# Twisted Frobenius Identities from Vertex Operator Superalgebras

Authors: Alexander Zuevsky
Journal: Advances in Mathematical Physics
 (2017)

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## Abstract

In consideration of the continuous orbifold partition function and a generating function for alln-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.


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## Body

## 1. Torus Intertwiningn-Point Functions

### 1.1. Torus Intertwiningn-Point Functions for M

In this section we recall the computation [1] of the torus intertwining n-point functions for M. 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] Y[u⊗eα,z]=Y(qzL(0)(u⊗eα),qz-1), for u⊗eα∈M and where qz=ez. As in [2] we find that (V,Y[,],1,ω~) and (V,Y(,),1,ω) are isomorphic generalized VOAs with ω~=ω-1/241, a new conformal vector with vertex operator Y[ω~,z]=∑n∈Z‍L[n]z-n-2. We let wt[u⊗eα]=wt[u]+1/2α2 denote the weight of an L[0] homogeneous vector u⊗eα. Let M⊗eα be an irreducible M-module for some α∈C with torus partition function (1)Zα1q=TrM⊗eαqL0-1/24=q1/2α2ηq,where η(q)=q1/24∏n≥1‍(1-qn) is the Dedekind eta-function for modular parameter q. In general, we define the genus one intertwining n-point correlation function on M⊗eα for n vectors u1⊗eβ1,…,vn⊗eβn∈M by (2)Zα1u1⊗eβ1,z1;…;un⊗eβn,zn;q=TrM⊗eαYq1L0u1⊗eβ1,q1⋯YqnL0un⊗eβn,qnqL0-1/24,for formal qi=ezi with i=1,…,n. Since eβq^M⊗eα=M⊗eα+β it follows that the n-point function vanishes when ∑i=1‍βi≠0.In [1] we describe a natural generalization of previous results in [3, 4]. Firstly, consider the n-point functions for n highest weight vectors 1⊗eβi, which we abbreviate below to eβi, for i=1,…,n.Proposition 1.
For∑i=1n‍βi=0 then (3)Zα1eβ1,z1;…;eβn,zn;q=q1/2α2ητexp⁡α∑i=1nβizi∏1≤r<s≤nKzrs,τβrβs,where zrs=zr-zs and K(z,τ) 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 n-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∑i=1n‍ζi=0 then (4)Zα1u1⊗eζ1,z1;…;un⊗eζn,zn;q=Qαζ1,…,ζnu1,z1;…;un,zn;qZα1eζ1,z1;…;eζn,zn;q,where Qαζ1,…,ζn(u1,z1;…;un,zn;q) is an explicit sum of elliptic and quasi-modular forms introduced in [3].


### 1.2. Torus Intertwinedn-Point Functions for VZ

Letg1, h1 be commuting automorphisms of VZ defined by (5)σf1=e2πiβ1a0,σg1=e-2πiα1a0,where σ=eπia(0) is the fermion number automorphism. We assume that α1,β1∈R so that θ1=-e-2πiβ1 and ϕ1=-e2πiα1 are of unit modulus.We then consider in [1] torus orbifold intertwining n+2-point functions for the σg1-twisted module VZ+α1=⊕m∈ZM⊗em+α1 for VZ defined by (6)ZVZ1f1g1u1⊗em1,z1;…,un⊗emn,zn;v1⊗en1+κ,w;v2⊗en2-κ,0;q=TrVZ+α1σf1Yq1L0u1⊗em1,q1⋯YqnL0un⊗emn,qn·YqwL0v1⊗en1+κ,qwYv2⊗en2-κ,1qL0-1/24=∑μ∈Z+α1e2πiμβ1Zμ1u1⊗em1,z1;…;un⊗emn,zn;v1⊗en1+κ,w;v2⊗en2-κ,0;q,where mi,n1,andn2∈Z for κ∈(-1/2,1/2) and qw=ew. In particular, we find (7)ZVZ1f1g1eκ,w;e-κ,0;q=∑μ∈Z+α1e2πiμβ1Zμ1eκ,w;e-κ,0;q=1ηqϑα1β1κw,τKw,τκ2,for genus one theta series (A.1).More generally, we define in [1] a generating function for all n+2-point functions (A.1) by the following formal differential form: (8)Gn1f1g1x1,y1,…,xn,yn≡ZVZ1f1g1ψ+,x1;ψ-,y1;…;ψ+,xn;ψ-,yn;eκ,w;e-κ,0;q∏i=1ndxi1/2dyi1/2,for VZ generators ψ±=e±1 alternatively inserted at xi, yi for i=1,…,n. 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 (9)Gn1f1g1x1,y1,…,xn,yn=ZVZ1f1g1eκ,w;e-κ,0;qdet⁡Sκ,where Sκ denotes the n×n matrix with components Sκxi,yj for i,j=1,…,n for Szegő kernel (A.9).Finally, we obtain in [1] the following generalization of Proposition 15 of [4] concerning the generating properties of (8).Proposition 4.
G
m
(
1
)
f
1
g
1
(
x
1
,
y
1
,
…
,
x
m
,
y
m
) is a generating function for all torus orbifold intertwining n+2-point functions (A.1). In particular, for a pair of square-bracket mode twisted Fock vectors (B.10) (10)Ψκk1,l2=eκq^ψ+-k11⋯ψ+-k1s1ψ--l21⋯ψ--l2t21,Ψ-κk2,l1=e-κq^ψ+-k21⋯ψ+-k2s2ψ--l11⋯ψ--l1t11,for p=s1+s2=t1+t2>0; then (11)ZVZ1f1g1Ψκk1,l2,w;Ψ-κk2,l1,0;q=ϵZVZ1f1g1eκ,w;e-κ,0;qdet⁡Cabka,lb,where ϵ=(-1)(t1+s2)t2+1/2peiπBκ(s2-t1), for the odd integer B fixed in (B.18) and (12)Cabka,lb=C11k1,l1C12k1,l2C21k2,l1C22k2,l2is a p×p block matrix with components Cab(kaia,lbjb) for ia=1,…,sa and jb=1,…,tb for a,b=1,2 for Sκ expansion coefficients (A.11).


