INSTITUTE of MATHEMATICS. ACADEMY of SCIENCES of the CZECH REPUBLIC. New modular form identities associated to generalized vertex operator algebras

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1 INSTITUTEofMATHEMATICS Academy of Sciences Czech Republic INSTITUTE of MATHEMATICS ACADEMY of SCIENCES of the CZECH REPUBLIC New modular form identities associated to generalized vertex operator algebras Alexander Zuevsky Preprint No PRAHA 204

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3 NEW MODULAR FORM IDENTITIES ASSOCIATED TO GENERALIZED VERTEX OPERATOR ALGEBRAS ALEXANDER ZUEVSKY Abstract. New identities appearing from consideration of higher genus characters for generalized vertex operator algebras with a formal VOSA parameter associated to a local coordinate on a self-sewn Riemann surface are considered. Genus two version of twisted product Jacobi identity is reviewed. Further generalization of classical number theory identities for modular forms are proposed.. Introduction This is the a continuation of the paper Z0 describing modular form identities obtained as a result of computation of characters for vertex operator algebra modules on genus two and higher) Riemann surfaces TZ,TZ2,TZ4, TZ5 TZ9. In thise part we announce new modular form identities proved in collaboration with M.P. Tuite) in TZ4 and obtained in the consideration of genus one and genus two characters for generalized vertex operator algebras with a formal VOSA parameter associated to a local coordinate on a self-sewn Riemann surface. In MTZ, TZ, TZ4 we have established modular properties of genus one and genus two partition and n-point functions for free fermionic vertex operator superalgebras. Modular properties of correlations functions for classes of vertex algebras on the torus were proven in Zh, DLM, Miy, MT MT7, MTZ, TZ TZ8 and others. In TZ we established that the genus two partition function for the rank two free fermion VOSA is modular invariant with respect to the group G = SL2, Z) SL2, Z)) Z 2, i.e., f 2) g 2) γ = e2) γ f 2) g 2) f 2) g 2) where γ denotes G-action on the partition function, and e 2) f 2) γ is a multiplier system. Note that modular invariance of the genus two partition function, g 2) 99 Mathematics Subject Classification. 7B69, 30F0, 32A25, F03. Key words and phrases. Vertex operator superalgebras, intertwining operators, Riemann surfaces, automorphic forms, theta-functions, Jacobi product identities.

4 2 ALEXANDER ZUEVSKY can also inferred from using modular properties of the Riemann theta function and the Heisenberg genus two partition function described in MT. Similarly, we can also obtain TZ, TZ4 modular invariance for the two rank two free fermion n-point generating differentials G 2) f 2),n x g 2), y,..., x n, y n ), 5.4) as well as for all n-point function differentials F 2) f 2),n v g 2), z ),..., v n, z n )) 5.3) see subsection 5.2 below) and extend this result to higher genus TZ. In more complicated cases at genus g TZ6 we can explicitly prove automorphic properties of generating functions by using their explicit representations following from vertex operator algebra computations. In these notes we mention new identities for modular forms which are right hand sides of normalized with respect to Heisenberg vertex operator algebra partition functions) partition and generating functions in the case of selfsewing of the torus to forma a genus two Riemann surface as well as some higher genus cases. For main definition related to vertex operator algebras and computation of their correlation functions see B, K, BZF, DL, MT7, MTZ. Acknowledgement We would like to express our gratitude to the organizers of the Conference Representation Theory XIII, Dubrovnik, Croatia, 203. This text is based on the author s talk on the Conference. 2. Genus one identities 2.. Self-sewing of the torus. Let M be the Heisneberg vertex operator algebra K, M e α be its irreducible M-module for some α C. The torus partition function is given by Z α ) q) = Tr M e α q L0) /24) = q 2 α2 ηq), where ηq) = q /24 n qn ) is the Dedekind eta for modular parameter q. Thus Z α ) q) Z ) M τ) = q 2 α2, where Z ) M τ) = /ηq) is the genus one Heisenberg rank one partition function. Twisted torus n-point functio is defined in TZ4 by ) Z α ) u e β, z ;... ; u n e βn, z n ; q ) ) = Tr M e α Y q L0) u e β ), q... Y qn L0) u n e βn ), q n q L0) /24), 2.)

