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Jeffrey A. Harvey

Publications and source records attributed to Jeffrey A. Harvey.

At least 19 recordsLinked to original sources
arXiv2025

Generalized Level-Rank Duality, Holomorphic Conformal Field Theory, and Non-Invertible Anyon Condensation

We study the interplay between holomorphic conformal field theory and dualities of 3D topological quantum field theories generalizing the paradigm of level-rank duality. A holomorphic conformal field theory with a Kac-Moody subalgebra implies a topological interface between Chern-Simons gauge theories. Upon condensing a suitable set of anyons, such an interface yields a duality between topological field theories. We illustrate this idea using the $c=24$ holomorphic theories classified by Schellekens, which leads to a list of novel sporadic dualities. Some of these dualities necessarily involve condensation of anyons with non-abelian statistics, i.e. gauging non-invertible one-form global symmetries. Several of the examples we discover generalize from $c=24$ to an infinite series. This includes the fact that Spin$(n^{2})_{2}$ is dual to a twisted dihedral group gauge theory. Finally, if $-1$ is a quadratic residue modulo $k$, we deduce the existence of a sequence of holomorphic CFTs at central charge $c=2(k-1)$ with fusion category symmetry given by $\mathrm{Spin}(k)_{2}$ or equivalently, the $\mathbb{Z}_{2}$-equivariantization of a Tambara-Yamagami fusion category.

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arXiv2025

Superconformal symmetry in a class of Schellekens theories

In 2023, Moore and Singh used the theory of orbifold vertex operator algebras to explicitly construct an $\mathcal{N} = 1$ supercurrent in the Beauty and the Beast module of Dixon, Ginsparg, and Harvey. Using their techniques, we show that $\mathcal{N} = 1$ superconformal symmetry exists in all VOAs constructed as spin lifts of canonical $\mathbb{Z}_2$ orbifolds of Niemeier lattice VOAs.

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arXiv2025

Modular invariance groups and defect McKay-Thompson series

It has been known since 1992 that the McKay-Thompson series $T_g(q)$ of the Moonshine module form Hauptmoduln for genus zero subgroups of $SL(2, \mathbb{R})$. In 2021, Lin and Shao constructed a series analogous to the McKay-Thompson series (a twined partition function of the Monster CFT), but using a non-invertible topological defect rather than an element of the Monster group $\mathcal{M}$. This "defect McKay-Thompson series" was found to be invariant under a genus zero subgroup of $SL(2, \mathbb{R})$, but was shown not to be the Hauptmodul of the subgroup. Nevertheless, one might wonder if a weaker version of Borcherds' theorem holds for non-invertible defects: perhaps defect McKay-Thompson series enjoy genus zero invariance groups in $SL(2, \mathbb{R})$, whether or not they are Hauptmoduln for those groups. Using the decompositions of the monster stress tensor found in Bae et al. (2021), we construct several new defect McKay-Thompson series, study their modular properties, and determine their invariance groups in $SL(2, \mathbb{R})$. We discover that many of the invariance groups are not genus zero.

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arXiv2025

Two New Avatars of Moonshine for the Thompson Group

The Thompson sporadic group admits special relationships to modular forms of two kinds. On the one hand, last century's generalized moonshine for the monster equipped the Thompson group with a module for which the associated McKay-Thompson series are distinguished weight zero modular functions. On the other hand, Griffin and Mertens verified the existence of a module for which the McKay-Thompson series are distinguished modular forms of weight one-half, that were assigned to the Thompson group in this century by the last two authors of this work. In this paper we round out this picture by proving the existence of two new avatars of Thompson moonshine: a new module giving rise to weight zero modular functions, and a new module giving rise to forms of weight one-half. We explain how the newer modules are related to the older ones by Borcherds products and traces of singular moduli. In so doing we clarify the relationship between the previously known modules, and expose a new arithmetic aspect to moonshine for the Thompson group. We also present evidence that this phenomenon extends to a correspondence between other cases of generalized monstrous moonshine and penumbral moonshine, and thereby enriches these phenomena with counterparts in weight one-half and weight zero, respectively.

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arXiv2025

One-loop Renormalization of BPS String Masses in Pseudo-anomalous Heterotic String

Compactification of heterotic string on a Calabi-Yau threefold can lead to a four-dimensional low-energy effective theory which contains a $U(1)$ gauge theory which is pseudo-anomalous, meaning that the fermion content is anomalous, but that the fermion anomaly is cancelled by a four-dimensional version of the Green-Schwarz mechanism involving a shift of the model-independent axion field. It is also well known that a field-dependent Fayet-Iliopoulos like term is induced at one-loop in string perturbation theory in such compactifications and that this leads to a mass for the $U(1)$ gauge field. Explicit one-loop computations of the mass shifts of massless charged scalars and fermions implied by this mechanism were originally carried out in the 1980's and have been revisited more recently using techniques of string field theory and super Riemann surfaces. We consider such theories further compactified on a circle of radius $R$. There is then an infinite tower of BPS states with winding and momenta on the circle. BPS strings which wind the circle can, at large $R$, be viewed as macroscopic or cosmic strings in four spacetime dimensions, or, at generic $R$, as BPS states in three dimensions. We compute the one-loop mass renormalization of such states and show that it is nonzero and proportional to their pseudo-anomalous $U(1)$ charge.

