Genus Two Virasoro Correlation Functions for Vertex Operator Algebras
aa r X i v : . [ m a t h . QA ] D ec Genus Two Virasoro Correlation Functions forVertex Operator Algebras
Thomas Gilroy ∗ and Michael P. Tuite † October 19, 2018
Abstract
We consider all genus two correlation functions for the Virasoro vacuum de-scendants of a vertex operator algebra. These are described in terms of explicitgenerating functions that can be combinatorially expressed in terms of a se-quence of globally defined differential operators on which the genus two Siegelmodular group Sp(4 , Z ) has a natural action. ∗ School of Mathematics and Statistics, University College Dublin, Ireland. email:[email protected] † School of Mathematics, Statistics and Applied Mathematics, NUI Galway, Ireland. email:[email protected]. Introduction
A Vertex Operator Algebra (VOA) (e.g. [FLM], [K], [LL], [MT1]) is an algebraic sys-tem related to Conformal Field Theory (CFT) in physics e.g. [DMS]. An essentialingredient of a VOA or CFT is the existence of a conformal Virasoro vector whose ver-tex operator modes generate the Virasoro algebra of central charge c . The connectionbetween VOAs, the Virasoro vector and genus one Riemann surfaces was establishedin the foundational work of Zhu [Z] giving a rigorous basis for many ideas in CFT.Zhu established a general recursion formula relating any genus one n -point correlationfunction to linear combination of ( n − V is C cofinite [Z]) the Ward identities implythe partition function for V and its ordinary modules satisfy a genus one modularordinary differential equation. This paper lays some of the ground work for the de-velopment of a new theory of genus two Sp(4 , Z ) Siegel modular partial differentialequation for the genus two partition function of a suitable VOA V and its ordinarymodules. This is illustrated in a sequel paper where we describe a new Sp(4 , Z ) mod-ular partial differential equation for the partition functions (for V and its ordinarymodules) for the (2 ,
5) minimal Virasoro model of central charge c = − [GT2].Correlation functions for VOAs on a genus two Riemann surface have been defined,and in some cases calculated based on an explicit sewing procedure for sewing twotori together [MT2, MT3]. We recently described a general Zhu recursion formulafor genus two n -point correlation functions which gives rise to new genus two Wardidentities when applied to Virasoro vector n -point functions [GT1]. The purpose of thispaper is to describe all genus two correlation functions for Virasoro descendants of thevacuum vector in terms of explicit generating functions. These generating functions,which satisfy the genus two Ward identity, are shown to be combinatorially expressedin terms of a sequence of globally defined differential operators on which the Siegelmodular group Sp(4 , Z ) has an natural action.We begin in Section 2 with a very brief review of VOAs on a genus two Riemannsurface. We review a sewing scheme for constructing a genus two surface from twopunctured tori and how to formally define the genus two partition and n -point corre-lation functions in terms of genus one VOA data [MT2, MT3].In Section 3 contains the main results of this paper. We first show that genus two n -point correlation functions for n Virasoro vectors are generating functions for thecorrelation functions for all Virasoro vacuum descendants in a similar fashion to thegenus zero and one cases [HT1]. These generating functions satisfy genus two Wardidentities (derived from genus two Zhu recursion) which involves genus two generalisedWeierstrass functions related to a global (2,1)-bidifferential Ψ( x, y ) holomorphic for x = y [GT1]. We also describe some analytic differential equations, which also involveΨ( x, y ), for the genus two bidifferential ω ( x, y ), normalised holomorphic 1-differentials ν ( x ) , ν ( x ) and the projective connection s ( x ). Using these differential equations, wedemonstrate in Theorem 3.11 how to express each generating function in a symmetricway as a sum of given weights of appropriate graphs. In particular, the Virasoro2ector n -point function is determined by the action of a specific symmetric differentialoperator O n on the normalised partition function Z (2) V / ( Z (2) M ) c (cf. (3.15)) where Z (2) V denotes the genus two partition function for V , Z (2) M denotes the genus two partitionfunction for the Heisenberg VOA M and c is the central charge.Lastly in Section 4 we consider general analytic and modular transformation prop-erties of the differential operator O n in any coordinate system on an arbitrary genustwo Riemann surface. In particular, in Theorem 4.6, we show how O n transforms underthe Siegel modular group Sp(4 , Z ) having established that the global (2,1)-bidifferentialΨ( x, y ) is Sp(4 , Z ) invariant in Theorem 4.2. We briefly review some concepts in genus two Riemann surface theory e.g. [FK, F,Mu1]. Let S (2) be a compact genus two Riemann surface with canonical homology basis α i , β i for i = 1 ,
