We establish a connection between three-dimensional self-mirror symmetric $$ \mathcal{N} $$ = 4 superconformal field theories (SCFTs) and time-reversal invariant topological quantum field theories (TQFTs) arising from universal mass deformations. Focusing on the Abelian case, the ultraviolet (UV) SCFT is characterized by the charge matrix $\textit{Q}$, while the infrared (IR) TQFT corresponds to an Abelian Chern-Simons theory with level matrix $\textit{K}$ = $\textit{QQ}$ . We derive constraints on the charge matrix for self-mirror symmetric SCFTs and demonstrate that the Coulomb and Higgs branch Hilbert series of these theories coincide. Additionally, we derive a general formula for the superconformal indices of Abelian $$ \mathcal{N} $$ = 4 SCFTs with arbitrary charge matrices. For SCFT with the constrained charge matrix, the superconformal index is argued to exhibit invariance under the inversion of fugacity associated with R-symmetry, providing further evidence of self-mirror symmetry. We explore various properties of time-reversal invariant Abelian Chern-Simons theories in detail and establish their connections to self-mirror symmetry in SCFTs from multiple perspectives. In particular, we introduce a quantity, dubbed Gauss generating function, which is real and thus invariant under complex conjugation for time-reversal symmetric TQFTs, in parallel with the superconformal index, which is invariant under the inversion of R-symmetry fugacity for self-mirror symmetric SCFTs.

United Kingdom

Hongliang Jiang

Time-reversal invariant TQFTs from self-mirror symmetric SCFTs

We ask when it is possible to construct arbitrary unitary multiplets of the superconformal algebra with eight Poincaré supercharges that are compatible with locality from (continuous deformations of) representations in free field theories. We answer this question in two, three, and five dimensions. In four dimensions, we resort to an intricate but self-consistent web of conjectures. If correct, these conjectures imply various new non-perturbative constraints on short multiplets in any local unitary 4d   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 superconformal field theory and on an unusual set of related vertex algebras. Throughout, we connect our results with properties of deformations in the space of theories.

Matthew Buican

From free fields to interacting SCFTs via representation theory

We consider a   <math> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </math>  $$ \frac{1}{2} $$ -BPS solution for a D3 brane probe in AdS5 × S 5 that has world-volume geometry of AdS3 × S 1. It intersects the boundary over a surface that represents a dimension 2 defect in the boundary   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 SYM theory. The effective action of the probe brane is proportional to the logarithmically divergent volume of AdS3 and may thus be interpreted as computing conformal anomaly of supersymmetric S 2 defect. The classical action scales as N. We compute the 1-loop correction to it due to quantum fluctuations of the D3 brane world-volume fields and compare the result to an earlier suggested expression for the defect anomaly. We also perform a similar analysis of a   <math> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </math>  $$ \frac{1}{2} $$ -BPS M5 brane probe solution in AdS7 × S 4 with the world-volume geometry of AdS5 × S 1 that represents a dimension 4 defect in the boundary (2,0) 6d theory. Here the classical M5 brane action computes the leading order N 2 term in a-anomaly of the supersymmetric S 4 defect. We perform a detailed computation of the 1-loop correction to the M5 brane effective action and thus provide a prediction for the subleading constant in the S 4 defect a-anomaly coefficient.

Arkady Tseytlin

On co-dimension 2 defect anomalies in   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 SYM and (2,0) theory via brane probes in AdS/CFT

We consider a family of Argyres-Douglas theories, which are 4D   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 strongly coupled superconformal field theories (SCFTs) but share many features with 4D   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 super-Yang-Mills theories. In particular, the two central charges of these theories are the same, namely a = c. We derive a simple and illuminating formula for the Schur index of these theories, which factorizes into the product of a Casimir term and a term referred to as the Schur partition function. While the former is controlled by the anomaly, the latter is identified with the vacuum character of the corresponding chiral algebra and is expected to satisfy the modular linear differential equation. Our simple expression for the Schur partition function, which can be regarded as the generalization of MacMahon’s generalized sum-of-divisor function, allows one to numerically compute the series expansions efficiently, and furthermore find the corresponding modular linear differential equation. In a special case where the chiral algebra is known, we are able to derive the corresponding modular linear differential equation using Zhu’s recursion relation. We further study the solutions to the modular linear differential equations and discuss their modular transformations. As an application, we study the high temperature limit or the Cardy-like limit of the Schur index using its simple expression and modular properties, thus shedding light on the 1/4-BPS microstates of genuine   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 SCFTs with a = c and their dual quantum gravity via the AdS/CFT correspondence.

