A uniform construction of non-supersymmetric 0-, 4-, 6- and 7-branes in heterotic string theory was announced and outlined in our letter [61]. In this full paper, we provide details on their properties. Among other things, we discuss the charges carried by the branes, their topological and dynamical stability, the exact worldsheet descriptions of their near-horizon regions, and the relationship of the branes to the mathematical notion of topological modular forms.

Japan

Justin Kaidi

Yuji Tachikawa

Kazuya Yonekura

On non-supersymmetric heterotic branes

It is known that the ’t Hooft anomalies of invertible global symmetries can be characterized by an invertible TQFT in one higher dimension. The analogous statement remains to be understood for non-invertible symmetries. In this note we discuss how the linking invariants in a non-invertible TQFT known as the Symmetry TFT (SymTFT) can be used as a diagnostic for ’t Hooft anomalies of non-invertible symmetries. When the non-invertible symmetry is non-intrinsically non-invertible, and hence the SymTFT is a Dijkgraaf-Witten model, the linking invariants can be computed explicitly. We illustrate this proposal through the examples of the abelian Higgs model in 2d, as well as adjoint QCD and   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 super Yang-Mills in 4d. We also comment on how the ’t Hooft anomalies of non-invertible symmetries impose new constraints on the dynamics.

United States

Emily Nardoni

Gabi Zafrir

Yunqin Zheng

Symmetry TFTs and anomalies of non-invertible symmetries

The common statement that any consistent quantum gravity theory contains dynamical objects with all possible charges suggests that there are still a number of hitherto-unidentified branes in string theory. Here we give the exact worldsheet description of near-horizon limits of nonsupersymmetric <math><mi>p</mi></math>-branes in ten-dimensional <math><mtext>Spin</mtext><mo>(</mo><mn>32</mn><mo>)</mo><mo>/</mo><msub><mi>Z</mi><mn>2</mn></msub></math> or <math><mo>(</mo><msub><mi>E</mi><mn>8</mn></msub><mo>×</mo><msub><mi>E</mi><mn>8</mn></msub><mo>)</mo><mo>⋊</mo><msub><mi>Z</mi><mn>2</mn></msub></math> heterotic superstring theories for <math><mi>p</mi><mo>=</mo><mn>7</mn></math>, 6, 4, 0.

Kantaro Ohmori

Nonsupersymmetric Heterotic Branes

We study the non-invertible symmetries of class   <math> <mi>S</mi> </math>  $$ \mathcal{S} $$  theories obtained by compactifying the type   <math> <msub> <mi>a</mi> <mrow> <mi>p</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </math>  $$ {\mathfrak{a}}_{p-1} $$  6d (2,0) theory on a genus g Riemann surface with no punctures. After setting up the general framework, we describe how such symmetries can be classified up to genus 5. Of central interest to us is the question of whether a non-invertible symmetry is intrinsic, i.e. whether it can be related to an invertible symmetry by discrete gauging. We then describe the higher-dimensional origin of our results, and explain how the Anomaly and Symmetry TFTs, as well as N-ality defects, of class   <math> <mi>S</mi> </math>  $$ \mathcal{S} $$  theories can be obtained from compactification of a 7d Chern-Simons theory. Interestingly, we find that the Symmetry TFT for theories with intrinsically non-invertible symmetries can only be obtained by coupling the 7d Chern-Simons theory to topological gravity.

Sweden

Vladimir Bashmakov

Michele Zotto

Azeem Hasan

Non-invertible symmetries of class S theories

Abstract The theory of topological modular forms (TMF) predicts that elliptic genera of physical theories satisfy a certain divisibility property, determined by the theory’s gravitational anomaly. In this note we verify this prediction in Duncan’s supermoonshine module, as well as in tensor products and orbifolds thereof. Along the way we develop machinery for computing the elliptic genera of general alternating orbifolds and discuss the relation of this construction to the elusive “periodicity class” of TMF.

Jan Albert

Ying-Hsuan Lin

Topological modularity of supermoonshine What is an elliptic object? What is an elliptic object? Elliptic Cohomology Global Anomalies in String Theory

It is expected on general grounds that the moduli space of 4d   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 3 theories is of the form ℂ3r  /Γ, with r the rank and Γ a crystallographic complex reflection group (CCRG). As in the case of Lie algebras, the space of CCRGs consists of several infinite families, together with some exceptionals. To date, no 4d   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 3 theory with moduli space labelled by an exceptional CCRG (excluding Weyl groups) has been identified. In this work we show that the 4d   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 3 theories proposed in [1], constructed via non-geometric quotients of type-  <math> <mi>e</mi> </math>  $$ \mathfrak{e} $$  6d (2,0) theories, realize nearly all such exceptional moduli spaces. In addition, we introduce an extension of this construction to allow for twists and quotients by outer automorphism symmetries. This gives new examples of 4d   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 3 theories going beyond simple S-folds.