2. There exists a unique holomorphic symmetric bidifferential (1 , ω ( x, y ), the normalised bidifferential of the second kind , where for x = y ∈ S (2) ω ( x, y ) = dxdy ( x − y ) + 16 s ( x ) − ( x − y )12 ∂ x s ( x ) + O (cid:0) ( x − y ) (cid:1) , (2.1) I α i ω ( x, · ) = 0 , i = 1 , .s ( x ) is the projective connection transforming under an analytic map x → φ ( x ) as s ( x ) = s ( φ ( x )) + { φ ( x ) , x } dx , (2.2)where { φ ( x ) , x } = φ ′′′ ( x ) φ ′ ( x ) − (cid:16) φ ′′ ( x ) φ ′ ( x ) (cid:17) is the Schwarzian derivative. Futhermore ν i ( x ) = I β i ω ( x, · ) , Ω ij = 12 πi I β i ν j , i, j = 1 , . for holomorphic differentials ν i ( x ) normalised by H β i ν j = 2 πiδ ij and period matrix Ω ∈ H , the genus two Siegel upper half plane i.e. Ω = Ω T and ℑ (Ω) > S (2) constructed by sewing two genus onetori S a = C / Λ a , for lattice Λ a = 2 πi ( Z τ a ⊕ Z ) with modular parameter τ a ∈ H for a = 1 , z a ∈ S a , ǫ ∈ C and define punctured tori b S = S \ { z , | z | ≤ | ǫ | /r } , b S = S \ { z , | z | ≤ | ǫ | /r } , | ǫ | ≤ r r . We identify the annular regions { z , | ǫ | /r ≤ | z | ≤ r } and { z , | ǫ | /r ≤ | z | ≤ r } via the sewing relation z z = ǫ . Then S (2) is parameter-ized by the sewing domain D sew = (cid:26) ( τ , τ , ǫ ) ∈ H × H × C : | ǫ | < D ( q ) D ( q ) (cid:27) , (2.3)where q a = e πiτ a and D ( q a ) = min λ a ∈ Λ a ,λ a =0 | λ a | . We may then obtain explicit expres-sions for ω ( x, y ), ν i ( x ) and Ω ij on S (2) for x, y ∈ b S ∪ b S described in [MT2]. We review aspects of vertex operator algebras (e.g. [FLM, K, LL, MT1]). A VertexOperator Algebra (VOA) is a quadruple (
V, Y, , ω ) consisting of a Z -graded complexvector space V = L n ∈ Z V ( n ) where dim V ( n ) < ∞ for each n ∈ Z , a linear map Y : V → End ( V )[[ z, z − ]] for a formal parameter z and pair of distinguished vectors:the vacuum ∈ V (0) and the conformal vector ω ∈ V (2) . For each v ∈ V , the imageunder the map Y is the vertex operator Y ( v, z ) = X n ∈ Z v ( n ) z − n − , with modes v ( n ) ∈ End ( V ), where Y ( v, z ) = v + O ( z ). Vertex operators satisfy locality i.e. for all u, v ∈ V there exists an integer k ≥ z − z ) k [ Y ( u, z ) , Y ( v, z )] = 0 . The vertex operator of the conformal vector ω is Y ( ω, z ) = P n ∈ Z L ( n ) z − n − wherethe modes L ( n ) satisfy the Virasoro algebra with central charge c [ L ( m ) , L ( n )] = ( m − n ) L ( m + n ) + c m − m δ m, − n Id V . Furthermore, L (0) v = kv for conformal weight wt ( v ) = k for all v ∈ V ( k ) and Y ( L ( − u, z ) = ∂ z Y ( u, z ).In order to describe VOAs on a torus, Zhu [Z] introduced an isomorphic VOA( V, Y [ , ] , , e ω ) with “square bracket” vertex operators Y [ v, z ] = X n ∈ Z v [ n ] z − n − = Y (cid:0) e L (0) v, e z − (cid:1) , and conformal vector e ω = ω − c with Virasoro modes L [ n ].Define the genus one partition function by the formal trace Z (1) V ( τ ) = Tr V (cid:0) q L (0) − c/ (cid:1) ,with q = e πiτ , the genus one correlation one point function by the formal trace Z (1) V ( v ; τ ) = Tr V (cid:0) o ( v ) q L (0) − c/ (cid:1) , v ∈ V, o ( v ) = v ( k −
1) for v ∈ V ( k ) and the genus one n -point correlation function for v , . . . , v n ∈ V inserted at z , . . . , z n ∈ C / (2 πi ( Z τ ⊕ Z )) by Z (1) V ( v , z ; . . . ; v n , z n ; τ ) = Z (1) V ( Y [ v , z ] . . . Y [ v n , z n ] ; τ ) . Zhu describes a general recursion formula for expressing any genus one n -point cor-relation function as a linear combination of ( n − Z (1) W ( . . . ) for an ordinary graded V -module W where thetrace is taken over W . We define the genus two partition function and n -point correlation function for a VOAbased on the sewing scheme for S (2) constructed from two tori S and S [MT3, GT1].The genus two partition function for V of strong CFT-type is defined by Z (2) V ( τ , τ , ǫ ) = X u ∈ V Z (1) V ( u ; τ ) Z (1) V ( u ; τ ) , (2.4)where the formal sum is taken over any V -basis and u is the dual of u with respectto an invariant invertible bilinear form h , i associated with the Mobius map z → ǫ/z (see [GT1] for more details).The genus two n -point correlation function for a , . . . , a L ∈ V and b , . . . , b R ∈ V formally inserted at x , . . . , x L ∈ b S and y , . . . , y R ∈ b S , respectively, is defined by Z (2) V ( a , x ; . . . ; a L , x L | b , y ; . . . ; b R , y R ; τ , τ , ǫ )= X u ∈ V Z (1) V ( Y [ a , x ] . . . Y [ a L , x L ] u ; τ ) Z (1) V ( Y [ b R , y R ] . . . Y [ b , y ] u ; τ ) . (2.5)Convergent expressions have been found for such correlation functions for particularVOAs such as the Heisenberg VOA and lattice VOAs [MT3]. A formal Zhu recursionformula for a genus two n -point function in terms of ( n − generalised Weierstrass func-tions , which depend on the conformal weight of the recursion vector but are otherwiseuniversal. For conformal weight 1 or 2, these series are holomorphic on appropriatedomains [GT1]. The above definition can be naturally extended to define Z (2) W ,W ( . . . )for a pair of ordinary graded V -modules W , W where the left or right hand trace istaken over W or W respectively. Consider the genus two Virasoro n -point correlation function for z , . . . , z n ∈ b S Z (2) V ( e ω, z ; . . . ; e ω, z n ) = X u ∈ V Z (1) V ( Y [ e ω, z ] . . . Y [ e ω, z n ] u ; τ ) Z (1) V ( u ; τ ) , (3.1)5where we suppress the dependence on τ , τ , ǫ ). We define the formal differential G n ( z ) = G n ( z , . . . , z