Abstract
In consideration of the continuous orbifold partition function and a generating function for all -point correlation functions for the rank two free fermion vertex operator superalgebra on the self-sewing torus, we introduce the twisted version of Frobenius identity.
1. Torus Intertwining -Point Functions
1.1. Torus Intertwining -Point Functions for
In this section we recall the computation [1] of the torus intertwining -point functions for . Here we recall several constructions from [1]. For the notions of free fermionic VOSA and its twisted modules see Appendix. Define the square-bracket vertex operator [2] , for and where . As in [2] we find that and are isomorphic generalized VOAs with , a new conformal vector with vertex operator . We let denote the weight of an homogeneous vector . Let be an irreducible -module for some with torus partition function where is the Dedekind eta-function for modular parameter . In general, we define the genus one intertwining -point correlation function on for vectors by for formal with . Since it follows that the -point function vanishes when .
In [1] we describe a natural generalization of previous results in [3, 4]. Firstly, consider the -point functions for highest weight vectors , which we abbreviate below to , for .
Proposition 1. For then where and is the genus one prime form (A.6).
It is a natural generalization of results developed in [3, 4]. In [1] generalizing results of [3] we obtain a closed form for the general -point function (2). In particular, due to (B.16), we may apply standard genus one Zhu recursion theory [2] to reduce (2) to an explicit multiple of (3) to find the following.
Proposition 2. For then where is an explicit sum of elliptic and quasi-modular forms introduced in [3].
1.2. Torus Intertwined -Point Functions for
Let , be commuting automorphisms of defined by where is the fermion number automorphism. We assume that so that and are of unit modulus.
We then consider in [1] torus orbifold intertwining -point functions for the -twisted module for defined by where for and . In particular, we find for genus one theta series (A.1).
More generally, we define in [1] a generating function for all -point functions (A.1) by the following formal differential form: for generators alternatively inserted at , for . Recall the notion of the Szegő kernel described in the Appendix. Then we prove in [1] the following.
Proposition 3. The generating form (8) is given by where denotes the matrix with components for for Szegő kernel (A.9).
Finally, we obtain in [1] the following generalization of Proposition of [4] concerning the generating properties of (8).
Proposition 4. is a generating function for all torus orbifold intertwining -point functions (A.1). In particular, for a pair of square-bracket mode twisted Fock vectors (B.10) for ; then where , for the odd integer fixed in (B.18) and is a block matrix with components for and for for expansion coefficients (A.11).
2. Twisted Frobenius Identity
In this subsection we finally derive the main formula of this paper, the twisted Frobenius identity. Recall the Frobenius identity (e.g., [4, 5]) where denotes the matrix with twisted Weierstrass function components of (A.8) for .
The trivial form of the twisted Frobenius identity follows from the consideration of the generating function for all -point functions
We obtain the following.
Proposition 5. The twisted two-point Frobenius identity is given by for (i.e., values of exponents in module elements), and correspondingly.
Proof. Consider the expression for the torus orbifold intertwining two-point function (11). Using (4) we derive the following: Thus we obtain the result.
Appendix
A. The Szegő Kernel on a Riemann Surface
Consider a compact connected Riemann surface of genus with canonical homology cycle bases , for . Let be a basis of holomorphic one-form with normalization and period matrix , the Siegel upper half plane. Define the theta function with real characteristics [5–7] for , and for . The Szegő kernel is defined for by [5, 8, 9] for and where , and for (where the factors are included for later convenience) and is the prime form [5, 6]. We use the convention for . The Szegő kernel is periodic in along the and cycles with multipliers and , respectively, and is a meromorphic -form (and is thus necessarily defined on a double-cover of the Riemann surface) satisfying where and .
A.1. The Genus Two Szegő Kernel in the -Formalism
Now we recall the construction of the Szegő kernel on a genus two Riemann surface in the -formalism [10]. The genus one prime form for and is where Let with . The genus one Szegő kernel is where where and are the periodicities of in on the standard and cycles, respectively.
It is convenient to define by (i.e., ) and introduce [1] for (with a different expression when given in [10]). We will assume throughout this paper. Note also that , the genus one Szegő kernel.
has an expansion in the neighborhood of the punctures at 0, in terms of local coordinates and , and and as follows [11]: where and for integer and . We may invert this to obtain the infinite block moment matrix
B. The Free Fermion VOSA and Its Twisted Modules
In this Appendix we recall [1] the notion of the free fermionic VOSA and its twisted modules.
B.1. The Free Fermion VOSA
We consider in this paper the rank two free fermion vertex operator superalgebra (VOSA) of central charge 1 (e.g., see [4, 12] for details). The weight space is spanned by , with vertex operator modes which satisfy the anticommutation relations is spanned by Fock vectors of the form for distinct and and of Virasoro weight with Virasoro vector The weight space is spanned by whose modes obey the Heisenberg commutation relations As is well known, we may decompose into irreducible -modules with eigenvalue so that , the lattice VOSA for the -lattice with trivial cocycle structure.
B.2. -Twisted -Modules and a Generalized VOA
generates continuous winding -automorphism for . In particular, the fermion number involution is . We define for all the following operators: Then we have [13] the following.
Proposition B.1. is a -twisted -module.
In Section of [11] an isomorphic construction is described whereby the -twisted module is determined by the action of the original vertex operators on a twisted vector space , where where is defined by In particular determines the -twisted grading operator Hence of (B.2) has -twisted Virasoro weight which is equal to the weight of the twisted Fock vector
In [11] we describe a generalized VOA with vector space formed as a direct sum of the Heisenberg VOA with all of its irreducible modules. is spanned by for all . The generalized vertex operators are Equation (B.12) reduces to the usual bosonized form of the vertex operators for for . A similar construction also appears in [14].
The generalized VOA leads to more general notions of locality, skew-symmetry, associativity, and commutivity than those for a VOSA as follows [1].
Proposition B.2. For , , , and for integer
It is convenient to define for formal parameter and where we choose, once and for all, an odd integer parametrizing the formal branch cut. Note some notational changes from [11]. Then generalized locality and skew-symmetry can be rewritten as
B.3. An Invariant Form on
In [11] we introduced an invariant bilinear form on associated with the Möbius map [8, 15, 16] for . We will later choose for the odd integer of (B.17). Thus we reformulate the sewing relationship as so that we get for .
Define the adjoint vertex operator A bilinear form on is said to be invariant if for all , , and we have Equation (B.22) reduces to the usual definition for a VOSA when [8, 16]. Choosing the normalization then on is symmetric, unique, and invertible with [11] The dual of the Fock vector with respect to , which we refer to as the -dual, is where denotes the integer part of [16]. Applying (B.21) and (B.24) it follows that of (B.10) has -dual
Conflicts of Interest
The author declares that they have no conflicts of interest.