We revisit 6d (2,0) SCFTs of type and their realization in M-theory, focusing on absolute variants of these theories and on their global finite 0- and 2-form symmetries. We derive the 7d SymTFT capturing these global symmetries from M-theory, both from the point of view of the low-energy supergravity action on × and from M2- and M5-branes giving rise to its topological operators. Along the way, results by Gukov, Hsin, and Pei are extended by keeping track of an additional 7d gauge field, associated to the outer automorphism of the algebra. In particular, we find an interplay of non-invertible symmetries and mixed anomalies for absolute 6d (2,0) SCFTs with ≥ 1. We highlight several subtle points related to the non-orientability of , the half-integral -flux that threads it, and the non-commutativity of fluxes. All these also play an essential role in a holographic derivation of the anomaly polynomial of 6d (2,0) SCFTs.

Spain

Federico Bonetti

Sweden

Michele Zotto

France

Ruben Minasian

SymTFTs and non-invertible symmetries of 6d (2,0) SCFTs of type

Mario Marco

Italy

Michele Graffeo

United Kingdom

Andrea Sangiovanni

Erratum to: 5d Conformal Matter

Geometric engineering is a collection of tools developed to establish dictionaries between local singularities in string theory and (supersymmetric) quantum fields. Extended operators and defects, as well as their higher quantum numbers captured by topological symmetries, can be encoded within geometric engineering dictionaries. In this paper we revisit and clarify aspects of these techniques, with special emphasis on ’t Hooft anomalies, interpreted from the SymTFT perspective as obstructions to the existence of Neumann boundary conditions. These obstructions to gauging higher symmetries are captured via higher link correlators for the SymTFT on spheres. In this work, we give the geometric engineering counterpart of this construction in terms of higher links of topological membranes. We provide a consistency check in the context of 5D SCFTs with anomalous 1-form symmetries, where we give two independent derivations of the anomaly in terms of higher links, one purely field theoretical and the other purely geometrical. Along the way, we also recover the construction of non-invertible duality defects in 4D   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 SYM from a geometric engineering perspective.

Shani Meynet

United States

Robert Moscrop

Remarks on geometric engineering, symmetry TFTs and anomalies

Six-dimensional superconformal field theories (SCFTs) have an atomic classification in terms of elementary building blocks, conformal systems that generalize matter and can be fused together to form all known 6d SCFTs in terms of generalized 6d quivers. It is therefore natural to ask whether 5d SCFTs can be organized in a similar manner, as the outcome of fusions of certain elementary building blocks, which we call 5d conformal matter theories. In this project we begin exploring this idea and we give a systematic construction of 5d generalized “bifundamental” SCFTs, building from geometric engineering techniques in M-theory. In particular, we find several examples of (  <math> <msub> <mi>e</mi> <mn>6</mn> </msub> </math>  $$ {\mathfrak{e}}_6 $$ ,   <math> <msub> <mi>e</mi> <mn>6</mn> </msub> </math>  $$ {\mathfrak{e}}_6 $$ ), (  <math> <msub> <mi>e</mi> <mn>7</mn> </msub> </math>  $$ {\mathfrak{e}}_7 $$ ,   <math> <msub> <mi>e</mi> <mn>7</mn> </msub> </math>  $$ {\mathfrak{e}}_7 $$ ) and (  <math> <msub> <mi>e</mi> <mn>8</mn> </msub> </math>  $$ {\mathfrak{e}}_8 $$ ,   <math> <msub> <mi>e</mi> <mn>8</mn> </msub> </math>  $$ {\mathfrak{e}}_8 $$ ) 5d bifundamental SCFTs beyond the ones arising from (elementary) KK reductions of the 6d conformal matter theories. We show that these can be fused together giving rise to 5d SCFTs captured by 5d generalized linear quivers with exceptional gauge groups as nodes, and links given by 5d conformal matter. As a first application of these models we uncover a large class of novel 5d dualites, that generalize the well-known fiber/base dualities outside the toric realm.

