arXiv subjects

Sarah M. Harrison

Publications and source records attributed to Sarah M. Harrison.

At least 19 recordsLinked to original sources

arXiv2025

Non-invertible defects from the Conway SCFT to K3 sigma models II: duality and Fibonacci defects

We continue the study, initiated in [hep-th:2504.18619], of topological defect lines (TDLs) in the Conway module $V^{f \natural}$ and K3 non-linear sigma models (NLSMs). In the case of $V^{f \natural}$, we fully classify the potential $N=1$ (and $N=4$)--preserving duality defects for cyclic Tambara--Yamagami categories TY$(\mathbb{Z}_N)$, noting a curious relation to genus zero groups of monstrous moonshine. We use the correspondence with Leech lattice endomorphisms, discovered in [hep-th:2504.18619], to construct a number of non-trivial examples of TDLs in $V^{f \natural}$, including examples of irrational quantum dimension. In particular, we fully classify and construct defects for the TY$(\mathbb{Z}_2)$ and TY$(\mathbb{Z}_3)$ cases, and provide examples of duality defects for TY$(\mathbb{Z}_2\times \mathbb{Z}_2)$ and Fibonacci fusion categories as well. In the case of K3 NLSMs, we describe a duality defect of irrational quantum dimension $\sqrt{2}$ for the category TY$(\mathbb{Z}_2, -1)$ in a particular torus orbifold, which exists on a 16-dimensional slice of the moduli space. We also provide a detailed analysis of spectral flow--preserving TDLs in Gepner models of K3, of independent interest, and use this to construct non-invertible defects for Fibonacci and $Rep(S_3)$ categories in particular examples. Finally we provide evidence for our conjecture in [hep-th:2504.18619] that special subcategories of such TDLs in $V^{f \natural}$ correspond to $N=(4,4)$ and spectral flow--preserving defect lines in a corresponding K3 NLSM. In particular, we compute defect--twined elliptic genera for all non-invertible defects constructed in this article, demonstrating that for each defect found in a K3 NLSM, there is a corresponding defect in $V^{f \natural}$ with coincident twining genus, and making a prediction for a number of TDLs in K3 NLSMs yet to be found.

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arXiv2025

Non-invertible defects from the Conway SCFT to K3 sigma models I: general results

We initiate the study of supersymmetry-preserving topological defect lines (TDLs) in the Conway moonshine module $V^{f \natural}$. We show that the tensor category of such defects, under suitable assumptions, admits a surjective but non-injective ring homomorphism into the ring of $\mathbb{Z}$-linear maps of the Leech lattice into itself. This puts strong constraints on possible defects and their quantum dimensions. We describe a simple construction of non-invertible TDLs from orbifolds of holomorphic (super)vertex operator algebras, which yields non-trivial examples of TDLs satisfying our main theorem. We conjecture a correspondence between four--plane--preserving TDLs in $V^{f\natural}$ and supersymmetry--preserving TDLs in K3 non-linear sigma models, which extends the correspondence between symmetry groups to the level of tensor category symmetry. We establish evidence for this conjecture by constructing non-invertible TDLs in special K3 non-linear sigma models.

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arXiv2023

Liouville theory and the Weil-Petersson geometry of moduli space: bordered, conic, and higher genus surfaces

Two-dimensional conformal field theory is a powerful tool to understand the geometry of surfaces. Here, we study Liouville conformal field theory in the classical (large central charge) limit, where it encodes the geometry of the moduli space of Riemann surfaces. Generalizing previous work, we employ this to study moduli spaces of higher genus surfaces, surfaces with boundaries, and surfaces with cone points. In each case, the knowledge of classical conformal blocks provides an extremely efficient approximation to the Weil-Petersson metric on moduli space. We find detailed agreement with analytic results for volumes and geodesic lengths on moduli space.

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arXiv2023

Liouville Theory and the Weil-Petersson Geometry of Moduli Space

Liouville theory describes the dynamics of surfaces with constant negative curvature and can be used to study the Weil-Petersson geometry of the moduli space of Riemann surfaces. This leads to an efficient algorithm to compute the Weil--Petersson metric to arbitrary accuracy using Zamolodchikov's recursion relation for conformal blocks. For example, we compute the metric on $\mathcal M_{0,4}$ numerically to high accuracy by considering Liouville theory on a sphere with four punctures. We numerically compute the eigenvalues of the Weil-Petersson Laplacian, and find evidence that the obey the statistics of a random matrix in the Gaussian Orthogonal Ensemble.

