We explore BPS strings in supergravity theories in six-dimensions and related Swampland Conjectures. We first propose a general modular ansatz for bootstrapping elliptic genera of 2d worldvolume theories on strings in the 6d theories. By employing mirror symmetry on F-theory examples, we explicitly compute the elliptic genera and validate our ansatz. We extend this approach to investigate BPS strings and their spectrum in non-geometric 6d theories which have no known F-theory constructions, and confirm the Swampland conjectures, including the Weak Gravity Conjecture, Distance Conjecture, and Emergent String Conjecture. We also discuss tensionless little strings that emerge near infinite-distance limits of strong gauge coupling in the moduli space of certain special theories.

Japan

Hirotaka Hayashi

South Korea

Hee-Cheol Kim

Minsung Kim

Spectra of BPS strings in 6d supergravity and the Swampland

We discuss two physics-inspired approaches to derivation of the eigenfunctions and eigenvalues of Ruijsenaars-Schneider model. First approach which was recently proposed by the authors relies on the computations of superconformal indices of class $$ \mathcal{S} $$ 4 $$ \mathcal{N} $$ = 2 theories with the insertion of surface defects. Second approach uses computations of Nekrasov-Shatashvili limit of 5 $$ \mathcal{N} $$ = 1 instanton partition functions in the presence of co-dimension two defect. We compare results of these two approaches for the low-lying levels of Ruijsenaars-Schneider model. We also discuss different previously proposed exact quantization conditions for the Coulomb branch parameters of the instanton partition functions and their interpretations in terms of index calculations.

United States

United Kingdom

Anton Nedelin

Israel

Shlomo Razamat

On Ruijsenaars-Schneider spectrum from superconformal indices and ramified instantons

We propose new constraints for 6d (1, 0) supergravity theories based on consistency conditions on the Kähler moduli spaces of their 5d reductions. The requirement that both the metric and the BPS string tensions in the Kähler moduli space are positive imposes specific restrictions on the Chern-Simons coefficients in the 5d effective Lagrangians that are derived from the Kaluza-Klein reductions of 6d theories. Moreover, the emergence of local interacting 5d CFTs when the moduli space metric degenerates introduces additional constraints coming from the analysis of 5d SCFTs. Focusing on the moduli spaces of 6d supergravity theories without a tensor multiplet and their Higgsings, we show that these constraints require the presence of certain primary states in the 2d worldvolume CFTs on 1/2 BPS strings. We specifically analyze a class of SU(2) models and infinite families of U(1) models using these constraints, and demonstrate that the theories featuring a 1-form symmetry in their massless spectra, unless the 1-form symmetry is gauged, fail to satisfy the constraints and therefore belong to the Swampland.

Cumrun Vafa

Exploring new constraints on Kähler moduli space of 6d   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 1 supergravity

We propose two novel methods for computing the superconformal index of 5d superconformal field theories that cannot be described by conventional Lagrangian descriptions under mass deformations. The first approach involves the use of Higgs branch flows from UV Lagrangian theories, guided by transitions in 5-brane webs in Type IIB string theory. The second method employs the relationship between O7+-plane and O7 − -plane with eight D7-branes, which applies to particular non-Lagrangian theories realized by brane configurations involving an O7+-plane. As a concrete application of our method, we compute the superconformal indices for all known rank-1 non-Lagrangian theories, which we also use to identify flavor symmetries and their global forms at the conformal field theory (CFT) fixed points.

China

Sung-Soo Kim

Gabi Zafrir

Superconformal indices for non-Lagrangian theories in five dimensions

We discuss 4D Lagrangian descriptions, across dimensions IR duals, of compactifications of the 6D (D, D) minimal conformal matter theory on a sphere with arbitrary number of punctures and a particular value of flux as a gauge theory with a simple gauge group. The Lagrangian has the form of a “star shaped quiver” with the rank of the central node depending on the 6D theory and the number and type of punctures. Using this Lagrangian one can construct across dimensions duals for arbitrary compactifications (any, genus, any number and type of USp punctures, and any flux) of the (D, D) minimal conformal matter gauging only symmetries which are manifest in the ultraviolet.

