### University of Manchester

### Informal Research Seminar on Geometry and
Mathematical Physics

### Special programme:

## Double and Multiple
Structures in Standard and Super Geometry

## Weekly details, 2007 - 08

#### Thursday February 21, 2008

At about 1.15 Ted will recall the Cattaneo-Felder action and its geometrical meaning
in terms of Lie algebroids and then start discussing its symmetries.
At about 2.00 Kirill will complete the description of the variational formula left incomplete last time.

#### Thursday November 29, 2007

At about 1.15 Ted will elaborate the statement made last time concerning the classical
equations of motion for the field-theoretical model considered by
Cattaneo and Felder. Namely, it will be explained that for two manifolds
endowed with vector fields, the condition that a smooth map relates (=
intertwines) these fields can be interpreted as the vanishing at this
map of a certain natural vector field on the (infinite-dimensional)
"manifold of smooth maps". This is, in fact, almost obvious. Less
obvious is that such a vector field on the space of maps can be
sometimes written in a "gradient" form, i.e., the above condition can be
expressed by variational equations. A heuristic statement when this
occurs will be given. It covers the CF case and other interesting
situations.
At about 2.15 Kirill will endeavour to explain the integral condition on
deformations with which he ended two weeks ago. If there is time, he
will start on the Lie algebroid case.

#### Thursday November 15, 2007

At about 1.10, Ted will start a discussion of the Cattaneo-Felder paper of 1999, where
they suggested a quantum-field theoretical explanation of
Kontsevich's formulas for deformation quantization of Poisson
manifolds. This should be related with another approach to
quantization of Poisson manifolds, namely, via symplectic groupoids.
Hence there is a hope of convergence of this line with the discussion
of integration of Lie brackets that has been started earlier by
Kirill.
At about 2.05pm, Kirill will resume the discussion of Duistermaat's proof
of the integrability of Lie algebras using path spaces. He will recall
some of the material from the meetings in spring and then go on to the
variational formula.

This page will be added to as the seminar develops.
February 20, 2008