##### Workshop

#### Geometry of Strings and Fields

**Aug 26, 2013 - Oct 20, 2013**

**Abstract**

Ever since the birth of string theory, interaction with geometry has been one of the primary driving forces that has led to progress in superstring theory. On one hand, string theory has generated many new geometrical concepts; and on the other hand new ideas from geometry have often found their first applications in string theory. These topics include vertex algebras, conformal field theory, mirror symmetry, topological field theory and string theory, exact solutions of supersymmetric gauge theory and noncommutative field theory. Recent exciting developments include the matrix model approach to N=1 gauge theory, open string mirror symmetry, the derived category approach to D-branes on Calabi-Yau manifolds, geometric transitions, proof of the N=2 Seiberg-Witten solution by instanton methods, wall crossing formulas, the relation between Langlands program and supersymmetric gauge theories, indications of integrable structures in super Yang-Mills theory and AdS string theory. The program will be devoted to geometrical subjects motivated by string theory, and to recent developments in string theory and related physical fields which are of strong mathematical interest. On the mathematical side, the aim is to foster interaction between such areas of mathematics as derived categories, elliptic cohomology, geometric Langlands correspondence, quantum cohomology. On the more physical side, the aim is to foster new progress in the continuing drive towards understanding the foundations of string/M-theory, and in the wealth of new ideas involving D-branes, BPS states and various dualities which are of great importance to mathematical subjects such as algebraic geometry. There will be included a training week with introductory lectures to some of these topics.

- Dates of the Conference:

September 8th-13th 2013

- Dates of the Training Weeks:

September 1th-6th 2013, October 6th-12th 2013

**Topics**

- Quantum field and string theoretical methods and higher geometrical structures

- Generalized complex geometry and supersymmetry

- Topological strings

- Extended topological field theories

- Vertex algebras

- Localization techniques in quantum field theory

**Organizers**

Francesco Bonechi, I.N.F.N. - Alberto Cattaneo, Zurich University- Sergei Gukov, Caltech University

Martin Rocek, Stony Brook University - Domenico Seminara, Universita di Firenze - Maxim Zabzine, Uppsala University

**Contact**

bonechi@fi.infn.it

**Website**

http://theory.fi.infn.it/seminara/Geometry_of_Strings_and_Fieds/Main.html

**Related events**

Geometry of Strings and Fields - Conference (Conference) - Sep 08, 2013

**Talks**

Aug 27, 2013 - 11:00 | A. Losev | String Einstein equations on schemes | Seminar | |||||

Aug 28, 2013 - 11:00 | S. Shatashvili | Quantization of integrable systems and supersymmetry | Seminar | |||||

Aug 29, 2013 - 11:00 | R. Szabo | Quantization of non-geometric flux backgrounds | Seminar | |||||

Aug 30, 2013 - 11:00 | P. Hekmati | Geometric transforms and differential K-theory | Seminar | |||||

Sep 02, 2013 - 11:30 | V. Pestun | Localization of supersymmetric quantum field theories on curved background I | Lecture | |||||

Sep 02, 2013 - 14:30 | V. Pestun | Localization of supersymmetric quantum field theories on curved background II | Lecture | |||||

Sep 02, 2013 - 16:00 | N. Rozenblyum | Branes from a derived point of view I | Lecture | The goal will be to describe the basics of derived categories of sheaves associated to algebraic varieties with a physically motivated point of view. We will start from the beginning and go on to describe various categorical invariants, such as Hochschild homology, which play an important role in field theory. Time permitting, we will explain integration in the context of derived algebraic geometry and describe its relationship to Feynman integrals. | ||||

Sep 03, 2013 - 10:00 | V. Pestun | Localization of supersymmetric quantum field theories on curved background III | Lecture | |||||

Sep 03, 2013 - 11:30 | V. Pestun | Localization of supersymmetric quantum field theories on curved background IV | Lecture | |||||

Sep 03, 2013 - 14:30 | N. Rozenblyum | Branes from a derived point of view II | Lecture | The goal will be to describe the basics of derived categories of sheaves associated to algebraic varieties with a physically motivated point of view. We will start from the beginning and go on to describe various categorical invariants, such as Hochschild homology, which play an important role in field theory. Time permitting, we will explain integration in the context of derived algebraic geometry and describe its relationship to Feynman integrals. | ||||