n ) = Z (2) V ( e ω, z ; . . . ; e ω, z n ) d z , (3.2)where d z = dz i . . . dz n . G n ( z ) is independent of whether we formally insert e ω at z i ∈ b S or at ǫ/z i ∈ b S (Proposition 6 of [MT3]). Similarly to [HT1] we find Proposition 3.1. G m ( z ) is symmetric in z i and is a generating function for all genustwo n -point correlation functions for Virasoro vacuum descendants.Proof. G m ( z ) is a symmetric in z , . . . , z m by locality. Consider the genus two n -pointfunction for n Virasoro vacuum descendants v i = L [ − k i ] . . . L [ − k im i ] inserted at z i ∈ b S for i = 1 , . . . , n and k ij ≥ Z (2) V ( v , z ; . . . ; v n , z n ) = X u ∈ V Z (1) V ( Y [ v , z ] . . . Y [ v n , z n ] u ; τ ) Z (1) V ( u ; τ ) .Z (1) V ( Y [ v , z ] . . . Y [ v n , z n ] u ; τ ) is the coefficient of Q ni =1 Q m i j =1 ( x ij ) k ij − in Z (1) V ( Y [ Y [ e ω, x ] . . . Y [ e ω, x m ] , z ] . . . Y [ Y [ e ω, x n ] . . . Y [ e ω, x nm n ] , z n ] u ; τ ) . Using associativity and lower truncation (e.g. [K, LL, MT1]) we find for N ≫ n Y i =1 m i Y j =1 ( x ij + z i ) N Y [ Y [ e ω, x ] . . . Y [ e ω, x m ] , z ] . . . Y [ Y [ e ω, x n ] . . . Y [ e ω, x nm n ] , z n ] u = n Y i =1 m i Y j =1 ( x ij + z i ) N Y [ e ω, z + x ] . . . Y [ e ω, z + x m ] . . . Y [ e ω, z n + x n ] . . . Y [ e ω, z n + x nm n ] u. Thus the genus two n -point function for v , . . . , v n is the coefficient of Q ni =1 Q m i j =1 ( x ij ) k ij − of the formal expansion of Z (2) V ( e ω, z + x ; . . . ; e ω, z n + x nm n ) for M = P ni =1 m i . Define a genus two modular derivative operator ∇ x = X ≤ a ≤ b ≤ ν a ( x ) ν b ( x ) ∂∂ Ω ab , (3.3)for period matrix Ω ab , and normalised holomorphic 1-differentials ν a . There exists aninjective but non-surjective holomorphic map F Ω from the sewing domain D sew intothe Siegel upper half plane [GT1] F Ω : D sew → H , ( τ , τ , ǫ ) Ω( τ , τ , ǫ ) , (3.4)6elow we will also denote by ∇ x the action of (cid:0) F Ω (cid:1) − ◦ ∇ x ◦ F Ω on D sew .In Section 5 of [GT1] we describe a genus two Ward identity for the genus twoVirasoro n -point correlation function. This is expressed in terms of generalised Weier-strass functions P k ( x, y ) for k ≥ ν ( x ) = [ ν ( x ) , ν ( x )] denotea row vector of holomorphic 1-differentials and defineΨ( x, y ) = P ( x, y ) dx ( dy ) − = − ω ( x, y ) (cid:12)(cid:12)(cid:12)(cid:12) ν ( x ) ν ( y ) (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) ν ( y ) ∇ x ν ( y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ν ( y ) ∂ y ν ( y ) (cid:12)(cid:12)(cid:12)(cid:12) dy . (3.5) Proposition 3.2 ([GT1]) . Ψ( x, y ) is a holomorphic (2 , − -bidifferential for x = y where, for any local coordinates x, y Ψ( x, y ) = (cid:18) x − y + regular terms (cid:19) dx ( dy ) − . We also define generalised Weierstrass functions for k ≥ P k ( x, y ) = 1( k − ∂ k − y (cid:0) P ( x, y ) (cid:1) = 1( x − y ) k + regular terms , (3.6)which is holomorphic for x = y . Proposition 3.3 ([GT1]) . G n ( z ) obeys the formal Ward identity for z , . . . , z n ∈ b S ∪ b S and ( τ , τ , ǫ ) ∈ D sew G n ( z ) = ∇ z + dz n X k =2 (cid:0) P ( z , z k ) ∂ z k + 2 · P ( z , z k ) (cid:1) ! G n − ( z , . . . , z n )+ c n X k =2 2 P ( z , z k ) G n − ( z , . . . , b z k , . . . , z n ) dz dz k , (3.7) where b z k denotes the omission of the given term. We note that the above expression for G n ( z ) is not manifestly symmetric in itsarguments. For the genus two bidifferential ω ( x, y ), normalised holomorphic 1-differentials ν a ( x )for a = 1 , s ( x ) we find from Section 6 of [GT1] that Proposition 3.4. ω ( x, y ) , ν a ( x ) for a=1,2 and s ( x ) satisfy the following analyticdifferential equations (cid:16) ∇ x + dx X r =1 (cid:0) P ( x, y r ) ∂ y r + P ( x, y r ) (cid:1) (cid:17) ω ( y , y ) = ω ( x, y ) ω ( x, y ) , (3.8) (cid:16) ∇ x + dx (cid:0) P ( x, y ) ∂ y + P ( x, y ) (cid:1) (cid:17) ν a ( y ) = ω ( x, y ) ν a ( x ) , (3.9) (cid:16) ∇ x + dx (cid:0) P ( x, y ) ∂ y + 2 P ( x, y ) (cid:1) (cid:17) s ( y ) + 6 P ( x, y ) dx dy = 6 ω ( x, y ) , (3.10)7 or all x, y , y ∈ b S ∪ b S and Ω ∈ F Ω ( D sew ) . We may generalise (3.8) and (3.9) in the following coordinate independent way:
Corollary 3.5. ω ( x, y ) and ν a ( x ) for a=1,2 satisfy the following coordinate indepen-dent analytic differential equations for all Ω ∈ H ∇ x ω ( y , y ) + X r =1 ∂ y r (Ψ( x, y r ) ω ( y , y )) dy r = ω ( x, y ) ω ( x, y ) , (3.11) ∇ x ν a ( y ) + ∂ y (Ψ( x, y ) ν a ( y )) dy = ω ( x, y ) ν a ( x ) . (3.12) Proof. Ψ( x, y ) ω ( y , y ) is a global (2 , , x, y , y ) (with a similar statementfor Ψ( x, y ) ω ( y , y )). Thus d y r (Ψ( x, y r ) ω ( y , y )) = ∂ y r (Ψ( x, y r ) ω ( y , y )) dy r , is a global (2 , , ∈ F Ω ( D sew ). But all parts of (3.11) are holomorphic forall Ω ∈ H and hence the identity can be analytically extended from F Ω ( D sew ) to H .(3.12) follows from (3.11) by integrating y along the β a homology cycle.We may also generalise (3.10) in the following way: Corollary 3.6. s ( y ) , for a given choice of local coordinate y , satisfies the followinganalytic differential equation for all Ω ∈ H (cid:16) ∇ x + dy (Ψ( x, y ) ∂ y + 2 ∂ y Ψ( x, y )) (cid:17) s ( y ) + dy ∂ y Ψ( x, y ) = 6 ω ( x, y ) . (3.13) Proof.
