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Michael P. Tuite

Publications and source records attributed to Michael P. Tuite.

At least 19 recordsLinked to original sources
arXiv2025

Genus $g$ Virasoro Correlation Functions for Vertex Operator Algebras

For a simple, self-dual, strong CFT-type vertex operator algebra (VOA) of central charge $c$, we describe the Virasoro $n$-point correlation function on a genus $g$ marked Riemann surface in the Schottky uniformisation. We show that this $n$-point function determines the correlation functions for all Virasoro vacuum descendants. Using our recent work on genus $g$ Zhu recursion, we show that the Virasoro $n$-point function is determined by a differential operator $\mathcal{D}_{n}$ acting on the genus $g$ VOA partition function normalised by the Heisenberg partition function to the power of $c$. We express $\mathcal{D}_{n}$ as the sum of weights over certain Virasoro graphs where the weights explicitly depend on $c$, the classical bidifferential of the second kind, the projective connection, holomorphic 1-forms and derivatives with respect to any $3g-3$ locally independent period matrix elements. We also describe the modular properties of $\mathcal{D}_{n}$ under a homology base change.

math.QA↗
arXiv2024

Genus $g$ Zhu Recursion for Vertex Operator Algebras and Their Modules

We describe Zhu recursion for a vertex operator algebra (VOA) and its modules on a genus $g$ Riemann surface in the Schottky uniformisation. We show that $n$-point (intertwiner) correlation functions are written as linear combinations of $(n-1)$-point functions with universal coefficients given by derivatives of the differential of the third kind, the Bers quasiform and certain holomorphic forms. We use this formula to describe conformal Ward identities framed in terms of a canonical differential operator which acts with respect to the Schottky moduli and to the insertion points of the $n$-point function. We consider the generalised Heisenberg VOA and determine all its correlation functions by Zhu recursion. We also use Zhu recursion to derive linear partial differential equations for the Heisenberg VOA partition function and various structures such as the bidifferential of the second kind, holomorphic $1$-forms, the prime form and the period matrix. Finally, we compute the genus $g$ partition function for any rational Euclidean lattice generalised VOA.

math.QA↗
arXiv2024

Inertial Transformations and the Nonexistence of Tachyons for Spacetime Dimension Greater than Two

We consider real linear transformations between two inertial frames with constant relative speed $v$ in a $d$-dimensional spacetime where light moves with constant speed $c=1$ (for some chosen units) in all frames. For $d=2$ we show that the standard relative velocity formula holds and that any associated anisotropic conformal factor is multiplicative under composition of inertial transformations for $|v|\neq 1$. Assuming that the inertial transformation matrix is continuous in a neighbourhood of $v=0$ and differentiable at $v=0$, we determine the conformal factor for all $|v|\neq 1$. For an isotropic spacetime, the general solution reduces to the standard $d=2$ Lorentz transformation for $|v|<1$ or to a Tachyonic transformation for $|v|>1$, first described by Parker in 1969. For $d>2$ we show that no Tachyonic-like inertial transformations exist which are compatible with constant light speed.

hep-th↗
arXiv2023

Some Properties and Applications of Bers Quasiforms on Riemann Surfaces

We describe some properties of the Bers quasiform on a compact Riemann surface in the Schottky sewing scheme. Our main results are: (i) the expansion of meromorphic differential forms in terms of holomorphic forms and derivatives of the Bers quasiform, (ii) power series expansions in the Schottky sewing parameters of the Bers quasiform and holomorphic forms, (iii) a novel differential operator which acts on meromorphic forms in several variables which we apply in deriving differential equations for classical objects such as the bidifferential of the second kind, the projective connection, holomorphic 1-forms and the prime form.

math.CV↗
arXiv2020

The Heisenberg Generalized Vertex Operator Algebra on a Riemann Surface

We compute the partition and correlation generating functions for the Heisenberg intertwiner generalized vertex operator algebra on a genus $g$ Riemann surface in the Schottky uniformization. These are expressed in terms of differential forms of the first, second and third kind, the prime form and the period matrix and are computed by combinatorial methods using a generalization of the MacMahon Master Theorem.

math.QA↗
arXiv2019

General Genus Zhu Recursion for Vertex Operator Algebras

We describe Zhu recursion for a vertex operator algebra (VOA) on a general genus Riemann surface in the Schottky uniformization where $n$-point correlation functions are written as linear combinations of $(n-1)$-point functions with universal coefficients. These coefficients are identified with specific geometric structures on the Riemann surface. We apply Zhu recursion to the Heisenberg VOA and determine all its correlation functions. For a general VOA, Zhu recursion with respect to the Virasoro vector is shown to lead to conformal Ward identities expressed in terms of derivatives with respect to the surface moduli. We derive linear partial differential equations for the Heisenberg VOA partition function and various structures such as the bidifferential of the second kind, holomorphic 1-forms and the period matrix. We also compute the genus $g$ partition function for an even lattice VOA.

math.QA↗
arXiv2019

Meromorphic Extensions of Green's Functions on a Riemann Surface

For a Riemann surface of genus $g\ge 2$ there exists a unique Green's function $G_{N}(x,y)$ which transforms as a weight $N\ge 2$ form in $x$ and a weight $1-N$ form in $y$ and is meromorphic in $x$, with a unique simple pole at $x=y$, but is not meromorphic in $y$. For a Schottky uniformized Riemann surface we consider meromorphic extensions of $G_{N}(x,y)$ called Green's Functions with Extended Meromorphicity or GEM forms. GEM forms are meromorphic in both $x$ and $y$ with a unique simple pole at $x=y$, transform as weight $N\ge 2$ forms in $x$ but as weight $1-N$ quasiperiodic forms in $y$. We give a reformulation of the bijective Bers map and describe a choice of GEM form with an associated canonical basis of normalized holomorphic $N$-forms. We describe an explicit differential operator constructed from $N=2$ GEM forms giving the variation with respect to moduli space parameters of a punctured Riemann surface. We also describe a new expression for the inverse Bers map.

