Nikita Markarian’s mathblog

Homological algebra of perturbation theory

Posted in Mathematics by nikitamarkarian on August 19, 2009

Below there are some obvious and elementary observations about D-modules and oscillating integrals. I suspect that they are widely known, but I could not find them anywhere in literature.

Consider the integral I(k)=\int e^{(f+kg) } \omega, where f and g are functions on a complex vector space V, \omega is the standard volume form on V, k is a formal parameter and the integral is taken along a cycle of the half dimension such, that f tends to -\infty along it. The question is what can we say about function I(k) without transcendental methods, that is only using the Stokes’ theorem.

In the first example f is negatively defined quadratic form, g is of degree more than 2 and the cycle is the real plane. Then I(k)=I(0)F(k), where F is a series of k with coefficients that are rational functions of coefficients of f and g. Note that the series F(k) does not converge, it is asymptotic. The reason is although f tends to -\infty along the cycle, f+k g does not for any k. This example is very important for quantum field theory and coefficients of F come from summing by Feynman diagrams.

In the second example f is an isolated singularity and g gives a versal deformation of it. Say, V is one-dimensional, f=z^{n+1} and g=a_{n-1}z^{n-1}+\dots+a_0. There are n linearly independent classes of cycles. Integrals by these cycles obey an ordinary differential equation of the first order. It means than the integral by a chosen cycle is managed by a differential equation of order n (the Picard-Fuchs equation). Its coefficients are rational functions of coefficients of f and g.

Given f as above, consider subspace \Gamma_f of the space \Gamma(O) of polynomial functions on V defined by: a\in \Gamma_f iff \int a e^f \omega=0 by the Stokes’ theorem. For example, f'\in \Gamma_f, because \int f' e^f \omega=\int d e^ f \omega. And also f''+f'^2\in \Gamma_f and so on. For example, if f=z^2, then \Gamma_f is generated by all Hermite polynomials except the first one (which is 1). If one infinitesimally changes f adding k g, changing of the integral depends only on projection of e^{k g} along \Gamma_f. If f is a simple singularity, then then the space \Gamma(O)/\Gamma_f is finite-dimensional. Its dimension is the Milnor number of singularity, that is the codimension of the Jacobian ideal. This is also equals to the dimension of the middle homology group of the non-singular fiber like in the second example above.

Given f as above, consider the local system given by connection is d+df on O. Denote corresponding left D-module by O_f and the conjugate right D-module by \omega_f. It is easy to check that \Gamma(O)/\Gamma_f=Tor_{D(V)}^0(O_{f/2}, \omega_{f/2}). D-modules O_f and O_{f+kg} are isomorphic, the isomorphism is multiplication by e^{k g}, which is a formal series with polynomial coefficients. Thus Tor_{D(V)}^0(O_{f/2}, \omega_{f/2}) is equipped by a connection over the space of f‘s. It is easy to see that this connection is encoded in the projection \Gamma(O)\to \Gamma(O)/\Gamma_f.

Note that the space Tor_{D(V)}^0(O_{f/2}, \omega_{f/2}) is isomorphic the top cohomology of the twisted de Rham complex (\Omega^*(V), d+df). The latter space also has a connection , but a different one. This is because the kernel of the natural projection of \Gamma(O) on this space differs from \Gamma_f. For example, for f=z^2 the projection of functions on the top cohomology of the twisted de Rham complex, which is 1-dimensional, is simply the value of a function at 0, but projection \Gamma(O)\to \Gamma(O)/\Gamma_f is less trivial as we have seen before.

As far as I understand, the connection given by the de Rham complex in a “physical” (TQFT?) context correspond to the “genus 0 terms” and terms given by the Feynman diagrams – to the “higher genera contribution”, but I may mistake at this point.

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