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 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.