We let y = y and y = y + ε and note from (2.1) that ω ( y , y ) = dy ε + 16 s ( y ) + O ( ε ) ,∂ y ω ( y , y ) = − dy ε + 112 ∂ y s ( y ) + O ( ε ) ,∂ y ω ( y , y ) = 2 dy ε + 112 ∂ y s ( y ) + O ( ε ) , Ψ( x, y ) = Ψ( x, y ) + ε∂ y Ψ( x, y ) + ε ∂ y Ψ( x, y ) + ε ∂ y Ψ( x, y ) + O ( ε ) ,∂ y Ψ( x, y ) = ∂ y Ψ( x, y ) + ε∂ y Ψ( x, y ) + ε ∂ y Ψ( x, y ) + O ( ε ) . The result follows by substituting the above into (3.11) and taking the ε → Z (2) M ( τ , τ , ǫ ) for the Heisen-berg VOA M obeys [GT1] 8 roposition 3.7. Z (2) M ( τ , τ , ǫ ) is holomorphic for ( τ , τ , ǫ ) ∈ D sew and satisfies ∇ x Z (2) M = 112 s ( x ) Z (2) M , (3.14) for x ∈ b S ∪ b S . Remark 3.8. Z (2) M ( τ , τ , ǫ ) can be considered as a holomorphic function on F Ω ( D sew ) but cannot be analytically continued to the full Siegel upper half plane H (cf. The-orem 7.2, [GT1]). In physics, this follows from the conformal anomaly (e.g. [FS])which for the Heisenberg VOA is believed to be related to the non-existence of a globalsection of certain determinant line bundles on the genus two Riemann surface [Mu2]. (3.8)-(3.14) are the genus two analogues of differential equations for elliptic andmodular functions described in [HT1]. Thus (3.14) corresponds to q ∂∂q (cid:18) η ( q ) (cid:19) = 12 E ( q ) (cid:18) η ( q ) (cid:19) , for the weight 2 quasi-modular Eisenstein series E ( q ) = − + 2 P m,n ≥ nq mn . We show below in Theorem 3.11 how to express G n ( z ) in a manifestly symmetricfashion as a sum of weights of appropriate graphs. The graph configurations are pre-cisely those exploited in [HT1] to describe genus one Virasoro n -point functions andmany of the arguments below mirror the genus one case. However, the graph weightsare differently defined in the genus two case and the technicalities are more involved.Furthermore, the genus two graph weights for G n ( z ) are described in terms of a lineardifferential operator O n ( z ) which is symmetric in its arguments and possesses funda-mental properties under analytic and genus two Sp(4 , Z ) modular transformations.Define, for central charge c , a formal normalised partition functionΘ V ( τ , τ , ǫ ) : = Z (2) M ( τ , τ , ǫ ) − c Z (2) V ( τ , τ , ǫ ) , (3.15)where Z (2) M ( τ , τ , ǫ ) is the genus two partition function for the Heisenberg VOA (whichis holomorphic on D sew ). Following Remark 3.8 and [FS] we conjecture: Conjecture 3.9. Θ V is holomorphic on H for a C -cofinite VOA V . If V is alsorational then Θ V is a component of a vector valued Siegel modular form of weight c/ . For example, for a lattice VOA V L for an even lattice L of rank c we find Θ V = Θ L (Ω),the genus two Siegel lattice theta function [MT3], a Siegel modular form of weight c/ | L ∗ /L | where L ∗ is the dual lattice. Thus, conjecturally, it is the normalisedpartition function Θ V on which the full genus two Sp(4 , Z ) modular group naturallyacts. One of the main purposes of this paper is to develop global differential operators We also note that Θ V has canonical properties in the one torus degeneration limite [HT2]. n ( z ) on which Sp(4 , Z ) acts. In the sequel [GT2] we show how these operators giverise to a Sp(4 , Z ) modular differential equation for the partition function for the (2 , − / O n ( z ) (which in general acts on differen-tiable functions of Ω) by O n ( z )Θ V : = Z (2) M ( τ , τ , ǫ ) − c G n ( z ) . (3.16)For n = 1 we find G ( z ) = ∇ z Z (2) V ( τ , τ , ǫ ) so that, using (3.14), we find O ( z ) = ∇ z + c s ( z ) . (3.17)In order to describe the n = 2 case, we also define the differential operator D z ,z = ∇ z + dz (cid:0) P ( z , z ) ∂ z + 2 P ( z , z ) (cid:1) . (3.18)Then for n = 2, the Ward identity (3.7) implies O ( z , z )Θ V = (cid:16) Z (2) M (cid:17) − c D z ,z ∇ z Z (2) V + c P ( z , z ) Θ V dz dz = D z ,z (cid:16) ∇ z Θ V + c s ( z )Θ V (cid:17) + c s ( z ) ∇ z Θ V + c s ( z ) s ( z )Θ V + c P ( z , z ) dz dz Θ V . From (3.9) we note that D z ,z ν a ( z ) ν b ( z ) = ω ( z , z ) ( ν a ( z ) ν b ( z ) + ν a ( z ) ν b ( z )) . (3.19)(3.19) together with (3.10) imply that O ( z , z ) = X ≤ a ≤ b ≤ X ≤ c ≤ d ≤ ν a ( z ) ν b ( z ) ν c ( z ) ν d ( z ) ∂ ∂ Ω ab ∂ Ω cd + c s ( z ) ∇ z + c s ( z ) ∇ z + c s ( z ) s ( z )+ 2 ω ( z , z ) X ≤ a ≤ b ≤ ν a ( z ) ν b ( z ) ∂∂ Ω ab + c ω ( z , z ) . (3.20)This expression is clearly symmetric in z , z in accordance with Proposition 3.1. Fur-thermore, each term in (3.20) is now written in coordinate independent way.Similarly to Section 3 of [HT1] we now develop a graphical/combinatorial approachfor computing O n ( z ) and hence G n for all n . We