**Applied Mathematics** Vol.4 No.10C(2013), Article ID:37612,20 pages DOI:10.4236/am.2013.410A3006

Higher Genus Characters for Vertex Operator Superalgebras on Sewn Riemann Surfaces

Max-Planck-Institut für Mathematik, Bonn, Germany

Email: zuevsky@mpim-bonn.mpg.de

Copyright © 2013 Alexander Zuevsky. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Received May 21, 2013; revised June 21, 2013; accepted June 28, 2013

**Keywords:** Vertex Operator Superalgebras; Intertwining Operators; Riemann Surfaces; Szegö Kernel; Modular Forms; Theta-Functions; Frobenius—Fay and Jacobi Product Identities

ABSTRACT

We review our recent results on computation of the higher genus characters for vertex operator superalgebras modules. The vertex operator formal parameters are associated to local parameters on Riemann surfaces formed in one of two schemes of (selfor tori- ) sewing of lower genus Riemann surfaces. For the free fermion vertex operator superalgebra we present a closed formula for the genus two continuous orbifold partition functions (in either sewings) in terms of an infinite dimensional determinant with entries arising from the original torus Szegö kernel. This partition function is holomorphic in the sewing parameters on a given suitable domain and possesses natural modular properties. Several higher genus generalizations of classical (including Fay’s and Jacobi triple product) identities show up in a natural way in the vertex operator algebra approach.

1. Vertex Operator Super Algebras

In this paper (based on several conference talks of the author) we review our recent results [1-5] on construction and computation of correlation functions of vertex operator superalgebras with a formal parameter associated to local coordinates on a self-sewn Riemann surface of genus which forms a genus surface. In particular, we review result presented in the papers [1-5] accomplished in collaboration with M. P. Tuite (National University of Ireland, Galway, Ireland).

A Vertex Operator Superalgebra (VOSA) [6-10] is a quadruple:

,

, is a superspace, is a linear map

so that for any vector (state) we have

, ,

, -parity.

The linear operators (modes) satisfy creativity

and lower truncation

conditions for and.

These axioms identity impy locality, associativity, commutation and skew-symmetry:

for and integers,

.

The vacuum vector is such that, , and the conformal vector satisfies

where form a Virasoro algebra for a central charge

satisfies the translation property

describes a grading with

, and

1.1. VOSA Modules

Definition 1 A -module for a VOSA is a pair, is a -graded vector space, , for all and.

,

for each,., and for the conformal vector

where,. The module vertex operators satisfy the Jacobi identity:

Recall that. The above axioms imply that satisfies the Virasoro algebra for the same central charge and that the translation property

1.2. Twisted Modules

We next define the notion of a twisted -module [8,11]. Let be a -automorphism, i.e., a linear map preserving and such that

for all. We assume that can be decomposed into -eigenspaces

where denotes the eigenspace of with eigenvalue.

Definition 2 A -twisted -module for a VOSA

is a pair, , ,

, for all, and., the vector space of ()-valued formal series in with arbitrary complex powers of. For

with, , sufficiently large.

,

where,. The -twisted vertex operators satisfy the twisted Jacobi identity:

for.

1.3. Creative Intertwining Operators

We define the notion of creative intertwining operators in [3]. Suppose we have a VOA with a -module.

Definition 3 A Creative Intertwining Vertex Operator for a VOA -module is defined by a linear map

for with modes; satisfies creativity

for and lower truncation

for, and. The intertwining vertex operators satisfy the Jacobi identity:

for all and.

These axioms imply that the intertwining vertex operators satisfy translation, locality, associativity, commutativity and skew-symmetry:

for, , and integers.

1.4. Example: Heisenberg Intertwiners

Consider the Heisenberg vertex operator algebra[10] generated by weight one normalized Heisenberg vector with modes obeying

.

In [3] we consider an extension of by its irreducible modules generated by a -valued continuous parameter automorphism.

We introduce an extra operator which is canonically conjugate to the zero mode, i.e.,

The state is created by the action of on the state. Using -conjugation and associativity properties, we explicitly construct in [3] the creative intertwining operators. We then prove:

Theorem 1 (Tuite-Z) The creative intertwining operators for are generated by -conjugation of vertex operators of. For a Heisenberg state,

The operators with some extra cocycle structure satisfy a natural extension from rational to complex parameters of the notion of a Generalized VOA as described by Dong and Lepowsky [7,12]. We then prove in [3].

