OO
O0
Vt*(O) --- £ n ~
--o0
Then O0
~(u,o)
=
Y'
O0
- i ( - ) g~""~n t "U
" ' )~n' l ,
0 ~ ']"- n ~ p , , "1-
E
(
- , )"( u
F ) .Iz( o F ) -~. n . .
.
(2.25) The degree g(u B) appears because the "bosonic" part u a can be either ordinary or Grassmann valued.S Once the two-form o~ is given the Poisson bracket of any two functions f and g can be evaluated as ( f , g ) -- -o~ (X/, Xg). In the phase space of the superstring one can define functions ~, and % that provide a representation for the super-Virasoro Poisson algebra (h,,,,)k. } = - i ( m - n ) X m + . , {X,.,%} = - i ( ½ m - n ) % . + . , {~%, % ) = - i 2 ) L , + , ,
(2.26)
where m, n ~ Z. We shall work out explicitly the R sector of the theory and just quote the results for the NS sector. In the NS sector the indices of the Grassmann valued quantities (like %) run over the set of half integers. The functions satisfying (2.26) are given by 1 X . = - 4-7
f],~ d°ei"°[(x')2
+ iq~/] ,
(2.27a)
~7
i
q~.= ~-~ _ doe~"Ü+x '
(2.27b)
After a mode expansion of x and ~p x . ( e ino -
x(o) = /I ~
~(o) =
ein~)
(2.28a)
--0C
~
q,.(e ~"°-e~"~),
(2.28b)
D. Harari et al. / Superstring
563
we can write a mode decomposition of )~n and % as oo ~rl ~
----
oo
21 ~,, p(n +p)xpX_n_ p 2 E p = - oe
t~n =
P~Pp~P-p-,
1
(2.29a)
p = - oo
f ( p q- ll ) @ p X _ n _ p . p = -oo
(2.29b)
The next step is the construction of the hamiltonian vector fields Xx, and X~o" and the corresponding operators Ox, and O~,. Afterwards, we need to introduce a polarization. Since we are dealing with a Kahler manifold, we will use its complex structure to define it. The complex structure J can be written as [ B. O v. J ( X ) = - i . . o E sgn(n) [X~[ --Ox~+ X~
O
]
- ~ . ].
(2.30)
Then J ( X ) = T-iX = X is of type (1, 0) and (0,1) (i.e. n < 0 or n > 0 respectively.) By construction j 2 = _ 1. Given the complex structure the natural (K~.hler) choice for the polarization is L~evg"= 0
for V ~ P = ( (1,0)-type vectors ).
(2.31)
In a coordinate system as the one used in (2.30) we can regard the polarization as spanned by all the vectors ( a/Ox,, O/O~pmwith n, m > 0). The wave function, q' can then be written as if' = e - r / 2 ~ with ~" a function of ft, a n d ~m only. The hamiltonian vector fields of functions t , , % preserve the polarization only if n >/0. In other words, ~vXx, and £PvX~. ~ P(y) only if n >/0. For all functions f which preserve the above polarization we may construct a quantum operator r(f) = Of on the Hilbert space, i.e.
r(f ). ~ = -i~Txi~ + f ~ .
