Supersymmetric theory of stochastic dynamics: Difference between revisions

Revision as of 13:15, 25 June 2025

Revision as of 13:20, 25 June 2025

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Another object of interest is the ''sharp'' trace of the GTO,

<math display="block" > W = Tr (-1)^{\hat k} \hat{\mathcal M }_{tt'} = \sum\nolimits_\alpha (-1)^{k_\alpha}e^{-(t-t')H_\alpha}, </math>

where <math> \hat k |\psi_\alpha\rangle = k_\alpha |\psi_\alpha\rangle</math> with <math>\hat k</math> being the operator of the degree of the differential form. This is a fundamental object of topological nature known in physics as the [[Witten index]]. From the properties of the eigensystem of GTO, only supersymmetric singlets contribute to the Witten index, <math>W=\sum\nolimits_{k=0}^D (-1)^k B_k=Eu.Ch(X)</math>, where <math>Eu.Ch.</math> is the [[Euler characteristic]] and ''B'' 's arteare the numbers of supersymmetric singlets of the corresponding degree. These numbers equal [[Betti numbers]] as follows from one of the [[#Eigensystem of GTO|properties of GTO]] that each de Rham cohomology class provides one supersymmetric singlet.

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