nLab homological perturbation theory

Idea

The aim of Homological Perturbation Theory is to construct small chain complexes from large ones. It was originally developed for the calculation of chain models of the total spaces of fiber bundles, but has since developed into a useful general computational tool.

Homological perturbation lemma

Let $(X,d), (Y,d)$ be chain complexes over a commutative ring $R$ and let $f: X \to Y, \nabla: Y \to X$ be chain maps, and $\Phi: X \to X$ be a chain homotopy such that

$$f \nabla=1, \quad \nabla f= 1 + d\Phi + \phi d,$$

$$f\Phi = 0, \Phi \nabla=0, \Phi^2=0, \Phi d \Phi= - \Phi.$$

Let $X,Y$ have filtrations $F^*$ bounded below by $0$ and preserved by $\nabla,f, \Phi$ and the differentials on $X,Y$. Suppose $X$ has another differential $d^\tau$ with the property that

$$(d^\tau -d)F^p X \subseteq F^{p-1} X$$

for all $p \geq 0$. The Homological Perturbation Lemma states that $Y$ can be given a new differential $d^\tau$ such that there is a quasi-isomorphism $(Y, d^\tau) \to (X, d^\tau)$.

The main point is that there is an explicit formula for the new chain homotopy as

$$\Phi^\tau= \sum _{r=0}^\infty \Phi (1+ d^\tau \Phi)^r.$$

There is considerable interest in describing the new differential in terms of a twisting cochain.

This result derived from earlier work of G. Hirsch, E.H. Brown, Weishu Shih, and has been widely developed into a useful theoretical and computational tool by Guggenheim, Lambe, Stasheff and others.

Applications

BV-complexes and Wick’s lemma

Homological perturbation theory is a key tool in the construction of BRST-BV complexes, where the quantum BV complex is a perturbation of a classical BV-complex. See (Gwilliam, section 2.5). In this context Wick's lemma in quantum field theory is a direct consequence of the homological perturbation lemma (Gwilliam, section 2.5.2).

References

Review includes

• Marius Crainic, On the perturbation lemma, and deformations (arXiv:math/0403266)

• Johannes Huebschmann, A survey on homological perturbation theory (pdf)

Other references include

• Ronnie Brown, The twisted Eilenberg-Zilber Theorem, Simposio di Topologia (Messina, 1964) pp. 33–37 Edizioni Oderisi, Gubbio. (pdf)

• Donald W. Barnes, Larry A. Lambe, A fixed point approach to homological perturbation theory, Proc. Amer. Math. Soc. 112 (1991), no. 3, 881–892.

• V Gugenheim, LA Lambe, JD Stasheff, Perturbation theory in differential homological algebra II, Illinois Journal of Mathematics 35:3 (1991) 357–373 euclid

For applications to “make computable” a bicategory of isolated hypersurface singularities and matrix factorisations comp that has been studied in the context of topological field theory, using the formulation in (Barnes-Lambe 91) is in

• Daniel Murfet, Computing with cut systems, (arXiv:1402.4541)

See also linear logic.

Discussion with an eye towards Hochschild cohomology and cyclic cohomology is in

• Larry A. Lambe, Homological perturbation theory, Hochschild homology and formal groups Contemp. Math. 189, AMS, 1992 (pdf)

• V. Álvarez, J.A. Armario, P. Real, B. Silva, Homological perturbation theory and computability Of Hochschild and cyclic homologies of cdgas (pdf)

Discussion in the context of BV-quantization is in section 2.5 of

• Owen Gwilliam, Factorization algebras and free field theories PhD thesis (pdf)