define an order n Virasoro graph to be a directed graph G n with n vertices labelled by z , . . . , z n . Each z i -vertex hasdegree deg( z i ) = 0 , G n consist of r -cycles, with r ≥ The operator (3.17) is a like a higher genus Serre derivative as discussed further in [GT1]. emark 3.10. The set of non-isomorphic order n Virasoro graphs is in one to onecorrespondence with the set of partial permutations of the label set { , . . . , n } . This isdescribed in further detail in [HT1]. We define a genus two weight W ( G n ) on G n as follows. For each directed edge E ij we define an edge weight W ( E ij ) = W ( (cid:16)(cid:23)(cid:17)(cid:22)(cid:18)(cid:21)(cid:19)(cid:20) z i (cid:16)(cid:23)(cid:17)(cid:22)(cid:18)(cid:21)(cid:19)(cid:20) z j / / ) = (cid:26) s ( z i ) for i = j,ω ( z i , z j ) for i = j. (3.21)Let C kℓ denote a chain in G n with end-vertices z k and z ℓ (cid:16)(cid:23)(cid:17)(cid:22)(cid:18)(cid:21)(cid:19)(cid:20) z m (cid:16)(cid:23)(cid:17)(cid:22)(cid:18)(cid:21)(cid:19)(cid:20) z n (cid:16)(cid:23)(cid:17)(cid:22)(cid:18)(cid:21)(cid:19)(cid:20) z k (cid:16)(cid:23)(cid:17)(cid:22)(cid:18)(cid:21)(cid:19)(cid:20) z ℓ · · · / / / / and assign a chain weight (including the degenerate chain) W ( C kℓ ) = W ( (cid:16)(cid:23)(cid:17)(cid:22)(cid:18)(cid:21)(cid:19)(cid:20) (cid:16)(cid:23)(cid:17)(cid:22)(cid:18)(cid:21)(cid:19)(cid:20)(cid:16)(cid:23)(cid:17)(cid:22)(cid:18)(cid:21)(cid:19)(cid:20) z k (cid:16)(cid:23)(cid:17)(cid:22)(cid:18)(cid:21)(cid:19)(cid:20) z ℓ · · · / / / / ) = A ( z k , z ℓ ) , (3.22)where A ( z k , z ℓ ) = P ≤ a ≤ b ≤ ν a ( z k ) ν b ( z ℓ ) α ab for free parameters α ab = α ba . Let K bethe number of cycles and define a weight for G n by W ( G n ) = (cid:16) c (cid:17) K Y E ij W ( E ij ) Y C kℓ W ( C kℓ ) , (3.23)where the first product ranges over all the edges and the second product ranges overall the chains of G n . Thus the weight depends on c , ω ( z i , z j ), s ( z i ), ν a ( z i ) and α ab . Wealso note that W is multiplicative on the disconnected components of G n .Lastly, define a linear map L α from C [ α ab ], the vector space of complex coefficientpolynomials in α ab , to the complex vector space spanned by ∂∂ Ω ab derivatives with L α ( α a b . . . α a M b M ) = ∂ M ∂ Ω a b . . . ∂ Ω a M b M . (3.24)Let p nKM be the number of inequivalent order n Virasoro graphs containing K cyclesand M chains. In [HT1] the following graph generating function is established p n ( α, β ) = X K ≥ ,M ≥ p nKM α M β K = ( − n n ! n X i =0 ( − α ) i i ! (cid:18) − β − in − i (cid:19) , (3.25)for chain and cycle counting parameters α and β respectively. Thus for n = 1 we find p ( α, β ) = α + β corresponding to two inequivalent graphs with weights W (cid:16) (cid:16)(cid:23)(cid:17)(cid:22)(cid:18)(cid:21)(cid:19)(cid:20) z (cid:17) = A ( z , z ) , W (cid:16) (cid:16)(cid:23)(cid:17)(cid:22)(cid:18)(cid:21)(cid:19)(cid:20) z (cid:28) (cid:28) (cid:17) = c s ( z )6 , whose weight sum under the action of L α is O ( z ) using (3.17).11or n = 2 we have p ( α, β ) = α + 2 αβ + β + β + 2 α for 7 graphs with weights: W (cid:16) (cid:16)(cid:23)(cid:17)(cid:22)(cid:18)(cid:21)(cid:19)(cid:20) z (cid:16)(cid:23)(cid:17)(cid:22)(cid:18)(cid:21)(cid:19)(cid:20) z (cid:17) = A ( z , z ) A ( z , z ) ,W (cid:16) (cid:16)(cid:23)(cid:17)(cid:22)(cid:18)(cid:21)(cid:19)(cid:20) z (cid:16)(cid:23)(cid:17)(cid:22)(cid:18)(cid:21)(cid:19)(cid:20) z (cid:28) (cid:28) (cid:17) = c s ( z )6 A ( z , z ) , W (cid:16) (cid:16)(cid:23)(cid:17)(cid:22)(cid:18)(cid:21)(cid:19)(cid:20) z (cid:16)(cid:23)(cid:17)(cid:22)(cid:18)(cid:21)(cid:19)(cid:20) z (cid:28) (cid:28) (cid:17) = c s ( z )6 A ( z , z ) ,W (cid:16) (cid:16)(cid:23)(cid:17)(cid:22)(cid:18)(cid:21)(cid:19)(cid:20) z (cid:16)(cid:23)(cid:17)(cid:22)(cid:18)(cid:21)(cid:19)(cid:20) z (cid:28) (cid:28) (cid:28) (cid:28) (cid:17) = (cid:16) c (cid:17) s ( z )6 s ( z )6 , W (cid:16) (cid:16)(cid:23)(cid:17)(cid:22)(cid:18)(cid:21)(cid:19)(cid:20) z (cid:16)(cid:23)(cid:17)(cid:22)(cid:18)(cid:21)(cid:19)(cid:20) z c c (cid:17) = c ω ( z , z ) ,W (cid:16) (cid:16)(cid:23)(cid:17)(cid:22)(cid:18)(cid:21)(cid:19)(cid:20) z (cid:16)(cid:23)(cid:17)(cid:22)(cid:18)(cid:21)(cid:19)(cid:20) z o o (cid:17) = W (cid:16) (cid:16)(cid:23)(cid:17)(cid:22)(cid:18)(cid:21)(cid:19)(cid:20) z (cid:16)(cid:23)(cid:17)(cid:22)(cid:18)(cid:21)(cid:19)(cid:20) z / / (cid:17) = A ( z , z ) , whose weight sum under the action of L α using (3.20) is X G L α (cid:0) W ( G ) (cid:1) = O ( z , z ) . These examples illustrate the general result:
Theorem 3.11.