Sep 04, 2013 - 10:00 | R. Von Unge | The secret life of sigma super-models I | Lecture | These lectures will give an introduction to sigma models for mathematicians and physicists. In the first lecture I will introduce sigma models and the various concepts needed to understand them. I will give several simple examples and in particular discuss the intriguing relation between properties of the field theory that defines the sigma model and the geometrical characteristics of its target space. The second lecture will be focussed on supersymmetric sigma models and their target space geometry. I will in particular discuss the relation between Generalized Kähler geometry and (2,2) supersymmetry. The third lecture will give some applications such as quotient constructions (reductions) and T-duality and also discuss the quantum properties of sigma models in more detail. | ||||

Sep 04, 2013 - 11:30 | N. Rozenblyum | Branes from a derived point of view III | Lecture | The goal will be to describe the basics of derived categories of sheaves associated to algebraic varieties with a physically motivated point of view. We will start from the beginning and go on to describe various categorical invariants, such as Hochschild homology, which play an important role in field theory. Time permitting, we will explain integration in the context of derived algebraic geometry and describe its relationship to Feynman integrals. | ||||

Sep 05, 2013 - 10:00 | R. Von Unge | The secret life of sigma super-models II | Lecture | These lectures will give an introduction to sigma models for mathematicians and physicists. In the first lecture I will introduce sigma models and the various concepts needed to understand them. I will give several simple examples and in particular discuss the intriguing relation between properties of the field theory that defines the sigma model and the geometrical characteristics of its target space. The second lecture will be focussed on supersymmetric sigma models and their target space geometry. I will in particular discuss the relation between Generalized Kähler geometry and (2,2) supersymmetry. The third lecture will give some applications such as quotient constructions (reductions) and T-duality and also discuss the quantum properties of sigma models in more detail. | ||||

Sep 05, 2013 - 11:30 | R. Heluani | (Sheaves of) vertex algebras and stringy manifold invariants I | Lecture | In these lectures we will introduce and play with the basic notions of vertex algebra theory like Poisson vertex algebras, quasi-classical limits, etc. We will give several examples reproducing the well known examples in the theory like universal enveloping vertex algebras, lattice vertex algebras and W-algebras. We will then explain how to construct a sheaf of vertex algebras attached to any smooth manifold M, known as the chiral de Rham complex of M. Finally, computing the cohomologies of these sheaves we will obtain geometric invariants of M. | ||||

Sep 05, 2013 - 14:30 | D. Calaque | Derived symplectic geometry and topological field theories I | Lecture | Derived geometry provides a way to deal with many problems that arise in geometry: 1) mapping space becomes somehow finite dimensional, 2) all fiber products and quotients become smooth. The goal of these lectures is to show that it provides a suitable framework for the so-called AKSZ construction (after Alexandrov-Kontsevich-Schwartz-Zaboronski). We'll start with the definition of n-shifted symplectic and Lagrangian structures, after Pantev-Toën-Vaquié-Vezzosi, and provide examples: - Shifted symplectic structures: BG, [g*/G], mapping stacks with symplectic target and "compact oriented" source - Lagrangian structures: moment maps, quasi-Hamiltonian structures, mapping stacks with boundary conditions. We will finally explain how this can be used to construct fully extended classical topological field theories |
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Sep 05, 2013 - 16:00 | D. Calaque | Derived symplectic geometry and topological field theories II | Lecture | Derived geometry provides a way to deal with many problems that arise in geometry: 1) mapping space becomes somehow finite dimensional, 2) all fiber products and quotients become smooth. The goal of these lectures is to show that it provides a suitable framework for the so-called AKSZ construction (after Alexandrov-Kontsevich-Schwartz-Zaboronski). We'll start with the definition of n-shifted symplectic and Lagrangian structures, after Pantev-Toën-Vaquié-Vezzosi, and provide examples: - Shifted symplectic structures: BG, [g*/G], mapping stacks with symplectic target and "compact oriented" source - Lagrangian structures: moment maps, quasi-Hamiltonian structures, mapping stacks with boundary conditions. We will finally explain how this can be used to construct fully extended classical topological field theories |
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Sep 06, 2013 - 10:00 | R. Von Unge | The secret life of sigma super-models II | Lecture | These lectures will give an introduction to sigma models for mathematicians and physicists. In the first lecture I will introduce sigma models and the various concepts needed to understand them. I will give several simple examples and in particular discuss the intriguing relation between properties of the field theory that defines the sigma model and the geometrical characteristics of its target space. The second lecture will be focussed on supersymmetric sigma models and their target space geometry. I will in particular discuss the relation between Generalized Kähler geometry and (2,2) supersymmetry. The third lecture will give some applications such as quotient constructions (reductions) and T-duality and also discuss the quantum properties of sigma models in more detail. | ||||

