Spring 2012

May 2012

The working group is devoted to a series of seminars “Towards the mathematics of quantum field theory” by Frédéric Paugam, University Paris 7.

Frédéric Paugam

May 3 at 2pm, May 4 at 10am

• Functorial geometry and analysis on spaces of fields: definitions and examples
• Examples of action functionals: the fermionic particle, pure Yang-Mills, Yang-Mills with matter, general relativity

Frédéric Paugam

May 10 at 2pm, May 11 at 10am

Frédéric Paugam

May 24 at 2pm, May 25 at 10am

Algebraic tools for the structural study of non-linear PDEs:

• D-modules and D-algebras. Differential D-geometry
• The Ran space: local and chiral operations

Frédéric Paugam

May 31 at 2pm, June 1 at 10am

The classical Batalin–Vilkovisky formalism:

• The derived critical D-space and gauge symmetries
• The classical master equation and the gauge fixing procedure

The working group is devoted to D-modules, with five lectures by Pierre Schapira. The references are the book by Masaki Kashiwara on D-modules and Microlocal Calculus, and the draft by Pierre Schapira From D-modules to Deformation Quantization Modules (Ch 1,2).

Pierre Shapira

on March 15, 22, 29 and April 12, 19 at 2pm (Room B14)

I will explain the basic notions of D-module theory in the analytic setting: construction and main properties of the sheaf of rings of differential operators on a complex manifold, characteristic variety of coherent D-modules, operations on D-modules (inverse and direct images), holomorphic solutions of D-modules. If time allows it, we will have a glance to microdifferential operators and the link with deformation quantization. I will use freely the language of derived categories and sheaves, but there is no need to know anything on derived categories. On the other hand, some familiarity with basic sheaf theory and basic complex geometry is welcome.

Fall 2011

The working group started with a lecture by a guest of the team.

Melchior Grützmann

H-twisted Lie and H-twisted Courant algebroids and their cohomology

We will introduce a new kind of algebroid similar to a Lie algebroid, but the Jacobi identity twisted by a 3-form with values in the anchor map. Already Lie algebras permit the construction of non-trivial examples. We will introduce three kinds of cohomology theory, two in the language of dg-supermanifolds. Another class of examples occurs as Dirac structures in (splittable) twisted Courant algebroids a generalization of Hansen and Strobl's idea of twisting Courant algebroids with a 4-form.

As from October, the working group has run a weekly research seminar on Covariant Field Theory. The lectures has been given by N. Poncin, they are mainly based on Frédéric Paugam’s recent works.

1. Algebraic Geometry

Generalized algebraic varieties, schemes, functor of points.

2. Points and coordinates in Geometry and Physics

‘Point’ and ‘function’ viewpoints, Grothendieck topology, spaces: sheaves on a site, specific spaces: varieties and schemes, geometric constructions on spaces, spaces described by equations: ‘point’ approach and algebraic building blocks.

3. Schemes relative to a symmetric monoidal category

Complements on closed monoidal categories, relative schemes, relative differential calculus.

4. Applications to Classical Mechanics and Field Theory

International workshop

Giuseppe Bonavolontà and Norbert Poncin organized an international workshop on Covariant Field Theory from 6 to 8 December 2011.