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 study the index of 4d <math><mi>N</mi><mo>=</mo><mn>4</mn></math> Yang-Mills theory with <math><mi>U</mi><mo>(</mo><mi>N</mi><mo>)</mo></math> gauge group, focusing on the physics of the dual Bogomolny-Prasad-Sommerfield black holes in <math><msub><mi>AdS</mi><mn>5</mn></msub><mo>×</mo><msup><mi>S</mi><mn>5</mn></msup></math>. Certain aspects of these black holes can be studied from finite <math><mi>N</mi></math> indices with reasonably large <math><msup><mi>N</mi><mn>2</mn></msup></math>. We make numerical studies of the index for <math><mi>N</mi><mo>≤</mo><mn>6</mn></math>, by expanding it up to reasonably high orders in the fugacity. The entropy of the index agrees very well with the Bekenstein-Hawking entropy of the dual black holes, say at <math><msup><mi>N</mi><mn>2</mn></msup><mo>=</mo><mn>25</mn></math> or 36. Our data clarifies and supports the recent ideas which allowed analytic studies of these black holes from the index, such as the complex saddle points of the Legendre transformation and the oscillating signs in the index. In particular, the complex saddle points naturally explain the <math><mfrac><mn>1</mn><mi>N</mi></mfrac></math>-subleading oscillating patterns of the index. We also illustrate the universality of our ideas by studying a model given by the inverse of the MacMahon function.

United Kingdom

Prarit Agarwal

South Korea

Sunjin Choi

United States

Joonho Kim

Seok Kim

June Nahmgoong

AdS black holes and finite <math><mi>N</mi></math> indices

We explore 6d (1, 0) superconformal field theories with SU(3) and SU(2) gauge symmetries which cascade after Higgsing to the E-string theory on a single M5 near an E 8 wall. Specifically, we study the 2d   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = (0, 4) gauge theories which describe self-dual strings of these 6d theories. The self-dual strings can be also viewed as instanton string solitons of 6d Yang-Mills theories. We find the 2d anomaly-free gauge theories for self-dual strings, amending the naive ADHM gauge theories which are anomalous, and calculate their elliptic genera. While these 2d theories respect the flavor symmetry of each 6d SCFT only partially, their elliptic genera manifest the symmetry fully as these functions as BPS index are invariant in strongly coupled IR limit. Our consistent 2d (0, 4) gauge theories also provide new insights on the non-linear sigma models for the instanton strings, providing novel UV completions of the small instanton singularities. Finally, we construct new 2d quiver gauge theories for the self-dual strings in 6d E-string theory for multiple M5-branes probing the E 8 wall, and find their fully refined elliptic genera.

Kimyeong Lee

Higgsing towards E-strings

We propose new ADHM-like methods to compute the Coulomb branch instanton partition functions of 5D and 6D supersymmetric gauge theories, with certain exceptional gauge groups or exceptional matters. We study <math><msub><mi>G</mi><mn>2</mn></msub></math> theories with <math><msub><mi>n</mi><mn>7</mn></msub><mo>≤</mo><mn>3</mn></math> matters in <math><mrow><mn>7</mn></mrow></math> and <math><mi>S</mi><mi>O</mi><mo>(</mo><mn>7</mn><mo>)</mo></math> theories with <math><msub><mi>n</mi><mn>8</mn></msub><mo>≤</mo><mn>4</mn></math> matters in the spinor representation <math><mrow><mn>8</mn></mrow></math>. We also study the elliptic genera of self-dual instanton strings of 6D SCFTs with exceptional gauge groups or matters, including all non-Higgsable atomic SCFTs with rank 2 or 3 tensor branches. Some of them are tested with topological vertex calculus. We also explore a D-brane-based method to study instanton particles of 5D <math><mi>S</mi><mi>O</mi><mo>(</mo><mn>7</mn><mo>)</mo></math> and <math><mi>S</mi><mi>O</mi><mo>(</mo><mn>8</mn><mo>)</mo></math> gauge theories with matters in spinor representations, which further tests our ADHM-like proposals.

Hee-Cheol Kim

Ki-Hong Lee

Jaemo Park

6D strings and exceptional instantons

We study the asymptotic behavior of the (modified) superconformal index for 4d   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 1 gauge theory. By considering complexified chemical potential, we find that the ‘high-temperature limit’ of the index can be written in terms of the conformal anomalies 3c − 2a. We also find macroscopic entropy from our asymptotic free energy when the Hofman-Maldacena bound 1/2 &lt; a/c &lt; 3/2 for the interacting SCFT is satisfied. We study   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 1 theories that are dual to AdS5 × Y  p,p  and find that the Cardy limit of our index accounts for the Bekenstein-Hawking entropy of large black holes.

