Supersymmetric theory of stochastic dynamics: Difference between revisions

Revision as of 19:48, 11 August 2025

Revision as of 21:58, 19 August 2025

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The idea of the Parisi–Sourlas method is to rewrite the partition function of the noise in terms of the dynamical variables of the model using [[BRST quantization|BRST]] gauge-fixing procedure.<ref name=":Baulieu_Grossman"/><ref name=":3"/> The resulting expression is the Witten index, whose physical meaning is (up to a topological factor) the partition function of the noise.

The pathintegralpath integral representation of the Witten index can be achieved in three steps: ''(i)'' introduction of the dynamical variables into the partition function of the noise; ''(ii)'' BRST gauge fixing the integration over the paths to the trajectories of the SDE which can be looked upon as the [[Gribov copies]]; and ''(iii)'' out integration of the noise. This can be expressed as the following

{{Equation box 1

|indent=:

|title='''supersymmetric pathintegralpath integral representation''' ''(of the Witten index)''

|equation=<math>

W = \langle \iint_{p.b.c} J(\xi) \left(\prod\nolimits_\tau \delta ^D (\dot x(\tau) - {\mathcal F}(x(\tau),\xi(\tau)))\right) {\mathcal D}x\rangle_{\text{noise}} = \iint_{p.b.c.} e^{(Q,\Psi(\Phi))}{\mathcal D}\Phi.

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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 pathintegralpath integral 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{ and } L_{G_a}=(Q,\imath_{G_a})</math> and <math display="inline"> \imath_A = i\bar\chi A</math> being the pathintegralpath integral versions of the Lie derivative and interior mutiplication.

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

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The Parisi-Sourlas method is peculiar in that sense that it looks like gauge fixing of an empty theory -- the gauge fixing term is the only part of the action. This is a definitive feature of [[topological quantum field theory|Witten-type topological field theories]]. Therefore, the Parisi-Sourlas method is a TFT <ref name=":3" /><ref name=":Baulieu_Grossman">{{Cite journal|last1=Baulieu|first1=L.|last2=Grossman|first2=B.|date=1988|journal=Physics Letters B|title=A topological interpretation of stochastic quantization|language=en|volume=212|issue=3|pages=351–356|doi=10.1016/0370-2693(88)91328-7|bibcode=1988PhLB..212..351B}}</ref><ref name=":4">{{Cite journal|last=Witten|first=E.|date=1988-09-01|title=Topological sigma models|journal=Communications in Mathematical Physics|language=en|volume=118|issue=3|pages=411–449|doi=10.1007/BF01466725|issn=0010-3616|bibcode=1988CMaPh.118..411W|s2cid=34042140|url=http://projecteuclid.org/euclid.cmp/1104162092|url-access=subscription}}</ref><ref name=":5">{{Cite journal|last=Witten|first=E.|date=1988-09-01|title=Topological quantum field theory|journal=Communications in Mathematical Physics|language=en|volume=117|issue=3|pages=353–386|doi=10.1007/BF01223371|issn=0010-3616|bibcode=1988CMaPh.117..353W|s2cid=43230714|url=http://projecteuclid.org/euclid.cmp/1104161738}}</ref><ref name=":6">{{Cite journal|last=Witten|first=E.|date=1982|title=Supersymmetry and Morse theory|journal=Journal of Differential Geometry|language=EN|volume=17|issue=4|pages=661–692|doi=10.4310/jdg/1214437492|issn=0022-040X|doi-access=free}}</ref><ref name=":7">{{Cite journal|last=Labastida|first=J. M. F.|date=1989-12-01|title=Morse theory interpretation of topological quantum field theories|journal=Communications in Mathematical Physics|language=en|volume=123|issue=4|pages=641–658|doi=10.1007/BF01218589|issn=0010-3616|bibcode=1989CMaPh.123..641L|citeseerx=10.1.1.509.3123|s2cid=53555484}}</ref>

and as a TFT it has got objects that are topological invariants.

The Parisi-Sourlas functional is one of them. It is essentially a pathintegralpath integral representation of the Witten index. The topological character of <math>W</math> is seen by noting that the gauge-fixing character of the functional ensures that only solutions of the SDE contribute. Each solution provides either positive or negative unity:

<math display="block">W =

\langle \iint_{p.b.c} J(\xi) \left(\prod\nolimits_\tau \delta ^D (\dot x(\tau) - {\mathcal F}(x(\tau),\xi(\tau)))\right) {\mathcal D}x \rangle_\text{noise} = \textstyle \left \langle I_N(\xi)\right \rangle_\text{noise},</math>