Sep 06, 2013 - 11:30 | D. Calaque | Derived symplectic geometry and topological field theories III | Lecture | Derived geometry provides a way to deal with many problems that arise in geometry: 1) mapping space becomes somehow finite dimensional, 2) all fiber products and quotients become smooth. The goal of these lectures is to show that it provides a suitable framework for the so-called AKSZ construction (after Alexandrov-Kontsevich-Schwartz-Zaboronski). We'll start with the definition of n-shifted symplectic and Lagrangian structures, after Pantev-Toën-Vaquié-Vezzosi, and provide examples: - Shifted symplectic structures: BG, [g*/G], mapping stacks with symplectic target and "compact oriented" source - Lagrangian structures: moment maps, quasi-Hamiltonian structures, mapping stacks with boundary conditions. We will finally explain how this can be used to construct fully extended classical topological field theories |
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Sep 06, 2013 - 14:30 | R. Heluani | (Sheaves of) vertex algebras and stringy manifold invariants II | Lecture | In these lectures we will introduce and play with the basic notions of vertex algebra theory like Poisson vertex algebras, quasi-classical limits, etc. We will give several examples reproducing the well known examples in the theory like universal enveloping vertex algebras, lattice vertex algebras and W-algebras. We will then explain how to construct a sheaf of vertex algebras attached to any smooth manifold M, known as the chiral de Rham complex of M. Finally, computing the cohomologies of these sheaves we will obtain geometric invariants of M. | ||||

Sep 06, 2013 - 16:00 | R. Heluani | (Sheaves of) vertex algebras and stringy manifold invariants III | Lecture | In these lectures we will introduce and play with the basic notions of vertex algebra theory like Poisson vertex algebras, quasi-classical limits, etc. We will give several examples reproducing the well known examples in the theory like universal enveloping vertex algebras, lattice vertex algebras and W-algebras. We will then explain how to construct a sheaf of vertex algebras attached to any smooth manifold M, known as the chiral de Rham complex of M. Finally, computing the cohomologies of these sheaves we will obtain geometric invariants of M. | ||||

Sep 08, 2013 - 14:30 | Edward Witten | Superstring Perturbation Theory Revisited I | Lecture | The main ideas of superstring perturbation theory were well-established by the mid-1980's, but some details important for understanding the infrared behavior were never fully clarified. To clarify these details, one primarily needs to formulate everything on the moduli space of super Riemann surfaces rather than trying to reduce everything to integration over the moduli space of ordinary Riemann surfaces. | ||||

Sep 16, 2013 - 14:30 | R. Garavuso | Analogues of Mathai-Quillen forms in sheaf cohomology and applications to topological field theory | Seminar | |||||

Sep 17, 2013 - 11:00 | M. Billo | Remarks on the ε-deformed prepotential of the N=2* theory | Seminar | |||||

Sep 18, 2013 - 11:00 | A. Tanzini | The Stringy Instanton Partition Function | Seminar | |||||

Sep 18, 2013 - 14:30 | G. Bonelli | N=1 Geoemetries via M-theory | Seminar | |||||

Sep 19, 2013 - 11:00 | N. Nekrasov | BPS/CFT correspondence: vertex operators and Bethe states, part I | Seminar | |||||

Sep 19, 2013 - 14:30 | N. Nekrasov | BPS/CFT correspondence: vertex operators and Bethe states, part II | Seminar | |||||

Sep 20, 2013 - 11:00 | S. Cremonesi | Coulomb branches of 3d N=4 gauge theories: monopole operators and Hilbert series | Seminar | |||||

Sep 23, 2013 - 11:00 | F. Benini | Elliptic general of two-dimensional N=2 gauge theories | Seminar | |||||

Sep 24, 2013 - 11:00 | M. Bochicchio | Large-N Yang-Mills as a Topological Field Theory | Seminar | |||||

Sep 25, 2013 - 11:00 | A. Morozov | TBA | Seminar | |||||

Sep 26, 2013 - 11:00 | A. Morozov | TBA | Seminar | |||||

Sep 27, 2013 - 11:00 | D. Waldram | The geometry of N=2 flux backgrounds | Seminar | |||||

Sep 30, 2013 - 11:00 | G. Festuccia | Supersymmetric Theories on curved spaces: part I | Seminar | |||||

Oct 01, 2013 - 11:00 | G. Festuccia | Supersymmetric Theories on curved spaces: part II | Seminar | |||||