The order n genus two Virasoro generating function is given by G n ( z ) = Z (2) M ( τ , τ , ǫ ) c O n ( z )Θ V ( τ , τ , ǫ ) , for linear differential operator O n ( z ) = X G n L α ( W ( G n )) , (3.26) where the sum is taken over all inequivalent order n Virasoro graphs G n .Proof. We prove the result by induction in n . We have already shown the result holdsfor n = 1 and n = 2 and employ the Ward identity (3.7) to inductively prove (3.26)for n ≥ n Virasoro graph G n can be characterized, according tothe nature of the z vertex, in terms of following five types:(i) deg( z ) = 0: (cid:16)(cid:23)(cid:17)(cid:22)(cid:18)(cid:21)(cid:19)(cid:20) z · · · (ii) deg( z ) = 1: (cid:16)(cid:23)(cid:17)(cid:22)(cid:18)(cid:21)(cid:19)(cid:20) z (cid:16)(cid:23)(cid:17)(cid:22)(cid:18)(cid:21)(cid:19)(cid:20) z a / / · · · or (cid:16)(cid:23)(cid:17)(cid:22)(cid:18)(cid:21)(cid:19)(cid:20) z (cid:16)(cid:23)(cid:17)(cid:22)(cid:18)(cid:21)(cid:19)(cid:20) z a o o · · · (iii) deg( z ) = 2 where the z -vertex forms a 1-cycle: (cid:16)(cid:23)(cid:17)(cid:22)(cid:18)(cid:21)(cid:19)(cid:20) z | | · · · (iv) deg( z ) = 2 where the z -vertex is an element of a 2-cycle: (cid:16)(cid:23)(cid:17)(cid:22)(cid:18)(cid:21)(cid:19)(cid:20) z (cid:16)(cid:23)(cid:17)(cid:22)(cid:18)(cid:21)(cid:19)(cid:20) z k $ $ d d · · · (v) deg( z ) = 2 where either the z -vertex is a non end-vertex of a chain or anelement of an r -cycle with r ≥ (cid:16)(cid:23)(cid:17)(cid:22)(cid:18)(cid:21)(cid:19)(cid:20) z (cid:16)(cid:23)(cid:17)(cid:22)(cid:18)(cid:21)(cid:19)(cid:20) z a (cid:16)(cid:23)(cid:17)(cid:22)(cid:18)(cid:21)(cid:19)(cid:20) z b · · · · · · / / / / O n ( z ) as follows: O n ( z ) = c s ( z ) O n − ( z , . . . , z n )+ ∇ z + dz n X k =2 (cid:0) P ( z , z k ) ∂ z k + 2 · P ( z , z k ) (cid:1) ! O n − ( z , . . . , z n )+ c n X k =2 2 P ( z , z k ) dz dz k O n − ( z , . . . , b z k , . . . , z n ) , (3.27)We now show how the parts of (3.27) relate to Virasoro graph weights by using in-duction in n . Thus given O n − and O n − satisfy (3.26), we see that the c s ( z ) O n − term of (3.27) arises from the sum over all G n graphs of type (iii).Let G n − denote an order n − z , . . . , z n of weight W ( G n − ). This gives a contribution to (3.27) of ∇ z L α (cid:0) W ( G n − ) (cid:1) = L α (cid:0) W ( G n − ) A ( z , z ) (cid:1) + L α (cid:0) ∇ z W ( G n − ) (cid:1) , (3.28)using the Leibniz rule for ∇ x . In particular, all terms of the form W ( G n − ) A ( z , z )arise as weights of G n graphs of type (i).Let us examine the contributions that arise from ∇ z W ( G n − ) in (3.28) and theremaining terms in (3.27) and show that these can be expressed in terms of a sum ofthe weights of graphs of type (ii), (iv) and (v). Let z k be a given vertex in G n − for k = 2 , . . . , n . Then, much as before, we can characterize G n − according to (a) z k is adegree 0 vertex (b) z k is a disconnected vertex of degree 2 (c) z k is a degree 1 vertexor (d) z k is a degree 2 vertex in a chain or an r -cycle for r ≥ Case (a). G n − consists of a z k vertex of degree 0 and an order n − G n − (with vertices z , . . . , b z k , . . . , z n ) of weight W ( G n − ) = A ( z k , z k ) W ( G n − ) . Using (3.19) this contributes to (3.27) the term D z ,z k A ( z k , z k ) W ( G n − ) = 2 A ( z , z k ) ω ( z , z k ) W ( G n − ) , the sum of the weights of two G n graphs of type (ii) where z and z k form a disconnectedchain of length 2. Case (b). G n − consists of a disconnected degree 2 vertex z k and an order n − G n − of weight W ( G n − ) = c s ( z k ) W ( G n − ) which contributes c D z ,z k ( s ( z k )) W ( G n − ) to (3.27). Summing with the c P ( z , z k ) W ( G n − ) contribu-tion to (3.27) gives c ω ( z , z k ) W ( G n − ) , using (3.10), the weight of a G n graph of type (iv) where z and z k form a 2-cycle.13 ase (c). z k is an end-vertex of a chain C kℓ so that W ( G n − ) = A ( z k , z ℓ ) ω ( z k , z m ) . . . ,where z k is joined to z m and the ellipsis denotes the factors independent of z k . Using(3.8) and (3.9) this contributes terms to (3.27) of the form (cid:16) ∇ z + dz (cid:16) P ( z , z k ) ∂ z k + 2 · P ( z , z k )+ P ( z , z m ) ∂ z m + P ( z , z m ) (cid:17)(cid:17) A ( z k , z ℓ ) ω ( z k , z m ) ! . . . = A ( z , z ℓ ) ω ( z , z k ) ω ( z k , z m ) . . . + A ( z k , z ℓ ) ω ( z k , z ) ω ( z , z m ) . . . . (3.29)Note that we have omitted in (3.29) contributions to (3.27) of the form: A ( z k , z ℓ ) ω ( z k , z m ) (cid:0) ∇ z + P ( z , z m ) ∂ z m + P ( z , z m ) (cid:1) ( . . . )which contribute to case (d) for z m . The first term in (3.29) is the weight of a G n graph of type (ii): (cid:16)(cid:23)(cid:17)(cid:22)(cid:18)(cid:21)(cid:19)(cid:20) z (cid:16)(cid:23)(cid:17)(cid:22)(cid:18)(cid:21)(cid:19)(cid:20) z k (cid:16)(cid:23)(cid:17)(cid:22)(cid:18)(cid:21)(cid:19)(cid:20) z m (cid:16)(cid:23)(cid:17)(cid:22)(cid:18)(cid:21)(cid:19)(cid:20) z ℓ · · · / / / / / / . . . and the second term is the weight of a graph of type (v): (cid:16)(cid:23)(cid:17)(cid:22)(cid:18)(cid:21)(cid:19)(cid:20) z k (cid:16)(cid:23)(cid:17)(cid:22)(cid:18)(cid:21)(cid:19)(cid:20) z (cid:16)(cid:23)(cid:17)(cid:22)(cid:18)(cid:21)(cid:19)(cid:20) z m (cid:16)(cid:23)(cid:17)(cid:22)(cid:18)(cid:21)(cid:19)(cid:20) z ℓ · · · / / / / / / . . . Case (d).
