Mathematics Seminar - Monday 18 July at 16:00

[Math Group]

M A T H E M A T I C S   S E M I N A R S  2016


Monday, 18 July, at 16.00 hrs.



Dana Fine (UMass Dartmouth)



Title:
Path integrals, supersymmetric quantum mechanics  and the Atiyah-Singer 
index theorem for twisted Dirac


Abstract:
In the 1980’s, Alvarez-Gaume, following Witten, used heuristic path 
integrals in supersymmetric quantum mechanics to give arguments for 
index theorems including the Gauss-Bonnet-Chern, Hirzebruch, and 
Atiyah-Singer theorems. In this talk, I summarize the arguments, and 
describe a mathematical construction of the relevant path integrals. 
This provides a new construction of the heat kernel for a class of 
Laplacians broad enough to include the square of the twisted Dirac 
operator, which corresponds to an extension of N=1/2 supersymmetric 
quantum mechanics. In this case, general results on the rate of 
convergence of the approximate path integrals suffice to derive the 
local version of the Atiyah-Singer index theorem.




VENUE:  Luigi Stasi Seminar Room (ICTP Leonardo da Vinci Building, first 
floor)