Unitarity of the Knizhnik-Zamolodchikov-Bernard connection and the Bethe Ansatz for the ell

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We work out finite-dimensional integral formulae for the scalar product of genus one states of the group G Chern-Simons theory with insertions of Wilson lines. Assuming convergence of the integrals, we show that unitarity of the elliptic KnizhnikZamolodchi

ESI

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Unitarity of the Knizhnik-Zamolodchikov-Bernard Connection and the Bethe Ansatz for the Elliptic Hitchin Systems

Fernando Falceto Krzysztof Gawedzki

Vienna, Preprint ESI 322 (1996)

April 18, 1996

Supported by Federal Ministry of Science and Research, Austria Available via anonymous ftp or gopher from FTP.ESI.AC.AT or via WWW, URL: http://www.esi.ac.at

Unitarity of the Knizhnik-Zamolodchikov-Bernard connection and the Bethe Ansatz for the elliptic Hitchin systemsFernando Falceto Depto. F sica Teorica, Univ. Zaragoza, E-50009 Zaragoza, Spain

Krzysztof Gawedzki I.H.E.S., C.N.R.S., F-91440 Bures-sur-Yvette, France

We work out nite-dimensional integral formulae for the scalar product of genus one states of the group G Chern-Simons theory with insertions of Wilson lines. Assuming convergence of the integrals, we show that unitarity of the elliptic KnizhnikZamolodchikov-Bernard connection with respect to the scalar product of CS states is closely related to the Bethe Ansatz for the commuting Hamiltonians building up the connection and quantizing the quadratic Hamiltonians of the elliptic Hitchin system.

Abstract

1 IntroductionThe present paper continues the program 17] 10] 19] 20] aimed at analysis of the scalar product of states in the Chern-Simons (CS) theory. It extends the considerations of ref. 9] were we treated the SU (2) CS theory on the elliptic curve (times the time-line and with insertions of time-like Wilson lines) to the case of a general group G. As in the previous papers of the series, the point is to express the formal scalar product of the CS theory, given by a functional integral over gauge elds, as a multiple nite-dimensional integral. The latter, if convergent for every state, provides the space of CS states W with a Hilbert space structure and the holomorphic vector bundle W, obtained by varying the modulus of the elliptic curve and the positions zn of insertions, with a hermitian structure. The integral expressions for the scalar product of the CS states are close cousins of the contour integral expressions for the conformal blocks of the corresponding WZW conformal theory. In the elliptic case, the contour integral representations were recently studied in ref. 12]. Our approach elucidates the origin of the complicated expressions which appear in such representations for a general group: they are induced by a simple trick, already used in 10], which handles a change of variables in the functional integral. 1

The WZW conformal blocks are holomorphic sections of the bundle W of the CS state spaces satisfying in the elliptic case the Knizhnik-Zamolodchikov-Bernard (KZB) equations 23, 3] (@+ 1 H0 )= 0; (@zn+ 1 Hn )= 0; n= 1;:::; N: (1) Above, Hn Hn (; z) are operators acting on the CS states and is a coupling constant. The KZB equations may be interpreted as the horizontality equations for a K

ZB connection in bundle W . The consistency of the equations (the atness of the KZB connection) requires that the operators Hn; n 0, commute for xed (; z). In fact, Hn; n 1; are quantum versions of the quadratic classical Hamiltonians hn of the elliptic Hitchin system which Poisson-commute. In the case of elliptic curve with no insertions the conformal blocks coincide with the (linear combinations of) characters of the integrable representations of the a ne algebras. H0 in eq. (1) is then proportional to the Laplacian on the Cartan algebra and the KZB equations reduce to the well known heat equation for the elliptic theta functions. In the case of elliptic curve with one insertion, H0( ) becomes a version of the Calogero-Sutherland Hamiltonian 6]. For many insertions, operators Hn (; z); n 1, are elliptic versions of the Gaudin Hamiltonians 15]. A hermitian structure on a holomorphic vector bundle induces a unique unitary connection of the 1,0 type. One expects 18] that the scalar product of the CS states induces this way the KZB connection. For the elliptic case with no insertions, that was shown implicitly in 21] where it was proven that the a ne characters form an orthonormal basis of the space of CS states. For group SU (2) in the elliptic case with one insertion the unitarity of the KZB connection w.r.t. the scalar product of the CS states was proven in 9]. In the present paper we study this problem for the general elliptic situation. We show that, assuming convergence of the integrals giving the scalar product, the unitarity of the elliptic KZB connection follows from a result announced in 12] which provides a basis for the Bethe-Ansatz diagonalization of the commuting operators Hn (; z); n 0. The relation of the integral representations for the genus zero conformal blocks of the WZW theory to the Bethe Ansatz was observed in 1], see also 2, 26, 11]. The elliptic SU (2) case counterparts of these relations go back to XIXth century works of Hermite on the Lame operator, as noticed in 6]. Recently, the integral representations for the elliptic conformal blocks were used in 12] to obtain the Bethe-Ansatz treatment of the elliptic CalogeroSutherland model, an open problem till then. Our work exhibits an intrinsic connection between the Bethe-Ansatz and the unitarity of the KZB connection. The paper is organized as follows. In Sect. 2, we describe brie y the space of CS states in the holomorphic quantization and identify the states on the elliptic curve as vectorvalued theta-functions. Sect. 3 is devoted to the scalar product of the elliptic CS states. As realized in 21] for the case with no insertions, the formal functional integral over the gauge elds giving the scalar product may be computed as an iterative Gaussian integral. The insertions for general group G are handled by combining the methods of refs. 21, 10]. In Sect. 4, we describe the KZB connection and recall its relations to the Hitchin integrable systems. Sect. 5 discusses the

unitarity of the KZB connection and nally, in Sect. 6, we consider the relations to the Bethe Ansatz. 2

