Supersymmetric theory of stochastic dynamics: Difference between revisions

Revision as of 12:58, 18 June 2025

Revision as of 16:35, 24 June 2025

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Here, the noise is assumed Gaussian white, p.b.c. signifies periodic boundary conditions, <math> \textstyle J(\xi) </math> is the Jacobian compensating (up to a sign) the Jacobian from the <math>\delta</math>-functional, <math> \Phi</math> is the collection of fields that includes, besides the original field <math> x</math>, the [[Faddeev–Popov_ghost|Faddeev–Popov ghosts]] <math> \chi, \bar\chi</math> and the Lagrange multiplier, <math> B</math>, the topological and/or BRST supersymmetry is,

<math display="block"> Q = \textstyle \int d\tau(\chi^i(\tau)\delta/\delta x^i(\tau) + B_i(\tau)\delta/\delta \bar\chi_i(\tau)), </math>

that can be looked upon as a pathintegral version of exterior derivative, and the gauge fermion
<math display="inline"> \Psi = \int d\tau (\imath_{\dot x} - \bar d )</math> with
<math display="inline"> \textstyle \bar d = \textstyle \imath_F - \Theta \imath_{G_a} L_{G_a}, \text{ withand } L_{G_a}=(Q,\imath_{G_a})</math> and <math display="inline"> \imath_A = i\bar\chi A</math> being the pathintegral versionversions of the Lie derivative and interior derivative.

=== STS as a topological field theory ===