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arXiv2022

Snowmass White Paper: Moonshine

We present a brief overview of Moonshine with an emphasis on connections to physics. Moonshine collectively refers to a set of phenomena connecting group theory, analytic number theory, and vertex operator algebras or conformal field theories. Modern incarnations of Moonshine arise in various BPS observables in string theory and, via dualities, invariants in enumerative geometry. We survey old and new developments, and highlight some of the many open questions that remain.

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arXiv2022

An Overview of Penumbral Moonshine

As Mathieu moonshine is a special case of umbral moonshine, Thompson moonshine (in half-integral weight) is a special case of a family of similar relationships between finite groups and vector-valued modular forms of a certain kind. We call this penumbral moonshine. We introduce and explain some features of this phenomenon in this work.

math.RT↗
arXiv2022

Modular Products and Modules for Finite Groups

Motivated by the appearance of penumbral moonshine, and by evidence that penumbral moonshine enjoys an extensive relationship to generalized monstrous moonshine via infinite products, we establish a general construction in this work which uses singular theta lifts and a concrete construction at the level of modules for a finite group to translate between moonshine in weight one-half and moonshine in weight zero. This construction serves as a foundation for a companion paper in which we explore the connection between penumbral Thompson moonshine and a special case of generalized monstrous moonshine in detail.

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arXiv2021

Conway Subgroup Symmetric Compactifications Redux

We extend the investigation of special toroidal compactifications of heterotic string theory for which the half-BPS states provide representations of subgroups of the Conway group. We also explore dual descriptions of these theories and find that they are all linked to either F-theory or type IIA string theory on K3 surfaces with symplectic automorphism groups that are the same Conway subgroups as those of the heterotic dual. The matching with type IIA K3 dual theories includes both the matching of symmetry groups and a comparison between the Narain lattice on the heterotic side and the cohomology lattice on the type IIA side. We present twelve examples where we can identify a type IIA dual K3 orbifold theory as the dual description of the heterotic theory. In addition, we include a Mathematica package that performs most of the computations required for these comparisons.

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arXiv2020

String Islands

We discuss string theories with small numbers of non-compact moduli and describe constructions of string theories whose low-energy limit is described by various pure supergravity theories. We also construct a D=4,N=4 compactification of type II string theory with 34 vector fields.

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arXiv2020

Galois Symmetry Induced by Hecke Relations in Rational Conformal Field Theory and Associated Modular Tensor Categories

Hecke operators relate characters of rational conformal field theories (RCFTs) with different central charges, and extend the previously studied Galois symmetry of modular representations and fusion algebras. We show that the conductor $N$ of a RCFT and the quadratic residues mod $N$ play an important role in the computation and classification of Galois permutations. We establish a field correspondence in different theories through the picture of effective central charge, which combines Galois inner automorphisms and the structure of simple currents. We then make a first attempt to extend Hecke operators to the full data of modular tensor categories. The Galois symmetry encountered in the modular data appears in the fusion and the braiding matrices as well, and yields isomorphic structures in theories related by Hecke operators.

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arXiv2020

Moonshine, Superconformal Symmetry, and Quantum Error Correction

Special conformal field theories can have symmetry groups which are interesting sporadic finite simple groups. Famous examples include the Monster symmetry group of a $c=24$ two-dimensional conformal field theory (CFT) constructed by Frenkel, Lepowsky and Meurman, and the Conway symmetry group of a $c=12$ CFT explored in detail by Duncan and Mack-Crane. The Mathieu moonshine connection between the K3 elliptic genus and the Mathieu group $M_{24}$ has led to the study of K3 sigma models with large symmetry groups. A particular K3 CFT with a maximal symmetry group preserving $(4,4)$ superconformal symmetry was studied in beautiful work by Gaberdiel, Taormina, Volpato, and Wendland. The present paper shows that in both the GTVW and $c=12$ theories the construction of superconformal generators can be understood via the theory of quantum error correcting codes. The automorphism groups of these codes lift to symmetry groups in the CFT preserving the superconformal generators. In the case of the $N=1$ supercurrent of the GTVW model our result, combined with a result of T. Johnson-Freyd implies the symmetry group is the maximal subgroup of $M_{24}$ known as the sextet group. (The sextet group is also known as the holomorph of the hexacode.) Building on \cite{gtvw} the Ramond-Ramond sector of the GTVW model is related to the Miracle Octad Generator which in turn leads to a role for the Golay code as a group of symmetries of RR states. Moreover, $(4,1)$ superconformal symmetry suffices to define and decompose the elliptic genus of a K3 sigma model into characters of the $N=4$ superconformal algebra. The symmetry group preserving $(4,1)$ is larger than that preserving $(4,4)$.