2 Elliptic CS statesLet us recall the description of the CS states on the elliptic curve T= C=(Z+ Z) with insertion points zn, given in 8, 9], with due modi cations required by the replacement of the group SU (2) by G. For simplicity, we assume G to be simple, connected and simply connected and will denote by g its Lie algebra. Let A denote the (complex) vector space of the 0; 1-components A Az dz of the g-valued gauge elds. The group G of complex gauge transformations g: T ! GC acts on A by

A 7! gA= gAg?1+ g@g?1;

(2)

with@= dz@z . The Chern-Simons states are holomorphic functionals on A. For the case with insertions of time-like Wilson lines, they take values in the tensor product n V n V of the irreducible representation spaces of G of highest weights n, associated to the insertions. States verify the chiral Ward identity (gA)= ekS (g?;A) n g (zn) n (A)1 ( )

(3)

where S (g; A) is the action of the Wess-Zumino-Witten model coupled to A, see 8], and the subsubscript (n) indicates that the group element acts in the factor V n of the tensor product space V . Restricting functionals to connections Au= udz= 2 for u in the Cartan algebra hC and 2 Im, we can assign to every state a holomorphic map: hC ! V related to by the equation (Au )= e kjuj=(22 2

) n (e (zn?zn )u=

) n (u)( )

(4)

where juj2 hu; ui with the Killing form h; i normalized so that j j2= 2 for long roots (we identify g with its dual). Holomorphic maps corresponding to states satisfy the following conditions (u+ q _ )= (u) for q _ in the coroot lattice Q_; (u+ q _)= e? ikhq_; q_+2ui n (e?2 izn q_ ) n (u);( )

(5,a) (5,b) (5,c) (5,d) (6)

0= n (h)(n) (u) for h 2 h; (wuw?1 ))= n (w)n (u) for w 2 G normalizing h; andX

for any root i= 1, u s.t. hu; i= m+ s with m; s integers, p= 1; 2;::: and t ! 0. Here and below e is the step generator of gC corresponding to root . 3

n, coweight p_ satisfying hp_;

e?2 iszn (e )(n)

(u+ tp_ )= O(tp )

We shall denote the space of maps (u) satisfying the above conditions by W;z; . Conditions (5,a) and (5,b) mean that is a vector of theta-functions which due to condition (5,c) take values in the zero weight subspace V 0 V and which by condition (5,d) are Weyl group covariant. Equations (6) specify their regularity on the hyperplanes hu; i 2 Z+ Z where 2 h are roots of the algebra g. They have been obtained for G= SU (2) in 8] and for general G in 7], see also 14]. They follow by demanding the regularity at t= 0 of the maps t 7! (hmsAu0;t )= (gmsAu+tp_ ) where t 7! Au0;t= (u0+ tp_+ e ) dz, with u0= u? (m+ s)p_, is a 1-parameter holomorphic family of gauge elds, hms (z )= exp ((m+ s)z? (m+ s)z )p_] is a multivalued gauge transformation and gms (z )= hms (z ) exp?t?1 e] h?1 (z ) is a univalued ms one. Since hu0; i= 0, Au0+tp_ may be gauge transformed to Au0;t for t 6= 0 by a

constant gauge transformation exp t?1 e]). Gauge elds hmsAu0;0 lie on codimension one strata in the space of gauge elds which cannot be attained by gauge transforming Au 's. Conditions (6) assure the global regularity of but, due to the properties (5,a,b), they are not independent for di erent m and s. For example, for all simple groups di erent from SU (2) or SO(2r+1) it is enough to take m= s= 0.2 2

3 Scalar product of CS statesThe scalar product of Chern-Simons states is formally given by the functional integral Z R 2= j (A)j2 e? ik tr(A A) DADA k k (7)2

We shall perform the above functional integration reducing the scalar product expression to a nite-dimensional integral which, if nite, will provide W;z; with a natural structure of a Hilbert space. The niteness has been proven in the special cases (general G with no insertions 21] or SU (2) with one insertion 9]) and has been conjectured to hold in general 18]. The strategy for the calculation of the integral (7) will be as in 21] and 9] so we shall be brief discussing in detail only the treatment of the insertions essentially borrowed from 10].3.1 Change of variables

We shall reparametrize the gauge elds in the functional integral (7) as A= g? Au (8) with g a complex gauge transformation and u in some fundamental domain of the action of translations u 7! u+ q _, u 7! u+ q _ on hC . ( hC=(Q_+ Q_ )=Weyl group is the space of gauge orbits of semistable gauge elds; ignoring the Weyl group action produces an overall factor in the scalar product equal to the order of the Weyl group.) We shall use the Iwasawa decomposition of the GC -valued eld g: X g= exp v e] exp=2] U n exp=2] U bU (9)1

where takes values in the Cartan algebra h and U in the compact group G. Upon the change of variables, eld U decouples from the functional integral (7) due to the gauge invariance leaving us with a WZW-type functional integral over the GC=G-valued elds b 21]

k k2= h (Au);Q

?1 n (bb )(n) (zn ) (Au )i ik R ekS (bb;Au+Au )? 2 tr(Au Au )

j (u; b) ( (0)) Db d2u

(10)

where Db= z db(z ) is the formal product of GC -invariant measures on GC=G, the delta function xes the remaining freedom in the parametrization (8) and the Jacobian of the change of variablesy j (u; b)= const: 2?2r e 2h_ S(bb; Au+Au ) det(@Au@Au ):