If deg( z k ) = 2 then W ( G n − ) = ω ( z a , z k ) ω ( z k , z b ) . . . , where z k is joinedto z a and z b and the ellipsis denotes the factors independent of z k . This contributesterms to (3.27) of the form (cid:16) ∇ z + dz (cid:16) P ( z , z k ) ∂ z k + P ( z , z a ) ∂ z a + P ( z , z b ) ∂ z b + P ( z , z a ) + 2 · P ( z , z k ) + P ( z , z b ) (cid:17)(cid:17) ω ( z a , z k ) ω ( z k , z b ) ! . . . = ω ( z a , z ) ω ( z , z k ) ω ( z k , z b ) . . . + ω ( z a , z k ) ω ( z k , z ) ω ( z , z b ) . . . (3.30)using (3.8). Note that we have omitted in (3.8) contributions to (3.27) of the form: ω ( z a , z k ) ω ( z k , z a ) (cid:16) ∇ x + P ( z , z a ) ∂ z a + P ( z , z b ) ∂ z b + P ( z , z a ) + P ( z , z b ) (cid:17) ( . . . )which contribute to case (c) and case (d) for z a or z b . The two terms in (3.30) areweights of a G n graphs of type (v): · · · (cid:16)(cid:23)(cid:17)(cid:22)(cid:18)(cid:21)(cid:19)(cid:20) z a (cid:16)(cid:23)(cid:17)(cid:22)(cid:18)(cid:21)(cid:19)(cid:20) z (cid:16)(cid:23)(cid:17)(cid:22)(cid:18)(cid:21)(cid:19)(cid:20) z k (cid:16)(cid:23)(cid:17)(cid:22)(cid:18)(cid:21)(cid:19)(cid:20) z b · · · / / / / / / , · · · (cid:16)(cid:23)(cid:17)(cid:22)(cid:18)(cid:21)(cid:19)(cid:20) z a (cid:16)(cid:23)(cid:17)(cid:22)(cid:18)(cid:21)(cid:19)(cid:20) z k (cid:16)(cid:23)(cid:17)(cid:22)(cid:18)(cid:21)(cid:19)(cid:20) z (cid:16)(cid:23)(cid:17)(cid:22)(cid:18)(cid:21)(cid:19)(cid:20) z b · · · / / / / / / Thus, altogether, we find that the weights of all G n graphs of type (i)-(v) contributeand hence (3.26) holds. 14 emark 3.12. O n ( z ) of (3.26) is symmetric in z , . . . , z n and is expressed in a co-ordinate free way in terms of ω ( z i , z j ) , ν a ( z i ) , s ( z i ) and ∂∂ Ω ab for all Ω ∈ H wherethe only dependence on the original VOA is the central charge c . Furthermore, usingCorollaries 3.5 and 3.6 it follows that O n ( z ) satisfies the recurrence relation: O n ( z ) = ∇ z + c s ( z ) + n X k =2 dz k (Ψ( z , z k ) ∂ z k + 2 ∂ z k Ψ( z , z k )) ! O n − ( z , . . . , z n )+ 3 c n X k =2 dz k (cid:0) ∂ z k Ψ( z , z k ) (cid:1) O n − ( z , . . . , b z k , . . . , z n ) , (3.31) for any choice of coordinates z and for all Ω ∈ H . Remark 3.13.
Theorem 3.11 can be readily generalized for any pair of ordinary V -modules W , W with genus two n -point function Z (2) W ,W ( e ω, z ; . . . ; e ω, z n ) d z = d z X u ∈ V Z (1) W ( Y [ e ω, z ] . . . Y [ e ω, z n ] u ; τ ) Z (1) W ( u ; τ )= O n ( z )Θ W ,W ( τ , τ , ǫ ) , where Θ W ,W ( τ , τ , ǫ ) = Z (2) M ( τ , τ , ǫ ) − c Z (2) W ,W ( τ , τ , ǫ ) . Following Conjecture 3.9 wefurther conjecture that if V is rational and C cofinite, then Θ W ,W for irreducible W , W form further components of a weight c/ vector valued Siegel modular form. By remark 3.12, we may express O n ( z ) in any coordinate system on an arbitrary genustwo Riemann surface. In particular, we can consider the behaviour of O n ( z ) under ageneral analytic transformation: Proposition 4.1.
Let z → φ ( z ) be an analytic map then we have O n ( z ) = O n ( z , . . . , φ ( z i ) , . . . , z n ) + c { φ ( z i ) , z i } dz i O n − ( z , . . . , b z i , . . . , z n ) , for i ∈ { , . . . , n } and { φ ( z ) , z } is the Schwarzian derivative.Proof. Choose i = 1 wlog by Proposition 3.1. ω ( z , z j ) , ν a ( z ) are invariant under ananalytic transformation whereas s ( z ) transforms as in (2.2). Such a s ( z ) term onlyarises in the Virasoro graphs G n of type (iii), where the z -vertex forms a 1-cycle ofweight W ( G n ) = c s ( z ) W ( G n − ). Thus the result follows.In order to describe the genus two modular properties of O n ( z ) we analyse themodular properties of the (2 , − x, y ) of (3.5). The genus two modular groupSp(4 , Z ) consists of integral block matrices γ := [ A BC D ] where
A, B, C, D obey: A T D − C T B = I, AB T = BA T , CD T = DC T ,A T C = C T A, B T D = D T B, (4.1)15or identity matrix I . It is convenient to define for γ ∈ Sp(4 , Z ) and Ω ∈ H M = C Ω +
D, N = ( C Ω + D ) − . The holomorphic differentials ν ( x ), the period matrix Ω, the derivative operator ∇ x , the bidifferential ω ( x, y ) and the projective connection s ( x ) transform under γ ∈ Sp(4 , Z ) as follows [F, Mu1, GT1] ν γ ( x ) = ν N, Ω γ = ( A Ω + B ) N, ∇ γx = ∇ x , (4.2) ω γ ( x, y ) = ω ( x, y ) − X ≤ a ≤ b ≤ ( ν a ( x ) ν b ( y ) + ν b ( x ) ν a ( y )) ∂∂ Ω ab log det M, (4.3) s γ ( x ) = s ( x ) − ∇ x log det M. (4.4)We now show that for the the (2 , −
1) bidifferential Ψ( x, y ) of (3.5)
Theorem 4.2. Ψ( x, y ) is Sp(4 , Z ) modular invariant. In order to prove Theorem 4.2 we need two lemmas. The first lemma concerns thesecond term on the right hand side of (4.3):
Lemma 4.3. X ≤ a ≤ b ≤ ( ν a ( x ) ν b ( y ) + ν b ( x ) ν a ( y )) ∂∂ Ω ab log det M = ν ( x ) N C ν T ( y ) . (4.5) Proof.