(2.32)
It is straightforward to show that r(f)q" satisfies the polarization condition if does. Upon computation we find, choosing a gauge for the connection A such that Ap is zero for p > 0. 0
r(X~)=-
E
(P+n)xp+~O~ p
p>0
0 71 E ( 2 p + n ) f f p + ~ 3 f f p p>0
+½ f P(n-p)xpxn-p+½ ~ (½n-p)~n-I,~Pp p=l
p=l
(2.33a)
D. Harari et aL / Superstring
564 and
0 r(q%) = - E ~ + , O--~p + Y', (n+p)~.+p p>0 p>0 +1
0
O~/p
i P~n-p'~p+1 i (n--P)~/pXn-p"
(2.33b)
p=l
p=l
Also for n = 0, we get 0 r(Xo) = - E n~, n>O O'¥n
~-~P~e
p>O
0
r(%) = -
0
~p
+ ia,
0
~P-~xp+ E pX--pO~p
(2.34a)
(2.34b)
p>O
The constant term (+ia) was added in eq. (2.34a) to account for possible normal ordering ambiguities, a is the ground state energy. Since the hamiltonian vector fields associated with the negatives n's do not satisfy (2.10), we need an indirect method to quantize those observables. By using a method similar to Bowick and Rajeev we can find the corresponding quantum operators for these observables. First note that since the polarization changes if we change the complex structure, so do the operators r(A,) and r(%). A change of the complex structure can be seen as a motion on super-Diff S1/S 1, since reparametrizations of the superloops, except for the rigid rotations, do not leave the complex structure invariant. The alternative Hilbert spaces for the superstring thus form a holomorphic vector bundle over super-Diff S1/S 1. Let L., F. be tangent vectors at the origin of super-Diff S1/S 1 that generate the superconforrnal algebra. If we require that r(X.) for a different complex structure be the left translation of r(X.) at the origin, we can define a connection by VL. =-oq°L~+ r(X.), and analogously VFn= ~ . + r(%). Now, since the superloop space is a KLlaler manifold, the metric is covariantly constant, i.e.
v,o(,, ,> =
VLo<*, *> = ~eLn<*,*> = <~eZp, *> + <*, ~ p > , we have
r(X_n) = --r(XO*. And similarly we get r(rp_.)= + r(%)*. Therefore we have quantum operators for
D. Harari et al. / Superstring
565
2~+. and ~ + n with n < 0. Explicitly 0
0
r()~_.) = E pX-p-Cg~p+. + ~1 E (2p q_ n )lpp O~p+n p>O
p>O
1
l
- ~ p=l 0~._~ 0 ~
o
e(
n_p
~ p=l 2
o{, ~ 0~._~ '
(2.35a)
o
l~p0--Xn+p p>0
r ( ep_ . ) = - E p x-'p-pV/aS'n + -'k E p>0
n
o
0
o
0
(2.35b)
p=l
The action of super-Diff S1/S 1 in the superloop space is tantamount to a change of the complex structure used to polarize the phase space, and thus to a change in the quantum operators r(~,n), r(cp.). As said before, such a change defines a connection through VL. =L#L. + r()%), VF. =LPF. + r(cp.). If the connection has curvature, the quantum theory has an anomaly. The curvature is given, in the R sector, by
F(L., Lm) = [VL, VLm] - VI/.., t-ml = [r(~n),r(Xm) ]-r((x.,~m})
= ( ~ d n 3 - 2 a n ) 8 . _m,
(2.36a)
F(Fn, Fro) = [r(e&),r(%,)]+-r({~p~.epm}) =(½dn2-2a)~n,_m,
(2.36b)
where d is the number of space-time dimensions. Recall that a is an arbitrary additive constant term in r(~,o). In the NS sector the result is
F( L,,Lm) = [~d(n 3 - n ) - 2an]8,,_m,
(2.36c)
F(Fr, Fs)= [ ½ d ( r 2 - ¼) - 2a] 8r,_s,
(2.36d)
where r, s are half integers. From eqs. (2.36) we see that the hermitian operators do not form a representation to super-Diff Sk The r.h.s, indeed represents the anomaly in the super-Virasoro
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algebra. However since this anomaly is a 2-cocycle these operators form a projective representation of super-Diff S1. Only by introducing ghosts are we able to construct a unitary representation of super-Diff S1. Throughout, the Kiihler potential K stands on special ground since it induces the symplectic structure. It is for this reason that K is identified with the closed string or superstring field. As it stands, however, we still do not have a gauge invariant description due to the anomaly. We must therefore find a geometric description of the ghost sector. In the next section we compute the curvature of the line bundle over super-Diff S1/S 1 This is interpreted as the ghost contribution to the representation. We then put the two bundles together to form the flat background solution to the super-Bowick-Rajeev equations of motion. A few remarks are in order. Since the superloop case also contains a bosonic sector, the constant maps (or rigid deformations) are lost. This renders ~0(u, v) to be degenerate for the zero modes. Bowick and Rajeev are able to remedy this by using the notion of a contact manifold [3]. Since the zero modes are not lost for the fermionic sector of the theory we can rely on this notion of a contact manifold just for the bosonic part of the theory, 3. Curvature of the line bundle