## 1.1. Torus Intertwiningn-Point Functions for M

In this section we recall the computation [1] of the torus intertwining n-point functions for M. 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] Y[u⊗eα,z]=Y(qzL(0)(u⊗eα),qz-1), for u⊗eα∈M and where qz=ez. As in [2] we find that (V,Y[,],1,ω~) and (V,Y(,),1,ω) are isomorphic generalized VOAs with ω~=ω-1/241, a new conformal vector with vertex operator Y[ω~,z]=∑n∈Z‍L[n]z-n-2. We let wt[u⊗eα]=wt[u]+1/2α2 denote the weight of an L[0] homogeneous vector u⊗eα. Let M⊗eα be an irreducible M-module for some α∈C with torus partition function (1)Zα1q=TrM⊗eαqL0-1/24=q1/2α2ηq,where η(q)=q1/24∏n≥1‍(1-qn) is the Dedekind eta-function for modular parameter q. In general, we define the genus one intertwining n-point correlation function on M⊗eα for n vectors u1⊗eβ1,…,vn⊗eβn∈M by (2)Zα1u1⊗eβ1,z1;…;un⊗eβn,zn;q=TrM⊗eαYq1L0u1⊗eβ1,q1⋯YqnL0un⊗eβn,qnqL0-1/24,for formal qi=ezi with i=1,…,n. Since eβq^M⊗eα=M⊗eα+β it follows that the n-point function vanishes when ∑i=1‍βi≠0.In [1] we describe a natural generalization of previous results in [3, 4]. Firstly, consider the n-point functions for n highest weight vectors 1⊗eβi, which we abbreviate below to eβi, for i=1,…,n.Proposition 1.
For∑i=1n‍βi=0 then (3)Zα1eβ1,z1;…;eβn,zn;q=q1/2α2ητexp⁡α∑i=1nβizi∏1≤r<s≤nKzrs,τβrβs,where zrs=zr-zs and K(z,τ) 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 n-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∑i=1n‍ζi=0 then (4)Zα1u1⊗eζ1,z1;…;un⊗eζn,zn;q=Qαζ1,…,ζnu1,z1;…;un,zn;qZα1eζ1,z1;…;eζn,zn;q,where Qαζ1,…,ζn(u1,z1;…;un,zn;q) is an explicit sum of elliptic and quasi-modular forms introduced in [3].