Shlomo S. Razamat

Star Shaped Quivers in Four Dimensions

It has been known for some time that the cosmological bound on the invisible axion scale can be avoided by an early phase of strong QCD. While most approaches rely on theories where the strong coupling constant is determined through the expectation value of some scalar field, we show that this mechanism can also be implemented into the benchmark Kim-Shifman-Vainshtein-Zakharov and Dine-Fischler-Srednicki-Zhitnitsky models when the early phase of strong QCD emerges by the modification of the running coupling during inflation. For both models the physics that are responsible for making QCD strong do not displace the axions minimum by too much, so that the efficiency of the relaxation is controlled by parameters of the theory and the number of inflationary <math><mi>e</mi></math>-folds. In particular, we consider the case of very efficient relaxation where the axion abundance is dominated by inflationary quantum and postinflationary thermal fluctuations. Within this situation we identify the parameter space compatible with all cosmological constrains and derive conditions on the reheating temperature and the QCD scale during inflation that result in the axion making up all the dark matter. Because of duality below the Peccei-Quinn scale and a minor influence of the Kim-Shifman-Vainshtein-Zakharov and Dine-Fischler-Srednicki-Zhitnitsky fields on the running, our findings also apply to a minimal two-form implementation of the axion.

Germany

Emmanouil Koutsangelas

Removing the cosmological bound on the axion scale in the Kim-Shifman-Vainshtein-Zakharov and Dine-Fischler-Srednicki-Zhitnitsky models

We propose blowup equations for 6d little string theories which generalize Nakajima-Yoshioka’s blowup equations for the 4d/5d instanton partition functions on Omega background. We find that unlike the blowup equations for standard SQFTs, we need to sum over auxiliary magnetic fluxes on the blown-up ℙ1 for a non-dynamical 2-form gauge field which plays a role in canceling the mixed anomalies of the gauge symmetries. We demonstrate with explicit examples that the blowup equations, when combined with the modular properties, can be solved in order to determine the elliptic genera of little strings.

Yuji Sugimoto

Blowup equations for little strings

ℤ n  S-folds of 5d SCFTs, including T  N , which lead to brane webs with E 6,7,8 7-branes were discussed recently in [1, 2]. We generalize the construction to ‘fractional quotients’, which are based on ℤ n  actions linking multiple copies of the seed theory and lead to H 0,1,2 7-branes. We provide the holographic duals for both classes. This expands the space of explicitly known Type IIB AdS6 solutions by incorporating F-theory 7-branes of type E 6,7,8 and H 0,1,2, extending previous constructions for O7-planes. We discuss observables including the free energies and link the results to matrix model descriptions.

Switzerland

Italy

Fabio Apruzzi

Oren Bergman

Christoph Uhlemann

Generalized quotients and holographic duals for 5d S-fold SCFTs

We study the IR phases of 3D class R theories associated with closed non-hyperbolic 3-manifolds. Non-hyperbolic 3-manifolds can be obtained by performing Dehn fillings on 1-cusped hyperbolic 3-manifolds along exceptional slopes. In 3D-3D correspondence, the ‘exceptional’ Dehn filling corresponds to the gauging of an SU(2) flavor symmetry in a superconformal field theory associated with a 1-cusped 3-manifold with ‘small’ Chern-Simons levels. With several explicit examples, we analyze various interesting non-perturbative IR phenomena (such as spontaneous SUSY breaking, generation of mass gap and supersymmetry enhancement) from the ‘exceptional’ gaugings. Interestingly, distinguished features of the IR phases can be captured by simple topological properties of non-hyperbolic 3-manifolds. We also find that 3D class R theories associated with certain classes of atoroidal non-hyperbolic 3-manifolds always exhibit supersymmetry enhancement at low energy and actually flow to 3D rank-0   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 SCFTs with trivial vacuum moduli space.

Sunjin Choi

Dongmin Gang

Infrared phases of 3D class R theories

We explore the ℤ2,3,4,6 S-foldings of some 5d superconformal field theories from the (p, q) 5-brane web perspective. The S-folding involves both a spatial quotient and an SL(2, ℤ) transformation on 5-branes simultaneously. The ℤ2,3,4,6 S-foldings are achieved by the insertion of the D 4 , E 6 , E 7 , E 8 7-branes, respectively. The deficit angles and monodromies of these 7-branes are exactly those necessary for the S-foldings. We explore the details of the S-folding process, especially the enhancement of global flavor symmetry in various simple cases. The characteristic of the S-folding depends sharply on whether the fixed point of the discrete symmetry is at the center of a compact face (or surface), at a 5-brane, or at a crossing point of 5 branes. The analysis of the prepotential greatly supports this view of the discrete gauging.