Oct 02, 2013 - 11:00 | B. de Wit | Deformations of special geometry: in search of the topological string | Seminar | |||||

Oct 03, 2013 - 11:00 | U Bruzzo | Stacky partial compactifications of moduli spaces of instantons | Seminar | |||||

Oct 04, 2013 - 11:00 | S. Cherkis | Monowall dynamics, melting crystals, and oriented matroids | Seminar | |||||

Oct 07, 2013 - 14:30 | Pavel Mnev | Around moduli spaces of flat connections I | Lecture | We will discuss the moduli spaces of flat connections on principal bundles over manifolds and various natural structures supported on them. The cases particularly rich in features are moduli spaces of flat bundles over surfaces and 3-manifolds, cases related to Chern-Simons topological field theory. We will also discuss supergeometric structures on moduli spaces, their cohomological description and, time permitting, geometric quantization of the moduli spaces of flat connections over a surface. | ||||

Oct 07, 2013 - 16:00 | Gregory Ginot | An introduction to differentiable stacks and moduli space I | Lecture | Available soon | ||||

Oct 08, 2013 - 10:00 | J. Qiu | Localization of 5D super Yang-Mills I | Lecture | In these lectures, I plan to go through the rough procedure of computing the perturbative partition of 5D SYM. The content of the lecture will be the following
1) The setup, in which I recall some basics of the 5D supersymmetry. 2) Sasaki-Einstein geometry and the Killing spinors, in which I will explain some basics of spin geometry and how to solve for the Killing spinors on such manifolds. 3) The cohomological complex, and the localization procedure, in which I will explain the localization arguments and investigate the localisation locus 4) The index theorem, where I will explain some basics of Atiyah?s K-theoretical formula of transversally elliptic operators. Depending on the time, I may also discuss the implication of the results, such as the N3 behaviour and comparison with perturbation theory. |
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Oct 08, 2013 - 11:30 | Pavel Mnev | Around moduli spaces of flat connections II | Lecture | We will discuss the moduli spaces of flat connections on principal bundles over manifolds and various natural structures supported on them. The cases particularly rich in features are moduli spaces of flat bundles over surfaces and 3-manifolds, cases related to Chern-Simons topological field theory. We will also discuss supergeometric structures on moduli spaces, their cohomological description and, time permitting, geometric quantization of the moduli spaces of flat connections over a surface. | ||||

Oct 08, 2013 - 14:30 | Edward Witten | Superstring Perturbation Theory Revisited I | Lecture | The main ideas of superstring perturbation theory were well-established by the mid-1980\\\'s, but some details important for understanding the infrared behavior were never fully clarified. To clarify these details, one primarily needs to formulate everything on the moduli space of super Riemann surfaces rather than trying to reduce everything to integration over the moduli space of ordinary Riemann surfaces. | ||||

Oct 08, 2013 - 16:00 | Andrea Cappelli | The Birth of String Theory | Lecture | An historical journey from the first version of the theory (the so-called dual resonance model) in the late sixties, as an attempt to describe the physics of strong interactions outside the framework of quantum field theory, to its reinterpretation around the mid-seventies as a quantum theory of gravity unified with the other forces, and its successive developments up to the superstring revolution in 1984. | ||||

Oct 09, 2013 - 10:00 | Edward Witten | Superstring Perturbation Theory Revisited II | Lecture | The main ideas of superstring perturbation theory were well-established by the mid-1980\'s, but some details important for understanding the infrared behavior were never fully clarified. To clarify these details, one primarily needs to formulate everything on the moduli space of super Riemann surfaces rather than trying to reduce everything to integration over the moduli space of ordinary Riemann surfaces. | ||||

Oct 09, 2013 - 11:30 | Pavel Mnev | Around moduli spaces of flat connections III | Lecture | We will discuss the moduli spaces of flat connections on principal bundles over manifolds and various natural structures supported on them. The cases particularly rich in features are moduli spaces of flat bundles over surfaces and 3-manifolds, cases related to Chern-Simons topological field theory. We will also discuss supergeometric structures on moduli spaces, their cohomological description and, time permitting, geometric quantization of the moduli spaces of flat connections over a surface. | ||||

Oct 09, 2013 - 14:30 | Marco Aldi | Hamiltonian Quantization of Loop Spaces and Vertex Algebras I | Lecture | In these lectures we review some recent results regarding the use of vertex algebras and Poisson vertex algebras in the Hamiltonian quantization of (connected) loop spaces. After recalling the basic constructions, we will explain how the Hamiltonian formalism can be used to study target-space geometry. Hamiltonian quantization of disconnected loop spaces will be addressed in the last lecture. | ||||