## 1.2. Torus Intertwinedn-Point Functions for VZ

Letg1, h1 be commuting automorphisms of VZ defined by (5)σf1=e2πiβ1a0,σg1=e-2πiα1a0,where σ=eπia(0) is the fermion number automorphism. We assume that α1,β1∈R so that θ1=-e-2πiβ1 and ϕ1=-e2πiα1 are of unit modulus.We then consider in [1] torus orbifold intertwining n+2-point functions for the σg1-twisted module VZ+α1=⊕m∈ZM⊗em+α1 for VZ defined by (6)ZVZ1f1g1u1⊗em1,z1;…,un⊗emn,zn;v1⊗en1+κ,w;v2⊗en2-κ,0;q=TrVZ+α1σf1Yq1L0u1⊗em1,q1⋯YqnL0un⊗emn,qn·YqwL0v1⊗en1+κ,qwYv2⊗en2-κ,1qL0-1/24=∑μ∈Z+α1e2πiμβ1Zμ1u1⊗em1,z1;…;un⊗emn,zn;v1⊗en1+κ,w;v2⊗en2-κ,0;q,where mi,n1,andn2∈Z for κ∈(-1/2,1/2) and qw=ew. In particular, we find (7)ZVZ1f1g1eκ,w;e-κ,0;q=∑μ∈Z+α1e2πiμβ1Zμ1eκ,w;e-κ,0;q=1ηqϑα1β1κw,τKw,τκ2,for genus one theta series (A.1).More generally, we define in [1] a generating function for all n+2-point functions (A.1) by the following formal differential form: (8)Gn1f1g1x1,y1,…,xn,yn≡ZVZ1f1g1ψ+,x1;ψ-,y1;…;ψ+,xn;ψ-,yn;eκ,w;e-κ,0;q∏i=1ndxi1/2dyi1/2,for VZ generators ψ±=e±1 alternatively inserted at xi, yi for i=1,…,n. 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 (9)Gn1f1g1x1,y1,…,xn,yn=ZVZ1f1g1eκ,w;e-κ,0;qdet⁡Sκ,where Sκ denotes the n×n matrix with components Sκxi,yj for i,j=1,…,n for Szegő kernel (A.9).Finally, we obtain in [1] the following generalization of Proposition 15 of [4] concerning the generating properties of (8).Proposition 4.
G
m
(
1
)
f
1
g
1
(
x
1
,
y
1
,
…
,
x
m
,
y
m
) is a generating function for all torus orbifold intertwining n+2-point functions (A.1). In particular, for a pair of square-bracket mode twisted Fock vectors (B.10) (10)Ψκk1,l2=eκq^ψ+-k11⋯ψ+-k1s1ψ--l21⋯ψ--l2t21,Ψ-κk2,l1=e-κq^ψ+-k21⋯ψ+-k2s2ψ--l11⋯ψ--l1t11,for p=s1+s2=t1+t2>0; then (11)ZVZ1f1g1Ψκk1,l2,w;Ψ-κk2,l1,0;q=ϵZVZ1f1g1eκ,w;e-κ,0;qdet⁡Cabka,lb,where ϵ=(-1)(t1+s2)t2+1/2peiπBκ(s2-t1), for the odd integer B fixed in (B.18) and (12)Cabka,lb=C11k1,l1C12k1,l2C21k2,l1C22k2,l2is a p×p block matrix with components Cab(kaia,lbjb) for ia=1,…,sa and jb=1,…,tb for a,b=1,2 for Sκ 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]) (13)ϑα1β1∑m=1nxm-ym,τϑα1β10,τ∏1≤i<j≤nKxi-xj,τKyi-yj,τ∏1≤i,j≤nKxi-yj,τ=det⁡P1,where P1 denotes the n×n matrix with twisted Weierstrass function components P1θ1ϕ1(xi,yj) of (A.8) for i,j=1,…,n.The trivial form of the twisted Frobenius identity follows from the consideration of the generating function for alln+2-point functions (14)ϑα1β1∑m=1nxm-ym+κw,τϑα1β1κw,τ∏1≤i<j≤nKxi-xj,τKyi-yj,τ∏1≤i,j≤nKxi-yj,τ·∏1≤i,j≤nKxi-w,τKyj,τKxi,τKyj-w,τκdxi1/2dyj1/2=det⁡Sκ.We obtain the following.Proposition 5.
The twisted two-point Frobenius identity is given by(15)∑μ∈Z+α1e2πiμβ1QμζΨκk1,l2,w;Ψ-κk2,l1,0;v1⊗en1+κ,w;v2⊗en2-κ,0;q·q1/2μ2exp⁡μ∑i=1ζiziKw,τκ2∏r<sKzrs,τζrζs=ϑα1β1κw,τdet⁡Cabka,lb,for ζ=ζ(Ψκ[k1,l2];Ψ-κ[k2,l1];v1⊗en1+κ;v2⊗en2-κ) (i.e., values of exponents in module elements), and z={w,0,w,0} correspondingly.Proof.
Consider the expression for the torus orbifold intertwining two-point function (11). Using (4) we derive the following: (16)ZVZ1f1g1Ψκk1,l2,w;Ψ-κk2,l1,0;v1⊗en1+κ,w;v2⊗en2-κ,0;q=∑μ∈Z+α1e2πiμβ1Zμ1Ψκk1,l2,w;Ψ-κk2,l1,0;v1⊗en1+κ,w;v2⊗en2-κ,0;q=∑μ∈Z+α1e2πiμβ1QμζΨκk1,l2,w;Ψ-κk2,l1,0;v1⊗en1+κ,w;v2⊗en2-κZμ1eζ1,z1;…;eζn,zn;q=∑μ∈Z+α1e2πiμβ1QμζΨκk1,l2,w;Ψ-κk2,l1,0;v1⊗en1+κ,w;v2⊗en2-κ,0;q·q1/2α2ητexp⁡μ∑i=1ζizi∏1≤r<s≤nKzrs,τβrβs.Thus we obtain the result.



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*Source: 2340410-2017-11-09.xml*