We continue the study, initiated in [1], of topological defect lines (TDLs) in the Conway module $\textit{V}$ and K3 non-linear sigma models (NLSMs). In the case of $\textit{V}$ , we fully classify the potential $$ \mathcal{N} $$ = 1 (and $$ \mathcal{N} $$ = 4) — preserving duality defects for cyclic Tambara-Yamagami categories TY($\textit{ℤ}$ ), noting a curious relation to genus zero groups of monstrous moonshine. We use the correspondence with Leech lattice endomorphisms, discovered in [1], to construct a number of non-trivial examples of TDLs in $\textit{V}$ , including examples of irrational quantum dimension. In particular, we fully classify and construct defects for the TY($\textit{ℤ}$ ) and TY($\textit{ℤ}$ ) cases, and provide examples of duality defects for TY($\textit{ℤ}$ × $\textit{ℤ}$ ) 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($\textit{ℤ}$ , –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 $\textit{Rep}$($\textit{S}$ ) categories in particular examples. Finally we provide evidence for our conjecture in [1] that special subcategories of such TDLs in $\textit{V}$ correspond to $$ \mathcal{N} $$ = (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 $\textit{V}$ with coincident twining genus, and making a prediction for a number of TDLs in K3 NLSMs yet to be found.

Germany

Roberta Angius

Italy

Stefano Giaccari

United States

Sarah Harrison

Roberto Volpato

Non-invertible defects from the Conway SCFT to K3 sigma models. Part II. Duality and Fibonacci defects

We initiate the study of supersymmetry-preserving topological defect lines (TDLs) in the Conway moonshine module $\textit{V}$ . 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 $\textit{V}$ 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.

Spain

Non-invertible defects from the Conway SCFT to K3 sigma models. Part I. General results

We consider the topological defect lines commuting with the spectral flow and the   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = (4, 4) superconformal symmetry in two dimensional non-linear sigma models on K3. By studying their fusion with boundary states, we derive a number of general results for the category of such defects. We argue that while for certain K3 models infinitely many simple defects, and even a continuum, can occur, at generic points in the moduli space the category is actually trivial, i.e. it is generated by the identity defect. Furthermore, we show that if a K3 model is at the attractor point for some BPS configuration of D-branes, then all topological defects have integral quantum dimension. We also conjecture that a continuum of topological defects arises if and only if the K3 model is a (possibly generalized) orbifold of a torus model. Finally, we test our general results in a couple of examples, where we provide a partial classification of the topological defects.

Topological defects in K3 sigma models

In quantum field theory, an orbifold is a way to obtain a new theory from an old one by gauging a finite global symmetry. This definition of orbifold does not make sense for quantum gravity theories, that admit (conjecturally) no global symmetries. In string theory, the orbifold procedure involves the gauging of a global symmetry on the world-sheet theory describing the fundamental string. Alternatively, it is a way to obtain a new string background from an old one by quotienting some isometry.

A fresh view on string orbifolds

Abstract We propose a correspondence between vertex operator superalgebras and families of sigma models in which the two structures are related by symmetry properties and a certain reflection procedure. The existence of such a correspondence is motivated by previous work on  supersymmetric non-linear sigma models on K3 surfaces, and on a vertex operator superalgebra with Conway group symmetry. Here we present an example of the correspondence for  supersymmetric non-linear sigma models on four-tori, and compare it to the K3 case.

Netherlands

Vassilis Anagiannis

Miranda C N Cheng

John Duncan

Vertex operator superalgebra/sigma model correspondences: The four-torus case A moonshine module for the Monster

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   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 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   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 1 supersymmetric version of the chiral CFT based on the E 8 lattice.

Canada

Natalie Paquette

Sweden

Daniel Persson

Fun with F24

We consider two dimensional   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = (4, 4) superconformal field theories in the moduli space of symmetric orbifolds of K3. We complete a classification of the discrete groups of symmetries of these models, conditional to a series of assumptions and with certain restrictions. Furthermore, we provide a partial classification of the set of twining genera, encoding the action of a discrete symmetry g on a space of supersymmetric states in these models. These results suggest the existence of a number of surprising identities between seemingly different Borcherds products, representing Siegel modular forms of degree two and level N &gt; 1. We also provide a critical review of various properties of the moduli space of these superconformal field theories, including the groups of dualities, the set of singular models and the locus of symmetric orbifold points, and describe some puzzles related to our (lack of) understanding of these properties.

Some comments on symmetric orbifolds of K3

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;  ℤ ). We comment on possible implications for Umbral moonshine and theories of AdS 3 gravity.

Shamit Kachru

Max Zimet

Heterotic sigma models on T 8 and the Borcherds automorphic form Φ 12

We determine the generating functions of 1/4 BPS dyons in a class of 4d N $$ \mathcal{N} $$ = 4 string vacua arising as CHL orbifolds of K 3 × T 2 , a classification of which has been recently completed. We show that all such generating functions obey some simple physical consistency conditions that are very often sufficient to fix them uniquely. The main constraint we impose is the absence of unphysical walls of marginal stability: discontinuities of 1/4 BPS degeneracies can only occur when 1/4 BPS dyons decay into pairs of 1/2 BPS states. Formally, these generating functions in spacetime can be described as multiplicative lifts of certain supersymmetric indices (twining genera) on the worldsheet of the corresponding nonlinear sigma model on K3. As a consequence, our procedure also leads to an explicit derivation of almost all of these twining genera. The worldsheet indices singled out in this way match precisely a set of functions of interest in moonshine, as predicted by a recent conjecture.

No more walls! A tale of modularity, symmetry, and wall crossing for 1/4 BPS dyons

We consider four dimensional CHL models with sixteen spacetime supersymmetries obtained from orbifolds of type IIA superstring on K3× T 2 by a ℤ N $$ {\mathrm{\mathbb{Z}}}_N $$ symmetry acting (possibly) non-geometrically on K3. We show that most of these models (in particular, for geometric symmetries) are self-dual under a weak-strong duality acting on the heterotic axio-dilaton modulus S by a “Fricke involution” S → −1 /NS . This is a novel symmetry of CHL models that lies outside of the standard S L 2 ℤ $$ \mathrm{S}\mathrm{L}\left(2,\mathrm{\mathbb{Z}}\right) $$ -symmetry of the parent theory, heterotic strings on T 6 . For self-dual models this implies that the lattice of purely electric charges is N -modular, i.e. isometric to its dual up to a rescaling of its quadratic form by N . We verify this prediction by determining the lattices of electric and magnetic charges in all relevant examples. We also calculate certain BPS-saturated couplings and verify that they are invariant under the Fricke S-duality. For CHL models that are not self-dual, the strong coupling limit is dual to type IIA compactified on T 6 / ℤ N $$ {T}^6/{\mathrm{\mathbb{Z}}}_N $$ , for some ℤ N $$ {\mathrm{\mathbb{Z}}}_N $$ -symmetry preserving half of the spacetime supersymmetries.