Jaewon Song

A 4d   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 1 Cardy Formula

Corrected the authors’ affiliations. Affiliation a and c were swapped.

China

Sung-Soo Kim

Erratum to: Instantons from blow-up

We generalize Nakajima-Yoshioka blowup equations to arbitrary gauge group with hypermultiplets in arbitrary representations. Using our blowup equations, we compute the instanton partition functions for 4d $$ \mathcal{N} $$ = 2 and 5d $$ \mathcal{N} $$ = 1 gauge theories for arbitrary gauge theory with a large class of matter representations, without knowing explicit construction of the instanton moduli space. Our examples include exceptional gauge theories with fundamentals, SO( ) gauge theories with spinors, and SU(6) gauge theories with rank-3 antisymmetric hypers. Remarkably, the instanton partition function is completely determined by the perturbative part.

Instantons from blow-up

We study the elliptic genera of 6d strings based on their modular properties. They are weak Jacobi forms of weight 0, whose indices are determined from the 2d chiral anomalies. We propose the ansatz for the elliptic genera which reflects the analytic structure of instanton partition functions. Given a finite amount of initial BPS data, we completely determine the elliptic genera of 6d strings in various 6d SCFTs. We also apply our ansatz to study   <math> <mi>N</mi> <mo>=</mo> <mfenced> <mn>2</mn> <mn>0</mn> </mfenced> </math>  $$ \mathcal{N}=\left(2,0\right) $$  and (1, 1) little strings as well as   <math> <mi>N</mi> <mo>=</mo> <mfenced> <mn>1</mn> <mn>0</mn> </mfenced> </math>  $$ \mathcal{N}=\left(1,0\right) $$  heterotic little strings, for which T-duality of little string theories supplies a sufficient number of initial BPS data. The anomaly polynomials of 6d little strings are worked out, which is needed for the elliptic genera bootstrap. In some little string theories, the elliptic genera must have the extra contributions from the Coulomb branch, which correspond to the additional zero modes for the full strings. The modified ansatze for such elliptic genera are also discussed.

On elliptic genera of 6d string theories

M5 branes probing D-type singularities give rise to 6d (1,0) SCFTs with SO × SO flavor symmetry known as D-type conformal matter theories. Gauging the diagonal SO-flavor symmetry leads to a little string theory with an intrinsic scale which can be engineered in F-theory by compactifying on a doubly-elliptic Calabi-Yau manifold. We derive Seiberg-Witten curves for these little string theories which can be interpreted as mirror curves for the corresponding Calabi-Yau manifolds. Under fiber-base duality these models are mapped to D-type quiver gauge theories and we check that their Seiberg-Witten curves match. By taking decompactification limits, we construct the curves for the related 6d SCFTs and connect to known results in the literature by further taking 5d and 4d limits.

Babak Haghighat

Wenbin Yan

Shing-Tung Yau

D-type fiber-base duality

We study Wilson surface defects in 6d  <math><mi>N</mi><mo>=</mo><mfenced><mn>2</mn><mn>0</mn></mfenced></math> $$ \mathcal{N}=\left(2,0\right) $$ superconformal field theories, engineered by semi-infinite M2-branes intersecting M5-branes. Two independent approaches are used to obtain the Wilson surface observables on the T 2 inside Ω-deformed R 4 × T 2. One approach is to compute 5d  <math><mi>N</mi><mo>=</mo><msup><mn>1</mn><mo>*</mo></msup></math> $$ \mathcal{N}={1}^{*} $$ instanton partition functions in the presence of Wilson lines, where the instanton corrections capture the 6d Kaluza-Klein momentum modes. The other approach is to study the elliptic genera of 2d  <math><mi>N</mi><mo>=</mo><mfenced><mn>0</mn><mn>4</mn></mfenced></math> $$ \mathcal{N}=\left(0,4\right) $$ gauge theories, which we propose as describing 6d self-dual strings in the presence of Wilson surface defects. We make a detailed comparison between these two independent computations, which precisely agree for Wilson surfaces in minuscule representations; for non-minuscule representations instead we only find partial agreement, due to technical problems which we comment about.