The physical processes of the string theory will of course be independent of the way the string is parametrized. As we have just seen, canonical transformations can induce a different parametrization and change the complex structure. Thus a particular complex structure carries with it a particular parametrization. By applying super-Diff S1 to J we can effectively generate all possible paramerizations. However super-Diff S1 (as Diff S1 for the bosonic case) contains generators which leave the complex structure invariant. Indeed the action of L 0 and F o on J leaves it invariant. By dividing out these rigid rotations (super S1) we may represent the space of all complex structures. This is the manifold super-Dill S1/S 1. We will consider super-Diff Sx/S 1 as an abstract manifold. By studing the properties of its curvature we can extract information on the anomaly structure of super-Diff S1/S 1. Then by combining the holomorphic vector bundle of the last section with the line bundle we are about to describe we will have a gauge, i.e. reparametrization invariant theory. The manifold M = super-Diff S1/S 1 is a homogeneous and KLlaler manifold since it is a coset space and has a KLlaler 2-form. The tangent vectors of M form the superconformal algebra, [Ln, L m ] = ( n - m ) L n + m,
[Fn, Fro]+= 2Ln+m,
(3.1)
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and carry the complex structure J given by
JG., =
iGm -iG.,
if m < 0 if m > 0,
(3 .2)
where G., ~ (L,., F., for m ~ Z} for the R sector, and Gm ~ {Lm, Fro+l~2 for the m ~ Z) for the NS sector. As in the previous section, we write all intermediate steps for the R sector and quote the results for both sectors. The most general solution to the closure condition of the Kahler 2-form is given by [9,10] o~( Lm, L , ) = (am 3 + brn)Sm, _ , , 0~(Lm, r , ) = 0, o~( F,,, F,) = (4am z + b )8,,,, _ , ,
(3.3)
o~ is not invertible for - b / a = m 2 for nonzero integers m. Let us consider M as a graded flag manifold G / H and exploit its complexification by splitting it into M+ and M . Then we can take the decomposition of the associated Lie algebra of G as
Gc=Hc+M++M_ where
H e = {G 0= Lo,F0}. Since M is a flag manifold, we can calculate the curvature of M by using the Toeplitz operator method [7]. For any tangent vector, X, we can define the Toeplitz operator ~(X) = Vx-~x, where V'x is a covariant derivative and -~x is the Lie derivative. Since cp(X)(fY) = f ~ p ( X ) Y , it preserves the tensoriality and is a linear operator. So the Toeplitz operator is an intrinsic quantity of the manifold. Since M is a homogeneous Kahler manifold, the Lie derivative preserves the Kahler structure of M (namely the complex and symplectic structure of M) and there is a natural connection W which preserves the KLlaler structure. The formulae for the Toeplitz operators in terms of the complexified quantities can be determined from the requirement that both ~7x and LPx preserve the Kahler structure [7]. They are ~(H) = -ad H
for H ~ He,
cp(X ) = - ~ r + . a d X
for X ~ M
¢p(X+) = _ ( _ ) g ( x + ) ~ ( ~ + ) t
for X + ~ M + ,
, (3.4)
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where g(X) is the grade of X (0 for L, and 1 for Fn) and we define ~(X+) as the adjoint of q0(X_) with respect to the metric, rt+ is the projection operator over M_. We need the phase factor for qo(X+), since Fn's anticommute with themselves. Since M is homogeneous, it is enough to calculate the curvature at one point. Then the curvature is, following the expression for a supermanifold Ill], for any vector Z on M and for any 1-form W on M,
w-R(x, Y)z=
Vrx,
}z
= (-)g(W)g(Z)w. ([¢p(X), ¢p(Y)] ± - ¢p([ X, Y]:L ) ) Z ,
(3.5)
where g(W) and g(Z) are the grades of W and Z respectively. Applied to (3.1) eq.