5 NEW MODULAR FORM IDENTITIES 3 for formal q i = e z i with i =,..., n. Since e β q M e α = M e α+β it follows that the n point function vanishes when i= β i 0. We next describe a natural generalization of previous results in MT,MTZ. Firstly, consider the n point functions for n highest weight vectors e β i, which we abbreviate below to e β i, for i =,..., n. Proposition. For n β i = 0 then i= Z α ) e β, z ;... ; e βn, z n ; q ) Z ) M τ) = q 2 α2 exp α ) n β i z i i= r<s n where z rs = z r z s and Kz, τ) is the genus one prime form. Kz rs, τ) βrβs,2.2) Similarly to MT, we may obtain a closed form for the general n point function 2.). In particular, we apply standard genus one Zhu recursion theory TZ to reduce 2.) to an explicit multiple of 2.2) to find Proposition 2. For n i= β i = 0 then Z ) α u e β, z ;... ; u n e β n, z n ; q ) Z ) α e β, z ;... ; e β n, zn ; q) = Q β,...,β n α u, z ;... u n, z n ; q), where Q β,...,β n α u, z ;... u n, z n ; q) is an explicit sum of elliptic and quasimodular forms see MT for details). In Propositions 4 and 5 of MTZ all torus orbifold n point functions for vectors in are computed by means of a generating function. In that analysis we made use of an explicit Zhu reduction formula developed for a VOSA with real grading. However, in the present case we do not have an intertwining Zhu reduction formula because of the absence of a commutator formula for interwiners DL. Instead, here we adopt an alternative approach for computing the intertwining n point functions by exploiting the bosonized formalism. Thus for example, we find Z ) f g e κ, w; e κ, 0; q) Z ) M τ) = = e Z 2πiµβ µ Z+α α ϑ κw, τ) β ) µ e κ, w; e κ, 0; q) Z ) M τ) Kw, τ) κ2, 2.3)

6 4 ALEXANDER ZUEVSKY for genus one theta series. More generally, we define a generating function for all n + 2 point functions by the following formal differential form G n ) f x g, y,..., x n, y n ) Z ) f ψ g +, x ; ψ, y ;... ; ψ +, x n ; ψ, y n ; e κ, w; e κ, 0; q) n i= dx 2 i dy 2 i, 2.4) for generators ψ ± = e ± alternatively inserted at x i, y i for i =,..., n. Proposition 3. The generating form 2.4) is given by G n ) f x g, y,..., x n, y n ) = det S κ, 2.5) Z ) f e g κ, w; e κ, 0; q) where S κ denotes the n n matrix with components S κ x i, y j ) for i, j =,..., n for Szegő kernel. Finally, we obtain the following generalization of Proposition 5 of MTZ concerning the generating properties of 2.4). Proposition 4. For a pair of square bracket mode twisted Fock vectors Ψ κ k, l 2 = e κ q ψ + k... ψ + k s ψ l 2... ψ l 2t2, Ψ κ k 2, l = e κ q ψ + k 2... ψ + k 2s2 ψ l... ψ l t, for p = s +s 2 = t +t 2 > 0, the generating function G m ) f g x, y,..., x m, y m ) for all torus orbifold intertwining n + 2 point functions is given by where Z ) f for some odd integer B and Ψ g κ k, l 2, w; Ψ κ k 2, l, 0; q) = ϵ det C ab k a, l b ), 2.6) Z ) f e g κ, w; e κ, 0; q) ϵ = ) t +s 2 )t p e iπbκs 2 t ), C ab k a, l b ) = C k, l ) C 2 k, l 2 ), C 2 k 2, l ) C 22 k 2, l 2 ) is a p p block matrix with components C ab k aia, l bjb ) for i a =,..., s a and j b =,..., t b for a, b =, 2 for S κ expansion coefficients TZ4.

7 NEW MODULAR FORM IDENTITIES 5 3. Genus two identities In TZ, Z we have proposed a way to derive higher genus generalizations of the classical identities for modular forms. In particular, in TZ we gave a genus two example of the Jacobi product identity. 3.. The self-sewing partition function. The partition or 0 point function is expressible in terms of the basic genus one twisted 2-point function 2.3) and the infinite Szegő moment matrix T = T τ, w, ρ θ, θ 2, ϕ, κ) of TZ4 as follows: Theorem. The genus two continuous orbifold partition function for on a Riemann surface in the ρ-sewing formalism is convergent on D ρ and is given by f τ, w, ρ) g = e 2iπβ2κ e iπb ρ) 2 κ2 det I T ). 3.) Z ) f e g κ, w; e κ, 0; q) 3.2. An identity for the Szegő kernel. The source genus two Szegő kernel on the torus is defined by TZ2 α ) κ ϑ x y + κw, τ) β S κ x, y) = ϑ x w, τ)ϑ y, τ) ϑ x, τ)ϑ y w, τ) α ϑ κw, τ) Kx y, τ) β dx 2 dy 2,3.2) for κ 2 with a different expression when κ = 2 given in TZ2). Recall also the definition of the Szegő kernel moments TZ2, TZ4, G ab k, l) = ρ 2 k a+l b ) C ab k, l), 3.3) h a x, k) = ρ 2 k a 2 ) d a x, k), 3.4) h a y, k) = ρ 2 ka 2 ) d a y, k), 3.5) with associated infinite matrix G = G ab k, l)) and row vectors hx) = h a x, k)) and hy) = h a y, k)). Here d a y, k) are half-order differentials d a x, k) = ya ka S κ x, y a )dy 2 a, 3.6) 2πi d a y, k) = 2πi C a y a ) C a x a ) x ka a S κ x a, y)dx 2 a, 3.7) and C ab k, l) is the infinite block moment matrix C ab k, l) = 2πi) 2 x a ) ka y b ) l b S κ x a, y b ) dx 2 a dy 2 b. 3.8) C a x a ) C b y b )

