Schwarz, Albert S.; Shapiro, Ilya Twisted de Rham cohomology, homological definition of the integral and “physics over a ring”. (English) Zbl 1192.81311 Nucl. Phys., B 809, No. 3, 547-560 (2009). Summary: We use the notion of the twisted de Rham cohomology to give meaning to an integral of the form \(\int g(x)e^{f(x)}\,dx\) over an arbitrary ring. We discuss also a definition of a family of integrals and some properties of the above homological definition of the integral. We show how to use the twisted de Rham cohomology to define the Frobenius map on the \(p\)-adic cohomology. Finally, we consider two-dimensional topological quantum field theories with general coefficients. Cited in 4 Documents MSC: 81T45 Topological field theories in quantum mechanics 14F40 de Rham cohomology and algebraic geometry 14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Keywords:topological quantum field theory; de Rham cohomology; Frobenius map; exponential integrals PDFBibTeX XMLCite \textit{A. S. Schwarz} and \textit{I. Shapiro}, Nucl. Phys., B 809, No. 3, 547--560 (2009; Zbl 1192.81311) Full Text: DOI arXiv References: [1] Baldassarri, F.; D’Agnolo, A., On Dwork cohomology and algebraic \(D\)-modules · Zbl 1061.32022 [2] Bernstein, J.; Leites, D. A., Integral forms and Stokes formula on supermanifolds, Funct. Anal. Appl., 11, 1, 55-56 (1977) · Zbl 0352.58004 [3] Bernstein, J., Algebraic Theory of \(D\)-modules, Notes available from [4] Borel, A., Algebraic \(D\)-modules (1987), Academic Press: Academic Press Boston · Zbl 0642.32001 [5] Dimca, A.; Maaref, F.; Sabbah, C.; Saito, M., Dwork cohomology and algebraic \(D\)-modules, Math. Ann., 318, 1, 107-125 (2000) · Zbl 0985.14007 [6] Kapustin, A.; Witten, E., Electric-magnetic duality and the geometric Langlands program · Zbl 1128.22013 [7] Katz, N. M., On the differential equations satisfied by period matrices, Publ. Math. Inst. Hantes Études Sci., 35, 71-106 (1968) · Zbl 0159.22502 [8] Kontsevich, M.; Schwarz, A.; Vologodsky, V., Integrality of instanton numbers and \(p\)-adic B-model, Phys. Lett. B, 637, 97-101 (2006) · Zbl 1247.14058 [9] Kontsevich, M.; Zagier, D., Periods, (Mathematics Unlimited—2001 and Beyond (2001), Springer: Springer Berlin), 771-808 · Zbl 1039.11002 [10] Ogus, A.; Vologodsky, V., Nonabelian Hodge theory in characteristic \(p\), Publ. Math. Inst. Hautes Études Sci., 106, 1-138 (2007) · Zbl 1140.14007 [11] Sabbah, C., On a twisted de Rham complex, Tohoku Math. J., 51, 1, 125-140 (1999) · Zbl 0947.14007 [12] Schwarz, A.; Shapiro, I., Supergeometry and arithmetic geometry, Nucl. Phys. B, 756, 3, 207-218 (2006) · Zbl 1215.14016 [13] Shapiro, I., Frobenius map for quintic threefolds · Zbl 1222.14089 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.