United Kingdom

Mario Martone

Exceptional moduli spaces for exceptional   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 3 theories

Non-invertible symmetries have recently been understood to provide interesting constraints on RG flows of QFTs. In this work, we show how non-invertible symmetries can also be used to generate entirely new RG flows, by means of so-called non-invertible twisted compactification. We illustrate the idea in the example of twisted compactifications of 4d   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 super-Yang-Mills (SYM) to three dimensions. After giving a catalogue of non-invertible symmetries descending from Montonen-Olive duality transformations of 4d   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 SYM, we show that twisted compactification by non-invertible symmetries can be used to obtain 3d   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 6 theories which appear otherwise unreachable if one restricts to twists by invertible symmetries.

Non-invertible symmetries of   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 SYM and twisted compactification

There is a well-known map from 4d   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 superconformal field theories (SCFTs) to 2d vertex operator algebras (VOAs). The 4d Schur index corresponds to the VOA vacuum character, and must be a solution with integral coefficients of a modular differential equation. This suggests a classification program for 4d   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 SCFTs that starts with modular differential equations and proceeds by imposing all known constraints that follow from the 4d → 2d map. This program becomes fully algorithmic once one specifies the order of the modular differential equation and the rank (complex dimension of the Coulomb branch) of the   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 theory. As a proof of concept, we apply the algorithm to the study of rank-two   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 SCFTs whose Schur indices satisfy a fourth-order untwisted modular differential equation. Scanning over a large number of putative cases, only 15 satisfy all of the constraints imposed by our algorithm, six of which correspond to known 4d SCFTs. More sophisticated constraints can be used to argue against the existence of the remaining nine cases. Altogether, this indicates that our knowledge of such rank-two SCFTs is surprisingly complete.

Leonardo Rastelli

Mitch Weaver

Needles in a haystack. An algorithmic approach to the classification of 4d   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 SCFTs

We introduce a class of noninvertible topological defects in <math><mrow><mo>(</mo><mn>3</mn><mo>+</mo><mn>1</mn><mo>)</mo><mstyle><mi>D</mi></mstyle></mrow></math> gauge theories whose fusion rules are the higher-dimensional analogs of those of the Kramers-Wannier defect in the <math><mrow><mo>(</mo><mn>1</mn><mo>+</mo><mn>1</mn><mo>)</mo><mstyle><mi>D</mi></mstyle></mrow></math> critical Ising model. As in the lower-dimensional case, the presence of such noninvertible defects implies self-duality under a particular gauging of their discrete (higher-form) symmetries. Examples of theories with such a defect include SO(3) Yang-Mills (YM) at <math><mrow><mi>θ</mi><mo>=</mo><mi>π</mi></mrow></math>, <math><mrow><mi>N</mi><mo>=</mo><mn>1</mn></mrow></math> SO(3) super YM, and <math><mrow><mi>N</mi><mo>=</mo><mn>4</mn></mrow></math> SU(2) super YM at <math><mi>τ</mi><mo>=</mo><mi>i</mi></math>. We also introduce an analogous construction in <math><mrow><mo>(</mo><mn>2</mn><mo>+</mo><mn>1</mn><mo>)</mo><mstyle><mi>D</mi></mstyle></mrow></math>, and give a number of examples in Chern-Simons–matter theories.

Kramers-Wannier-like Duality Defects in <math><mrow><mo>(</mo><mn>3</mn><mo>+</mo><mn>1</mn><mo>)</mo><mstyle><mi>D</mi></mstyle></mrow></math> Gauge Theories

In this work we revisit the “holomorphic modular bootstrap”, i.e. the classification of rational conformal field theories via an analysis of the modular differential equations satisfied by their characters. By making use of the representation theory of PSL(2, ℤ n ), we describe a method to classify allowed central charges and weights (c, h  i ) for theories with any number of characters d. This allows us to avoid various bottlenecks encountered previously in the literature, and leads to a classification of consistent characters up to d = 5 whose modular differential equations are uniquely fixed in terms of (c, h  i ). In the process, we identify the full set of constraints on the allowed values of the Wronskian index for fixed d ≤ 5.