8 6 ALEXANDER ZUEVSKY Thus the genus two Szegő kernel on the self-sewn torus is given by S 2) x, y) = S κ x, y) + ξhx)d θ I T ) h T y), 3.9) where h T y) denotes the transpose of the infinite row vector hy). Using 3.9) we define matrices T = ξgd θ 2, 3.0) D θ 2 θ k, l) = 2 0 δk, l). 3.) 0 θ 2 S 2) = S 2) x i, y j ) ), S κ = S κ x i, y j )), 3.2) H = hx i )) k, a)), H T = hy i ) ) l, b) ) T. S 2) and S κ are finite matrices indexed by i, j =,..., n; H is semi-infinite with n rows indexed by i and columns indexed by k and a =, 2 and H T is semi-infinite with rows indexed by l and b =, 2 and with n columns indexed by j. Similarly to Proposition 3 in TZ2 we have Proposition 5. det Sκ ξhd θ 2 H T I T with T, D θ 2 of 3.0) and 3.). = det S 2) deti T ), 3.3) 3.3. The genus two n-point generating form. Similarly to the genus one situation 2.4), we define a genus two continuous orbifold generating differential form by G 2) n Then we have f g x, y,..., x n, y n ) = f ψ g +, x ; ψ, y ;..., ψ +, x n ; ψ, y n ; τ, w, ρ) Theorem 2. The generating form 3.4) is given by G n 2) f x g, y,..., x n, y n ) = det S 2), f τ, w, ρ) g n i= dx 2 i dy 2 i. 3.4) 3.5)

9 NEW MODULAR FORM IDENTITIES 7 for n n matrix S 2) of 3.2) and fg τ, w, ρ) is the genus two twisted partition function 3.). Normalized relative to the genus two partition function, the 2-point function for ψ + and ψ is thus given by the Szegő kernel and more generally, the generating function is a Szegő kernel determinant. This agrees with the assumed form in R or as found by string theory methods using a Schottky parametrization in DVPFHLS and in the genus two ϵ-sewing scheme TZ Genus two Heisenberg partition function. We may compute the genus two twisted partition and generating functions in an alternative way by use of a bosonic basis u e ν +κ for u M and ν Z + κ. In particular, we can immediately exploit and extend the results for lattice VOAs in MT3. Let us firstly recall from MT2, MT3 the following definitions: P 2 τ, z) = τ, z) + E 2 τ) = z 2 + k )E k τ)z k 2, 3.6) for Weierstrass function τ, z) and Eisenstein series E k τ). We define for k 2 and for k, l k=2 P k+ τ, z) = k zp k τ, z), Ck, l, τ) = ) k+ k + l )! k )!l )! E k+lτ), Dk, l, τ, z) = ) k+ k + l )! k )!l )! P k+lτ, z), R ab k, l) = ρk+l)/2 kl Dk, l, τ, w) Ck, l, τ) Ck, l, τ) Dl, k, τ, w) Theorems 5. and 5.6 of MT3 tell us that: Theorem 3. The genus two normalized partition function for the Heisenberg VOA M in the self-sewing scheme is where M τ, w, ρ) is holomorphic on Dρ.. M τ, w, ρ) Z ) M τ) = det R)) /2, 3.7)