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arXiv2022

3-Manifolds and VOA Characters

By studying the properties of $q$-series $\widehat Z$-invariants, we develop a dictionary between 3-manifolds and vertex algebras. In particular, we generalize previously known entries in this dictionary to Lie groups of higher rank, to 3-manifolds with toral boundaries, and to BPS partition functions with line operators. This provides a new physical realization of logarithmic vertex algebras in the framework of the 3d-3d correspondence and opens new avenues for their future study. For example, we illustrate how invoking a knot-quiver correspondence for $\widehat{Z}$-invariants leads to many infinite families of new fermionic formulae for VOA characters.

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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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arXiv2021

BPS Algebras in 2D String Theory

We discuss a set of heterotic and type II string theory compactifications to 1+1 dimensions that are characterized by factorized internal worldsheet CFTs of the form $V_1\otimes \bar V_2$, where $V_1, V_2$ are self-dual (super) vertex operator algebras. In the cases with spacetime supersymmetry, we show that the BPS states form a module for a Borcherds-Kac-Moody (BKM) (super)algebra, and we prove that for each model the BKM (super)algebra is a symmetry of genus zero BPS string amplitudes. We compute the supersymmetric indices of these models using both Hamiltonian and path integral formalisms. The path integrals are manifestly automorphic forms closely related to the Borcherds-Weyl-Kac denominator. Along the way, we comment on various subtleties inherent to these low-dimensional string compactifications.

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arXiv2020

Fun with $F_{24}$

We study some special features of $F_{24}$, the holomorphic $c=12$ superconformal field theory (SCFT) given by 24 chiral free fermions. We construct eight different Lie superalgebras of "physical" states of a chiral superstring compactified on $F_{24}$, and we prove that they all have the structure of Borcherds-Kac-Moody superalgebras. This produces a family of new examples of such superalgebras. The models depend on the choice of an $\mathcal{N}=1$ supercurrent on $F_{24}$, with the admissible choices labeled by the semisimple Lie algebras of dimension 24. We also discuss how $F_{24}$, with any such choice of supercurrent, can be obtained via orbifolding from another distinguished $c=12$ holomorphic SCFT, the $\mathcal{N}=1$ supersymmetric version of the chiral CFT based on the $E_8$ lattice.

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arXiv2020

$\mathcal{N}=2$ Minimal Models: A Holographic Needle in a Symmetric Orbifold Haystack

We explore large-$N$ symmetric orbifolds of the $\mathcal N=2$ minimal models, and find evidence that their moduli spaces each contain a supergravity point. We identify single-trace exactly marginal operators that deform them away from the symmetric orbifold locus. We also show that their elliptic genera exhibit slow growth consistent with supergravity spectra in AdS$_3$. We thus propose an infinite family of new holographic CFTs.

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arXiv2020

3d Modularity

We find and propose an explanation for a large variety of modularity-related symmetries in problems of 3-manifold topology and physics of 3d $\mathcal{N}=2$ theories where such structures a priori are not manifest. These modular structures include: mock modular forms, $SL(2,\mathbb{Z})$ Weil representations, quantum modular forms, non-semisimple modular tensor categories, and chiral algebras of logarithmic CFTs.

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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

A Borcherds-Kac-Moody superalgebra with Conway symmetry

We construct a Borcherds Kac-Moody (BKM) superalgebra on which the Conway group Co$_0$ acts faithfully. We show that the BKM algebra is generated by the BRST-closed states in a chiral superstring theory. We use this construction to produce denominator identities for the chiral partition functions of the Conway module $V^{s\natural}$, a supersymmetric $c=12$ chiral conformal field theory whose (twisted) partition functions enjoy moonshine properties and which has automorphism group isomorphic to Co$_0$. In particular, these functions satisfy a genus zero property analogous to that of monstrous moonshine. Finally, we suggest how one may promote the denominators to spacetime BPS indices in type II string theory, which might thus furnish a physical explanation of the genus zero property of Conway moonshine.

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arXiv2017

Symmetries of the refined D1/D5 BPS spectrum

We examine the large $N$ 1/4-BPS spectrum of the symmetric orbifold CFT Sym$^N(M)$ deformed to the supergravity point in moduli space for $M= K3$ and $T^4$. We consider refinement under both left- and right-moving $SU(2)_R$ symmetries of the superconformal algebra, and decompose the spectrum into characters of the algebra. We find that at large $N$ the character decomposition satisfies an unusual property, in which the degeneracy only depends on a certain linear combination of left- and right-moving quantum numbers, suggesting deeper symmetry structure. Furthermore, we consider the action of discrete symmetry groups on these degeneracies, where certain subgroups of the Conway group are known to play a role. We also comment on the potential for larger discrete symmetry groups to appear in the large $N$ limit.