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arXiv2020

Conformal Field Theories with Sporadic Group Symmetry

The monster sporadic group is the automorphism group of a central charge $c=24$ vertex operator algebra (VOA) or meromorphic conformal field theory (CFT). In addition to its $c=24$ stress tensor $T(z)$, this theory contains many other conformal vectors of smaller central charge; for example, it admits $48$ commuting $c=\frac12$ conformal vectors whose sum is $T(z)$. Such decompositions of the stress tensor allow one to construct new CFTs from the monster CFT in a manner analogous to the Goddard-Kent-Olive (GKO) coset method for affine Lie algebras. We use this procedure to produce evidence for the existence of a number of CFTs with sporadic symmetry groups and employ a variety of techniques, including Hecke operators, modular linear differential equations, and Rademacher sums, to compute the characters of these CFTs. Our examples include (extensions of) nine of the sporadic groups appearing as subquotients of the monster, as well as the simple groups ${}^2{E}_6(2)$ and ${F}_4(2)$ of Lie type. Many of these examples are naturally associated to McKay's $\widehat{E_8}$ correspondence, and we use the structure of Norton's monstralizer pairs more generally to organize our presentation.

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arXiv2019

Ramanujan's influence on string theory, black holes and moonshine

Ramanujan influenced many areas of mathematics, but his work on q-series, on the growth of coefficients of modular forms, and on mock modular forms stands out for its depth and breadth of applications. I will give a brief overview of how this part of Ramanujan's work has influenced physics with an emphasis on applications to string theory, counting of black hole states and moonshine. This paper contains the material from my presentation at the meeting celebrating the centenary of Ramanujan's election as FRS and adds some additional material on black hole entropy and the AdS/CFT correspondence.

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arXiv2018

Hecke Relations in Rational Conformal Field Theory

We define Hecke operators on vector-valued modular forms of the type that appear as characters of rational conformal field theories (RCFTs). These operators extend the previously studied Galois symmetry of the modular representation and fusion algebra of RCFTs to a relation between RCFT characters. We apply our results to derive a number of relations between characters of known RCFTs with different central charges and also explore the relation between Hecke operators and RCFT characters as solutions to modular linear differential equations. We show that Hecke operators can be used to construct an infinite set of possible characters for RCFTs with two independent characters and increasing central charge. These characters have multiplicity one for the vacuum representation, positive integer coefficients in their $q$ expansions, and are associated to a two-dimensional representation of the modular group which leads to non-negative integer fusion coefficients as determined by the Verlinde formula.

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arXiv2018

Attractive Strings and Five-Branes, Skew-Holomorphic Jacobi Forms and Moonshine

We show that certain BPS counting functions for both fundamental strings and strings arising from fivebranes wrapping divisors in Calabi--Yau threefolds naturally give rise to skew-holomorphic Jacobi forms at rational and attractor points in the moduli space of string compactifications. For M5-branes wrapping divisors these are forms of weight negative one, and in the case of multiple M5-branes skew-holomorphic mock Jacobi forms arise. We further find that in simple examples these forms are related to skew-holomorphic (mock) Jacobi forms of weight two that play starring roles in moonshine. We discuss examples involving M5-branes on the complex projective plane, del Pezzo surfaces of degree one, and half-K3 surfaces. For del Pezzo surfaces of degree one and certain half-K3 surfaces we find a corresponding graded (virtual) module for the degree twelve Mathieu group. This suggests a more extensive relationship between Mathieu groups and complex surfaces, and a broader role for M5-branes in the theory of Jacobi forms and moonshine.

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arXiv2018

An Uplifting Discussion of T-Duality

It is well known that string theory has a T-duality symmetry relating circle compactifications of large and small radius. This symmetry plays a foundational role in string theory. We note here that while T-duality is order two acting on the moduli space of compactifications, it is order four in its action on the conformal field theory state space. More generally, involutions in the Weyl group $W(G)$ which act at points of enhanced $G$ symmetry have canonical lifts to order four elements of $G$, a phenomenon first investigated by J. Tits in the mathematical literature on Lie groups and generalized here to conformal field theory. This simple fact has a number of interesting consequences. One consequence is a reevaluation of a mod two condition appearing in asymmetric orbifold constructions. We also briefly discuss the implications for the idea that T-duality and its generalizations should be thought of as discrete gauge symmetries in spacetime.

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