Seminar Quantenphysik und Geometrie
**Batalin-Vilkovisky Formalism**

Bahns, Cortes, Fredenhagen, Louis, Richter und Schweigert

The Batalin-Vilkovisky formalism provides the modern approach to field theories with gauge symmetries. It is indispensable for supergravity theories. Mathematically, it provides an interesting combination of the Koszul complex with the Chevalley Eilenberg complex. Our goal is to understand these relations, see e.g. [St].

1. Talk: BRST and BV Quantization
(Vid Stojevic, Geomatikum, 25.10.2007)

An overview over the BV formalism in the language of classical field
theory.

Literature: the talk can be based on [FHM], supplementary
material [He1] and [He2]

2. Talk: BV formalism: a mathematical (p)review
(Christoph Schweigert, DESY, 8.11.2007 and Geomatikum 22.11.2007)

An overview over mathematical aspects of the BV formalism

Literature: [Li,St]

3. Talk: Cohomology of Lie algebras (Fridolin Roth, Geomatikum, 20.12.2007)

Show how the cohomology of Lie algebras arises in the study
of invariants.

There can be a crash course on Tor and Ext, or one can
simply reduce it to the Chevalley-Eilenberg resolution.

Examples. Possibly mention the relation to de Rham cohomology of
the assocaited Lie group.

Literature: the talk can be based on Chapter 10 of [Lo]

4. Talk: Koszul complex (Urs Schreiber, DESY,
17.01.2008)

The Koszul complex, its definition, description of quotients

Examples

Literature: Chapter 17 in [Eis]

4. Talk: Homological perturbation theory
(Birgit Richter, DESY, 17.4.2008)

Handout and
full version
of the talk.

Literature

- [Ei] D. Eisenbud: Commutative Algebra with a view toward algebraic geometry. Springer GTM 150.
- [Fi] D. Fiorenza: An introduction to the Batalin-Vilkovisky formalism. math/0402057
- [FHM] A. Fuster, M. Henneaux, A. Maas: BRST-antifield Quantization: a short review hep-th/0506098
- [He1] M. Henneaux: On the algebraic structure of the BRST Symmetry. Physics, Geometry, and Topology. H.C. Lee Ed, Penum, New York, 1990. link
- [He2] M. Henneaux: Lectures on the antifield-BRST Formalism for Gauge Theories. Nuclear Physics B (Proc. Suppl.) 18A (1990) 47-106 link
- [Lo] J.-L. Loday: Cyclic Homology. Springer Grundlehren 301.
- [St] J. Stafheff: The secret homological algebra of the
Batalin-Vilkovisky approach
hep-th/9712157

Seminar Quantenphysik und Geometrie,*email:*schweigert@math.uni-hamburg.de