math.CV↗
arXiv2017

Zhu reduction for Jacobi $n$-point functions and applications

We establish precise Zhu reduction formulas for Jacobi $n$-point functions which show the absence of any possible poles arising in these formulas. We then exploit this to produce results concerning the structure of strongly regular vertex operator algebras, and also to motivate new differential operators acting on Jacobi forms. Finally, we apply the reduction formulas to the Fermion model in order to create polynomials of quasi-Jacobi forms which are Jacobi forms.

math.QA↗
arXiv2016

Genus Two Virasoro Correlation Functions for Vertex Operator Algebras

We consider all genus two correlation functions for the Virasoro vacuum descendants of a vertex operator algebra. These are described in terms of explicit generating functions that can be combinatorially expressed in terms of a sequence of globally defined differential operators on which the genus two Siegel modular group $Sp(4,\mathbb{Z})$ has a natural action.

math.QA↗
arXiv2016

Genus Two Zhu Theory for Vertex Operator Algebras

We consider correlation functions for a vertex operator algebra on a genus two Riemann surface formed by sewing two tori together. We describe a generalisation of genus one Zhu recursion where we express an arbitrary genus two $n$--point correlation function in terms of $(n-1)$--point functions. We consider several applications including the correlation functions for the Heisenberg vertex operator algebra and its modules, Virasoro correlation functions and genus two Ward identities. We derive novel differential equations in terms of a differential operator on the genus two Siegel upper half plane for holomorphic $1$--differentials, the normalised bidifferential of the second kind, the projective connection and the Heisenberg partition function. We prove that the holomorphic mapping from the sewing parameter domain to the Siegel upper half plane is injective but not surjective. We also demonstrate that genus two differential equations arising from Virasoro singular vectors have holomorphic coefficients.

math.QA↗
arXiv2014

On Exceptional Vertex Operator (Super) Algebras

We consider exceptional vertex operator algebras and vertex operator superalgebras with the property that particular Casimir vectors constructed from the primary vectors of lowest conformal weight are Virasoro descendents of the vacuum. We show that the genus one partition function and characters for simple ordinary modules must satisfy modular linear differential equations. We show the rationality of the central charge and module lowest weights, modularity of solutions, the dimension of each graded space is a rational function of the central charge and that the lowest weight primaries generate the algebra. We also discuss conditions on the reducibility of the lowest weight primary vectors as a module for the automorphism group. Finally we analyse solutions for exceptional vertex operator algebras with primary vectors of lowest weight up to 9 and for vertex operator superalgebras with primary vectors of lowest weight up to 17/2. Most solutions can be identified with simple ordinary modules for known algebras but there are also four conjectured algebras generated by weight two primaries and three conjectured extremal vertex operator algebras generated by primaries of weight 3, 4 and 6 respectively.

math.QA↗
arXiv2013

Genus Two Partition and Correlation Functions for Fermionic Vertex Operator Superalgebras II

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.

math.QA↗
arXiv2013

On the Torus Degeneration of the Genus Two Partition Function

We consider the partition function of a general vertex operator algebra $V$ on a genus two Riemann surface formed by sewing together two tori. We consider the non-trivial degeneration limit where one torus is pinched down to a Riemann sphere and show that the genus one partition function on the degenerate torus is recovered up to an explicit universal $V$-independent multiplicative factor raised to the power of the central charge.

math.QA↗
arXiv2011

Some Generalizations of the MacMahon Master Theorem

We consider a number of generalizations of the $β$-extended MacMahon Master Theorem for a matrix. The generalizations are based on replacing permutations on multisets formed from matrix indices by partial permutations or derangements over matrix or submatrix indices.

math.CO↗
arXiv2011

Virasoro Correlation Functions for Vertex Operator Algebras

We consider all genus zero and genus one correlation functions for the Virasoro vacuum descendants of a vertex operator algebra. These are described in terms of explicit generating functions that can be combinatorially expressed in terms of graph theory related to derangements in the genus zero case and to partial permutations in the genus one case.

math.QA↗
arXiv2011

A Generalized Vertex Operator Algebra for Heisenberg Intertwiners

We consider the extension of the Heisenberg vertex operator algebra by all its irreducible modules. We give an elementary construction for the intertwining vertex operators and show that they satisfy a complex parametrized generalized vertex operator algebra. We illustrate some of our results with the example of integral lattice vertex operator superalgebras.

math.QA↗
arXiv2011

Free Bosonic Vertex Operator Algebras on Genus Two Riemann Surfaces II

We continue our program to define and study $n$-point correlation functions for a vertex operator algebra $V$ on a higher genus compact Riemann surface obtained by sewing surfaces of lower genus. Here we consider Riemann surfaces of genus 2 obtained by attaching a handle to a torus. We obtain closed formulas for the genus two partition function for free bosonic theories and lattice vertex operator algebras $V_L$. We prove that the partition function is holomorphic in the sewing parameters on a given suitable domain and describe its modular properties. We also compute the genus two Heisenberg vector $n$-point function and show that the Virasoro vector one point function satisfies a genus two Ward identity. We compare our results with those obtained in the companion paper, when a pair of tori are sewn together, and show that the partition functions are not compatible in the neighborhood of a two-tori degeneration point. The \emph{normalized} partition functions of a lattice theory $V_L$ \emph{are} compatible, each being identified with the genus two theta function of $L$.

math.QA↗