(3.4) gives, cP(Lo)L_ p = - [ L o , L p ] = - p L _ p , ¢P(Lo)F_ p = - [ L o , F_p] = - p F _ p , ¢P(ro)L_ p = - [ g o , L_p]-- - ½ P r p , ¢p( go)F_p = - { g o , F p } = - 2 L _ p , ¢P(Lm)L_ p = - O ( p - m)(p + m)Lm_p,
m > O,
¢p(L,,,)F_ p = - O ( p - m ) ( ½ m + p ) F m _ p,
m>0,
¢p( Fm)L_ p = - O ( p - m ) ( m + ½ p ) r m _ p,
m>0,
~(Fm)F_ p = --20(p-- m)Lm_p, =
m > O,
ap3 + bp ( p + 2m) a(p + m) 3 + b(p + m) L-m-P'
m>0,
(p + ~m)(4ap z + b) ep( L_m)F_ p =
¢p(F ,,,) L_ p = _
4a(p + rn)2 + b
F_p_m,
m>0,
2( ap3 + bp ) 4a( p + m)2 + b
g-(p+m),
m>0,
(3.6)
and cp(F_m)g p =
½(p+ 3m)(4ap 2 +b) a(p+m)3+b(p+m)L_(m+p),
re>O,
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569
where p > 0. From eq. (3.5) we get
R( L ,,,, L,,)L_p= - O ( p - m)(p + m) 2a(p
+
m) 3 +
b(p ap3 + bp
m)
(p + 2m)2(ap3 + bp) -2mp}8~,,8~L q, (3.7a) a(p+m)3+b(p+m)
t l m124a(p- m)2 + b R(L_m,L,,)F - e = ~t O(p-m)[p , + -~ ] -4-ap2--+
( p + 3m)2(4ap2 + b ) + 2rap} ~m,n~qF-q, 4a(p+m)Z+b R(F_m,F.)L_r=
(3.7b)
4a(p_m)2+b O(p-m)(½P+m) 2 -~p'~p
4(apa +bP)
+ 4a( p + m )2 + b
+ 2p)~m n~qZ_q,
(3.7C)
and R(F
-~'
F.)F- e
= f-40(p-
m) a ( p -
I
m) 3 +
b ( p - m)
4ap2 + b 1 (p+ 3m)E(4ap2 + b) -2p~Sm,SqF q, (3.7d) 4 a(p+m)3+b(p+m) ] ' -
where O is the step function. All other components vanish. To get the Ricci two-form we have to take trace over positive p of the Riemann curvature. Performing the trace one finds, in the R sector
RJc( L_m, L.) =
1°m36
Ric(F_,,,, F,,) = - 2t°m28 re,el
•
(3.8a)
In the NS sector the result is Ric(L_m, Ric(F_r,
L.) = ( - ~m 3+ ¼m)3.... 1 ~ V~) = ( - ~ r z + ~) r,~ ,
(3.8b)
D. Harariet al. / Superstring
570
with r, s half integers. When calculating the Ricci form, we find that the infinite series converge remarkably. So again we do not need any regularization to calculate the trace. We combine the result of this section and the curvature of the holomorphic vector bundle, eq. (2.36), to write the curvature of the twisted bundle in the R sector as
G( Lm, L _ , ) = F( Lm, L _ , ) + Ric(L_m, L + , ) = [ 1 ( d - 10)m 3 - 2arn]8 . . . .
G( F,,, V_,) = F( Fm, V_,) + R i c ( F _ m, F,) = [ ½ ( d - 10)m 2 - 2a]/Jm, . .
(3.9a)
Thus for d = 10 and a = 0 we have a gauge invariant description of the theory. In passing we note that the vanishing of the ground state energy in the R sector signals the presence of space-time supersymmetry. In the NS sector the curvature of the twisted bundle is given by G ( L m, L _ , ) = [ l ( d _ 10)m 3 - ( I d - ¼ + 2a)m]/Jm,,, G(F~, F_s) = [ ½ ( d - 10)r 2 - ( I d - j + 2a )] 6r, s . Thus d = 10 and a = sector, as expected.
