We study the Higgs branch moduli space of certain 6D (1, 0) superconformal field theories (SCFTs) after gauging their <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi mathvariant="double-struck">Z</mi><mn>2</mn></msub></math> Green-Schwarz automorphism. We explain how to read the flavor symmetry of such SCFTs directly from the 6D construction, and we confirm the expectation by computing the Coulomb branch Hilbert series of their <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi mathvariant="double-struck">Z</mi><mn>2</mn></msub></math>-wreathed 3d <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="script">N</mi><mo>=</mo><mn>4</mn></math> magnetic quiver. To perform the latter computation, we explicitly introduce a methodology to determine such Hilbert series for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi mathvariant="double-struck">Z</mi><mn>2</mn></msub></math>-wreathed orthosymplectic quivers.

Germany

Craig Lawrie

Italy

Thekla Lepper

United States

Alessandro Mininno

Discrete gauging of 6D SCFTs and wreathed 3D <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="script">N</mi><mo>=</mo><mn>4</mn></math> quivers

We present an algorithm to extract the Coulomb branch Hasse diagram of orthosymplectic three-dimensional (3D) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="script">N</mi><mo>=</mo><mn>4</mn></math> quiver gauge theories. The algorithm systematically predicts all descendant theories arising from Coulomb branch Higgsing, thereby detailing the stratification of the symplectic singularity defined by the initial Coulomb branch. Leveraging the Lie algebra isomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi mathvariant="fraktur">su</mi><mo stretchy="false">(</mo><mn>4</mn><mo stretchy="false">)</mo><mo>≅</mo><mi mathvariant="fraktur">so</mi><mo stretchy="false">(</mo><mn>6</mn><mo stretchy="false">)</mo></mrow></math>, we validate our algorithm via the 3D mirror of the 4D theories of class <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="script">S</mi></math> of such type. This comparison involves moduli spaces that admit both orthosymplectic and unitary quiver realizations, the latter being well understood via standard techniques such as decay and fission. Higgsing on the Coulomb branch of the 3D mirror or the magnetic quiver translates to Higgs branch renormalization group flows of the corresponding higher-dimensional superconformal field theories. Thus, we benchmark our method via Higgsing 6D <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="script">N</mi><mo>=</mo><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></math> D-type orbi-instanton theories, predicting novel Higgsing patterns involving products of interacting fixed points, and class <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="script">S</mi></math> theories of type <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="fraktur">so</mi><mo stretchy="false">(</mo><mn>2</mn><mi>N</mi><mo stretchy="false">)</mo></math>, demonstrating Higgsing to products of theories of types specified by Levi subalgebras of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="fraktur">so</mi><mo stretchy="false">(</mo><mn>2</mn><mi>N</mi><mo stretchy="false">)</mo></math>.

Lorenzo Mansi

Austria

Marcus Sperling

United Kingdom

Zhenghao Zhong

Pathway to decay and fission of orthosymplectic quiver theories

We explore the consequence of generalized symmetries in four-dimensional <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi mathvariant="script">N</mi><mo>=</mo><mn>1</mn></mrow></math> superconformal field theories. First, we classify all possible supersymmetric gauge theories with a simple gauge group that have a nontrivial one-form symmetry and flows to a superconformal field theory. Upon identifying unbroken discrete zero-form symmetries from the Adler-Bell-Jackiw (ABJ) anomaly, we find that many of these theories have mixed zero-form or one-form ’t Hooft anomalies. Then we classify the relevant deformations of these SCFTs that preserve the anomaly. From this mixed anomaly together with the anomalies of the discrete zero-form symmetries, we find obstructions for the relevant deformations of these SCFTs to flow to a trivially gapped phase. We also study non-Lagrangian SCFTs formed by gauging copies of Argyres-Douglas theories and constrain their deformations. In particular, we explore a new duality between the diagonal gauging of two <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi mathvariant="script">D</mi><mn>3</mn></msub><mo stretchy="false">(</mo><mi>S</mi><mi>U</mi><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></math> theories and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>S</mi><mi>U</mi><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">)</mo></math> gauge theory with two adjoints. We also repeat our analysis for a host of nonsupersymmetric gauge theories having nontrivial one-form symmetry including examples that appear to flow to Bank-Zaks type CFTs.