Above, r stands for the rank of G, h_ for its dual Coxeter number and@Au The last determinant was computed in 21]:y 2 det(@Au@Au )= const: 2 r e h_ ju?u j= j (u)j42 2

@+ Au;].

where

is the Kac-Weyl denominator:Y Y Y (u)= e id=12 (e ihu; i? e? ihu; i ) (1? q l )r (1? q l e 2 ihu; i )

l>0

with d denoting the dimension of the group and q parametrization (9) takes the form

e2 i . The WZW action in the

S (bb; Au+ Au )=? 4iwhere2

h@;@ i? 2i2

he u (n?1@nu ) e? u; n?1@nu i u u

nu= e? (z?z )u= n e (z?z )u=;Note that

=? (z? z )(u? u)= 2: expX

nu= exp

e? hu; i(z?z )= v e]2

v0 e]:

In terms of

the Iwasawa variables, the invariant measure on GC=G is

db=

j=1

d2(e?h; i=2v )

where j= hhj; i are the coordinates of w.r.t. an orthonormal basis (hj ) of h. Using the holomorphic functions (u) to represent and the parametrization (9), we obtain

h (Au); (bb )?1(zn ) (Au) i= e nY

( )

k Re(juj2 )

h (u); n((nu)?1e? u n?1 ) n (zn) (u) i: u( )

Finally,

? 2ik s tr(AuAu )=? khu; ui= 2:With all these ingredients,

= const: h (u); n((nu )?1 e? u n?1 ) n (zn ) (u) i exp? i(k+2h ) sh@;@ i] u 4 i(k+2h_ ) she u (n?1@n ) e? u; n?1@n i+ (k+2h_ ) ju? uj2] exp? 2 u u u u 2( )

j (u)j4 ( (0)) d2ru

j=1

>0;z

d2(e?h

u (z); i=2v 0 (z )):

(11)

In order to render the nu -dependent terms in the action of the last functional integral quadratic, we shall introduce new variables ( 0 )>0 de ned by

n?1@nu= uSince nu (z+ 1)= nu (z ) and nu (z+0 (z+ 1)= 0 (z )

>0 )= e?2 iu nu (z )e 2 iu, it

@ 0e:

(12) follows that (13)

0 (z+ )= e?2 ihu; i 0 (z )

and that the change of variables is well de ned since@ is invertible (for generic u) on functions satisfying the periodicity conditions (13). It is easy to see that the change of variables (v 0 ) 7! ( 0 ) has a triangular nature: 0= v 0+ F ((v 0 )< ) where< if? is a positive root. Hence, the formal volume element does not change:Y

>0; z

d2(e?h

u (z); i=2v 0 (z ))

>0;z

d2(e?h

u (z); i=2

0 (z )):

dependence of the insertions in the functional integral (11). Thus, we have to invert relation P 0 e . This will be done generalizing (12) in order to express nu as a function of 0>0 a trick of 10]. L Let n= C e be the nilpotent subalgebra of gC . The corresponding group N GC>0 may be mapped into the enveloping algebra U (n) (no completion problem arises if we work in highest weight irreducible representations of GC ). U (n) is graded by the positive cone Q+ in the root lattice Q:

v0 (and their derivatives) becomes quadratic in (derivatives of) 0, there persists the nu -

Although the action in the functional integral (11), which was polynomial in variables

U (n)=

q2Q+

U (n)q1 1

e m 2 U (n)+:::+ m . according to the eigenvalues of the adjoint action of h, with e?1 (z ) The map z ! nu (z ) may be viewed as taking values in U (n) and it satis es then the twisted periodicity conditionsq (z+ 1)= q (z ); q (z+

)= e?2 ihu;qi q (z )

(14)

where q denotes the U (n)q -component of . Relation (12) may be now rewritten as an equation

@z n?1=?(@z 0) n?1; u ufor U (n)-valued functions and solved with use of the Green functions of the twisted@z operator. Since (n?1 )0= 1, we obtain u1

n?1= u

K=0

? (?1)K@z 1 (@z 0)

K times

@z?1 (@z 0) 1:2

(15)

The Green function of@z acting on functionsP obeying conditions (14) may be easily expressed by the Jacobi theta function#1(z )= l (?1)l e i(l+1=2)+2 iz(l+1=2) satisfying

#1(z+ 1)=?#1 (z);#1 (z+ )=?e?Explicitly,

it is equal to0 x Px (z)=#1(0)(#1)(#+zz)#1 x 1 ( )

i(+2z)#

1(z ):

(16) (17)

with x= hu; q i. Hence eq. (15) takes the form

n?1 (z)= u

K=0 0 )(y2) (@z 2

(?1)K

( 1;:::; K ) Phu; K i(yK?1? yK )(@z 0 K )(yK ) e

Phu;

+:::+ K i(z? y1 )(@z

0 )(y1) P hu; 2+:::+ K i (y1? y2 ) 11

:::e K d2y1P

d 2 yK:

(18)

In the (u) matrix element, we may insert the partition of unity n ( j an; an ih an; an j) an where vectors j an; an i corresponding to weights an form an orthonormal basis in the representation spaces V n . This gives

h (u); n((nu)?1e? u n?1 ) n (zn) (u) i u( )

a n

e?h u (zn ); an i j n h an; an j n n?1 (zn ) (u) ij2 u

into which we may insert the expressions (18) for n?1 (zn ). As in 10], the nal renormalu ization of the functional integral over the Cartan algebra-valued eld will kill most of the terms obtained this way leaving only the ones with j an; ani equal to the highest weight vectors j n i and the sequences of positive roots ( 1;:::; p) composed uniquely of simple roots. We may then write

h (u);

