**Monday** (chairs: Reffert; Orlando)

### Amihay Hanany

**Branes and the Kraft–Procesi transition. ** In their work from 1976, Kraft and Procesi computed minimal singularities in nilpotent orbits for A type algebras, extending on the work of Brieskorn from 1971. We will present the brane version for this computation.

### Alberto Zaffaroni

**AdS4 black holes and 3d gauge theories. ** One of the great success of string theory is the microscopical explanation of the entropy of a class of asymptotically flat black holes. Much less is known about asymptotically AdS black holes. In this talk I explain how to derive the entropy of a class of asymptotically AdS supersymmetric black holes in four dimensions using holography. The counting of black hole micro-states is related to a counting of states in the dual 3d gauge theory which can be explicitly performed using localization.

### Johannes Walcher

**Exponential networks and representations of quivers. ** I will report on work in progress with Richard Eager and Sam Selmani, in which we adapt the spectral network machinery of Gaiotto–Moore–Neitzke to give a B-model description of the (known) BPS spectrum of some simple Calabi–Yau three-folds.

### Simone Giacomelli

**T-branes through 3d mirror symmetry. ** T-branes are exotic bound states of D-branes, characterized by mutually non-commuting vacuum expectation values for the worldvolume scalars, and the M/F-theory geometry lifting D6/D7-brane configurations is blind to the T-brane data. In this talk I will explain how to make this data manifest by looking at the effect of T-branes on a probe two brane. I will show that, exploiting 3d mirror symmetry, we can understand in detail how the wordvolume brane theory is modified, uncovering a new class of \(N=2\) quiver gauge theories, whose Higgs branches mimic those of membranes at ADE singularities.

### Iñaki García Etxebarria

**N=3 four dimensional field theories. ** I describe a class of four dimensional field theories constructed by quotienting ordinary \(N=4\) \(U(N)\) SYM by particular combinations of R-symmetry and \(SL(2,\mathbb{Z})\) automorphisms. These theories appear naturally on the worldvolume of D3 branes probing terminal singularities in F-theory, where they can be thought of as non-perturbative generalizations of the O3 plane. I will focus on cases preserving only 12 supercharges, where the quotient gives rise to theories with coupling fixed at a value of order one. These constructions possess an unconventional large \(N\) limit described by a non-trivial F-theory fibration with base \(AdS_5 \times (S^5/\mathbb{Z}_k)\). Upon reduction on a circle the \(N=3\) theories flow to \(N=6\) ABJM theories.

### Peter Koroteev

**Elliptic algebras and large-N supersymmetric gauge theories. ** We discuss the duality between supersymmetric gauge theories in various dimensions and elliptic integrable systems such as Ruijsenaars-Schneider model and periodic intermediate long wave hydrodynamics. These models arise in instanton counting problems and are described by certain elliptic algebras. We discuss the correspondence between the two types of models by employing the large-N limit of the dual gauge theory.

### Richard Eager

**Two-dimensional N = (0, 2) theories and Calabi-Yau 4-algebras. ** We determine the world-volume theory of a stack of N D1-branes transverse to a Calabi-Yau fourfold in type IIB string theory from a tilting bundle on the fourfold. When the Calabi-Yau fourfold is toric, the construction provides a three-dimensional analog of the theory of brane tilings.

**Tuesday** (chairs: Amariti; Siampos)

### Vasily Pestun

**Quiver W-algebras. ** I will explain how the geometry of instanton moduli spaces of quiver gauge theories leads naturally to q-deformed W-algebras associated to the same quiver, and, in parallel, to the quantization of the algebraic integrable system of periodic monopoles.

### Alberto Lerda

**Resumming Instantons in N=2* Theories. ** We study the prepotential of 4d gauge theories with \(N=2\) susy coupled to a massive adjoint hypermultiplet (\(N=2^*\) theories) and show that it satisfies a modular anomaly equation whose validity is related to S-duality. The recursion relations that follow from the modular anomaly equation allow one to write the prepotential in terms of (quasi)-modular forms, thus resumming all instanton contributions. These results can be checked against the microscopic multi-instanton calculus in the case of classical algebras, but are valid also for the exceptional algebras, where direct computations are not available. We also comment on the extensions and applications of these results.

### Ignatios Antoniadis

**Topological amplitudes in string theory and Omega deformations. ** I will give an overview of topological amplitudes in string theory and describe how they compute actual physical couplings of the superstring effective action. In particular I will present a two-parameter class of amplitudes that reproduce the Nekrasov partition function of \(N=2\) supersymmetric gauge theories in the low energy (field theory) limit.

### Jörg Teschner

**Comments on instanton partition functions for non-Lagrangian SUSY field theories. ** It is known that the geometric engineering of SUSY field theories within string theory can be used to calculate certain field theoretical quantities such as the instanton partition functions exactly. This continues to hold in cases where the resulting SUSY field theories do not have a Lagrangian description. We will revisit some of the results obtained for these cases by means of the refind topological vertex from the point of view of possible generalisations of the AGT-correspondence, clarifying in particular their meaning from the point of view of conformal field theory.