Modularity in Argyres-Douglas theories with a = c

In quantum field theory above two spacetime dimensions, one is usually only able to construct exact operator maps from UV to IR of strongly coupled renormalization group flows for the most symmetry-protected observables. Famous examples include maps of chiral rings in 4D <math><mi>N</mi><mo>=</mo><mn>2</mn></math> supersymmetry. In this Letter, we construct the first nonperturbative UV-IR map for less protected operators: starting from a particularly “simple” UV strongly coupled non-Lagrangian 4D <math><mi>N</mi><mo>=</mo><mn>2</mn></math> quantum field theory, we show that a universal nonchiral quarter–Bogomol’nyi-Prasad-Sommerfield ring can be mapped exactly and bijectively to the IR. In particular, strongly coupled UV dynamics governing infinitely many null states manifest in the IR via Fermi statistics of free gauginos. Using the concept of arc space, this bijection allows us to compute the exact UV Macdonald index in the IR.

Chinmaya Bhargava

Exact Operator Map from Strong Coupling to Free Fields: Beyond Seiberg-Witten Theory

We study ultraviolet (UV) incarnations of deep infrared (IR) physics on the Coulomb branch of the simplest interacting 4D   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 superconformal field theory: the minimal Argyres-Douglas (MAD) theory. One of the most basic properties of the Coulomb branch is an emergent infinite-dimensional higher-spin symmetry. While the MAD theory is interacting and therefore does not have such a symmetry, we find UV operators that encode the emergent complex higher-spin symmetry on the Coulomb branch. Moreover, we show that cousins of these UV operators give rise to cousins of the IR higher-spin multiplets. In terms of superconformal representation theory, we are led to a conjecture on the exact spectrum of   <math> <msub> <mover> <mi>C</mi> <mo>¯</mo> </mover> <mrow> <mi>R</mi> <mo>,</mo> <mi>r</mi> <mfenced> <mi>j</mi> <mover> <mi>j</mi> <mo>¯</mo> </mover> </mfenced> </mrow> </msub> </math>  $$ {\overline{\mathcal{C}}}_{R,r\left(j,\overline{j}\right)} $$  multiplets in the MAD theory for all R, r, j, and   <math> <mover> <mi>j</mi> <mo>¯</mo> </mover> </math>  $$ \overline{j} $$  satisfying   <math> <mi>R</mi> <mo>+</mo> <mover> <mi>j</mi> <mo>¯</mo> </mover> <mo>−</mo> <mi>j</mi> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mn>0</mn> </math>  $$ R+\overline{j}-j+1=0 $$ , thereby making progress towards a full characterization of the protected spectrum. Along the way, we give a geometrical interpretation of these operators and include them in an extension of the Coulomb branch /   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 chiral operator correspondence.

United States

Argyres-Douglas avatars of Coulomb branch physics

Celestial holography provides a promising avenue to studying bulk scattering in flat spacetime from the perspective of boundary celestial conformal field theory (CCFT). A key ingredient in connecting the two sides is the celestial amplitude, which is given by the Mellin transform of momentum space scattering amplitude in energy. As such, celestial amplitudes can be identified with the correlation functions in celestial conformal field theory. In this paper, we introduce the further notion of celestial Mellin amplitude, which is given by the Mellin transform of celestial amplitude in coordinate. For technical reasons, we focus on the celestial Mellin amplitudes for scalar fields in three dimensional flat spacetime dual to 1D CCFT, and discuss the celestial Mellin block expansion. In particular, the poles of the celestial Mellin amplitude encode the scaling dimensions of the possible exchanged operators, while the residues there are related to the OPE coefficient squares in a linear and explicit way. We also compare the celestial Mellin amplitudes with the coefficient functions which can be obtained using inversion formulae. Finally, we make some comments about the possible generalizations of celestial Mellin amplitudes to higher dimensions.

