Nikita Markarian’s mathblog

Oscillating integrals and spherical homology

Posted in Mathematics by nikitamarkarian on November 16, 2009

In this post we discussed the following situation. Let V be a vector space, f be a family of functions on V and \omega be a volume form on V. We describe what one can say “without analysis” about the family of integrals \int e^f \omega. It appears that this is possible to execute in terms of the following differential Z/2Z-graded BV algebra: complex is polyvector fields on V, differential is \wedge df and \Delta=d+ \wedge df (d is the de Rham differential, differential forms is identified with polyvector fields by means of the volume form \omega). It allows to generalize the perturbation theory for arbitrary BV algebras.

The main observation: 2-algebra associated with the Gerstenhaber algebra of polyvector fields is Hochschild cochains, and its Hochschild homology as an algebra is the ring of differential operators on total differential forms \mathop{Diff}(\Omega^*, \Omega^*). This fact is contained in R.Nest, B.Tsygan “On the Cohomology Ring of an Algebra” and in D.Tamarkin, B.Tsygan, “The ring of differential operators on forms in noncommutative calculus”, see also the draft note from this post. \mathop{Diff}(\Omega^*, \Omega^*) is not exactly differential operators, but a thing Morita equivalent to it. Consider the differential 2-algebra B that corresponds to the BV algebra produced in the previous paragraph. Its Hochschild homology as an algebra is (\mathop{Diff}(\Omega^*, \Omega*), [\cdot, df]) ([,] is the usual commutator), that is a thing which is Morita equivalent to differential operators. Left and right modules over it provided by the statement correspond to O_f and \omega_f under the Morita equivalence. As a result one has Tor_{D(V)}^*(O_f, \omega_f)=HS_*(B).

Differential operators D(V)

HH(B)

D-modules O_f and \omega_f

B as left and right HH(B)-module

Tor_{D(V)}^*(O_f, \omega_f)

HS_*(B)

Tor_{D(V)}^*(O_f, \omega_f)
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