10 8 ALEXANDER ZUEVSKY 3.5. Lattice VOA. Theorem 6. of MT3 concerns the genus two partition function for a lattice VOA. We obtain a natural generalization for a twisted genus two partition function defined by µ,ντ, w, ρ) = Z µ ) Ψ ν M e ν Ψν, w; Ψ ν, 0; q ), 3.8) for µ, ν C where the sum is taken over a M e ν basis, Ψ ν is the square bracket dual with respect to a non-generate form, see TZ, TZ4) and the summand is a genus one 2-point function 2.) for M. We find that the normalized partition function is given by the theta-function. Theorem 4. µ,ντ, w, ρ) = eiπ M τ, w, ρ) where Ω 2) is the genus two period matrix. ) µ 2 Ω 2) +2µνΩ2) 2 +ν2 Ω 2) 22, 3.9) We are now able to compute the genus two twisted partition function 3.) by use of a bosonic basis to obtain Theorem 5. The genus two normalized twisted partition function for on a Riemann surface in the self-sewing formalism is given by f τ, w, ρ) g α ϑ Ω β 2)), 3.20) = M τ, w, ρ) for genus two Riemann theta function with some characteristics α, α 2 = κ,β,β 2 and where M τ, w, ρ) is the genus two Heisenberg partition function Generalizations of classical identities for modular forms. In TZ we proved a generalization of the classical Jacobi identity suitable for genus two. By comparison of the direct computation via algebraic technique MT MT7, TZ TZ8 of the partition and n-point functions with the bosonization technique MT, we obtain TZ,TZ4 the genus two version of the Jacobi product and Frobenius Fay trisecant identities. Comparing with the fermionic expression 3.) and noting that e iπb ρ ) Z) κ2 2 f g e κ, w; e κ, 0; q) Z ) M τ) = e iπb ) ρ 2 κ2 Kw, τ) 2 ϑ ) α κw, τ), β 3.2) we obtain a genus two analogue of the classical Jacobi triple product identity for the elliptic theta function separate to a similar identity shown in TZ) as follows:

11 NEW MODULAR FORM IDENTITIES 9 Theorem 6. The ratio of genus two and genus one Riemann theta functions on D ρ is given by α Ω ϑ 2) 2) ) β = e 2iπβ 2κ e iπb ) ρ 2 κ2 ϑ ) α Kw, τ) κw, τ) 2 det I T ) det I R) 2. β As suggested in R, we also remark that Fay s Trisecant Identity at genus two follows by comparing the generating function G n 2) fg x, y,..., x n, y n ) of 3.4) to its form in terms of a bosonic basis. Thus we see that even at the partition function level, automorphic objects in particular, ratios of θ-functions) can be related to certain functions expressed via determinants MT4, MTZ, TZ, TZ3 containing data coefficients in expansions of regular parts of ordinary or half-order e.g., Szegő kernel) differentials) coming from sewing of lower genus Riemann surfaces. These identities reflect automorphic differential outcome of the boson fermion correspondence. 4. Higher genus identities 4.. The partition functions for Heisenberg VOA. Let M Ω 2) ) be the genus two partition function for the rank one free Heisenberg VOA M MT4. It has been computed in MT4, MT5 M Ω 2) ) ) /2. Z ) M τ ) Z ) M τ 2) = det I A ) A) 2 4.) Here A ) a for a =, 2 are infinite matrices with components indexed by k, l MT3, MT4, A a k, l, τ a, ϵ) = ϵ k+l)/2 )k+ k + l )! klk )!l )! E k+l τ a ), 4.2) where ϵ is Riemann surface sewing scheme parameter Y, MT3, and E n τ) are the Eisenstein series S. Note that E n τ) appear explicitly as a result of vertex operator algebra computation. At genus g, in two curves sewing formalism, we obtain TZ6 similar formula for the rank two Heisenberg VOA module Z g) M 2 Z g ) M 2 Z g 2) M 2 ) = det I A g,g 2 ), 4.3) where Z g) M 2 is a non-vanishing holomorphic function on the sewing domain, and is automorphic with respect to SL2, Z)... SL2, Z) Sp2g, Z) with

12 0 ALEXANDER ZUEVSKY automorphy factor detcω g) + D), and a multiplier system. Here A g,g 2 ) is an infinite moment matrix containing genus g sewing data, and Ω g) is the genus g period matrix. Automorphic properties of the above mentioned genus two partition functions follow TZ,TZ4 from the structure of the determinant and lower genus partition functions in 4.3). An alternative construction of the genus g partition function for the Heisenberg vertex operator algebra is performed in TZ5, TZ8 via the Schottky parameterization The free fermion partition functions. The genus one free fermion twisted partition function is given by K, MTZ Z ) f τ) = q α2 /2 /24 θ q l +α) 2 θq l α) 2, 4.4) g l which represents a form of the genus one Jacobi triple identity K. It is clearly a modular invariant object. This partition function can be also obtained TZ2, TZ4 in the form analogous to 4.3) as a determinant of an infinite matrix containing moments of corresponding genus zero differentials when considering a self-sewing of the sphere. Genus g formula for the partition function can be also conjectured TZ6: Z g) f g) Ω g) ) g g) Z g ) f g ) Ω ) g ) Z g 2) f g 2 ) Ω ) = det g 2 ) g g ) g g 2 ) I Q g,g 2 ) ), 4.5) where g = g + g 2, Q g,g 2 ) is an infinite matrix, and Ω g), Ω g ), Ω g 2) are period matrices for a sewn genus g curve and lower genus curves of genuses g, g 2. In particular in TZ we proved that the genus two twisted partition function for the rank two fermion VOSA is given by Z ) f 2) f g g 2) τ, τ 2, ϵ) τ ) Z ) f2 g 2 τ 2 ) = det I Q,)). 4.6) 5. Differential side: Ward identities Another group of identities are used to prove a autmorphic differential correspondence Z2, Z3. Let X be a compact connected simply connected algebraic cirve. As it is explained in F05, BZF, BD2, twisted D-modules on the moduli stack Bun G of G-bundles over X arise in conformal field theories as sheaves of conformal blocks on the moduli space M g,n of pointed complex curves of genus g. These D-modules encode chiral correlation functions of a model. For CFT s with Lie algebraic symmetries F05, correlation functions satisfy Ward identities which involve BPZ, FS projectively flat connections on a bundle of conformal