Oct 09, 2013 - 16:00 | Henrique Bursztyn | A graded-geometric viewpoint to generalized geometry II | Lecture | These lectures will offer a basic introduction to the geometry of graded manifolds. The main goal is discussing how graded symplectic geometry provides a natural approach to \"generalized geometry\", including the study of objects such as Courant algebroids, Dirac structures, generalized complex structures, their symmetries and reduction. | ||||

Oct 10, 2013 - 10:00 | J. Qiu | Localization of 5D super Yang-Mills II | Lecture | In these lectures, I plan to go through the rough procedure of computing the perturbative partition of 5D SYM. The content of the lecture will be the following
1) The setup, in which I recall some basics of the 5D supersymmetry. 2) Sasaki-Einstein geometry and the Killing spinors, in which I will explain some basics of spin geometry and how to solve for the Killing spinors on such manifolds. 3) The cohomological complex, and the localization procedure, in which I will explain the localization arguments and investigate the localisation locus 4) The index theorem, where I will explain some basics of Atiyah?s K-theoretical formula of transversally elliptic operators. Depending on the time, I may also discuss the implication of the results, such as the N3 behaviour and comparison with perturbation theory. |
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Oct 10, 2013 - 11:30 | Edward Witten | Superstring Perturbation Theory Revisited III | Lecture | The main ideas of superstring perturbation theory were well-established by the mid-1980s, but some details important for understanding the infrared behavior were never fully clarified. To clarify these details, one primarily needs to formulate everything on the moduli space of super Riemann surfaces rather than trying to reduce everything to integration over the moduli space of ordinary Riemann surfaces. | ||||

Oct 10, 2013 - 14:30 | Marco Aldi | Hamiltonian Quantization of Loop Spaces and Vertex Algebras II | Lecture | In these lectures we review some recent results regarding the use of vertex algebras and Poisson vertex algebras in the Hamiltonian quantization of (connected) loop spaces. After recalling the basic constructions, we will explain how the Hamiltonian formalism can be used to study target-space geometry. Hamiltonian quantization of disconnected loop spaces will be addressed in the last lecture. | ||||

Oct 10, 2013 - 16:00 | Gregory Ginot | An introduction to differentiable stacks and moduli space II | Lecture | Available soon | ||||

Oct 11, 2013 - 10:00 | Gregory Ginot | An introduction to differentiable stacks and moduli space III | Lecture | Available soon | ||||

Oct 11, 2013 - 11:30 | J. Qiu | Localization of 5D super Yang-Mills III | Lecture | In these lectures, I plan to go through the rough procedure of computing the perturbative partition of 5D SYM. The content of the lecture will be the following
1) The setup, in which I recall some basics of the 5D supersymmetry. 2) Sasaki-Einstein geometry and the Killing spinors, in which I will explain some basics of spin geometry and how to solve for the Killing spinors on such manifolds. 3) The cohomological complex, and the localization procedure, in which I will explain the localization arguments and investigate the localisation locus 4) The index theorem, where I will explain some basics of Atiyah?s K-theoretical formula of transversally elliptic operators. Depending on the time, I may also discuss the implication of the results, such as the N3 behaviour and comparison with perturbation theory. |
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Oct 11, 2013 - 14:30 | Marco Aldi | Hamiltonian Quantization of Loop Spaces and Vertex Algebras III | Lecture | In these lectures we review some recent results regarding the use of vertex algebras and Poisson vertex algebras in the Hamiltonian quantization of (connected) loop spaces. After recalling the basic constructions, we will explain how the Hamiltonian formalism can be used to study target-space geometry. Hamiltonian quantization of disconnected loop spaces will be addressed in the last lecture. | ||||

Oct 15, 2013 - 14:30 | R. Minasian | Higher derivative corrections, B-field and N=2 theories | Seminar | |||||

Oct 16, 2013 - 11:00 | J. Minahan | Three-point functions for short operators | Seminar | |||||

Oct 16, 2013 - 14:30 | A Cattaneo | BV around the corner: A gentle introduction to the BV formalism from scratch to field theories on manifolds with boundary and corners [Part I] | Seminar | |||||

Oct 17, 2013 - 11:00 | A. Cattaneo | BV around the corner: A gentle introduction to the BV formalism from scratch to field theories on manifolds with boundary and corners [Part II] | Seminar |