Using the Sp(4 , Z ) relations (4.1) we find N = A T − C T Ω γ so that( N C ) T = N C. (4.6)The result follows by direct calculation using (4.6) where we find ∂ log det M = ( M C − M C ) / det M = ( N C ) ,∂ log det M = ( N C ) ,∂ log det M = 2( N C ) . Lemma 4.4.
For ν γ ( x ) = ν N of (4.2) we have (cid:12)(cid:12)(cid:12)(cid:12) ν γ ( x ) ν γ ( y ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) ν ( x ) ν ( y ) (cid:12)(cid:12)(cid:12)(cid:12) det N, (4.7) (cid:12)(cid:12)(cid:12)(cid:12) ν γ ( y ) ∂ y ν γ ( y ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) ν ( y ) ∂ y ν ( y ) (cid:12)(cid:12)(cid:12)(cid:12) det N, (4.8) (cid:12)(cid:12)(cid:12)(cid:12) ν γ ( y ) ∇ γx ν γ ( y ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) ν ( y ) ∇ x ν ( y ) (cid:12)(cid:12)(cid:12)(cid:12) det N + ν ( x ) N C ν T ( y ) (cid:12)(cid:12)(cid:12)(cid:12) ν ( x ) ν ( y ) (cid:12)(cid:12)(cid:12)(cid:12) det N. (4.9)16 roof. (cid:12)(cid:12)(cid:12)(cid:12) ν γ ( x ) ν γ ( y ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) ν ( x ) N ν ( y ) N (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) ν ( x ) ν ( y ) (cid:12)(cid:12)(cid:12)(cid:12) det N and similarly for (4.8). To prove (4.9), wefirst note that ∇ x M = C ν ( x ) T ν ( x ) . Furthermore, ( ∇ x N ) M = − N ∇ x M so that ∇ x N = − N ( ∇ x M ) N = − N C ν ( x ) T ν ( x ) N. (4.10)Hence, using (4.2) and (4.6), we find that (cid:12)(cid:12)(cid:12)(cid:12) ν γ ( y ) ∇ γx ν γ ( y ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) ν ( y ) N ∇ x ( ν ( y )) N + ν ( y ) ∇ x N (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) ν ( y ) ∇ x ν ( y ) (cid:12)(cid:12)(cid:12)(cid:12) det N + (cid:12)(cid:12)(cid:12)(cid:12) ν ( y ) N − ν ( y ) N C ν ( x ) T ν ( x ) N (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) ν ( y ) ∇ x ν ( y ) (cid:12)(cid:12)(cid:12)(cid:12) det N + ν ( x ) N C ν T ( y ) (cid:12)(cid:12)(cid:12)(cid:12) ν ( x ) ν ( y ) (cid:12)(cid:12)(cid:12)(cid:12) det N. Proof of Theorem 4.2.
Combining Lemmas 4.3 and 4.4 we immediately obtain ω γ ( x, y ) (cid:12)(cid:12)(cid:12)(cid:12) ν γ ( x ) ν γ ( y ) (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) ν γ ( y ) ∇ γx ν γ ( y ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:20) ω ( x, y ) (cid:12)(cid:12)(cid:12)(cid:12) ν ( x ) ν ( y ) (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) ν ( y ) ∇ x ν ( y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:21) det N. Thus using (4.8) we find Ψ( x, y ) is Sp(4 , Z ) modular invariant. Corollary 4.5.
The differential equations (3.11) - (3.13) are Sp(4 , Z ) invariant.Proof. Under the action of γ ∈ Sp(4 , Z ), the change in the left hand side of (3.11) is − ∇ x (cid:0) ν ( y ) N C ν T ( y ) (cid:1) − X r =1 ∂ y r (cid:0) Ψ( x, y r ) ν ( y ) N C ν T ( y ) (cid:1) dy r = − X r =1 ω ( x, y r ) ν ( x ) N C ν T ( y r ) − ν ( y )( ∇ x N ) C ν T ( y ) , using (3.12). But (4.6) and (4.10) imply − ν ( y )( ∇ x N ) C ν T ( y ) = (cid:0) ν ( x ) N C ν T ( y ) (cid:1) (cid:0) ν ( x ) N C ν T ( y ) (cid:1) . Thus the total change in the left hand side of (3.11) is ω γ ( x, y ) ω γ ( x, y ) − ω ( x, y ) ω ( x, y ) , as required. A similar method shows that (3.12) is modular invariant. Lastly, ∂ ky Ψ( x, y )is modular invariant so that modular invariance of (3.13) follows by considering the y → y limit in the above analysis as described in the proof of Corollary 3.6.17et us now consider the modular properties of the operator O n ( z ). FollowingRemark 3.12 we know that O n ( z ) depends on ω ( x, y ), ν ( x ), ν ( x ), s ( x ) and ∂∂ Ω ab forall Ω ∈ H . These terms transform under γ ∈ Sp(4 , Z ) as in (4.2)-(4.4) so that O n ( z ) → O γn ( z ) . We then find
Theorem 4.6.
For differentiable F = F (Ω) we have for all γ ∈ Sp(4 , Z ) that O γn ( z ) (cid:0) det( M ) c/ F (cid:1) = det( M ) c/ O n ( z ) F. (4.11) Proof.
We prove the result by induction in n . The result is trivially true for n = 0.For n = 1 we use (3.17) to find O γ ( z ) (cid:0) det( M ) c/ F (cid:1) = (cid:16) ∇ γz + c s γ ( z ) (cid:17) (cid:0) det( M ) c/ F (cid:1) = det( M ) c/ O ( z ) F, using (4.2) and (4.4). (3.31) implies by induction that for n ≥ O γn ( z ) (cid:0) det( M ) c/ F (cid:1) = ∇ γz + c s γ ( z ) + n X k =2 dz k (Ψ( z , z k ) ∂ z k + 2 ∂ z k Ψ( z , z k )) ! (cid:0) det( M ) c/ O n − F (cid:1) + 3 c det( M ) c/ n X k =2 dz k (cid:0) ∂ z k Ψ( z , z k ) (cid:1) O n − F = det( M ) c/ O n ( z ) F, using (4.2) and (4.4) again. Thus the result follows. References [DMS] Di Francesco, P., Mathieu, P. and Senechal, D.:
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