The scientifc program of the trimester will generally consist of mini-courses Monday, Wednesday, and Friday, and seminars Tuesday and/or Thursday. You can find the schedule for the trimester by clicking on the link CALENDAR to the left, or on semparis.

That's all, folks!

Hope you enjoyed the program.

See you again soon in Paris!

**Mini-courses**

Ron Donagi: Super Riemann surfaces and foundations of superstring perturbation theory

Edward Frenkel: Langlands Duality in Math and Physics

Marco Gualtieri: Constructions and Deformations of Generalized Kähler structures

Anton Kapustin: Anomalies and symmetry protected phases of matter

Andrei Negut: Exts and AGT with matter

Vasily Pestun: Localization in gauge theory

Stefan Theisen and Adam Schwimmer: Trace anomalies and their applications

Daniel Waldram: Supersymmetric backgrounds and generalized geometry

Dimitri Zvonkine: Cohomological relations on the moduli space of stable curves

**Past events**

**WEEK: July 11th - July 15th**

**Seminar: Artan Sheshmani**

**Nested Hilbert schemes and local Donaldson-Thomas theory**

Monday 3PM - 5:15PM

Abstract: We provide a rigorous and general construction of deformation obstruction theories and virtual classes for nested (flag) Hilbert scheme of one dimensional subschemes of a smooth projective algebraic surface. This will provide us with a general framework to compute a large class of invariants, such as Poincare invariants of Okonek et al, or the reduced local invariants of Kool and Thomas in the context of their local surface theory. We then compute the generating series of deformation invariants associated to these flag Hilbert schemes, and by exploiting the properties of vertex operators, show that they are modular. We finally establish a connection between certain DT invariants of threefolds (given as local surfaces) and such nested Hilbert schemes. This eventually (after using some of Mochizuki’s wallcrossing techniques) enables us to compute the generating series of Seiberg-Witten invariants of the surface with respect to modular forms. This is joint work with Amin Gholampour and Shing-Tung Yau.

** WEEK: July 4th - July 8th**

**Seminar: Ken Intrilgator**

**Aspects of susy CFTs and RG flows**

Monday 3PM - 5:15PM

Abstract: We tabulate and discuss unitary representations of superconformal algebras in d=3,4,5,6 dimensions, with various numbers of supercharges, and discuss some curiosities and subtleties. We discuss susy-preserving renormalization group flows, and classify the UV and IR deformations of SCFTs. We discuss the supersymmetry connection between the 6d conformal anomaly and the 6d analog of ’t Hooft anomalies for the supergravity background fields, and its implication for the 6d a-theorem.

******************

**Seminar: Andrei Okounkov**

**Elliptic stable envelopes**

Tuesday 3PM - 5:15PM

Abstract: This is aimed as a gentle introduction to the theory of stable envelopes, how they manifest itself in elliptic cohomology, and where are they useful. Based on joint work with Davesh Maulik and Mina Aganagic.

******************

The Summer School on Symplectic geometry, sheaves and mirror symmetry at Jussieu may also be of interest to some participants.

**WEEK**: **June 27th - July 1st**

***** STRING-MATH 2016 *****

** WEEK: June 20th - June 24th**

Mini-course by **Edward Frenkel **

**Langlands Duality in Math and Physics**

Mon, Wed, Th 3PM - 5:15PM

I will survey the origins of the Langlands Program and its geometric version in mathematics as well as the links to the S-duality of 4D supersymmetric gauge theories uncovered in the works of Witten and others. Time permitting, I will also discuss some more recent work related to the AGT correspondence.

******************

Seminar: **Artan Sheshmani**

**Donaldson-Thomas theories and modular forms and S-duality conjecture**

Tuesday 3PM - 5:15PM

Abstract: I will start by an introduction to Donaldson Thomas theory and some of the statements about its modularity properties, as well as its connection to S-duality conjecture in superstring theory, made formerly by physicists Gaiotto, Strominger, Yin. I will then provide an algebraic geometric approach to prove this conjecture for DT invariants of sheaves supported on hyperplane sections of the quintic Calabi-Yau threefold.

******************

The Memorial Coference at ENS in honor of Louis Boutet de Monvel may also be of interest to some participants.

** WEEK: June 13th - June 17th**

Mini-course by **Anton Kapustin **

**Anomalies and symmetry protected phases of matter**

Mon, Wed, Fr 10AM - 12:15PM

**Note location: Amphithéâtre Perrin, across from IHP building**

Topological Quantum Field Theory (TQFT) can be fruitfully applied to the study of gapped phases of quantum matter. I will discuss how it can be used to classify and understand properties of short-range entangled phases of interacting bosonic matter with symmetries. The classification turns out to be expressed in terms of Thom bordism groups. I will also explain how TQFT can be used to understand anomalies for finite group symmetries. Recently it was noted that symmetry in d-dimensional QFT is described by a d-group rather than a group. I will explain how d-groups for d>1 naturally arise in the study of anomalies as well as the study of fermionic phases of matter.