13 NEW MODULAR FORM IDENTITIES blocks. For sheaves of conformal blocks on Bun G, one obtains a projective connection in the bundle or the structure of a twisted D-module on the sheaf of conformal blocks. Based on algebraic properties of VOA modules, we show TZ3 that certain normalized i.e., divided by appropriate Heisenberg VOA partition function, e.g., 5.), 5.2) and 5.5), see subsections 5. and 5.2) generating differentials for n-point functions can be represented as differential operators acting on functions with nice automorphic properties such as θ-functions). Then such differential operators turn out to be connections in bundles over Riemann surfaces. This construction can be extended to differentials associated to arbitrary n-point functions e.g., 5.3), see subsection 5.2) even on higher genus Riemann surfaces. This construction represents another side of the geometric correspondence for vertex algebras. 5.. The partition functions for free fermionic VOSA. Let be a free fermionic vertex operator superalgebra V module MTZ. The normalized form of the genus one twisted partition function see 4.4) in subsection 4.2) for the rank two free fermionic vertex operator superalgebra see, e.g., K, MTZ) can be also expressed as Z ) f g τ) Z ) M τ) = e 2πiαβ ϑ ) α τ), 5.) β for the torus theta function with characteristics α, β. The genus two normalized twisted partition function 4.6) can be computed in the bosonization formalism to obtain MT, TZ f 2) Ω 2) ) g 2) M Ω 2) ) = e 2πiα2) β 2) ϑ 2) α 2) β 2) Ω 2)), 5.2) for the genus two Riemann theta function with characteristics α 2) = α, α 2 ), β 2) = β, β 2 ) where M Ω 2) ) is the genus two Heisenberg partition function. Natural formulas relating the free fermion normalized twisted partition functions and corresponding theta-functions are also available at higher genus TZ6, TZ Free fermion genus two twisted generating n-point functions. For v i V, and points z i, i =,... n, on a genus g curve, let us introduce TZ

14 2 ALEXANDER ZUEVSKY the following differentials: F g) f g),n g g) v, z ),..., v n, z n )) F g),n f g) g g) v, z ),..., v n, z n )) n i= dz wtv i) i, 5.3) where wtv i ) are weights of the states v i, K. In TZ we proved that for g = 2 the generating differential for all rank two free fermion VOSA n-point functions is given by the differential G 2),n f 2) g 2) F 2),n x, y,..., x n, y n ) 5.4) f 2) g 2) ψ +, x ), ψ, y )..., ψ +, x n ), ψ, y n )) n i,j= dx /2 i dy /2 j, where F 2) f 2),n ψ +, x g 2) ), ψ, y )..., ψ +, x n ), ψ, y n )) is the genus two 2n-point function in coordinates x i, y i, i =,..., n; ψ ± are generating states, and f 2), g 2) being vectors containing pairs of V -twisting automorphisms. Then its normalized form is given by G 2),n f g x, y,..., x n, y n ) M where elements of the matrix S 2) = genus two Szegő kernels, i.e., S 2) θ 2) x i, y j ) = K 2) and K 2) θ ϕ ϕ 2) = e 2πiα β det S 2) ϑ 2) α β Ω 2)), 5.5) S 2) θ ϕ x i, y j ), i, j =,..., n are θ 2) ϕ 2) x i, y j ) dx /2 i dy /2 j, 5.6) z, z 2 ) is the functional part of the genus two prime form Fay Genus two one-point Virasoro vector function. In TZ see also TZ4) we gave an example of a genus two generating function relation containing a flat connection over a Riemann surface. In TZ we computed the one-point function for the Virasoro vector ω = 2a a, where a is the Heisenberg element in the bosonized version of the rank two free fermion VOSA on a genus two Riemann surface obtained in the sewing procedure of two tori.