?1??1 n((nu ) e u nu )(n) (zn ) (u) i=YY

e?h u (zn ); nin;1

FK; (; u; z; y)+::: (19)

n i=1

(@z 0 n;i )(yn;i ) d2yn;i h j n (e

n;Kn )

(u) i

where h j n h n j,= ( 1;1;:::; 1;K; 2;1;::::::; N;KN ) P quence of K= n Kn simple roots satisfying1

( 1;:::; K ) is a se(20)

s=11

n=1

y= (y1;1;:::; y1;K; y2;1;::::::; yN;KN ) (y1;:::; yK ) is a sequence of K points in T and the kernels Fp; are composed of the Green functions of twisted@ FK; (; u; z; y)=Y

Phu;

n;1+:::+ n;Kn i (zn

? yn;1 ) Phu;

Phu;

n;2+:::+ n;Kn i (yn;1 n;Kn i (yn;Kn?1

? yn;Kn ): (21)

? yn;2)

" . . ." contains the terms that will drop under the renormalization of the -integral.3.2 Functional integration

The above use of eq.(15) to express the 0 dependence of the insertions reduces the 0 integral to the formZ Y

s=1

+2 _ X (@z 0 s )(ys )(@z 0 s )(vs ) exp?i(k2 h ) s e?h u; i@ 0@ 0]

= ( k+2h_ )K

K Y

2SK s=1

>0 e h u (ys ); si s; (s)

>0; zY

d2(e?h

u; i=2

(ys? v (s) )

det(@ y@ )?1 (22)

with running over the permutations of K points and with@ determinants are well knownY2

e?h u; i=2@ e h u; i=2 . The

Y det(@ y@ )?1= const: e? r=6 j (u)j?2 j1? q l j2r

_ _ exp ih sh@;@ i? 2h ju? uj2]: 42

l=1

After the 0 -integration and an easy combinatorial manipulation, see 10], trading the sum over root sequences into sums over permutations (two 's satisfying eq. (20) di er necessarily only by a permutation), the scalar product formula (11) becomes

k k2= const: e? r

Z 1 K N=6 Y j1? q l j2r Y e?h u (zn ); n i Y e h u (ys ); si s=1 n=1 l=1 r Y i exp? 4 sh@;@ i+ 2 2 ju? uj2] ( (0)) D j j (u)j2 j=1 K 2 Y X X FK; (; u; z; y) h j n (e( )n;1 e( )n;Kn )(n) (u) i d2r u d2ys s=1 K 2SK2

(23)

where

k+ h_ and is a xed sequence of K simple roots satisfying (2

0).8

The remaining -integral is of the Gaussian (Coulomb gas) type and may be easily performed:

where we should take f=? n n zn+ s s ys which corresponds in the Coulomb gas jargon to external charges at points zn and screening charges at points ys . The Green function isP P

D j j=1 P R 1 j 0 j 0 r== const: 2 2 det0(@ y@ )?r=2 e? 2(k+ h_ ) j f (y)G (y?y )f (y ) P R 1 1?r=2 e r 2=6 Q j1? q l j?2r e? 2(k+ h_ ) j f j (y)G (y?y0 )f j (y0 ) const: 2 l=1R

R P

j (y)f j (y)

exp? i sh@;@ i] ( (0))4

(24)

G (z)= ln j#1(z)j2+ 2 (z? z )2+ const:2

(25)

Renormalizing the divergences due to the singularities at coinciding points by pointsplitting, as explained in 10] and 9], we nally obtain for the scalar product of CS states the following nite-dimensional integral expression:

k2=

? const: 2 r=2

2 2

jw?wj2 e? 1 S (;z;y) h G(

; u; z; y); (u) i d2r u

s=1

d2ys (26)

where (u)

(u) (u),

u+ 1

n=1

zn n? 1

s=1

ys s;

(27)

S is a multivalued holomorphic functions of and non-coincident zn 's and ys 's and G a holomorphic multivalued map with values in the dual space V of V given, respectively, by S (; z; y)=X

n<n0

h n; n0 i ln#1(zn? zn0 )?e

h n; si ln#1(zn? ys ) n;s+ h s; s0 i ln#1(ys? ys0 )e X e

e with#1 (z )#1 (z )=#01(0) and

s<s0

(28) (29)

G(; u; z; y)=1

X X

K 2SK

FK; (; u; z; y) h j

n (e( )n;1

)n;Kn )(n):

Using the transformation properties (16), it is straightforward to verify that for H e? S hG; i, n= (0;::: 1:::; 0) and s de ned similarly,

H( H( H( H( H(

b n _; z; y);u+ q; u+ q _; z; y); u; z+ n; y); u; z+ n; y); u; z; y+ s )

=====

H (; u; z; y);

e? i hq_; q_+2wi H (; u; z; y ); (?1)h n; n i= H (; u; z; y); (?1)h n; n i= e? i h n; ni=+ 2 ihw; n i H (; u; z; y); (?1)h s; si= H (; u; z; y); 9

(30)

H (; u; z; y+

= (?1)h s; si= e? i h s; si=? 2 ihw; si H (; u; z; y):

It is then easy to see that the under-integral expression in (26) is univalued under the u 7! u+ ( )q _, zn 7! zn+ ( )1, ys 7! ys+ ( )1 transformations. It is useful to compare the above expression for the scalar product of the CS states with the genus zero ones obtained in 10]. The genus zero states are determined by their values (0) at A= 0 belonging to the G-invariant subspace V G of the the tensor product V of the representation spaces. Adapting the notations to the ones of the present paper, eqs. (19.20) of 10] read

kwhere

= const:

1 e? S 0 (z;y) h G0(z; y);

K 2Y 2 d ys s=1

(31)

S 0(z; y)=and

n<n0

h n; n0 i ln(zn? zn0 )?

n;s

h n; si ln(zn? ys )+ h s; s0 i ln(ys? ys0 ) (32)X

s<s0

G0 (z; y)=with

X X

k 2SK1

FK (z; y) h jY

n (e( )n;1

)n;Kn )(n)

(33) (34)

FK (z; y)=

1 1 zn?yn;1 yn;1?yn;2

1 yn;Kn?1?yn;Kn

The similarity to the genus one case is obvious.