******************

Mini-course by **Andrei Negut **(part IV)

**Exts and AGT with matter**

Mon 3PM - 5:15PM

The AGT relations for N = 2 super-symmetric U(r) gauge theory with adjoint matter predict a relation between a certain integral over the moduli space of instantons (a geometric object) and the trace of an intertwiner for the W_r algebra (a representation theoretic object). The main purpose of this mini-course is to explain one proof of this statement in the r = 2 case, by showing how to compute the commutation relations between the Carlsson-Okounkov Ext operator and the Heisenberg-Virasoro algebra. The main technical tool is the shuffle algebra incarnation of the Schiffmann-Vasserot Yangian, and in fact, the proof of the relations holds in arbitrary rank. I also wish to ask the audience for help in proving the general r case, which essentially boils down to connecting the W_r algebra with the shuffle algebra. The breakdown of the lectures is the following:

1) An overview of the problem, including certain analogues such as the case of finite-dimensional Lie algebras.

2) The Schiffmann-Vasserot Yangian via shuffle algebras. The Heisenberg and Virasoro subalgebras.

3) The moduli space M of rank r sheaves on the plane and the Ext operator.

4) The shuffle algebra acts on the cohomology ring of M. The Ext operator is an intertwiner.

5) From intertwiners to conformal blocks. Ideas for generalizing to arbitrary W_r.

******************

Seminar: **Marc Henneaux**

**Twisted Self-Duality for Gravity and Higher Spin Gauge Fields**

Tuesday, 11AM - noon

We provide manifestly duality-symmetric formulations of linearized gravity and higher spin gauge fields. This requires introducing appropriate prepotentials and leads to a loss of manifest spacetime covariance. The prepotentials intriguingly exhibit higher spin Weyl symmetry for all spins. The analysis is motivated by the study of the conjectured hidden symmetries of gravity (such as E(10)), which have duality buit in.

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Seminar: **Alessandro Tomasiello**

**Fivebranes and T-branes**

Tueday, 3PM - 4PM

I will discuss a class of CFTs obtained by Higgsing the (2,0) ADE theory by a pair of nilpotent elements in the ADE group. In the A_k and D_k case, the nilpotent elements are labeled by Young diagrams, and the holographic dual is known explicitly in IIA. T-dualizing to F-theory is necessary to understand the D_k case. In the E_k case, only the F-theory description is available, via the theory of so-called Bala-Carter labels.

**WEEK: June 6th - June 10th**

Mini-course by **Andrei Negut **(part II and III)

**Exts and AGT with matter**

Mon, Wed 10AM - 12:15PM

The AGT relations for N = 2 super-symmetric U(r) gauge theory with adjoint matter predict a relation between a certain integral over the moduli space of instantons (a geometric object) and the trace of an intertwiner for the W_r algebra (a representation theoretic object). The main purpose of this mini-course is to explain one proof of this statement in the r = 2 case, by showing how to compute the commutation relations between the Carlsson-Okounkov Ext operator and the Heisenberg-Virasoro algebra. The main technical tool is the shuffle algebra incarnation of the Schiffmann-Vasserot Yangian, and in fact, the proof of the relations holds in arbitrary rank. I also wish to ask the audience for help in proving the general r case, which essentially boils down to connecting the W_r algebra with the shuffle algebra. The breakdown of the lectures is the following:

1) An overview of the problem, including certain analogues such as the case of finite-dimensional Lie algebras.

2) The Schiffmann-Vasserot Yangian via shuffle algebras. The Heisenberg and Virasoro subalgebras.

3) The moduli space M of rank r sheaves on the plane and the Ext operator.

4) The shuffle algebra acts on the cohomology ring of M. The Ext operator is an intertwiner.

5) From intertwiners to conformal blocks. Ideas for generalizing to arbitrary W_r.

******************

Mini-course by **Ron Donagi**** **

**Super Riemann surfaces and foundations of superstring perturbation theory**

Mon 3PM - 5:15PM, Tu 10AM - 12:15PM, Wed 3PM - 5:15PM

Superstring perturbation theory expresses scattering amplitudes as integrals over the moduli of punctured super Riemann surfaces. The moduli space of super Riemann surfaces has many analogies with the moduli space of ordinary Riemann surfaces. For example, there is a super analog of the Deligne-Mumford compactification, and it is important in understanding the qualitative properties of superstring scattering amplitudes. There is also a superanalog of the Mumford isomorphism between certain line bundles over moduli space, and it plays a role analogous to the role that the ordinary Mumford isomorphism plays in bosonic string theory.