### Johannes Schmude

**On the Chiral Ring of Warped AdS(3) Compactifications of Type IIB Supergravity. ** In this talk I will discuss recent results regarding the Kaluza-Klein spectrum of a large class of AdS(3) compactifications of type IIB supergravity. The geometries admit a Cauchy-Riemann (CR) structure and the problem shares thus some features with the familiar case of AdS(5) compactifications with Sasaki--Einstein factors. Exploiting the CR structure, it is possible to show that (a subset) of the chiral ring of the dual two-dimensional sCFTs corresponds to holomorphic sections on the complex cone over the internal manifold. An especially interesting class of these geometries arises from compactification of four-dimensional \(N=1\) theories (such as the \(Y^{p,q}\) theories) on Riemann surfaces.

### Davide Cassani

**On supersymmetric regularization in field theory and holography. **
When computing an a priori divergent supersymmetric observable it is crucial to work in a renormalization scheme that preserves supersymmetry. Two powerful approaches are provided by localization and holography, however in both cases identifying a supersymmetric scheme can be non-trivial. I will elaborate on this problem focusing on a new intrinsic observable of \(d=4\) SCFT's in curved space. This is defined as a supersymmetric version of the Casimir energy on \(S^1 \times M_3\) (where \(M_3\) has for instance the topology of a three-sphere), and can be computed using various techniques, including localization. After having discussed scheme-independence of the supersymmetric Casimir energy in field theory, I will show how this is reproduced by holography. I will emphasize that the usual holographic renormalization violates a BPS condition, and will prove that restoring this requires certain non-standard counterterms.

**Wednesday** (chairs: Reffert; Amariti)

### Chris Hull

**Gravity and Duality. ** Duality has played a crucial role in our understanding of supersymmetric gauge theories. In the talk, possibility of a similar role for duality in theories of gravity is explored.

### Neil Lambert

**(2,0) to 2 M2's. **
We present a generalization of the \((2,0)\) superalgebra based on a 3-algebra to include a constant abelian 3-form. The resulting system of equations describes two M5-branes in the absence of the 3-form or two M2-branes when the 3-form is turned on.

### Stefano Cremonesi

**Hilbert series of 3d N=2 gauge theories. **
The Hilbert series enumerates gauge invariant chiral operators of theories with 4 supercharges and an R-symmetry. It encodes precious information about the chiral ring and the moduli space of supersymmetric vacua of the theory. In this talk I will present a formula for the Hilbert series of general \(3d\) \(N=2\) gauge theories and explain how this formalism is related to the old semiclassical analysis of the moduli space.

### Noppadol Mekareeya

**The Moduli Space of Instantons from 3d N=3 and N=4 Gauge Theories. **
The moduli spaces of \(N=3\) and \(N=4\) gauge theories in three dimensions are hyperKähler cones. For a large class of such theories, the moduli space is isomorphic to the moduli space of instantons on flat or ALE spaces. I will discuss those theories as well as their dualities in the talk. This approach has led to several new descriptions of the instanton moduli space, complementing the well-known ADHM and Kronheimer--Nakajima constructions.

### Marcus Sperling

**Coulomb branches for rank 2 gauge groups in 3d, N = 4 gauge theories. **
The Coulomb branch of 3-dimensional \(N = 4\) gauge theories is the space of bare and dressed BPS monopole operators. We utilise the conformal dimension to define a fan which, upon intersection with the weight lattice of a GNO-dual group, gives rise to a collection of semi-groups. It turns out that the unique Hilbert bases of these semi-groups are a sufficient, finite set of monopole operators which generate the entire chiral ring. Moreover, the knowledge of the properties of the minimal generators is enough to compute the Hilbert series explicitly. The techniques of this paper allow an efficient evaluation of the Hilbert series for general rank gauge groups. As an application, we provide various examples for all rank two gauge groups to demonstrate the novel interpretation.

### Ilmar Gahramanov

**Progress in integrable statistical models inspired by supersymmetric gauge theories. **
Recently, there has been observed several connections of integrable models to supersymmetric gauge theories. One of such connections is a correspondence between supersymmetric quiver gauge theories and integrable lattice models such that the integrability emerges as a manifestation of supersymmetric dualities. Particularly, partition functions of three-dimensional \(N = 2\) supersymmetric quiver gauge theories on different manifolds can be identified with partition functions of two-dimensional exactly solvable statistical models. Using this relationship one can obtain new solutions of the star-triangle relation and other forms of the Yang-Baxter equation.

### Frederic Brünner

**SQCD and a network of linear quivers. **
I present evidence for a new duality network of linear quiver gauge theories. Applying the Bailey lemma to elliptic hypergeometric integrals on the \(A_n\) root system, it can be shown that the superconformal index of SQCD is identical to that of a large number of \(N=1\) linear quiver gauge theories. It is possible to construct both electric and magnetic parts of Seiberg duality with arbitrary colours and flavours by quivering an s-confining theory. (based on work with V.P. Spiridonov, 1605:xxxxx, see also 0910.5944).