Letters in Mathematical Physics

, Volume 105, Issue 11, pp 1605–1632 | Cite as

Homological Perturbation Theory for Nonperturbative Integrals

  • Theo Johnson-FreydEmail author


We use the homological perturbation lemma to produce explicit formulas computing the class in the twisted de Rham complex represented by an arbitrary polynomial. This is a non-asymptotic version of the method of Feynman diagrams. In particular, we explain that phenomena usually thought of as particular to asymptotic integrals in fact also occur exactly: integrals of the type appearing in quantum field theory can be reduced in a totally algebraic fashion to integrals over an Euler–Lagrange locus, provided this locus is understood in the scheme-theoretic sense, so that imaginary critical points and multiplicities of degenerate critical points contribute.

Mathematics Subject Classification

Primary 81S40 Secondary 18G40 


homological perturbation theory path integrals Feynman diagrams Jacobian varieties 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Albert, C., Bleile, B., Fröhlich, J.: Batalin-Vilkovisky integrals in finite dimensions. J. Math. Phys. 51(1), 015213, 31 (2010). doi: 10.1063/1.3278524
  2. 2.
    Alexandrov, M., Kontsevich, M., Schwarz, A., Zaboronsky, O.: The geometry of the master equation and topological quantum field theory. Intern. J. Mod. Phys. A. 12(7), 1405–1429 (1997). doi: 10.1142/S0217751X97001031
  3. 3.
    Costello, K., Gwilliam, O.: Factorization algebras in perturbative quantum field theory. (2011).
  4. 4.
    Costello, K.: Renormalization and effective field theory, vol. 170. Mathematical Surveys and MonographsAmerican Mathematical Society, Providence (2011)Google Scholar
  5. 5.
    Crainic, M.: On the perturbation lemma, and deformations. (2004). arXiv:math/0403266
  6. 6.
    Douai, A.: Construction de variétés de Frobenius via les polynômes de Laurent: une autre approche. In: Singularités, volume 18 of Inst., Élie Cartan, pp. 105–123. Univ. Nancy, Nancy (2005)Google Scholar
  7. 7.
    Dimca A., Saito M.: On the cohomology of a general fiber of a polynomial map. Compos. Math. 85(3), 299–309 (1993)zbMATHMathSciNetGoogle Scholar
  8. 8.
    Fiorenza, D.: An introduction to the Batalin-Vilkovisky formalism. Lecture given at the “Rencontres Mathematiques de Glanon”. (2003). arXiv:math/0402057
  9. 9.
    Gwilliam, O., Johnson-Freyd, T.: How to derive Feynman diagrams for finite-dimensional integrals directly from the BV formalism. (2012). arXiv:1202.1554
  10. 10.
    Goodwillie, T.: Answer to “Is the pairing between contours and functions perfect (modulo the kernel given by Stokes’ theorem)?” MathOverflow. (2012).
  11. 11.
    Hien M., Roucairol C.: Integral representations for solutions of exponential Gauss-Manin systems. Bull. Soc. Math. France 136(4), 505–532 (2008)zbMATHMathSciNetGoogle Scholar
  12. 12.
    Huebschmann, J.: Origins and breadth of the theory of higher homotopies. In: Higher structures in geometry and physics, volume 287 of Progr. Math., pp. 25–38. Birkhäuser/Springer, New York (2011)Google Scholar
  13. 13.
    Isserlis L.: On a formula for the product-moment coefficient of any order of a normal frequency distribution in any number of variables. Biometrika 12, 134–139 (1918)CrossRefGoogle Scholar
  14. 14.
