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 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

Israel

Gabi Zafrir

Superconformal indices for non-Lagrangian theories in five 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

We generalize Nakajima-Yoshioka’s blowup formula to calculate the partition functions counting the spectrum of bound states to half-BPS Wilson loop operators in 5d (and 6d) supersymmetric field theories. The partition function in the presence of a Wilson loop operator on the Ω-background is factorized when put on the blowup   <math> <mover> <mi>ℂ</mi> <mo>̂</mo> </mover> </math>  $$ \hat{\mathbb{C}} $$  2 into two Wilson loop partition functions under the localization. This structure provides a set of blowup equations for Wilson loop operators. We explain how to formulate the blowup equations and solve them to compute the partition functions of Wilson loop operators. We test this idea by explicitly calculating the Wilson loop partition functions in various 5d/6d field theories and comparing them against known results and expected dualities.

5d/6d Wilson loops from blowups

We propose a systematic approach to computing the BPS spectrum of any 5d/6d supersymmetric quantum field theory in Coulomb phases, which admits either gauge theory descriptions or geometric descriptions, based on the Nakajima-Yoshioka’s blowup equations. We provide a significant generalization of the blowup equation approach in terms of both properly quantized magnetic fluxes on the blowup   <math> <mover> <mi>ℂ</mi> <mo>̂</mo> </mover> </math>  $$ \hat{\mathrm{\mathbb{C}}} $$  2 and the effective prepotential for 5d/6d field theories on the Omega background which is uniquely determined by the Chern-Simons couplings on their Coulomb branches. We employ our method to compute BPS spectra of all rank-1 and rank-2 5d Kaluza-Klein (KK) theories descending from 6d   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = (1, 0) superconformal field theories (SCFTs) compactified on a circle with/without twist. We also discuss various 5d SCFTs and KK theories of higher ranks, which include a few exotic cases such as new rank-1 and rank-2 5d SCFTs engineered with frozen singularity as well as the 5d SU(3)8 gauge theory currently having neither a brane web nor a smooth shrinkable geometric description. The results serve as non-trivial checks for a large class of non-trivial dualities among 5d theories and also as independent evidences for the existence of certain exotic theories.

Ki-Hong Lee

Bootstrapping BPS spectra of 5d/6d field theories

We compute the partition function for 6d   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 1 SO(2N) gauge theories compactified on a circle with ℤ2 outer automorphism twist. We perform the computation based on 5-brane webs with two O5-planes using topological vertex with two O5-planes. As representative examples, we consider 6d SO(8) and SU(3) gauge theories with ℤ2 twist. We confirm that these partition functions obtained from the topological vertex with O5-planes indeed agree with the elliptic genus computations.