4 Knizhnik-Zamolodchikov(-Bernard) connection and the Hitchin systems0 The trivial bund

le with the ber V over the space XN C N n where contains vectors z with coinciding components carries a 1-parameter family of at holomorphic connections de ned by 0 rzn=@zn; rzn=@zn+ 1 Hn(z) 0 where Hn (z) are the Gaudin Hamiltonians 15]: 0 Hn (z)=X X

with (ta ) forming an orthonormal basis of Lie algebra g. Connection r appeared (implicitly) for the rst time in ref. 23]: it was shown there that the genus zero conformal 10

tan tan0 a=1 n0 6=n zn0? znd( ) ( )

(35)

blocks of the WZW theory satisfy the (Knizhnik-Zamolodchikov) equations r= 0. In fact, the WZW conformal blocks are horizontal sections of a (generally proper) subbundle 0 W 0 XN V G. The bers Wz0 of W 0 may be identi ed by the assignment ! (0) with the genus zero CS state spaces. The subbundle W 0 may be described by giving explicit algebraic conditions, depending holomorphically on z, on the invariant tensors in V G 0 29, 16]. The KZ connection r preserves the subbundles XN V G and W 0 of the trivial 0 bundle XN V . The extension of the Knizhnik-Zamolodchikov connection to the genus one case was rst obtained in ref. 3] and elaborated further in 8, 6, 14, 12]. We shall use the description of the genus one CS states by the theta functions (u)= (u) (u). The spaces W;z of states form a holomorphic bundle over the space of pairs (; z) with no coincidences in zn 's viewed as points in T . The holomorphic sections of W correspond to holomorphic families (; z) ! (; z; ). The KZB connection is given by the formulae

r=@; rzn=@zn; r=@+ 1 H0(; z); rzn=@zn+ 1 Hn(; z); (36)compare to eqs. (1). Explicitly 6, 14, 12],

H0 (; z)=

i 4

Hn (; z)=?

i X@ P x hu; i (zn? zn0 ) (e )(n) (e? )(n0 ) 4 n;n0=1 r X+ 8i ( (zn? zn0 )2+ 0 (zn? zn0 )) hjn) hjn0 ) ( ( j=1X X X

; (37)

j=1

hjn@uj?( )

n0 6=n

Phu; i(zn? zn0 ) (e ) n (e? ) n0( ) (

)

j=1

(zn? zn0 )) hjn hjn0( ) (

)

(38)

2 where u#01=#1. The expressions for Hn; n 1; reduce to the j (@uj ) and ones for the Gaudin Hamiltonians (35) in the limit ! i1. The operators Hn; n 0; acting, say, on meromorphic functions of u 2 hC with values in V 0 commute, see 14]. Their commutation forms part of the conditions assuring the atness of the KZB connection (36). Although the coe cient functions in Hn have poles on the hyperplanes hu; i 2 Z+ Z the, connection maps holomorphic families of genus one CS states into families with the same property, due to the increased regularity (6) of the CS states on the singular hyperplanes, see 8, 14]. 0 The commuting operators Hn for genus zero or Hn at genus one are quantizations of classical Poisson-commuting Hamiltonians of the, respectively, genus zero and genus one Hitchin integrable systems 22, 4, 24, 5]. Let us brie y recall this relation. Given a Riemann surface and a group G, let A denote the corresponding space of 0; 1-gauge elds and G the group of complex gauge transformations, as in the beginning of Sect. 2. Both are (in ni

te dimensional) complex manifolds and we shall work in the holomorphic category. The cotangent bundle T A is composed of pairs (A; ) where is a 1; 0 form on with values in Lie algebra gC . The action (2) of G on A lifts to the symplectic action

(A; ) 7! (gA; g

g g?1) on T A with the moment map M (A; )=@A+ A^+^ A

(the 2-forms on with values in gC form the space dual to the Lie algebra of G ). If z is a sequence of N insertion points in and a corresponding sequence of Cartan algebra P elements, then we may symplectically reduce T A w.r.t. the G -coadjoint orbit of n n zn de ning the reduced phase space

Pz;= M?1(fP

where Gz; is the subgroup of G xing n n zn . The symplectic form on T A descends to (the non-singular part of) Pz; turning it into a nite dimensional (complex) symplectic manifold (we shall ignore here the singularities of Pz; ). For a GC invariant homogeneous polynomial p on gC of degree dp, the assignment (A; ) 7?! p( ) de nes a map on T A with values in the space of dp -di erentials (sections of the dp-th symmetric power of the canonical bundle on ). All those vector-valued maps Poisson-commute since they depend only on . They descend to Pz; giving a system of Poisson-commuting maps hp with values in the ( nite-dimensional) spaces of meromorphic dp-di erentials with poles of order dp at the insertion points zn . Their components form a maximal set of classical Hamiltonians in involution turning Pz; into an integrable system introduced and analyzed in 22] for the case without insertions, see 24, 5] for the generalization including the insertions. Let us specify rst the above construction to the genus zero case. In that case, (almost) each gauge eld A is in the gauge orbit of A= 0, i.e. it is of the form

n zn g)

Gz;