15 NEW MODULAR FORM IDENTITIES 3 The Virasoro vector can be written as ω = 2 ψ+ 2ψ +ψ 2ψ + ) for the rank two free fermion generators ψ ± in the Zhu equivalent VOSA formulation. Let w, z be two distinct points on a genus two Riemann surface. To compute a one point function we use the normalized generating form Then we find TZ3 w F 2),2 G 2), f 2) g 2) w, z) f 2) g 2) = S 2) f 2) g 2) ψ +, w; ψ, z) f 2) g 2) = F = w z) 2 + 2) f 2), θ 2) ϕ 2) w, z). w F 2) f 2), Y ψ +, w zψ, z) g 2) f 2) g 2) ψ + 2ψ, z) g 2) +..., f 2) g 2) and similarly for z. Due to 5.6) it follows that the Virasoro vector one-point differential form is F 2) f 2), ω, z) ) g 2) f 2) g 2) = dz 2 lim w z w z) w z ) K 2) w, z) An alternative expression TZ for this is given below in 5.8).. 5.7) 5.4. Flat connections. Introduce the differential operator U, MT, TZ D g) = ν g) i x) ν g) j x). 2πi i j g Ω g) ij It includes holomorphic -forms ν g) i as well as derivative with respect to the genus g Riemann surface period matrix Ω g). The genus g projective connection s g) is defined by Gu s g) x) = 6 lim x y ω g) x, y) dxdy ) x y) 2, where ω g) is the meromorphic differential of the second kind on a Riemann surface. The projective connection s g) x) is not a global 2-form but rather transforms under a general conformal transformation x ϕx) as follows s g) ϕx)) = s g) x) {ϕ; x}dx 2, where {ϕ; x} = ϕ ϕ 3 2 derivative. ϕ ϕ ) 2 is the Schwarzian

16 4 ALEXANDER ZUEVSKY 5.5. Genus two Ward identity. It turns out that the Virasoro one-point differential form on genus two Riemann surface can be represented by means of the Ward identity as a flat connection in certain bundle. Let M be the genus two partition function for the rank one Heisenberg VOA M see explicit formula 4.)). Using results of MT5 we proved in TZ that the Virasoro one-point normalized differential form for the rank two fermion VOSA satisfies the genus two Ward identity F 2) f 2), ω, z) g 2) M = e 2πiα2) β 2) D 2) + 2 s2) z) ϑ 2) α 2) β 2) Ω 2)). 5.8) Here the expression for the normalized genus two one-point differential form is represented as the action of a differential operator on an automorphic function. One can generalize TZ9 5.8) to genus g formula. Note that the Ward identities coherent with the above formula appeared previously in EO, KNTY. In contrast to the pure algebraic geometry FS, TUY, U, KNTY and physics approaches BPZ, FS, the Ward identity shows up from algebraic properties of corresponding vertex algebra. The Ward identities were also used in F to establish a version of the geometric Langlands correspondence for Kac Moody Lie algebras at critical level. One can also represent the genus two free fermion generating differentials 5.4) see subsection 5.2) using projective connections TZ. In particular, due to definitions of the genus g Szegő kernel and prime form Fay, it is possible to rewrite more general formula for the normalized twisted generating differential 5.5) for all n-point free fermion functions as a quite complicated operator acting on a theta function in a form of 5.8). Similar Ward identities are also available at higher genus cases GT, TZ9. Another interesting way to represent the formula 5.5) and thus 5.8), 5.7), see subsection 5.3) is by using an alternative formulation of the prime form given in Fay2. This point will be discussed in Z2. Computations involving boson fermion correspondence compared to direct computations of generating functions give MTZ us various higher genus generalizations of the Frobenius Fay identities Fay. Using the Ward identity form e.g., 5.8)) for generating differentials, one is also able to deduce projective connection form for those identities by comparison with pure vertex operator algebra computations. In examples given above we see that for fermionic VOSAs there is some sense to write expressions for the partition and generating functions in normalized form sinse then differential projective connection structure) is clear. On the other hand, unnormalized form reflects the automorphic side of the correspondence.

