[Paper Review] Genus Two Partition and Correlation Functions for Fermionic Vertex Operator Superalgebras II

Michael P. Tuite, Alexander Zuevsky|arXiv (Cornell University)|Aug 12, 2013

Algebraic structures and combinatorial models4 citations

TL;DR

This paper computes the genus two partition function and generating function for n-point correlation functions in the ρ-formalism for the rank two free fermion vertex operator superalgebra. Using bosonization and genus one intertwiner correlation functions, it derives results involving infinite and finite determinants of genus one Szegő kernels, proving holomorphicity and modular invariance under a subgroup of Sp(4,Z), with an analogue of the Jacobi triple product identity for genus two theta series.

ABSTRACT

We define and compute the continuous orbifold partition function and a generating function for all $n$-point correlation functions for the rank two free fermion vertex operator superalgebra on a genus two Riemann surface formed by self-sewing a torus. The partition function is proportional to an infinite dimensional determinant with entries arising from torus Szego kernel and the generating function is proportional to a finite determinant of genus two Szego kernels. These results follow from an explicit analysis of all torus $n$-point correlation functions for intertwiners of the irreducible modules of the Heisenberg vertex operator algebra. We prove that the partition and $n$-point correlation functions are holomorphic on a suitable domain and describe their modular properties. We also describe an identity for the genus two Riemann theta series analogous to the Jacobi triple product identity.

Motivation & Objective

• To compute the genus two partition function and generating function for n-point correlation functions in the ρ-formalism for the rank two free fermion vertex operator superalgebra.
• To establish holomorphicity and modular invariance of the partition and correlation functions on a suitable covering space of the sewing domain.
• To derive an analogue of the Jacobi triple product identity for genus two Riemann theta series.
• To extend genus one intertwiner correlation function results to generalized VOAs and apply them to genus two constructions via self-sewing a torus.
• To analyze the multi-valued nature of the correlation functions on the sewing domain and their lift to holomorphic functions on a covering space.

Proposed method

• Uses the ρ-formalism to construct a genus two Riemann surface by self-sewing a torus at two points separated by w, with modular parameter τ and sewing parameter ρ.
• Employs bosonization to identify the rank two free fermion VOSA with the Z-lattice VOSA, enabling treatment of intertwiners for all α ∈ ℂ via the generalized VOA M ⊗ e^α.
• Computes genus one n-point correlation functions for intertwiners in the generalized VOA M ⊗ e^α using elliptic prime forms and quasi-modular forms.
• Derives a generating function for genus one n-point functions as a finite determinant of genus one Szegő kernels.
• Constructs the genus two partition and correlation functions using an infinite matrix T derived from genus one Szegő kernel data and multipliers, with key dependence on det(I - T).
• Leverages an explicit expression for the genus two Szegő kernel S^(2) in terms of genus one data and multipliers from [TZ1], with (I - T)^{-1} appearing in the kernel formula.

Experimental results

Research questions

• RQ1 How can the genus two partition function be explicitly computed for the rank two free fermion VOSA in the ρ-formalism?
• RQ2 What is the structure of the generating function for all genus two n-point correlation functions, and how does it relate to genus one data?
• RQ3 How do the partition and correlation functions behave holomorphically and modularly on the sewing domain D^ρ?
• RQ4 What is the genus two analogue of the Jacobi triple product identity for Riemann theta series?
• RQ5 How do the multi-valued properties of the correlation functions on D^ρ resolve upon lifting to a covering space?

Key findings

• The genus two partition function is proportional to det(I - T), where T is an infinite matrix constructed from genus one Szegő kernel data and multipliers, and is holomorphic on the sewing domain D^ρ.
• The generating function for genus two n-point correlation functions is proportional to a finite determinant of genus two Szegő kernels, derived from genus one intertwiner correlation functions.
• The partition and correlation functions are multi-valued on D^ρ but lift to holomorphic functions on a suitable covering space.
• The functions exhibit modular invariance under a subgroup of Sp(4,Z), the genus two symplectic group.
• An identity for the genus two Riemann theta series is derived, analogous to the Jacobi triple product identity.
• The genus one n-point correlation functions for intertwiners in the generalized VOA M ⊗ e^α are computed explicitly using shifted Virasoro modes and trace formulas involving q-series and elliptic functions.