5d Conformal matter

We compute the equivariant partition function of the six-dimensional M-string SCFTs on a background with the topology of a product of a two-dimensional torus and an ALE singularity. We determine the result by exploiting BPS strings probing the singularity, whose worldvolume theories we determine via a chain of string dualities. A distinguished feature we observe is that for this class of background the BPS strings’ worldsheet theories become relative field theories that are sensitive to finer discrete data generalizing to 6d the familiar choices of flat connections at infinity for instantons on ALE spaces. We test our proposal against a conjectural 6d   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = (1, 0) generalization of the Nekrasov master formula, as well as against known results on ALE partition functions in four dimensions.

CERN

Germany

Guglielmo Lockhart

The ALE partition functions of M-strings

We begin a study of the Higgs branch of six-dimensional (1, 0) little string theories governing the worldvolumes of heterotic ALE instantons. We give a description of this space by constructing the corresponding magnetic quiver. The latter is a three-dimensional   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 quiver gauge theory that flows in the infrared to a fixed point whose quantum corrected Coulomb branches is the Higgs branch of the six-dimensional theory of interest. We present results for both types of heterotic strings, and mostly for ℂ2 /ℤ k  ALE spaces. Our analysis is valid both in the absence and in the presence of small instantons. Along the way, we also describe small SO(32) instanton transitions in terms of the corresponding magnetic quiver, which parallels a similar treatment of the small E 8 instanton transitions in the context of the E 8 × E 8 heterotic string.

Marco Fazzi

Suvendu Giri

The Higgs branch of heterotic ALE instantons

This work is the second of a series of papers devoted to revisiting the properties of Heterotic string compactifications on ALE spaces. In this project we study the geometric counterpart in F-theory of the T-dualities between Heterotic ALE instantonic Little String Theories (LSTs) extending and generalising previous results on the subject by Aspinwall and Morrison. Since the T-dualities arise from a circle reduction one can exploit the duality between F-theory and M-theory to explore a larger moduli space, where T-dualities are realised as inequivalent elliptic fibrations of the same geometry. As expected from the Heterotic/F-theory duality the elliptic F-theory Calabi-Yau we consider admit a nested elliptic K3 fibration structure. This is central for our construction: the K3 fibrations determine the flavor groups and their global forms, and are the key to identify various T-dualities. We remark that this method works also more generally for LSTs arising from non-geometric Heterotic backgrounds. We study a first example in detail: a particularly exotic class of LSTs which are built from extremal K3 surfaces that admit flavor groups with maximal rank 18. We find all models are related by a so-called T-hexality (i.e. a 6-fold family of T-dualities) which we predict from the inequivalent elliptic fibrations of the extremal K3.

Muyang Liu

Paul-Konstantin Oehlmann

Back to heterotic strings on ALE spaces. Part II. Geometry of T-dual little strings

Quantum field theories with identical local dynamics can admit different choices of global structure, leading to different partition functions and spectra of extended operators. Such choices can be reformulated in terms of a topological field theory in one dimension higher, the symmetry TFT. In this paper we show that this TFT can be reconstructed from a careful analysis of the infrared Coulomb-like phases. In particular, the TFT matches between the UV and the IR. This provides a purely field theoretical counterpart of several recent results obtained via geometric engineering in various string/M/F theory setups for theories in four and five dimensions that we confirm and extend.

Iñaki García Etxebarria

Global structures from the infrared

It is well-known that six-dimensional superconformal field theories can be exploited to unravel interesting features of lower-dimensional theories obtained via compactifications. In this short note we discuss a new application of 6d (2,0) theories in constructing 4d theories with Kramers-Wannier-like non-invertible symmetries. Our methods allow to recover previously known results, as well as to exhibit infinitely many new examples of four dimensional theories with “M-ality” defects (arising from operations of order M generalizing dualities). In particular, we obtain examples of order M = p  k , where p &gt; 1 is a prime number and k is a positive integer.

Vladimir Bashmakov

Azeem Hasan

On the 6d origin of non-invertible symmetries in 4d

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.

Japan

Justin Kaidi

Non-invertible symmetries of class S theories