Jeffrey A. Harvey

A bstract: 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 Spin( k ) ₂ or equivalently, the ℤ ₂ -equivariantization of a Tambara-Yamagami fusion category.

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 S L ( 2 , R ) (Conway et al 1979 Bull. London Math. Soc. 11 308; Thompson 1979 Bull. London Math. Soc. 11 352; Frenkel et al 1989 Vertex Operator Algebras and the Monster (Academic Press); Borcherds 1992 Invent. Math. 109 405). In Lin and Shao (2021 J. Phys. A: Math. Theor. 54 065201), 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 M . This ‘defect McKay–Thompson series’ was found to be invariant under a genus zero subgroup of S L ( 2 , 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 S L ( 2 , R ) , whether or not they are Hauptmoduln for those groups. Using the decompositions of the monster stress tensor found in Bae et al (2021 Commun. Math. Phys. 388 1), we construct several new defect McKay–Thompson series, study their modular properties, and determine their invariance groups in S L ( 2 , R ) . We discover that many of the invariance groups are not genus zero.

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.

We extend the investigation in [1] 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 supplementary Mathematica package that performs most of the computations required for these comparisons.

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.

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 an RCFT and the quadratic residues modulo 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 transforms the fusion and the braiding matrices as well, and yields isomorphic structures in theories related by Hecke operators.

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 [41]. 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 [41] 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).