Theorem 2 (Tuite-Z) satisfy the generalized Jacobi identity

for all.

1.5. Invariant Form for the Extended Heisenberg Algebra

The definitions of invariant forms [8,13] for a VOSA and its -twisted modules were given by Scheithauer [14] and in [2] correspondingly. A bilinear form on is said to be invariant if for all, , we have

We are interested in the Möbius map

associated with the sewing condition so thatwith. We prove in [3]

Theorem 3 (Tuite-Z) The invariant form on is symmetric, unique and invertible with

1.6. Rank Two Free Fermionic Vertex Operator Super Algebra

Consider the Vertex Operator Super Algebra (VOSA) generated by

for two vectors with modes satisfying anti-commutation relations

The VOSA vector space is a Fock space with basis vectors

of weight

where and with for all.

1.7. Rank Two Fermionic Vertex Operator Super Algebra

The conformal vector is

whose modes generate a Virasoro algebra of central charge 1. has -weight. The weight subspace of is, for normalized Heisenberg bosonic vector, the conformal vector, and the Virasoro grading operator are

2. Sewing of Riemann Surfaces

2.1. Basic Notions

For standard homology basis, with on a genus Riemann surface [15,16] consider the normalized differential of the second kind which is a symmetric meromorphic form with, has the form

A normalized basis of holomorphic 1-forms, the period matrix, and normalized differential of the third kind are given by

where, for,.

2.2. Period matrix

is symmetric with positive imaginary part i.e., the Siegel upper half plane. The canonical intersection form on cycles is preserved under the action of the symplectic group where

This induces the modular action on

2.3. Sewing Two Tori to Form a Genus Two Riemann Surface

Consider two oriented tori with

for for, the complex upper half plane. For the closed disk

is contained in provided

where

Introduce a sewing parameter and excise the disks and where

Identify the annular regions and via the sewing relation

gives a genus two Riemann surface parameterized by the domain

2.4. Torus Self-Sewing to Form a Genus Two Riemann Surface

In [1] we describe procedures of sewing Riemann surfaces [17]. Consider a self-sewing of the oriented torus, ,.

Define the annuli, centered at and of with local coordinates and respectively. We use the convention,

. Take the outer radius of to be

.

Introduce a complex parameter,. Take inner radius to be, with., must be sufficiently small to ensure that the disks do not intersect. Excise the disks

to form a twice-punctured surface

Identify the annular regions,

as a single region via the sewing relation

to form a compact genus two Riemann surface

parameterized by

3. Elliptic Functions

3.1. Weierstrass Function

The Weierstrass -function periodic in with periods and is

for,. We define for,

Then

.

has periodicities

3.2. Eisenstein Series

The Eisenstein series is equal to for odd, and for

where is the th Bernoulli number. If

then is a holomorphic modular form of weight on

for all, where. is a quasimodular form

having the exceptional transformation law.

3.3. The Theta Function

We recall the definition of the theta function with real characteristics [18]

for

,

for.

3.4. Twisted Elliptic Functions

Let denote a pair of modulus one complex parameters with for. For and we define “twisted” Weierstrass functions for [19,20]

for where means we omit if

.

converges absolutely and uniformly on compact subsets of the domain [20].

Lemma 1 (Mason-Tuite-Z) For,

is periodic in with periods and with multipliers and respectively.

3.5. Modular Properties of Twisted Weierstrass Functions

Define the standard left action of the modular group for

on with

We also define a left action of on

Then we obtain:

Theorem 4 (Mason-Tuite-Z) For we have

3.6. Twisted Eisenstein Series

We introduce twisted Eisenstein series for,

where means we omit if and where is the Bernoulli polynomial defined by

In particular

.

Note that

the standard Eisenstein series for even, whereas

for odd.

Theorem 5 (Mason-Tuite-Z) We have

Theorem 6 (Mason-Tuite-Z) For,

is a modular form of weight where

3.7. Twisted Elliptic Functions

In particular,

where

and

4. The Prime Form

There exists a (nonsingular and odd) character such that [18,21,22]

Let

be a holomorphic 1-form, and let denote the form of weight on the double cover of.

