We revisit the instanton partition function for 5d $$ \mathcal{N} $$ = 1 SO($\textit{N}$) gauge theories compactified on S , computed from the topological vertex formalism with the O-vertex based on a 5-brane web diagram with an O5-plane. We introduce an identity that enables us to rewrite the unrefined partition function into a new expression in terms of the Nekrasov factors summed over Young diagrams, which can be interpreted as the freezing of an O7-plane. Based on this, we propose topological vertex formalism with an O7 -plane.

China

Sung-Soo Kim

Xiaobin Li

Futoshi Yagi

Rui-Dong Zhu

O-vertex, O7

We report a novel BPS jumping phenomenon of 5d   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 1 supersymmetric gauge theories whose brane configuration is equipped with an O7-plane. The study of the relation between O7+-plane and O7 − -plane reveals that such BPS jumps take place when the Higgsing is triggered near the O7-plane upon a particular parameter tuning of the theories. We propose two types of gauge theories whose BPS spectra jump. One is the SU(2N + 8) gauge theory with a symmetric hypermultiplet converted to the SU(2N) gauge theory with an antisymmetric hypermultiplet. The other is pure SO(2N + 8) gauge theory jumping to pure Sp(N) gauge theory. We explicitly confirm our proposal through the (un)refined instanton partition functions. Furthermore, we discuss feasible generalizations involving an Op-plane for supersymmetric gauge theories of eight supercharges in four and three dimensions (p = 6, 5 respectively).

Satoshi Nawata

Freezing and BPS jumping

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.

South Korea

Hee-Cheol Kim

Minsung Kim

Israel

Gabi Zafrir

Superconformal indices for non-Lagrangian theories in five dimensions

We construct Seiberg-Witten curves for 5d   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 1 gauge theories whose Type IIB 5-brane configuration involves an O7-plane and discuss an intriguing relation between theories with an O7+-plane and those with an O7 − -plane and 8 D7-branes. We claim that 5-brane configurations with an O7+-plane can be effectively understood as 5-brane configurations with a set of an O7 − -plane and eight D7-branes with some special tuning of their masses such that the D7-branes are frozen at the O7 − -plane. We check this equivalence between SU(N) gauge theory with a symmetric hypermultiplet and SU(N) gauge theory with an antisymmetric with 8 fundamentals, and also between SO(2N) gauge theory and Sp(N) gauge theory with eight fundamentals. We also compute the Seiberg-Witten curves for non-Lagrangian theories with a symmetric hypermultiplet, which includes the local ℙ2 theory with an adjoint.

Japan

Hirotaka Hayashi

Kimyeong Lee

Seiberg-Witten curves with O7 ± -planes

We propose brane web configurations for D-type and E-type   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  =(1,0) little string theories based on a trivalent or quadrivalent gluing of 5-brane web diagrams. Tri-/quadri-valent gluing is a powerful way of computing 5d/6d partition functions for supersymmetric gauge theories based on the topological vertex. We generalize the gluing techniques to little string theories by introducing a new compact direction and compute their supersymmetric partition functions on Omega-deformed ℝ4 ×T 2. As concrete examples, we consider little string theories arising from Type IIB NS5-branes probing D 4 or D 5 singularity. Their effective gauge theory descriptions as the affine D 4 or D 5 quiver gauge theory can be realized with quadrivalent or trivalent gluing, respectively. Based on these gluings of 5-brane webs, we compute their refined partition functions and compare them with the known results. We extend the computation of the partition function to little string theory engineered from IIB NS5-branes probing E 6 singularity based on a trivalent gluing. We also discuss the generalization to higher rank cases and the symmetries of the partition functions.

Xing-Yue Wei

Yuji Sugimoto

DE-type little strings from glued brane webs

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 refined topological vertex formalism for 5-brane systems with ON-planes by introducing a new vertex associated with reflection over an ON-plane, which gives rise to new vertex and edge factors. We test our proposal against various 5d 𝒩 = 1 gauge theories which can be realized as 5-brane webs with ON-planes, which include D-type quiver theories. In particular, we compute the refined partition functions for 6d E-string theory on a circle as well as 5d SU(3) theory at the Chern-Simons level 9, which can be realized as 5-brane webs with two ON-planes. Our results completely agree with the known results.

Refined topological vertex with ON-planes

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.

S-foldings of 5d SCFTs

We study the Higgs branch of the SCFT limit of 5d SO(6) and SO(8) gauge theory with hypermultiplets in the spinor and vector representations. In the case of SO(6) gauge theories, we contrast the magnetic quivers obtained with those of SU(4) gauge theory with hypermultiplets in the fundamental and second rank antisymmetric representations. Since SU(4) gauge theories admit several different values of the Chern-Simons level, we make some observations about how to distinguish those theories from the brane webs of the SO(6) theories. In the case of SO(8) gauge theories, we use SO(8) triality to propose (naively) inequivalent magnetic quivers, which will turn out to have the same moduli spaces of vacua, at least locally around their most singular loci. We encounter several interesting new phenomena occurring in the magnetic quivers, such as hypermultiplets between neighbouring symplectic gauge nodes and matter in two-index representations of unitary nodes. We also give a prescription for computing the local Coulomb branch Hilbert series for quivers involving bad USp(2) gauge nodes.

United Kingdom

Mohammad Akhond

Federico Carta

Siddharth Dwivedi

Exploring the orthosymplectic zoo

We consider type IIB 5-brane web diagrams for a 5D <math><mi>S</mi><mi>p</mi><mo>(</mo><mi>N</mi><mo>)</mo></math> gauge theory with an antisymmetric hypermultiplet and <math><msub><mi>N</mi><mi>f</mi></msub></math> fundamental hypermultiplets. The corresponding 5-branes can be obtained by Higgsing a 5-brane web for quiver gauge theory. We use the refined topological vertex formalism to compute Nekrasov partition functions of 5D <math><mi>S</mi><mi>p</mi><mo>(</mo><mn>2</mn><mo>)</mo></math> theories with one antisymmetric hypermultiplet and flavors. Our results agree with the known results obtained from the Atiyah-Drinfeld-Hitchin-Manin method. We also discuss a particular tuning of Kähler parameters associated with this Higgsing.