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arXiv2017

Properties of extremal CFTs with small central charge

We analyze aspects of extant examples of 2d extremal chiral (super)conformal field theories with $c\leq 24$. These are theories whose only operators with dimension smaller or equal to $c/24$ are the vacuum and its (super)Virasoro descendents. The prototypical example is the monster CFT, whose famous genus zero property is intimately tied to the Rademacher summability of its twined partition functions, a property which also distinguishes the functions of Mathieu and umbral moonshine. However, there are now several additional known examples of extremal CFTs, all of which have at least $\mathcal N=1$ supersymmetry and global symmetry groups connected to sporadic simple groups. We investigate the extent to which such a property, which distinguishes the monster moonshine module from other $c=24$ chiral CFTs, holds for the other known extremal theories. We find that in most cases, the special Rademacher summability property present for monstrous and umbral moonshine does not hold for the other extremal CFTs, with the exception of the Conway module and two $c=12, ~\mathcal N=4$ superconformal theories with $M_{11}$ and $M_{22}$ symmetry. This suggests that the connection between extremal CFT, sporadic groups, and mock modular forms transcends strict Rademacher summability criteria.

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arXiv2017

K3 Elliptic Genus and an Umbral Moonshine Module

Umbral moonshine connects the symmetry groups of the 23 Niemeier lattices with 23 sets of distinguished mock modular forms. The 23 cases of umbral moonshine have a uniform relation to symmetries of $K3$ string theories. Moreover, a supersymmetric vertex operator algebra with Conway sporadic symmetry also enjoys a close relation to the $K3$ elliptic genus. Inspired by the above two relations between moonshine and $K3$ string theory, we construct a chiral CFT by orbifolding the free theory of 24 chiral fermions and two pairs of fermionic and bosonic ghosts. In this paper we mainly focus on the case of umbral moonshine corresponding to the Niemeier lattice with root system given by 6 copies of $D_4$ root system. This CFT then leads to the construction of an infinite-dimensional graded module for the umbral group $G^{D_4^{\oplus 6}}$ whose graded characters coincide with the umbral moonshine functions. We also comment on how one can recover all umbral moonshine functions corresponding to the Niemeier root systems $A_5^{\oplus 4}D_4$, $A_7^{\oplus 2}D_5^{\oplus 2}$ , $A_{11}D_7 E_6$, $A_{17}E_7$, and $D_{10}E_7^{\oplus 2}$.

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arXiv2017

Landau-Ginzburg Orbifolds and Symmetries of K3 CFTs

Recent developments in the study of the moonshine phenomenon, including umbral and Conway moonshine, suggest that it may play an important role in encoding the action of finite symmetry groups on the BPS spectrum of K3 string theory. To test and clarify these proposed K3-moonshine connections, we study Landau-Ginzburg orbifolds that flow to conformal field theories in the moduli space of K3 sigma models. We compute K3 elliptic genera twined by discrete symmetries that are manifest in the UV description, though often inaccessible in the IR. We obtain various twining functions coinciding with moonshine predictions that have not been observed in physical theories before. These include twining functions arising from Mathieu moonshine, other cases of umbral moonshine, and Conway moonshine. For instance, all functions arising from $M_{11} \subset 2.M_{12}$ moonshine appear as explicit twining genera in the LG models, which moreover admit a uniform description in terms of its natural 12-dimensional representation. Our results provide strong evidence for the relevance of umbral moonshine for K3 symmetries, as well as new hints for its eventual explanation.

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arXiv2017

K3 String Theory, Lattices and Moonshine

In this paper we address the following two closely related questions. First, we complete the classification of finite symmetry groups of type IIA string theory on $K3\times \mathbb R^6$, where Niemeier lattices play an important role. This extends earlier results by including points in the moduli space with enhanced gauge symmetries in spacetime, or, equivalently, where the world-sheet CFT becomes singular. After classifying the symmetries as abstract groups, we study how they act on the BPS states of the theory. In particular, we classify the conjugacy classes in the T-duality group $O^+(Γ^{4,20})$ which represent physically distinct symmetries. Subsequently, we make two conjectures regarding the connection between the corresponding twining genera of $K3$ CFTs and Conway and umbral moonshine, building upon earlier work on the relation between moonshine and the $K3$ elliptic genus.

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arXiv2017

Heterotic sigma models on $T^8$ and the Borcherds automorphic form $Φ_{12}$

We consider the spectrum of BPS states of the heterotic sigma model with $(0,8)$ supersymmetry and $T^8$ target, as well as its second-quantized counterpart. We show that the counting function for such states is intimately related to Borcherds' automorphic form $Φ_{12}$, a modular form which exhibits automorphy for $O(2,26;{\mathbb Z})$. We comment on possible implications for Umbral moonshine and theories of AdS$_3$ gravity.

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