17 NEW MODULAR FORM IDENTITIES 5 Another interesting direction in generating identities for modular form is related to Jacobi forms via vertex operator algebras KM, KM2, Kr. We find in this way analogues of the Frobenius Fay identities Z4 for Jacobi forms. References B Borcherds, R.E.: Vertex algebras, Kac-Moody algebras and the Monster, Proc. Nat. Acad. Sc., ), BPZ Belavin A., Polyakov A., Zamolodchikov A. Infinite conformal symmetries in two-dimensional quantum field theory, Nucl. Phys. B , 984. BZF Ben-Zvi, D., Frenkel, E.: Vertex algebras and algebraic curves. Mathematical Surveys and Monographs, 88. American Mathematical Society, Providence, RI, 200. Second ed.: BD Beilinson, A., Drinfeld, V.: Quantization of Hitchins integrable system and Hecke eigensheaves. Preprint, 997. BD2 Beilinson, A., Drinfeld, V.: Chiral algebras, Cooloquim Publications Vol 5 Amer. Math. Soc., Providence, RI, DG Dolan, L., Goddard, P.: Current algebra on the torus. Comm. Math. Phys. 285 no., , DL Dong, C., Lepowsky, J.: Generalized Vertex Algebras and Relative Vertex Operators, Progress in Mathematics 2, Birkhäuser, Boston, MA, 993. DLM Dong, C., Li, H., Mason G.: Modular-invariance of trace functions in orbifold theory and generalized moonshine, Comm. Math. Phys. 24, 56, DVPFHLS Di Vecchia, P., Pezzella, F., Frau, M., Hornfeck, K., Lerda, A., Sciuto, S.: N point g loop vertex for a free fermionic theory with arbitrary spin. Nucl. Phys. B , 990. EO Eguchi, T., Ooguri, H.: Conformal and current algebras on a general Riemann surface, Nucl. Phys. B282, , 987. Fay Fay, J.D.: Theta Functions on Riemann Surfaces. Lecture Notes in Mathematics, Vol Berlin-New York: Springer-Verlag, 973. Fay2 Fay, J.D.: Kernel functions, analytic torsion, and moduli spaces. Mem. Amer. Math. Soc. 96 no. 464, 992. FS Freidan, D., Shenker, S.: The analytic geometry of two dimensional conformal field theory, Nucl. Phys. B F05 Frenkel, E.: Lectures on the Langlands Program and Conformal Field Theory Frontiers in number theory, physics, and geometry. II, , Springer, Berlin, F Frenkel, E.: Langlands Correspondence for Loop Groups, Cambridge Studies in Advanced Mathematics, 03, Cambridge University Press, FHL Frenkel, I. B., Huang, Y.-Z., Lepowsky, J., On axiomatic approaches to vertex operator algebras and modules, preprint, 989; Memoirs Amer. Math. Soc. 04, 993. FLM Frenkel, I. B., Lepowsky J., Meurman A., Vertex Operator Algebras and the Monster, Pure and Appl. Math., Vol. 34, Academic Press, Boston, 988. GT Gilroy, T., Tuite, M.P.: To appear, 204. Gu Gunning, R.C. : Lectures on Riemann Surfaces, Princeton Univ. Press, Princeton, 966. Hu Huang, Y.-Z.: Two-dimensional conformal geometry and vertex operator algebras. Progress in Mathematics, 48. Birkhauser Boston, Inc., Boston, MA, 997. HL Huang, Y.-Z., Lepowsky, J.: Tensor categories and the mathematics of rational and logarithmic conformal field theory, arxiv: , 203.

18 6 ALEXANDER ZUEVSKY K Kac, V.: Vertex Operator Algebras for Beginners, University Lecture Series 0, AMS, Providence 998. KM Krauel, M., Mason G.: Vertex operator algebras and weak Jacobi forms. Internat. J. Math. 23, no. 6, , 202. KM2 Krauel, M., Mason G.: Jacobi trace functions in the theory of vertex operator algebras. arxiv: , 203. KNTY Kawamoto, N., Namikawa, Y., Tsuchiya, A., Yamada, Y.: Geometric realization of conformal field theory on Riemann surfaces, Commun. Math. Phys. 6, , 988. Kr Krauel, M.: A Jacobi theta series and its transformation laws, arxiv: , 203. KW Kac, V., Wakimoto, M.: Integrable highest weight modules over affine superalgebras and Appell s function. Comm. Math. Phys. 25 no. 3, , 200. Miy Miyamoto, M.: A modular invariance on the theta functions defined on vertex operator algebras, Duke Math. J. Vol. 0 2) , MT Mason, G., Tuite, M.P.: Torus chiral n-point functions for free boson and lattice vertex operator algebras. Comm. Math. Phys. 235, no., 47 68, MT2 Mason, G., Tuite, M.P.: Chiral algebras and partition functions. Lie algebras, vertex operator algebras and their applications, 40 40, Contemp. Math., 442, Amer. Math. Soc., Providence, RI, MT3 Mason, G., Tuite, M.P.: On genus two Riemann surfaces formed from sewn tori. Comm. Math. Phys. 270 no. 3, , MT4 Mason, G., Tuite, M.P.: The Genus Two Partition Function for Free Bosonic and Lattice Vertex Operator Algebras. arxiv: , MT5 Mason, G., Tuite, M.P.: Free bosonic vertex operator algebras on genus two Riemann surfaces I, Commun. Math. Phys. 300, , 200. MT6 Mason, G., Tuite, M.P.: Free bosonic vertex operator algebras on genus two Riemann surfaces II, To appear in Proceedings of Conformal Field Theory, Automorphic Forms and Related Topics, Heidelberg, arxiv:.2264, 20. MT7 Mason, G., Tuite, M.P.: Vertex operators and modular forms. A window into zeta and modular physics, 83278, Math. Sci. Res. Inst. Publ., 57, Cambridge Univ. Press, Cambridge, 200. MTZ Mason, G., Tuite, M. P., Zuevsky, A.: Torus n-point functions for R-graded vertex operator superalgebras and continuous fermion orbifolds. Commun. Math. Phys. 283 no. 2, , R Raina, A.K.: Fay s trisecant identity and conformal field theory. Comm. Math. Phys. 22, , 989. Ra Raina, A.K.: Fays trisecant identity and conformal field theory. Commun. Math. Phys. 22, , 989. Ra2 Raina, A.K.: An algebraic geometry study of the b-c system with arbitrary twist fields and arbitrary statistics. Commun. Math. Phys. 40, , 99. RaS Raina, A.K., Sen, S.: Grassmannians, multiplicative ward identities and thetafunction identities. Phys. Lett. B203, , 988. S Serre, J.-P.: A course in arithmetic, Springer-Verlag, Berlin, 978. TUY Tsuchiya A., Ueno K., Yamada Y.: Conformal field theory on universal family of stable curves with gauge symmetries, in: Advanced Studies in Pure Math., Vol. 9, Kinokuniya Company Ltd., Tokyo, , 989. TZ Tuite, M. P., Zuevsky, A.: Genus two partition and correlation functions for fermionic vertex operator superalgebras I, Commun. Math. Phys., 306 no. 2, , 20.

