University of Manchester

Informal Research Seminar on Geometry and Mathematical Physics

Special programme:

Double and Multiple Structures in Standard and Super Geometry


Weekly details, 2006 - 07

Thursday March 15, 2007

TV will speak about deformation theory, continuing from the previous time. A geometric setting will be introduced. Namely, the main equation of deformation theory (the "master equation" or "Maurer-Cartan" equation) will be related with homological fields on supermanifolds. This leads to an explanation of the statement of Kontsevich's formality theorem.

KM will describe Duistermaat's proof of the integrability of Lie algebras, on which Crainic and Fernandes based their generalization to Lie algebroids.

Thursday March 1, 2007

TV will continue the discussion begun last time, on quantization as deformation, and relate it to the "formality" approach of Kontsevich.

KM will continue with approaches to the integration of Lie algebras and Lie algebroids.

Thursday February 15, 2007

At 1.30 T.V. will give a brief introduction to the problem of quantization understood as a non-commutative deformation of a commutative algebra. Then results of Fedosov (and others) and Kontsevich about symplectic and general Poisson manifolds will be stated. From there we plan to start analyzing the approach that has been put forward by A. Cattaneo and G. Felder. Our eventual intention is to clarify links between quantization of non-linear Poisson brackets and integration of Lie algebroids (which is the topic of K.M. lectures).

At 2.30, KM will speak on the background to the work of Cattaneo and Felder. Cattaneo and Felder set out to construct, for any Poisson manifold, a symplectic realization and it appears that they did not at first appreciate that they had constructed a symplectic groupoid for the cotangent Lie algebroid. I will describe the relationship between these problems and, if time permits, also recall the integrability results known prior to Cattaneo and Felder: the most general result known was for transitive Lie algebroids, where there is a single cohomological obstruction, a kind of `non-abelian first Chern class'.

Thursday December 14, 2006

KM: I will finish speaking on connections, and then describe Lie groupoids and the process of passing from a Lie groupoid to its Lie algebroid.

TV: I'll move on to the relation between morphisms of Q-manifolds, on the one hand, and morphisms of Lie algebroids and L-infinity maps of L-infinity algebras, on the other. The idea is to show how introducing Q-manifolds simplifies things. Time permitting, I can start discussing other topics promised long ago (Lie bialgebroids, double Lie algebroids).

Thursday December 7, 2006

KM: I will outline connection theory in transitive Lie algebroids. If there is time, I will begin a brief description of Lie groupoids. These are the global objects which stand to Lie algebroids as Lie groups do to Lie algebras.

TV: I will continue with Q-manifolds as started two weeks ago, and consider their maps. This will naturally lead as to morphisms of Lie algebroids (alternative description) and L-infinity maps as examples. After that, the plan will be to move to to bialgebroids and multiple structures, as long promised.

Thursday November 30, 2006

12.00: KM will finish talking about general morphisms of Lie algebroids and will go on to cover connections in transversal Lie algebroids and Gauss-Manin connections.

2.30pm (notice unusual time and location!) in MSS/C13a: Dmitry Leykin will speak on `A canonical flat meromorphic connection on the universal bundle of Jacobians'

Thursday November 23, 2006

KM: Last week I finished the main classes of examples of Lie algebroids, and said that I would go on to give a quick account of connection theory in Lie algebroids, and in particular of the Gauss-Manin connection. I still plan to do this but I'll postpone it to the 30th or to December 6th.

This week I will cover general morphisms of Lie algebroids. These are much more involved than one might expect: they include for example Maurer-Cartan forms and sigma models.

TV: I am planning to add a few more algebraic and geometric examples of brackets coming about from the "derived" or "higher derived" construction, discussed at the previous lectures. Then my plan is to turn to Q-manifolds, that is, supermanifolds (possibly, with an extra Z-grading) endowed with a homological vector field. Q-manifolds, together with Poisson manifolds and Schouten manifolds, may be considered as equally important "non-linear" generalizations of Lie algebras.

I will show how morphisms of Q-manifolds supply, in particular, morphisms of Lie algebroids over possibly different bases (introduced by KM in his lecture), as well as L-infinity maps of L-infinity algebras.

Thursday November 16, 2006

KM: I will finish the description of examples of Lie algebroids which I began on the 9th. I'll include some comments on the Gauss-Manin connection. I also expect to be able to say something about morphisms of Lie algebroids. Morphisms of Lie algebroids are more involved than one would expect: they include for example Maurer-Cartan forms and sigma models.

TV: I shall continue discussing strongly homotopy Lie algebras (L-infinity algebras) from the viewpoint of homological vector fields, as started last time. Then I shall probably come back to Lie algebroids. One point where the supermanifold approach seems very helpful, is the notion of morphisms of Lie algebroids over different bases. I hope to supplement the treatment of morphisms that will be given in KM's lecture. Then, time permitting, I can start discussing Lie bialgebroids and double Lie algebroids.

Thursday November 9, 2006

KM will talk about the notion of Lie algebroid, covering the examples mentioned at TV's talk last week and describing some general constructions. There will be a few minutes on the duality of double vector bundles (continuing from last week) at the start.

TV will continue with the application of homological vector fields. This week the plan is to come back to Lie algebras and, using them as an example, introduce the derived bracket construction. Then we will generalize to "strongly homotopy Lie algebras" (= L-infinity algebras) and discuss higher derived brackets. More examples will follow. Then we can return to Lie algebroids and touch on their morphisms (introduced by KM from a different viewpoint). Then probably Lie bialgebroids and double Lie algebroids (all from the viewpoint of supermanifolds).

Thursday November 2:

Both speakers will continue from the previous week.

Thursday October 26:

At 12 noon, Kirill Mackenzie will talk on double vector bundles and their duality. This underlies everything which will follow, but is also of interest in its own right. It can be appreciated by anyone who knows what a vector bundle is.

A double vector bundle is a manifold D with two vector bundle structures over bases A and B, each of which is a vector bundle over a manifold M; there is a compatibility condition. Since D has two vector bundle structures, it can be dualized in two ways. These dualizations do not commute, but generate the dihedral group of order 6. This is thus a kind of `non-abelian duality'.

At 2pm, Ted Voronov will talk on the concept of `neighbors' and application of homological vector fields for describing structures such as Lie (bi)algebras and Lie (bi)algebroids, and others.

A neighbor of a vector space or a vector bundle (or a double, or multiple, vector bundle) is obtained by applying operations of taking dual and reversing parity. A homological vector field on a supermanifold is an odd vector field commuting with itself, i.e., satisfying [Q,Q]=0.


March 14, 2007