We define the prime form

The prime form is anti-symmetic,

and a holomorphic differential form of weight

onand has multipliers and along the

and cycles in [21]. The normalized differentials of the second and third kind can be expressed in terms of the prime form [18]

Conversely, we can also express the prime form in terms of by [22]

Torus Prime Form

The prime form on torus [18]

for and and where

.

We have

has periodicities

5. The Szegö Kernel

The Szegö Kernel [18,21,22] is defined by

with, , ,

, where is the genus prime form. The Szegö kernel has multipliers along the and cycles in given by and respectively and is a meromorphic -form on

where and.

Finally, we describe the modular invariance of the Szegö kernel under the symplectic group where we find [21]

with, ,

where denotes the diagonal elements of a matrix.

5.1. Modular Properties of the Szegö Kernel

Finally, we describe the modular invariance of the Szegö kernel under the symplectic group where we find [21]

where, for

where denotes the diagonal elements of a matrix.

5.2. Torus Szegö Kernel

On the torus the Szegö kernel for is

where

for

, and

for.

6. Structures on Constructed from Genus One Data

Yamada (1980) described how to compute the period matrix and other structures on a genus Riemann surface in terms of lower genus data.

6.1. on the Sewn Surface

can be determined from on each torus in Yamada’s sewing scheme [17,23]. For a torus the differential is

for Weierstrass function

and Eisenstein series for

vanishes for odd and is a weight modular form for. is a quasi-modular form. Expanding

we compute in the sewing scheme in terms of the following genus one data,

6.2. A Determinant and the Period Matrix

Consider the infinite matrix where is the infinite identity matrix and define by

as a formal power series in [23].

Theorem 7 (Mason-Tuite)

a) The infinite matrix

is convergent for.

b) is non-vanishing and holomorphic on.

Furthermore we may obtain an explicit formula for the genus two period matrix on [23].

Theorem 8 (Mason-Tuite) is holomorphic on and is given by

Here refers to the -entry of a matrix.

6.3. Genus Two Szegö Kernel on in the -Formalism

We may compute for in the sewing scheme in terms of the genus one data

is described in terms of the infinite matrix for

Theorem 9 (Tuite-Z)

a) The infinite matrix is convergent forb) is non-vanishing and holomorphic on.

6.4. Genus Two Szegö Kernel in the -Formalism

It is convenient to define by.

Then we prove [1] the following Theorem 10 (Tuite-Z) is holomorphic in for with

for where is defined for, by

with similar expression for for.

Let, for and integer. We introduce the moments for:

with associated infinite matrix. We define also half-order differentials

and let and, denote the infinite row vectors indexed by,. From the sewing relation we have

for, depending on the branch of the double cover of chosen. It is convenient to define

with an infinite diagonal matrix

Defining by the formal power series in

we prove in [1].

Theorem 11 (Tuite-Z)

a) is convergent forb) is non-vanishing and holomorphic in on.

Theorem 12 (Tuite-Z) is given by

7. Genus One Partition and n-Point Functions

7.1. The Torus Partition Function for a Heisenberg VOA

For a VOA of central charge define the genus one partition (trace or characteristic) function by

for the Heisenberg VOA commutation relations with modes

7.2. Genus One Twisted Graded Dimension

We define the genus one partition function for the VOSA by the supertrace

where.

More generally, we can construct a -twisted module for any automorphism generated by the Heisenberg state. We introduce the second automorphism and define the orbifold -twisted trace by

to find for,

7.3. Genus One Fermionic One-Point Functions

Each orbifold 1-point function can found from a generalized Zhu reduction formulas as a determinant.

Theorem 13 (Mason-Tuite-Z) For a Fock vector

where for

7.4. Genus One n-Point Functions for VOA

In general, we can define the genus one orbifold n-point function for by

Every orbifold n-point function can be computed using generalized Zhu reduction formulas in terms of a determinant with entries arising from the basic 2-point function for [19].

