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Tuesday Seminar on Topology
Seminar information archive ～04/23｜Next seminar｜Future seminars 04/24～
| Date, time & place | Tuesday 17:00 - 18:30 056Room #056 (Graduate School of Math. Sci. Bldg.) |
|---|---|
| Organizer(s) | KAWAZUMI Nariya, KITAYAMA Takahiro, SAKASAI Takuya |
2015/12/15
17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)
Constantin Teleman (University of California, Berkeley)
The Curved Cartan Complex (ENGLISH)
Constantin Teleman (University of California, Berkeley)
The Curved Cartan Complex (ENGLISH)
[ Abstract ]
The Cartan model computes the equivariant cohomology of a smooth manifold X with
differentiable action of a compact Lie group G, from the invariant polynomial
functions on the Lie algebra with values in differential forms and a deformation
of the de Rham differential. Before extracting invariants, the Cartan differential
does not square to zero and is apparently meaningless. Unrecognised was the fact
that the full complex is a curved algebra, computing the quotient by G of the
algebra of differential forms on X. This generates, for example, a gauged version of
string topology. Another instance of the construction, applied to deformation
quantisation of symplectic manifolds, gives the BRST construction of the symplectic
quotient. Finally, the theory for a X point with an additional quadratic curving
computes the representation category of the compact group G, and this generalises
to the loop group of G and even to real semi-simple groups.
The Cartan model computes the equivariant cohomology of a smooth manifold X with
differentiable action of a compact Lie group G, from the invariant polynomial
functions on the Lie algebra with values in differential forms and a deformation
of the de Rham differential. Before extracting invariants, the Cartan differential
does not square to zero and is apparently meaningless. Unrecognised was the fact
that the full complex is a curved algebra, computing the quotient by G of the
algebra of differential forms on X. This generates, for example, a gauged version of
string topology. Another instance of the construction, applied to deformation
quantisation of symplectic manifolds, gives the BRST construction of the symplectic
quotient. Finally, the theory for a X point with an additional quadratic curving
computes the representation category of the compact group G, and this generalises
to the loop group of G and even to real semi-simple groups.