    Kashaev, R.M.: The hyperbolic volume of knots from the quantum dilogarithm. Lett. Math. Phys. 39(3),269–275 (1997). doi: 10.1023/A:1007364912784
  15. 15.
    Kouchnirenko A.G.: Polyèdres de Newton et nombres de Milnor. Invent. Math. 32(1), 1–31 (1976)MathSciNetCrossRefADSGoogle Scholar
  16. 16.
    Malgrange, B.: Méthode de la phase stationnaire et sommation de Borel. In: Complex analysis, microlocal calculus and relativistic quantum theory (Proc. Internat. Colloq., Centre Phys., Les Houches, 1979), volume 126 of Lecture Notes in Phys., pp. 170–177. Springer, Berlin (1980)Google Scholar
  17. 17.
    Morozov, A., Shakirov, Sh.: Introduction to integral discriminants. J. High Energy Phys. (12), 002, 39, (2009). doi: 10.1088/1126-6708/2009/12/002
  18. 18.
    Pham, F.: Vanishing homologies and the n variable saddlepoint method. In: Singularities, Part 2 (Arcata, Calif., 1981), volume 40 of Proc. Sympos. Pure Math., pp. 319–333. Amer. Math. Soc., Providence, RI (1983)Google Scholar
  19. 19.
    Pantev, T., Toën B., Vaquié, M., Vezzosi, G.: Shifted symplectic structures. Publ. Math. Inst. Hautes Études Sci. 117, 271–328 (2013). doi: 10.1007/s10240-013-0054-1. arXiv:1111.3209
  20. 20.
    Reshetikhin, N.: Topological quantum field theory: 20 years later. In: European Congress of Mathematics, pp. 333–377. Eur. Math. Soc., Zürich (2010)Google Scholar
  21. 21.
    Reshetikhin, N.Yu., Turaev, V.G.: Ribbon graphs and their invariants derived from quantum groups. Commun. Math. Phys. 127(1), 1–26 (1990)Google Scholar
  22. 22.
    Sabbah, C.: Hypergeometric periods for a tame polynomial. Port. Math. (N.S.) 63(2), 173–226 (2006)Google Scholar
  23. 23.
    Schulze, M.: Good bases for tame polynomials. J. Symbolic Comput. 39(1), 103–126 (2005). doi: 10.1016/j.jsc.2004.10.001
  24. 24.
    Shakirov, Sh.R.: A nonperturbative approach to finite-dimensional non-Gaussian integrals. Teoret. Mat. Fiz. 163(3), 495–504 (2010). doi: 10.1007/s11232-010-0064-9
  25. 25.
    Schwarz, A., Shapiro, I.: Twisted de Rham cohomology, homological definition of the integral and “physics over a ring”. Nucl. Phys. B. 809(3), 547–560 (2010). doi: 10.1016/j.nuclphysb.2008.10.005
  26. 26.
    Stasheff, J.: The (secret?) homological algebra of the Batalin-Vilkovisky approach. Talk given at Conference on Secondary Calculus and Cohomological Physics, Moscow, Russia, pp. 24–31. (1997). arXiv:hep-th/9712157
  27. 27.
    Weibel, C.A.: History of homological algebra. In: History of topology, pp. 797–836. North-Holland, Amsterdam (1999). doi: 10.1016/B978-044482375-5/50029-8
  28. 28.
    Witten E.: Quantum field theory and the Jones polynomial. Commun. Math. Phys. 121(3), 351–399 (1989)zbMATHCrossRefADSGoogle Scholar
  29. 29.
    Witten E.: A note on the antibracket formalism. Mod. Phys. Lett. A 5(7), 487–494 (1990)zbMATHCrossRefADSGoogle Scholar
  30. 30.
    Witten, E.: Analytic continuation of Chern-Simons theory. In: Chern-Simons gauge theory: 20 years after, volume 50 of AMS/IP Stud. Adv. Math., pp. 347–446. Amer. Math. Soc., Providence (2011)Google Scholar
  31. 31.
    Zworski, M.: Semiclassical analysis, volume 138 of Graduate Studies in Mathematics. American Mathematical Society, Providence (2012)Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.Mathematics DepartmentNorthwestern UniversityEvanstonUSA

Personalised recommendations