7.5. Zhu Reduction Formula

To reduce an -point function to a sum of -point functions we need:

The supertrace property

Borcherds commutation formula:

expansions for -functions:

Theorem 14 (Mason-Tuite-Z) For any we have

where is given by

7.6. General Genus One Fermionic n-Point Functions

The generating two-point function (for) is given by

Theorem 15 (Mason-Tuite-Z)

Theorem 16 (Mason-Tuite-Z) For Fock vectors

and

for and with

. Then for the corresponding -point functions are non-vanishing provided

and

where is certain parity factor. Here is the block matrix

with

for with and

for with and. is the sign of the permutation associated with the reordering of to the alternating ordering.

Furthermore, the -point function is an analytic function in and converges absolutely and uniformly on compact subsets of the domain.

7.7. Torus Intertwined n-Point Functions

As in ordinary (non-intertwined) case [2,19,20,24-29] we construct in [4] the partition and -point functions [30-39] for vertex operator algebra modules.

Let, , be VOSA automorphisms commuting with. For and the states we define the intertwined -point function [4] on the torus by

where, , ,

;, for variables associated to the local coordinates on the torus, and is dual for with respect to the invariant form on. The supertrace over a -module is defined by

For an element of a VOSA -twisted -module we introduce also the differential form

associated to the torus intertwined -point function.

7.8. Torus Intertwined Two-Point Function

The rank two free fermionic VOSA, [10]

is generated by with

The rank two free fermion VOSA intertwined torus

-point function is parameterized by, , and, [2, 4] where

for real valued, , ,.

For and,

we obtain [4] the basic intertwined two-point function on the torus

We then consider the differential form

associated to the torus intertwined -point function

with alternatively inserted states and states distributed on the resulting genus two Riemann surface at points. We then prove in [4].

Theorem 17 (Tuite-Z) For the rank two free fermion vertex operator superalgebra and for the generating form is given by

is the basic intertwined two-point function on the torus, and -matrix

with elements given by parts of the Szegö kernel.

8. Genus Two Partition and n-Point Functions

8.1. Genus Two Partition Function in -Formalism

We define the genus two partition function in the earlier sewing scheme in terms of data coming from the two tori, namely the set of 1-point functions for all. We assume that has a nondegenerate invariant bilinear form—the Li-Zamolodchikov metric. Define

The inner sum is taken over any basis and is dual to wrt to the Li-Zamolodchikov metric.

8.2. Genus Two Partition Function for the Heisenberg VOA

We can compute using a combinatorial-graphical technique based on the explicit Fock basis and recalling the infinite matrices.

Theorem 18 (Mason-Tuite) a) The genus two partition function for the rank one Heisenberg VOA is

;

b) is holomorphic on the domain;

c) is automorphic of weight;

d) has an infinite product formula.

8.3. Genus Two Fermionic Partition Function

Following the definition for the bosonic VOA we define for,

The inner sum is taken over any basis and is dual to with respect to the Li-Zamolodchikov square bracket metric. is the genus one orbifold 1-point function. Recall that the non-zero 1-point functions arise for Fock vectors

such that

,

The Li-Zamolodchikov metric dual to the Fock vector is

Recalling the infinite matrix we find

Theorem 19 (Tuite-Z)

a) The genus two orbifold partition function is

b) is holomorphic on the domain;

c) has natural modular properties under the action of.

8.4. Genus Two Partition and -Point Functions in -Formalism

Let be automorphisms, and be twisted -modules of a vertex operator superalgebra. For

with and,

, we define the genus two -point function [4] in the -formalism by

where, where (respectively)

denotes the pair, (respectively,). The sum is taken over any -basis.

In particular, we introduce the genus two partition function

where is the genus one intertwined two point function.

Remark 1 We can generalize the genus two -point function by introducing and computing the differential form associated to the torus -point function containing several intertwining operators in the supertrace as well as corresponding genus two -point functions.