Integration over supermoduli space gives a powerful framework for understanding important properties of string theory such as spacetime supersymmetry. But in practice the understanding of superstring perturbation theory via super Riemann surfaces has been relatively little-developed, beyond elementary examples in genus 0 and 1. Only in the early 2000s did DHoker and Phong succeed in determining some genus 2 amplitudes explicitly, by exploiting the super period matrix of a super Riemann surface. Likewise there has been only limited mathematical work on super Riemann surfaces in recent years, one recent result being that despite the analogies between them, super moduli space cannot be reduced to the ordinary moduli space in the sense that there is no holomorphic projection from super moduli space to its underlying reduced space.

Main topics will include: supermanifolds, supersymmetry, super Riemann surfaces, moduli spaces, Deligne-Mumford compactifications, novel issues related to Ramond punctures, super Mumford isomorphisms, super period matrices, and how these all fit into perturbative superstring theory: the measure on super moduli space, the Neveu-Schwarz and Ramond sectors, and other topics including the spontaneous breaking of supersymmetry at 1 loop.

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Workshop: **String Theory and Gender**

Th, Fr

**WEEK: May 30th - June 3rd**

Mon - Fr

**WEEK: May 23rd - May 27th**

**Workshop "Number theory and Physics"**

Mon, Tue, Wed

******************

Mini-course by **Andrei Negut**

**Exts and AGT with matter**

Friday 10AM - 12:15PM

The AGT relations for N = 2 super-symmetric U(r) gauge theory with adjoint matter predict a relation between a certain integral over the moduli space of instantons (a geometric object) and the trace of an intertwiner for the W_r algebra (a representation theoretic object). The main purpose of this mini-course is to explain one proof of this statement in the r = 2 case, by showing how to compute the commutation relations between the Carlsson-Okounkov Ext operator and the Heisenberg-Virasoro algebra. The main technical tool is the shuffle algebra incarnation of the Schiffmann-Vasserot Yangian, and in fact, the proof of the relations holds in arbitrary rank. I also wish to ask the audience for help in proving the general r case, which essentially boils down to connecting the W_r algebra with the shuffle algebra. The breakdown of the lectures is the following:

1) An overview of the problem, including certain analogues such as the case of finite-dimensional Lie algebras.

2) The Schiffmann-Vasserot Yangian via shuffle algebras. The Heisenberg and Virasoro subalgebras.

3) The moduli space M of rank r sheaves on the plane and the Ext operator.

4) The shuffle algebra acts on the cohomology ring of M. The Ext operator is an intertwiner.

5) From intertwiners to conformal blocks. Ideas for generalizing to arbitrary W_r.

******************

The talks Thursday morning at the IHP by Stephane Detournay and Ilarion Melnikov of the bimonthly Rencontres series will also be of interest to some participants.

**WEEK: May 16th - May 20th**

Seminar:** Nikolai Reshetikhin **

**Gauge field theories on space time manifolds with boundary**

Tuesday 3 PM - 5:15 PM

This talk is a review of recent developments in perturbative quantization of gauge field theories on manifolds with boundaries. First part is an overview of BV quantization (closed manifolds) and of BV-BFV quantization, the extension to the case when the space time has a boundary. In the second part we will focus on examples of BF theory and nonlinear Poisson sigma models.

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Mini-course by **Marco Gualtieri**

**Constructions and Deformations of Generalized Kahler structures**

Wed and Fr 10 AM - 12:15 PM, Fr 3 PM - 5:15 PM

I will describe some of the recent advances in constructing generalized Kahler structures, focusing especially on their relationship to holomorphic Poisson geometry. I will also describe a new approach to generalized Kahler geometry which places the Poisson structures at the heart of the story. This is helpful for understanding deformation theory of generalized Kahler structures in general.

******************

Seminar: **Sungjay Lee**

**Three-Charge Black Holes and 1/4 BPS States in Little String Theory**

Thursday 3 PM - 4 PM

We show that the system of k NS5-branes wrapping T^4×S^1 has non-trivial vacuum structure. Different vacua have different spectra of 1/4 BPS states that carry momentum and winding around the S^1. In one vacuum, such states are described by black holes; in another, they can be thought of as perturbative BPS states in Double Scaled Little String Theory. In general, both kinds of states are present. We compute the degeneracy of perturbative BPS states exactly, and show that it differs from that of the corresponding black holes. We comment on the implication of our results to the black hole microstate program, UV/IR mixing in Little String Theory, string thermodynamics, the string/black hole transition, and other issues.

**WEEK: May 9th - May 13th**

Mini-course by **Dimitri Zvonkine**

**Cohomological relations on the moduli space of stable curves**

Mon, Tue and Fr, 10AM - 12:15PM

We construct a family of relations between tautological cohomology classes on the moduli space Mbar_{g,n} of stable curves. This family contains all relations known to this day and is expected to be complete and optimal. The construction uses the Frobenius manifold of the A_2 singularity, the 3-spin Witten class and the Givental-Teleman classification of semi-simple cohomological field theories (CohFTs). The plan of the three talks will be as follows.

