- Vertex Algebras and their modules — Carina Boyallian
- Calabi-Yau manifolds and Landau-Ginzburg models — Alessandro Chiodo
- String compactifications and fluxes — Mariana Graña
- Representations of infinite dimensional Lie algebras — Vyacheslav Futorny
- Quantum field theory on curved spacetimes (after R. Borcherds) — Estanislao Herscovich
- Generalized geometry and supersymmetric spaces — Dan Waldram

Vertex Algebras and their modulesCarina Boyallian

The course will provide an introduction to vertex algebras and their modules.

Course contents:

1. Motivations. Formal calculus, formal series and delta function. Derivations and formal Taylor theorem. Expansions of zero and applications.

2. Vertex algebras. Definition and fundamental properties.

3. Commutativity and associativity properties in vertex algebras.

4. Construction of families of vertex algebras associated to Virasoro and affine Lie algebras.

5. Modules over vertex algebras.Notesfor this course are available here.

Calabi-Yau manifolds and Landau-Ginzburg modelsAlessandro ChiodoThe aim of this course is to introduce a theory counting curves in the context of Landau-Ginzburg models. The idea was inspired by Witten and it was translated into mathematical language by several authors among which Fan-Jarvis-Ruan, Polishchuk-Vaintrob, and Chang-Li-Li-Liu. The first step is to complete a corned of the mirror symmetry picture. Indeed, mirror symmetry is often presented as a correspondence matching the curve-counting theory of Gromov--Witten invariants of complex varieties to the geometry of Landau--Ginzburg models. Switching the two sides, one would like to set up a theory counting curves of Landau--Ginzburg models. This can be stated at several levels. Already in the most elementary terms, this leads to understand the simplest mirror symmetry phenomena.

The students attending this course will gain experience in very closely related subjects such as moduli of curves, orbifold Chen-Ruan cohomology and quantum cohomology.

Course contents:

1. A basic definition of orbifold Chen-Ruan cohomology.

2. A basic introduction to the enumerative geometry of curves.

3. Some quantum cohomology computations. The Landau-Ginzburg model.

4. The generalized mirror symmetry theorem.

5. Applications and generalizations of mirror symmetry.

Notesfor this course are available here.

String compactifications and fluxesMariana Graña

String theory is consistently defined in ten dimensions, six of which should be curled up in some small internal compact manifold. The procedure of linking this manifold to four-dimensional physics is called string compactification, and in these lectures, I will review it quite extensively. I will start with a very brief introduction to string theory; in particular, I will work out its massless spectrum and show how the condition on the number of dimensions arises. I will then dwell on the different possible internal manifolds, starting from the simplest to the most relevant phenomenologically. I will show that these are most elegantly described by an extension of ordinary Riemannian geometry termed generalized geometry, first introduced by Hitchin. I shall finish by discussing (partially) open problems in string phenomenology, such as the embedding of the Standard Model and obtaining de Sitter solutions.

Course contents:

1. Introduction to String Theory. Bosonic Strings and superstrings. Compactifications on tori. Kaluza-Klein compactifications on S1 (in Field Theory). Compactifications of the Bosonic String on S1. Toroidal string compactifications.

2. Calabi-Yau compactifications. The geometry of Calabi-Yau manifolds. Effective Theory for compactifications of Type II theories on CY3.

3. Fluxes and Generalized Geometry. Charges and fluxes. Generalized Geometry. Flux compactifications and Generalized Complex Geometry.

4. 4D effective actions. Compactifications on twisted tori and N = 8 Gauged Supergravity. Compactifications on Generalized Complex Geometries and N=2 Gauged Supergravity. Exceptional Generalized Geometry.

5. Open problems in phenomenology. Calabi-Yau orientifold reductions. Moduli stabilization in Type IIB. Moduli stabilization including non perturbative effects and de Sitter vacua.Exercisesfor this course can be found here.

Representations of infinite dimensional Lie algebrasVyacheslav Futorny

Representation theory of algebras is a very active mainstream branch of modern mathematics. It has numerous connections and applications to various other fields such as geometry, mathematical physics, combinatorics, topology, PDE´s, etc. The course will provide an introduction to the representation theory of various classes of infinite dimensional Lie algebras including affine Kac-Moody algebras, Krichever-Novikov algebras associated with elliptic curves and Lie algebras of vector fields on algebraic varieties.

Course contents:

1. Basics of the classical Lie theory. Classification of simple complex finite dimensional Lie algebras.

2. Some classes of infinite dimensional Lie algebras and the theory of Kac-Moody algebras.

3. The Virasoro algebra and Lie algebras of vector fields and their representation theory.

4. Construction and properties of elliptic affine Lie algebras and their applications.

5. Free field realizations and vertex structures.Exercisesfor this course can be found here.

Basics on Calabi-Yau manifoldsEstanislao Herscovich

The aim of these lectures is to present the basics on Calabi-Yau manifolds, which are the one of the main topics of this school.

Course contents:

1. Basics on smooth and holomorphic manifolds.

2. Vector bundles over manifolds.

3. Almost complex structures and their relation to complex structures.

4. Riemannian manifolds.

5. Covariant derivatives, torsion and curvature. Holonomy and curvature.

6. Kähler manifolds. Characteristic classes. Calabi-Yau manifolds.Notesfor this course can be found here.

Generalized geometry and supersymmetric spacesDan Waldram

The goal of these lectures is first to introduce the notion of generalized geometry and discuss its relation to supergravity and string theory, and then to show how it can be used to describe natural string theory generalizations of special holonomy manifolds. These later spaces are central to defining string compactifications and examples of the AdS/CFT correspondence.