Monica Jinwoo Kang

South Korea

Ki-Hong Lee

Jaewon Song

Generalized symmetry constraints on deformed 4D SCFTs

We study a landscape of four-dimensional <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="script">N</mi><mo>=</mo><mn>1</mn></math> superconformal field theories (SCFTs) with identical central charges. These theories are obtained by renormalization group flows triggered by supersymmetry-preserving superpotential deformations of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="script">N</mi><mo>=</mo><mn>1</mn></math> gauging of the flavor symmetry of a collection of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="script">N</mi><mo>=</mo><mn>2</mn></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi mathvariant="script">D</mi><mi>p</mi></msub><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></math> Argyres-Douglas SCFTs. In this work, we focus on the fixed points in the landscape of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>S</mi><mi>U</mi><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo></math> gauging of three copies of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi mathvariant="script">D</mi><mn>2</mn></msub><mo stretchy="false">(</mo><mi>S</mi><mi>U</mi><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><msub><mi>H</mi><mn>2</mn></msub></math> theory together with an adjoint-valued chiral multiplet. We catalog the network of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>a</mi><mo>=</mo><mi>c</mi></math> fixed points, and, along the way, we find a variety of dualities and instances of supersymmetry enhancement.

Israel

Landscape of 4d <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="script">N</mi><mo>=</mo><mn>1</mn></math> SCFTs with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>a</mi><mo>=</mo><mi>c</mi></math>

We detail the Higgs branches of 6D (1, 0) superconformal field theories (SCFTs) and little string theories (LSTs) that exhibit supersymmetry-enhancing Higgs branch renormalization group flows to the 6D (2, 0) SCFTs and LSTs of type DE. Generically, such theories are geometrically engineered in F-theory via a configuration of (<math><mo>−</mo><mn>2</mn></math>)-curves, arranged in an (affine) DE-type Dynkin diagram, and supporting special unitary gauge algebras; this describes the effective field theory on the tensor branch of the SCFT. For the Higgsable to D-type (2, 0) SCFTs/LSTs, there generically also exists a type IIA brane description, involving a Neveu-Schwarz orientifold plane, which allows for the derivation of a magnetic quiver for the Higgs branch. These are 3D <math><mi>N</mi><mo>=</mo><mn>4</mn></math> unitary-orthosymplectic quivers whose Coulomb branch is isomorphic to the Higgs branch of the 6D theories. From this magnetic quiver, together with an extended quiver subtraction algorithm that we explain, the foliation structure of the Higgs branch as a symplectic singularity is unveiled. For this class of 6D SCFTs, we observe a simple rule, which we refer to as “slice subtraction,” to read off the transverse slice in the foliation from the tensor branch. Based on this slice subtraction observation, we conjecture the transverse slices in the Higgsable to E-type (2, 0) Hasse diagram, where the SCFTs lack any known magnetic quiver for their Higgs branches.

Higgs branch of 6D (1, 0) SCFTs and little string theories with Dynkin DE-type SUSY enhancement

Many six-dimensional (1, 0) superconformal field theories (SCFTs) are known to fall into families labeled by nilpotent orbits of certain simple Lie algebras. For each of the three infinite series of such families, we show that the anomalies for the continuous zero-form global symmetries of a theory labelled by a nilpotent orbit <math><mi>O</mi></math> of <math><mi>g</mi></math> can be determined from the anomalies of the theory associated to the trivial nilpotent orbit (the parent theory), together with the data of <math><mi>O</mi></math>. In particular, knowledge of the tensor branch field theory is bypassed completely. We show that the known anomalies, previously determined from the geometric/atomic construction, are reproduced by analyzing the Nambu-Goldstone modes inside of the moment map associated to the <math><mi>g</mi></math> flavor symmetry of the parent SCFT. This provides further evidence for the physics underlying the labeling of the SCFTs by nilpotent orbits. We remark on some consequences, such as the reinterpretation of the 6D <math><mi>a</mi></math>-theorem for such SCFTs in terms of group theory.

Florent Baume

Bestiary of 6D (1, 0) SCFTs: Nilpotent orbits and anomalies

We study the Higgs branches of the 6D (1, 0) little string theories that live on the world volume of NS5-branes probing an ADE-singularity in the heterotic <math><msub><mi>E</mi><mn>8</mn></msub><mo>×</mo><msub><mi>E</mi><mn>8</mn></msub></math> and <math><mtext>Spin</mtext><mo>(</mo><mn>32</mn><mo>)</mo><mo>/</mo><msub><mi>Z</mi><mn>2</mn></msub></math> string theories. On the <math><msub><mi>E</mi><mn>8</mn></msub><mo>×</mo><msub><mi>E</mi><mn>8</mn></msub></math> side, such little string theories (LSTs) are obtained via fusion of orbi-instanton SCFTs. For the <math><msup><mi>C</mi><mn>2</mn></msup><mo>/</mo><msub><mi>Z</mi><mi>K</mi></msub></math> orbifolds, we determine a magnetic quiver for the Higgs branch from the alternative type IIA brane system engineering the LST; we show that the magnetic quiver obtained in this way is the same as the Coulomb gauging of the 3D mirrors associated with the orbi-instanton building blocks. Using quiver subtraction, we determine the Hasse diagram of Higgs branch renormalization group flows between the LSTs, and we analyze how the structure constants of the generalized global symmetries vary along the edges of the Hasse diagram. From the Hasse diagram of the Higgs branch, we are immediately able to identify LSTs with the same T-duality-invariant properties, and thus propose candidate T-dual pairs. We perform a similar analysis of the Higgs branch Hasse diagram and putative T-dual families for particular <math><msub><mi>E</mi><mn>6</mn></msub></math>-orbifold LSTs by taking advantage of a duality between a rank zero orbi-instanton theory and a rank one conformal matter theory.