1. An introduction to moduli space and its tautological cohomology ring; simplest examples of tautological relations.

2. Cohomological field theories and Witten's r-spin class. Witten's r-spin class is actually a family of cohomology classes on the space of stable maps, defined using the space of tensor r-th roots of the canonical line bundle. I will explain why these classes satisfy the axioms of a cohomological field theory (CohFT).

3. The Givental-Teleman classification of semi-simple CohFTs. I will explain the classification theorem, show how it applies to Witten's class and how one can deduce tautological relations from it. In the end I will compute several cohomological relations using our method.

This is a joint work with R. Pandharipande and A. Pixton.

******************

Mini-course by **Vasily Pestun **(continued)

**Localization in gauge theory**

Wednesday 3PM - 5:15PM

1. Equivariant cohomology. Atiyah-Bott localization formula theorem.

2. Supersymmetric, cohomological and topological field theories.

3. Gauge theory partition function on a sphere.

******************

The talks Thursday morning at the IHP by Stefan Hohenegger and Emery Sokatchev of the bimonthly Rencontres series will also be of interest to some participants.

**WEEK: May 2nd - May 6th**

Mini-course by **Daniel Waldram** (continued)

**Supersymmetric backgrounds and generalised geometry**

Mon and Wed, 10AM - 12:15PM

Supersymmetric backgrounds of type II and eleven dimensional supergravity including flux degrees of freedom provide natural generalisations of the notion of a manifold with special holonomy. The classic example is that of a generalised Calabi—Yau manifold introduced by Hitchin and Gualtieri, which defines an integrable structure on the sum of the tangent and cotangent spaces. We will discuss how a version of generalised geometry with an E_d structure group gives a unified description of the supergravity fields such that supersymmetric backgrounds correspond to torsion-free generalised structures. This defines the generic string theoretic extension of Calabi—Yau and Sasaki—Einstein geometries, among others. As an application, we discuss the moduli spaces of the “exceptional Sasaki—Einstein” geometries that are the generic duals of N=1 superconformal field theories.

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Mini-course by **Vasily Pestun**

**Localization in gauge theory**

Wednesday 3PM - 5:15PM

1. Equivariant cohomology. Atiyah-Bott localization formula theorem.

2. Supersymmetric, cohomological and topological field theories.

3. Gauge theory partition function on a sphere.

**WEEK: April 25th - April 29th**

**Workshop "String Geometry and BPS state counting"**

Mon, Tue, Wed

******************

Mini-course by **Daniel Waldram**

**Supersymmetric backgrounds and generalised geometry**

Friday 10AM - 12:15PM

Supersymmetric backgrounds of type II and eleven dimensional supergravity including flux degrees of freedom provide natural generalisations of the notion of a manifold with special holonomy. The classic example is that of a generalised Calabi—Yau manifold introduced by Hitchin and Gualtieri, which defines an integrable structure on the sum of the tangent and cotangent spaces. We will discuss how a version of generalised geometry with an E_d structure group gives a unified description of the supergravity fields such that supersymmetric backgrounds correspond to torsion-free generalised structures. This defines the generic string theoretic extension of Calabi—Yau and Sasaki—Einstein geometries, among others. As an application, we discuss the moduli spaces of the “exceptional Sasaki—Einstein” geometries that are the generic duals of N=1 superconformal field theories.

**WEEK: April 18th - 22nd**

Mini-course by **Adam Schwimmer** and **Stefan Theisen**

**Trace anomalies and their applications**

Mon 10AM - 11AM and 2PM - 3PM, Wed and Fr 10AM - 12:15PM

The following topics will be covered:

- Relevant features of conformal field theories

- Weyl anomalies, their properties and classification

- Spontaneous breaking of conformal symmetry, anomaly matching and the a-theorem

- Holography and Weyl anomalies

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Seminar: **Richard Eager**

**Derived Equivalence from non-Abelian Gauged Linear Sigma Models**

Thursday, 3PM - 4PM

Abstract:

Different phases of two dimensional (2,2) supersymmetric gauged linear sigma models (GLSMs) have equivalent categories of B-branes. Special non-Abelian gauged linear sigma models can have two distinct phases that describe birationally distinct Calabi-Yau varieties. The equivalence of categories of B-branes implies that the two distinct Calabi Yau varieties have equivalent derived categories of coherent sheaves. In this talk, I explain how the convergence of the hemi-sphere partition function determines a grade-restriction rule. The grade-restriction rule determines how branes can be transported between two different phases. In the special case of the Rodland GLSM, the grade-restriction rule naturally recovers the Borisov-Caldararu equivalence between the derived category of coherent sheaves on both sides of the Pfaffian-Grassmannian correspondence.