A= h?1@h for h 2 G with h determined modulo h 7! h0 h with constant h0 . Equation M (A; )= Pn n zn

becomes now

@ (h )=which has a unique solution

h(zn) n h?1 (zn )?1

n (h )(z )= 21 i h(zn )?hz (zn ) dz z n n provided that the sum of the residues vanishes:X X

h(zn ) n h?1 (zn)= 0:

The group Gz; is composed of arbitrary gauge transformations g s.t. g (zn ) 2 GC n, acting on h by h 7! hg?1. Above, GC n denotes the subgroups of the GC stabilizing n under the 12

0 adjoint action. For the symplectically reduced phase space Pz; (the superscript 0 referring to the genus zero case) we obtain 0 Pz;= f j

2O n;

n=0g

where n h(zn ) n h?1 (zn ) run through the (co)adjoint orbits O n . The latter are naturally complex symplectic manifolds and it is not di cult to check that, as a symplectic 0 manifold, Pz; is the reduction of n O n by the diagonal action of GC . Since for an invariant 0 polynomial p, p( )= p(h ), the corresponding Poisson-commuting Hamiltonians on Pz; are

h0 ( )(z)= p p

1 2 i

dp z? zn (dz):

In particular, for the quadratic polynomial given by the Killing form p2= h; i, we obtain the quadratic meromorphic di erential (dz

) n;n0 (z? zn )(z? zn0 ) X 1 n? 4 h2(z;? ni )2+ z? z h0 (dz)2= n zn n n where the residues at z= zn are2 2

h0 ( )(z)=? 41 p

h n; n0 i

h0= 2 1 n

h n; n0 i: n0 6=n zn0? znX

These are, up to normalization, the classical versions of the Gaudin Hamiltonians of eq. (35). The latter may be obtained from h0 's by replacing the coordinates a h n; tai on n n the coadjoint orbit O n by the generators tan of g acting in the irreducible representation V n obtained by geometric quantization of O n .( )

Similarly at genus one, for= T, (almost) each gauge eld is in the gauge orbit of the gauge elds Au, i.e. it is of the form

A= h? Au= (hu h)?1@ (hu h)1

with u 2 hC and hu e z (u?u)= . Hence the gauge elds A may be parametrized by pairs (u; h) with the identi cations2

(u; h)= (wuw?1; wh)= (u+ q _; h?_1 h)= (u+ q _; h?1 h) q_ q for q _ in the coroot lattice Q_ and w in the normalizer N (hC ) M (A; )= Pn n zn becomes now X@ (hu h)= (huh)(zn) n (huh)?1 (zn ) zn:n

(39)

GC of hC . Equation(40)

Upon decomposing (hu h)(zn ) n (hu h)?1 (zn )=X

? n

0 n

with 0 in the Cartan subalgebra hC, eq. (40) may be solved, provided that n 0= 0, n n with use of the Green functions of the twisted and untwisted@ -operator on T:(hu h)

(z )=

0 dz

1 2 i

Phu; i(z? zn )? n

e+ (z? zn)

0 n

where 0 is an arbitrary constant in becomes

hC . The symplectically reduced phase space Pz;X

Pz;= f (u; 0; ) 2 T hC ( n O n ) jwith the identi cations

0 n

= 0 g N (hC ) o (Q_+ Q_)

(u; 0; )= (wuw?1; w 0 w?1; w w?1 )= (u+ ( )q _; 0; (h?1)q_ (zn ) n h( )q_ (zn ))): ( As a symplectic manifold, Pz; is a symplectic reduction of T N (hC ) o (Q_+ Q_). The Hitchin Hamiltonians become

hC ( n O n ) by the group0 n

hp (u; 0; )(z)=p0

+ 21 i n

Phu; i (z? zn )? e+ (z? zn ) n2 2

(dz )dp

h which for p= p2 and upon writing hp (u; 0; )(z )= e p (u; 0; )(z )(dz )2, reduces to hp (u; 0; )(z)= h 0; 0i+e2

1 i

Phu; i (z? zn ) P?hu; i (z? zn0 )? n=4 1X2

(z? zn ) h 0; nn0

0i? 1 4 2

+ (z? zn ) (z? zn0 ) h 0; 0 0 i n n

n;n0

0 (z? zn )h n; n i+ X (z? zn ) hn (u; 0; )+ h0 (u; 0; ): n n1 h 0; 0i? 1 X i n 2 2 0 n 6=nX

with the residues at zn

hn (u; 0; )=

Phu; i(zn? zn0 )

? n n0

+ (zn? zn0 ) h 0; 0 0 i n nX

and the holomorphic (actually z independent) piece

h0 (u; 0; )= h 0; 0i? 81

n;n0

@x Phu; i(zn? zn0 )

? n n0

+ ( (zn? zn0 )2+ 0 (zn? zn0 )) h 0; 0 i: n n Hamiltonians hn, n 0 are the classical versions of the elliptic Gaudin Hamiltonians Hn of eqs. (37) and (38), see 24] 25]. The SU (n) elliptic Hitchin system corresponding to one insertion has unexpectedly appeared recently in the description of the low energy sector of supersymmetric YangMills theories 4].