19 Powered by TCPDF NEW MODULAR FORM IDENTITIES 7 TZ2 Tuite, M. P., Zuevsky, A. The Szegő kernel on a sewn Riemann surface. Commun. Math. Phys., 306 no.3, , 20. TZ3 Tuite, M. P., Zuevsky, A.: A generalized vertex oerator algebra for Heisenberg intertwiners. J. Pure and Applied Alg., 26 no. 6, , 202. TZ4 Tuite, M. P., Zuevsky, A.: Genus two partition and correlation functions for fermionic vertex operator superalgebras II. To appear in Commun. Math. Phys., TZ5 Tuite, M. P., Zuevsky, A.: The bosonic vertex operator algebra on a genus g Riemann surface. RIMS Kokyuroko , 20. TZ6 Tuite, M. P., Zuevsky, A.: The Heisenberg vertex operator algebra on a genus g Riemann surface. To appear, 204. TZ7 Tuite, M. P., Zuevsky, A.: Torus intertwined correlation functions for fermionic vertex operator superalgebras. To appear, 204. TZ8 Tuite, M. P., Zuevsky, A.: Higher genus partition and n-point functions for vertex operator algebras via Schottky uniformization. To appear, 204. TZ9 Tuite, M. P., Zuevsky, A.: Higher genus free fermionic partition and n-point functions. To appear, 204. U Ueno, K.: Introduction to conformal field theory with gauge symmetries. Geometry and Physics - Proceedings of the Conference at Aarhus Univeristy, Aaarhus, Denmark, New York: Marcel Dekker, 997. Y Yamada, A.: Precise variational formulas for abelian differentials. Kodai Math. J. 3, 4 43, 980. W Witten, E: Quantum field theory, Grassmannians, and algebraic curves, Comm. Math. Phys , 988. Zh Zhu, Y.: Modular invariance of characters of vertex operator algebras: J. Amer. Math. Soc , 996. Z Zuevsky, A.: Twisted correlation functions on self-sewn Riemann surfaces via generalized vertex algebra of intertwiners. Springer vol. on Conformal Field Theory, Automorphic Forms and Related Topics, Heidelberg, 20; Preprint 2-63, Max-Planck-Inst. fur Mathematik, Bonn, Z0 Zuevsky, A.: Identities for modular forms generated by vertex algebras. Miskolc Mathematical Notes, 4, No. 2, pp , 203. Z Zuevsky, A.: To appear, 204. Z2 Zuevsky, A.: Geometric correspondence for vertex algebras. To appear, 204. Z3 Zuevsky, A.: Towards automorphic to geometric correspondence for vertex algebras. Submitted, 204. Z3 Zuevsky, A.: Current algebras at higher genus. To appear, 204. Z4 Zuevsky, A.: Frobenius Fay identities for Jacobi forms. To appear, 204. Institute of Mathematics, Czech Academy of Sciences, Prague address: zuevsky@mpim-bonn.mpg.de