Similar to the ordinary genus two case [2], we define the differential form [4] associated to the -point function on a sewn genus two Riemann surface for and, with

, ,

9. Generalizations of Classical Identities

9.1. Bosonization

The genus one orbifold partition function can be alternatively computed by decomposing the VOSA into Heisenberg modules indexed by integer eigenvalues, i.e., a lattice [26]. Let be lattice elements of the rank one even lattice, , and -cocycle. Then

Theorem 20 (Tuite-Mason)

Then ther genus one twisted partition function is given by

Comparing to the fermionic product formula we obtain the classical Jacobi triple product formula:

9.2. Genus Two Jacobi Triple Product Formula

The genus two partition function can similarly be computed in the bosonized formalism to obtain a genus two version of the Jacobi triple product formula for the genus two Riemann theta function [19]

for an appropriate character valued genus two Riemann theta function

Comparing with the fermionic result we thus find that on

9.3. Fay’s Trisecant Identity

Recall Fay’s trisecant identity [21]

for, , where is the Jacobian of the curve.

9.4. Bosonized Generating Function and Trisecant Identity

In a similar fashion we can compute the general - generating function in the bosonic setting to obtain:

Theorem 21 (Mason-Tuite-Z)

Comparing this to fermionic expressions for we obtain the classical Frobenius elliptic function version of generalized Fay’s trisecant identity

[21]:

Corollary 1 (Mason-Tuite-Z) For we have

9.5. Generalized Fay’s Trisecant Identity

We may generalize these identities using [26]. Consider the general lattice -point function. We have [19], For integers satisfying, we have

Comparing this to the expression for -point functions we obtain a new elliptic generalization of Fay’s trisecant identity:

Corollary 2 (Mason-Tuite-Z) For we have

Here is the block matrix

with the matrix

for and, and -functions are given by the expansion

10. Genus Two Intertwined Partition and n-Point Functions

In [4] we then prove:

Theorem 22 (Tuite-Z) Let be twisted -modules for the rank two free fermion vertex operator superalgebra. Let. Then the partition function on a genus two Riemann surface obtained in the -self-sewing formalism of the torus is a non-vanishing holomorphic function on given by

where is the intertwined module torus basic two-point function, and is some function.

We may similarly compute the genus two partition function in the -formalism for the original rank one fermion VOSA in which case we can only construct a -twisted module. Then we have [4] the following:

Corollary 3 (Tuite-Z) Let be the rank one free fermion vertex operator superalgebra and, , be automorphisms. Then the partition function for -module on a genus two Riemann surface obtained from formalism of a self-sewn torus is given by

where is the rank one fermion intertwined partition function on the original torus.

10.1. Genus Two Generating Form

In [4] we define matrices

and are finite matrices indexed by, for; is semi-infinite with rows indexed by and columns indexed by and and is semi-infinite with rows indexed by and and with columns indexed by. We then prove

Lemma 2 (Tuite-Z)

with,.

Introduce the differential form

associated to the rank two free fermion VOSA genus two -point function

with alternatively inserted states and states. The states are distributed on the genus two Riemann surface at points. Then we have Theorem 23 (Tuite-Z) All -point functions for rank two free fermion VOSA twisted modules on self-sewn torus are generated by the differential form

where the elements of the matrix

and is the genus two partition function.

10.2. Modular Invariance Properties of Intertwined Functions

Following the ordinary case [20,27,40] we would like to describe modular properties of genus two “intertwined” partition and -point generating functions. As in [27], consider with elements

is generated by, and with relations

.

We also define where with elements

Together these groups generate

.

From [27] we find that acts on the domain of as follows:

We then define [4] a group action of on the torus intertwined two-point function

for:

with the standard action and, and

and the torus multiplier, [1,19]. Then we have [4]

Theorem 24 (Tuite-Z) The torus intertwined twopoint function for the rank two free fermion VOSA is a modular form (up to multiplier) with respect to

where.

The action of the generators, and is given by [1]

.

In a similar way we may introduce the action of on the genus two partition function [4]

We may now describe the modular invariance of the genus two partition function for the rank two free fermion VOSA under the action of. Define a genus two multiplier for in terms of the genus one multiplier as follows

for the generator. We then find [4].

Theorem 25 (Tuite-Z) The genus two partition function for the rank two VOSA is modular invariant with respect to with the multiplier system, i.e.,

Finally, we can also obtain modular invariance for the generating form

for all genus two -point functions [4].

Theorem 26 (Tuite-Z)

is modular invariant with respect to with a multiplier.

11. Acknowledgements

The author would like to express his deep gratitude to the organizers of the Conference “Algebra, Combinatorics, Dynamics and Applications”, Belfast, UK, August 27-30, 2012.

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