Higgs branch of heterotic little string theories: Hasse diagrams and generalized symmetries

Within the framework of relative and absolute quantum field theories (QFTs), we present a general formalism for understanding polarizations of the intermediate defect group and constructing noninvertible duality defects in theories in <math><mrow><mn>2</mn><mi>k</mi></mrow></math> spacetime dimensions with self-dual gauge fields. We introduce the polarization pair, which fully specifies absolute QFTs as far as their (<math><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></math>)-form defect groups are concerned, including their (<math><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></math>)-form symmetries, global structures (including discrete <math><mi>θ</mi></math>-angle), and local counterterms. Using the associated symmetry topological field theory (TFT), we show that the polarization pair is capable of succinctly describing topological manipulations, e.g., gauging (<math><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></math>)-form global symmetries and stacking counterterms, of absolute QFTs. Furthermore, automorphisms of the (<math><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></math>)-form charge lattice naturally act on polarization pairs via their action on the defect group; they can be viewed as dualities between absolute QFTs descending from the same relative QFT. Using this formalism, we present a prescription for building noninvertible symmetries of absolute QFTs. A large class of known examples, e.g., noninvertible defects in 4D <math><mi>N</mi><mo>=</mo><mn>4</mn></math> super-Yang-Mills, can be reformulated via this prescription. As another class of examples, we identify and investigate in detail a family of noninvertible duality defects in 6D superconformal field theories (SCFTs), including from the perspective of the symmetry TFT derived from type IIB string theory.

Xingyang Yu

Hao Y. Zhang

Intermediate defect groups, polarization pairs, and noninvertible duality defects

We discover a four-dimensional <math><mi>N</mi><mo>=</mo><mn>1</mn></math> supersymmetric field theory that is dual to the <math><mi>N</mi><mo>=</mo><mn>4</mn></math> super Yang-Mills theory with gauge group <math><mrow><mi>SU</mi><mo>(</mo><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow></math> for each <math><mi>n</mi></math>. The dual theory is constructed through the diagonal gauging of the <math><mrow><mi>SU</mi><mo>(</mo><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow></math> flavor symmetry of three copies of a strongly coupled superconformal field theory (SCFT) of Argyres-Douglas type. We find that this theory flows in the infrared to a strongly coupled <math><mi>N</mi><mo>=</mo><mn>1</mn></math> SCFT that lies on the same conformal manifold as <math><mi>N</mi><mo>=</mo><mn>4</mn></math> super Yang-Mills with gauge group <math><mrow><mi>SU</mi><mo>(</mo><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow></math>. Our construction provides a hint on why certain <math><mi>N</mi><mo>=</mo><mn>1</mn></math>, 2 SCFTs have identical central charges (<math><mi>a</mi><mo>=</mo><mi>c</mi></math>).

Emergent <math><mi>N</mi><mo>=</mo><mn>4</mn></math> Supersymmetry from <math><mi>N</mi><mo>=</mo><mn>1</mn></math>

Nearly all 6D superconformal field theories (SCFTs) have a partial tensor branch description in terms of a generalized quiver gauge theory consisting of a long one-dimensional spine of quiver nodes with links given by conformal matter; a strongly coupled generalization of a bifundamental hypermultiplet. For theories obtained from M5-branes probing an ADE singularity, this was recently leveraged to extract a protected large R-charge subsector of operators, with operator mixing controlled at leading order in an inverse large R-charge expansion by an integrable spin s Heisenberg spin chain, where s is determined by the <math><mrow><mi>su</mi></mrow><msub><mrow><mo>(</mo><mn>2</mn><mo>)</mo></mrow><mrow><mi>R</mi></mrow></msub></math> R-symmetry representation of the conformal matter operator. In this work, we show that this same structure extends to the full superconformal algebra <math><mrow><mi>osp</mi></mrow><mo>(</mo><mn>6</mn><mo>,</mo><mn>2</mn><mo>|</mo><mn>1</mn><mo>)</mo></math>. In particular, we determine the corresponding Bethe ansatz equations which govern this super-spin chain, as well as distinguished subsectors which close under operator mixing. Similar considerations extend to 6D little string theories (LSTs) and 4D <math><mi>N</mi><mo>=</mo><mn>2</mn></math> SCFTs with the same generalized quiver structures.