5 Unitarity of the KZB connectionOne of the essential features of the structures discussed above should be the compatibility of the KZ and K

ZB connections with the scalar product of CS states. The integrals in eq. (31) have been conjectured in 17] (and proved in many cases) to converge precisely for invariant tensors 2 Wz0 and to equip bundle W 0 with the hermitian structure preserved by the KZ connection. The latter condition means that for all local holomorphic sections z 7! (z) of bundles W 0, corresponding to holomorphic families z 7! (z) of CS states, (41) In order to see why one should expect such a relation, it will be convenient to express the scalar product integral in the language of di erential forms, following refs. 27, 28]. Let !(y) 1 y?1 dy. Introduce the V -valued K forms0 K;

@zn k k2= (; rzn ):

(z; y)= ! (z1? y1;1 )^ ! (y1;1? y1;2)^:::^ ! (y1;K?1? y1;K )1 1 1

and

^:::::::::::::::::::::::::^!(zN? yN;1)^ !(yN;1? yN;2)^:::^ !(yN;KN?1? yN;KN ) h j n (e n;::: e n;Kn ) n0(z; y)

( )

X X

K 2SK

(?1)j j K; (z; y): e? S h1 0

We may rewrite the scalar product formula (31) as

k2=

const:

where Yz stands for the space of y 's with ys 's not coinciding among themselves and with zn 's. We use the conventions that j 0j2 (?1)K(K?1)=2( i )K 0^ 0 and that the integral of the forms of degree lower than the dimension of the cycle is zero. Assuming a su ciently strong convergence of the integrals, we may enter with the holomorphic exterior derivative under the integral so that

@ k k2= const:@ e?Yz

h 0; i 215

= const:@ e? S h1 0

i^ e?

h 0; i

where the@ -operator on the l.h.s. acts on the z -variables and the one under the integral may be taken as acting on both z - and y -variables. The V -valued holomorphic multivalued forms 0 have two basic properties:

and

(@S 0)^

n=1

0 dzn^ Hn

(42)

with the contragradient action of the Gaudin Hamiltonians on the V -valued form 0. The rst relation is trivial. The second, more involved one, has been proven in 28]. Using eqs. (42), we obtain

@ k k2= const:

e? S1

dzn^ h 0; (@zn+ 1 Hn ) i^ e?

h 0; i

which implies the relation (41). The above analysis has its counterpart for the elliptic case. We again conjecture that the integrals in eq. (26) converge for corresponding to CS states (in fact exactly when the regularity conditions (6) are ful lled) and that the resulting hermitian structure on the bundle W renders the KZB connection unitary, see 9] for the proof of this conjecture for the case of G= SU (2) and one insertion. In order to substantiate the conjecture for the case of general G and arbitrary insertions, let us rewrite the genus one scalar product integral (26) in the language of di erential forms. De ne

!q (z) Phu;qi (z) dz?K;

i 2

@x Phu;qi(z) d;

see eq. (17). It is easy to see that@ !q= 0, if@ di erentiates only the variables (; z ). Set (; u; z; y )= !1

; (z1? y1;1 )^ !; (y1;1? y1;2)^:::^ !;K (y1;K?1? y1;K )^

:::::::::::::::::::::::::::::::::^! N; (zN? yN;1)^ ! N; (yN;1? yN;2)^:::^ ! N;KN (yN;KN?1? yN;KN ) h j n (e n;::: e n;Kn ) n (43)11 12 1 1 1 1 2 1 ( )

with n;i=

Kn P i0=i

n;i0

and (; u; z; y )=X X

K 2SK

(?1)j j K; (; u; z; y):

We still have to dress de ne the form1

with the duj di erentials. The convenient way to do this is toX

= e? S dw1^:::^ dwr^ h; i+ 2 1i 16

j=1

dw1^:::d:::^ dwr@uj h; ib

(44)

where wj hhj; wi with w given by eq. (27) and, as before, eq. (26) may be rewritten as

. It is easy to see that

k2=

? const: 2 r=2

UY;z

2 2

jw?w j2 e? 1 S h

; i d2r uZ

?= const: 2 r=2

UY;z

2 2

jw?w j2 j

j2: (45)

This holds since the terms with d di erentials do not contribute to the integral over the cycle UY;z composed of points (u; y) with identi cations u= u+ ( )q _, ys= ys+ ( )1 which form the (r+K )-dimensional torus T r+K (with coincidences of ys 's among themselves and with zn 's removed). The gain from adding the terms with d 's is that the form transforms under the maps u 7! u+ ( )q _, ys 7! ys+ ( )1 (and zn 7! zn+ ( )1) exactly as the functions H, see eq. (30), and, as the result, the form under the last integral in eq. (45) is well de ned on UY;z . We would like to compute the holomorphic di erential of the norms squared k k2 of a holomorphic family of states. Due to the fact that the integration cycles UY;z depend nontrivially on the di erentiation variables, this requires a more subtle consideration than in the genus zero case where we could ignore this problem (assuming enough convergence at the coinciding points). Ignoring again the coinciding points, the geometric setup is as follows. We are given a bundle$: N ! M of multidimensional tori over the space M of pairs (; z). Let us forget for a moment the complex structures on these spaces considering them as real manifolds. On N we are given a di erential form of degree equal to the dimension of the ber (we shall not need an obvious generalization of the story to forms of arbitrary degree) and we are studying a function f on the base obtained by ber-wise integration of

f (n)=

$?1 (fng)

In local trivialization of the bundle, only the components of with di erentials along the ber contribute to f . If there are no convergence problems, then obviously

df (n)=

$?1 (fng)

d?=

$?1 (fng)

(46)

where the di erential d? in the transverse directions is de ned using a trivialization and the second equality follows since dk (d? d? ) is a closed form when restricted to the bers so that its ber-wise integral vanishes (again assuming no convergence problems). Let us rst see how this argument may be used in the simplest case with no insertions where we have

k k2= const:

