Branes from a derived point of view I

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.

Branes from a derived point of view II

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.

The secret life of sigma super-models I

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.

Branes from a derived point of view III

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.

The secret life of sigma super-models II

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.

(Sheaves of) vertex algebras and stringy manifold invariants I

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.

Derived symplectic geometry and topological field theories I

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

Derived symplectic geometry and topological field theories II

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

The secret life of sigma super-models II

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.

Derived symplectic geometry and topological field theories III

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

(Sheaves of) vertex algebras and stringy manifold invariants II

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.

(Sheaves of) vertex algebras and stringy manifold invariants III

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.

Superstring Perturbation Theory Revisited I

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.

Around moduli spaces of flat connections I

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.

An introduction to differentiable stacks and moduli space I

Available soon

Localization of 5D super Yang-Mills I

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.

Around moduli spaces of flat connections II

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.

Superstring Perturbation Theory Revisited I

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.

The Birth of String Theory

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.

Superstring Perturbation Theory Revisited II

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.

Around moduli spaces of flat connections III

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.

Hamiltonian Quantization of Loop Spaces and Vertex Algebras I

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.

A graded-geometric viewpoint to generalized geometry II

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.

Localization of 5D super Yang-Mills II

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.

Superstring Perturbation Theory Revisited III

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.

Hamiltonian Quantization of Loop Spaces and Vertex Algebras II

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.

An introduction to differentiable stacks and moduli space II

Available soon

An introduction to differentiable stacks and moduli space III

Available soon

Localization of 5D super Yang-Mills III

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.

Hamiltonian Quantization of Loop Spaces and Vertex Algebras III

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.