Course contents:

1. Notion of G-structures in conventional geometry: intrinsic torsion, examples of Riemannian geometry, complex and symplectic geometry, special holonomy spaces, relation to Killing spinor equations.

2. Introduction to generalized geometry with O(d,d)×R+ structure group: extended tangent space, generalized Lie derivative, generalized connections and torsion, analogue of Levi-Civita connection.

3. Review of type II supergravity, and rewriting of NS-NS sector as generalized Einstein theory: generalized Ricci tensor, supersymmetric transformations in terms of generalized Levi-Civita connection.

4. Construction of gravity theory from generic generalized geometry: case of exceptional structure groups, relation to R-R sector of type II and d=11 supergravity.

5. Generalized special holonomy as torsion-free generalized G-structure: relation to Killing spinor equations, embedding of conventional special holonomy, simple flux examples, possible holonomy groups. Specific example of Exceptional Calabi-Yau manifolds: definition in terms of invariant generalized tensors, integrability conditions as moment maps, deformations and cohomology.Notesfor this course are available here.Exercisesfor this course are available here.

Introduction to homological algebra (mini course)Andrea Solotar

This course aims at introducing the students to the basic notions of homological algebra, approached from an elementary viewpoint using derived functors.

Course contents:

1. We will start with a review of functors and categories including several examples and emphasizing the definitions of kernel and cokernel, product and coproduct, as well as the notions of right and left exact functors.

2. We will talk about complexes in a module category and cohomology theories.

3. Finally, we will analyze different kinds of examples and applications.

Loday algebras and derived brackets (mini course)María Ofelia Ronco

The aim of the course is to provide a brief introduction to Loday algebras and derived brackets.

Course contents:

1. Leibniz algebras, a non-symmetric version of Lie algebras: examples and homology. Leibniz algebras as Lie algebras in a category of linear homomorphisms. Diassociative algebras, enveloping algebra of a Leibniz algebra. Koszul duality for diassociative algebras. Dendriform algebras. Homology of diassociative algebras. Loday algebras spanned by the faces of generalized associahedra.

2. Baxter operators and dendriform algebras. Derived brackets: Leibniz algebras and non-commutative Poisson brackets, Nijenhuis algebras, Gerstenhaber algebras and Batalin-Vilkovisky algebras. Pre-Lie algebras: the Connes-Kreimer Hopf algebra.

3. Formal deformations of a cotensor algebra, Baues´s B∞ algebras. Gertenhaber algebras up-to-homotopy. Higher algebraic structures on cohomology. Loday algebras and deformation quantization.

Hopf algebras and quantum field theory (mini course)Alessandra Frabetti

Affine group schemes form a very rich theory, which extends that of compact Lie groups and is deeply related to representation theory. Such groups are equivalent to their spaces of coordinate functions, on which they induce the structure of a finitely generated commutative Hopf algebra, or an inductive limit of such algebras.

In quantum field theory, several extensions of such Hopf algebras appear in connection with the renormalization of Green's functions, after the pioneering idea of D. Kreimer.

On the one side, these algebras present new type of generators, like Feynman graphs or trees, which may be seen as finer (and virtual) coordinate functions suitable for a ``quantum'' geometrical world. On the other side, non-scalar field theories demand non-commutative algebras, and thus a relaxed notion of affine groups defined on categories of not necessarily commutative algebras.In these lectures, I will introduce the so-called renormalization Hopf algebras, all examples of Loday-Ronco's combinatorial Hopf algebras, some associated standard groups of formal series and finally their generalization to affine loops on associative and even non-associative algebras.

Course contents:

1. QFT and renormalization Hopf algebras.

2. Formal diffeomorphisms and invertible series.

3. Generalized bialgebras for groups and loops.Notesfor this course can be found here.

Wess-Zumino-Witten models (mini course)Sergio Iguri

Wess-Zumino-Witten models are conformal field theories with the particularity that they can be formulated directly in terms of an action. The goal of these lectures is to introduce them and to show how a Lie-algebraic structure naturally arises this way. Affine primary fields, the Sugawara construction and the Knizhnik-Zamolodchikov equations for the four-point functions will be discussed in detail. Free-field representations will also be addressed.Course contents:

1. Some basics on conformal field theories. WZW models. Conserved currents and the emergence of affine Lie algebras. The Sugawara construction.

2. Affine primary fields. Correlation functions and the Knizhnik-Zamolodchikov equations. Four-point functions and the crossing-symmetry constraint. The bootstrap approach.

3. Free-fermion representations. Vertex representations of simply-laced algebras. Parafermions. The Wakimoto representation. Screening operators and correlation functions.

Calabi-Yau algebras (mini course)Mariano Suárez-ÁlvarezCalabi-Yau algebras arose naturally in the non-commutative geometry of Calabi-Yau manifolds but nowadays play important roles in many areas, most notably in representation theory and combinatorics, as it turns out that the Calabi-Yau nicely encodes very useful properties.

This short course will start with a survey on the subject, and then proceed to describe on one hand some important constructions of Calabi-Yau algebras and, on the other, to present results on their representation theory.

Course contents:

1. A survey of Calabi-Yau algebras2. Constructions and examples.

3. Some representation theory.