Celestial Mellin amplitude

Despite the power of supersymmetry, finding exact closed-form expressions for the protected operator spectra of interacting superconformal field theories (SCFTs) is difficult. In this paper, we take a step towards a solution for the “simplest” interacting 4D   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 SCFT: the minimal Argyres-Douglas (MAD) theory. We present two results that go beyond the well-understood Coulomb branch and Schur sectors. First, we find the exact closed-form spectrum of multiplets containing operators that are chiral with respect to any   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 1 ⊂   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 superconformal subalgebra. We argue that this “full” chiral sector (FCS) is as simple as allowed by unitarity for a theory with a Coulomb branch and that, up to a rescaling of U(1) r  quantum numbers and the vanishing of a finite number of states, the MAD FCS is isospectral to the FCS of the free   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 Abelian gauge theory. In the language of superconformal representation theory, this leaves only the spectrum of the poorly understood   <math> <msub> <mover> <mi>C</mi> <mo>¯</mo> </mover> <mrow> <mi>R</mi> <mo>,</mo> <msub> <mi>r</mi> <mfenced> <mi>j</mi> <mover> <mi>j</mi> <mo>¯</mo> </mover> </mfenced> </msub> </mrow> </msub> </math>  $$ {\overline{\mathcal{C}}}_{R,{r}_{\left(j,\overline{j}\right)}} $$  multiplets to be determined. Our second result sheds light on these observables: we find an exact closed-form answer for the number of   <math> <msub> <mover> <mi>C</mi> <mo>¯</mo> </mover> <mrow> <mn>0</mn> <mo>,</mo> <msub> <mi>r</mi> <mfenced> <mi>j</mi> <mn>0</mn> </mfenced> </msub> </mrow> </msub> </math>  $$ {\overline{\mathcal{C}}}_{0,{r}_{\left(j,0\right)}} $$  multiplets, for any r and j, in the MAD theory. We argue that this sub-sector is also as simple as allowed by unitarity for a theory with a Coulomb branch and that there is a natural map to the corresponding sector of the free   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 Abelian gauge theory. These results motivate a conjecture on the full local operator algebra of the MAD theory.

On the protected spectrum of the minimal Argyres-Douglas theory

It was recently shown that the   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 string topological amplitudes in the heterotic weak coupling limit generate a six-dimensional Melvin space, providing a description of the Ω-background in string theory, where string propagation can be exactly studied. In this work, we generalise the analysis to the refined case of the Ω-background with two independent deformation parameters. The Melvin space is now ten-dimensional and is extended by an action on the internal K3 compactification manifold of the heterotic superstring, corresponding to an SU(2) R  rotation in the field theory description. We identify the class of heterotic topological amplitudes realising this background as the scattering of two anti-self-dual gravitons and arbitrary numbers of anti-self-dual graviphotons, self-dual vector fields of the dilaton multiplet, together with self-dual magnetic fluxes along the K3. In the field theory limit, our result correctly reproduces the perturbative part of the Nekrasov free energy in the case where both equivariant parameters are turned on.

Italy

Carlo Angelantonj

France

Sweden

Ignatios Antoniadis

Greece

Ioannis Florakis

Refined topological amplitudes from the Ω-background in string theory

Long ago, Argyres and Douglas discovered a particularly simple interacting 4D <math><mi>N</mi><mo>=</mo><mn>2</mn></math> superconformal field theory (SCFT) on the Coulomb branch of <math><mi>S</mi><mi>U</mi><mo>(</mo><mn>3</mn><mo>)</mo></math> <math><mi>N</mi><mo>=</mo><mn>2</mn></math> super Yang-Mills. Further hints of the theory’s simplicity arise due to the fact that it has the smallest possible value of the <math><mi>c</mi></math> central charge among unitary interacting <math><mi>N</mi><mo>=</mo><mn>2</mn></math> SCFTs. The main purpose of this paper is to uncover additional aspects of this minimal Argyres-Douglas (MAD) theory’s simplicity. In particular, we argue that: (1) the MAD theory shares an infinite set of large spin thresholds in part of its operator spectrum with the free <math><mi>N</mi><mo>=</mo><mn>2</mn></math> Maxwell theory (this data is therefore invariant under generic <math><mi>N</mi><mo>=</mo><mn>2</mn></math>-preserving renormalization group flows to the IR), and (2) the MAD theory has, at every order in the natural grading, the smallest number of “Schur” operators of any unitary <math><mi>N</mi><mo>=</mo><mn>2</mn></math> theory (interacting or free). We then show that property (1) has a suitable generalization for all <math><mo>(</mo><msub><mi>A</mi><mn>1</mn></msub><mo>,</mo><msub><mi>A</mi><mrow><mn>2</mn><mi>k</mi></mrow></msub><mo>)</mo></math> cousins of the MAD theory. In particular, the corresponding large spin thresholds encode generic renormalization group flows within this class. This construction therefore gives a different handle on these flows from the one provided by the Seiberg-Witten description. To emphasize the importance of these spin thresholds, we abstractly study theories with “enough matter” to form Higgs branches and argue that infinitely many spin thresholds are small or vanishing.