Higher Genus Characters for Vertex Operator Superalgebras on Sewn Riemann Surfaces ()
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 that
with
. 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
on
and 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 for
b) 
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 for
b) 
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.