(3.98)
½ gives a reparametrization invariant theory in the NS
4. Conclusions and remarks
As we remarked earlier by combining the holomorphic vector bundle with the line bundle corresponding to the ghost sector we are able to describe a gauge, i.e. reparametrization, invariant theory. We have shown that the twisted bundle with curvature G given in eq. (3.9) is precisely this combination for d = 10. Throughout we have focused only on the flat space case. It would be interesting to perform an adiabatic approximation for the super manifold to find the graded analog of the bosonic case. Bowick and Rajeev found that such an approximation for the bosonic case yields the vacuum Einstein equations. The more general setting realizes a manifold endowed with a KLlaler potential K. From here one can construct the covariant derivatives and all other geometric objects. The equation GK = 0 is to be interpreted as the equation of motion for the closed superstring field. Also we remark that in computing the curvature for super-Diff S1/S x we relied on techniques which go through without regularization. It has been shown that coset space techniques designed for finite-dimensional graded Lie algebras may be used if
D. Harariet al. / Superstring
571
one regulates with zeta functions [12]. This places the computation of the curvature in a more familiar setting. Since the superstring enjoys a global supersymmetry, we may exploit this fact to argue that M describes the ghost sector of the superstring just as in the bosonic case where here this sector also admits commuting ghost for the fermions. Let us make a further remark about supersymmetry in general. Consider a 2n-dimensional graded vector space with 2n bosonic and 2n fermionic vector fields. For the bosonic sector the complex structures are parametrized by O(2n)/U(n), since we consider only those orthogonal rotations which do not leave the complex structure invariant. For O(2n) its overlap with SP(2n) is U(n) therefore we arrive at O(2n)/U(n). However for the fermionic sector the complex structure is symmetric (given by O(2n)) and its vector transformations are SP(2n). Therefore the space of fermionic complex structure is parameterized by SP(2n)/U(n). One can see that dimensionally these spaces are different, i.e. Dim(bosonic complex structure) =
2n(2n - 1)
2 2n (2n + 1)
Dim(fermion complex structure) =
n2=n2-n,
n2=n2+n.
A global supersymmetry transformation maps the bosonic and fermionic structures into each other but this mapping is not 1-1. It is interesting to ask what further structure is needed to define the complex structure of a graded vector space which has supersymmetry. In the ~ dimensional case the mapping may be 1-1. Rajeev [13] has found a prescription for taking n to oo in the bosonic case (namely by considering finite energy requirements). Perhaps similar reasoning is needed for the graded case. Finally in ref. [14], Pilch and Warner have used still other methods to compute the curvature due to the holomorphic vector bundle and the contribution to the ghost. We would like to thank the high energy theory group at the University of Florida especially P. Oh. and E. Piard for very fruitful discussions. We would also like to thank M. Bowick and S.G. Rajeev for explaining their work. One of us (D.K.H.) is supported by the Division of Sponsored Research at the University of Florida.
References
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M. Bowickand S.G. Rajeev,MIT preprint CTP.~1450; Phys. Rev. Lett. 58 (1987) 535 S. Goldberg, Curvature and homology(Dover, NY, 1982) V.I. Arnold, Mathematicalmethodsof classicalmechanics(Springer, New York, 1978) NJ.M. Woodhouse,Geometricquantization(ClarendonPress, Oxford, UK, 1980)
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[5] J. Sniatycki, Geometric quantization and quantum mechanics (Springer, New York, 1980) [6] A. Ashtekar and M. StiUerman, Syracuse University preprint (1985) [7] D. Freed, in Infinite dimensional groups with applications, ed. V. Kac (Springer, Berlin, 1985); Ph.D. thesis, University of California at Berkeley (1985) [8] B. Kostant, Graded manifolds, graded Lie theory and prequantization, in Differential geometrical methods in mathematical physics, Lecture Notes in Mathematics, vol. 570 (Springer, Berlin, 1975) [9] P. van Alstine, Phys. Rev. D12 (1975) 1834 [10] G. Segal, Commun. Math. Phys. 80 (1981) 30 [11] B. DeWitt, Supermanifolds (Cambridge University Press, 1984) [12] P. Oh and P. Ramond, University of Florida preprint (April 1987) [13] S.G. Rajeev, private communication [14] K. Pilch and N.P. Warner, MIT preprint CTP#1457 (Feb. 1987)