?r=2

2 2

ju?uj2 j (u)j2 d2r u= const:?r=2 2

2 2

ju?uj2 j j2

where (u+ q _ )= (u) and (u+ q _)= e? i hq_; q_+2ui (u), i.e. is an r-d

imensional theta-function of degree 2, and= du1^:::^ dur+ 2 1iX

j=1

@uj du1^:::d:::^ durb

is a holomorphic r-form which under u 7! u+ ( )q _ transforms the same way as the theta-function . Note the relations:

duj^=

d^@uj;2

duj^@uj=

(47)

The (2r)-form= e ju?uj j j2 is well de ned on the bundle N of tori obtained by varying T r with . Applying the above geometrical considerations to the case at hand, we obtain2 2

k2=

? const: 2 r=2

(d+ 4ri (d? d )^) e2

2 2

ju?uj2 j j2

@ k k2 . Explicit di erentiation gives:@k

Clearly, only the holomorphic exterior derivative@ will contribute under the integral to

?= const: 2 r=2

TrZ

2 2

ju?uj2

2 2

ju? uj2 d^+

j j

(uj? uj ) duj^

+ 4ri d^+ d^@+2

duj^@ujX

?= const: 2 r=2

2 2

ju?uj2

2 2

ju? uj2 d^+ 2i d^ (uj? uj )@uj2

+ 4ri d^+ d^@+ 2 i d^ u2

(48)

where to obtain the last equality we have used the relations (47). The latter expression may be simpli ed if we notice that4

d^ Pj@uj e ju?uj (@uj+ (uj? uj )2 2 2 2 4 2 2

2 2

ju?uj2

ju? uj2d^+ i d^2 2 2 2

j j j (u? u )@uj

u+ 4ri2 d

(49)

with e ju?uj (@uj+ (uj? uj )^ being a (2r)-form well de ned on N . It follows that the ber-wise integral of both sides of eq. (49) vanishes and, consequently, that2 2

@ k k2= const: 2?r=2?= const: 2 r=2Z

2 2

ju?uj2

@+@+184

i u

^(u) d2r u= (; r )

2 2

ju?uj2 (u)

which proves the unitarity of the KZB connection for the case with no insertions in a way that maybe was not the simplest (two straightforward integrations by parts would do) but which has the virtue of generalizing to the case with arbitrary insertions (modulo convergence problems). To treat the general case, we shall apply the formula (46) to the di erential form= 2 with given by eq. (44) and satisfying the generalization of relations e jw?w j e? S j replaced by dwj . As before, we obtain, denoting by@ 0 the holomorphic (47) with du exterior derivative in all but u directions,2 2 2 1

k2=

? const: 2 r=2Z

?= const: 2 r=2

UY;zZ

UY;z 2 e 2 2 jw?w j

(@+ 4ri d^) e2

2 2

jw?wj2 j2

j2X

2 2

jw? wj2 d^+

(wj? wj ) dwj^+ 4ri d^2

+@0+2 2

?= const: 2 r=2

+@0+ 2 i d^ u?

UY;z

jw?wj21X

2 2

jw? wj2 d^+ 2i d^ (wj? wj )@uj+ 4ri d^X2

duj^@ujj

j dz^@ n n uj

j dy^@ s s uj

A large part of the right hand side vanishes by integration by parts with use of the relation generalizing (49):4

d^ Pj@uj e jw?wj (@uj+ (wj? wj )2 2 2 2 4 2 2

2 2

jw?w j2

jw? wj2d^+ i d^2 2

j (w

j? wj )@ j u

d^ d^+1

u+ 4ri2 d

(50)

and we obtain

?= const: 2 r=2

UY;z

2 2

jw?w j2

@0+i1

u s

1 X j dz^@ n n uj n

j dy^@ s s

uj j dzn^@u n j

^(51)

?= const: 2 r=2

UY;z

2 2

jw?w j2

@0+

j dy^@ s s uj

e? S h; i^ e? S h; i d2r u:

The@uj -terms may be transformed with use of relationXX

n i=1

j n;i d(yn;i?1? yn;i )^

= 2i d^@uj;

(52)

in the notations of eq. (43) and with yn;0 zn; which follows easily form the de nitions of the forms and !q . Eq. (52) may be rewritten asX

j dzn n

j dys s

^=( )

i 2

d^@uj?

dzn^ hjn;( )

(53)

with the contragradient action of hjn on the V -valued form . The last relation, upon substitution to eq. (51), yields

@ k k2= const: 2?r=2

UY;z+ 4i dZ

2 2

jw?wj2 e? 1 S 22

@ 0 h; i? 1@S^ h; i d^X

^ uh; i?X

+1?= const: 2 r=2UY;z

n2 2

jw?wj e? 1 S 22

dzn^@uj ( h hjn; i )^ d2r u( )

@uj (h@uj; i)

@ 0? 1@S^+1X

? 4i d^4

+(?1)K

;@0+and

n dzn^ hjn)@uj (

dzn^ hjn@uj;( )

^ d2r u:(54)

The crucial result are the following equalities:

@0= 0

(@S )^+ d^ H0+

dzn^ Hn= 0

with the contragradient action of Hn 's. The rst of these equalities is a straightforward consequence of the closedness of the forms !q from which is built. The second, more technical one, has been announced in 12] (as Prop. 9). Using these relations, we nally obtain

k2=

? const: 2 r=2

UY;z

2 2

jw?w j2 e? 1 S 2 (?1)K h

;@ 0+ 1 d^ H0

dzn^ Hn i^ d2ru= (; r )

which proves the unitarity of the KZ connection w.r.t. the scalar product (26) modulo the control of convergence of the integrals.

6 Bethe AnsatzThe basic algebraic relations (42) responsible for the unitarity of the KZ connection turn out to be a disguised (and compact) form of the Bethe Ansatz solution of the eigenvalue problem for the Gaudin Hamiltonians (35). This was remarked in 1] in the context of contour 20

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