×

Archimedean \(L\)-factors and topological field theories. I. (English) Zbl 1251.14011

In number theory, \(L\)–functions are primary objects of interest. The global \(L\)–function ‘decomposes’ as an analytic continuation of the product of local Archimedean and non-Archimedian \(L\)–factors. Non-Archimedian \(L\)–factors are better understood than Archimedean \(L\)–factors, reflecting our greater understanding of ‘finite primes’ than of ‘primes at infinity’.
The paper under review represents one step in an ambitious and exciting programme by the authors, to draw on physical techniques and insights in order to develop a unified geometric understanding of Archimedean and non-Archimedean objects in arithmetic geometry, and to interpolate between real and non-Archimedean Whittaker functions. The Macdonald polynomials, which interpolate between Archimedean and non-Archimedean zonal spherical functions, provide a precedent for such a programme.
The main result is a direct analytic construction of Archimedean \(L\)–factors as functional integrals in an equivariant type A topological linear sigma model over a disk (Theorem 2.1). The underlying action is quadratic, which means that the resulting formula (expressing Archimedean \(L\)–factors as products of \(\Gamma\)–functions) makes mathematical sense by way of \(\zeta\)–function regularization. Theorem 2.1 in particular leads to an interpretation of the \(\Gamma\)–function as an equivariant symplectic volume of an infinite dimensional space of holomorphic maps of the disk to \(\mathbb{C}\), and this description can be viewed as mirror dual to the classical Euler integral representation of the \(\Gamma\)–function see [A. Gerasimov, D. Lebedev, and S. Oblezin, Commun. Number Theory Phys. 5, No. 1, 101–134 (2011; Zbl 1251.14012)]. General \(q\)–deformed \(\Gamma\)–functions admit an analogous physical interpretation by means of a three–dimensional equivariant topological sigma model, discussed in Section 4, which has an ultraviolet completion which gives rise to a description of quantum \(K\)–theory invariants in the sense of Givental.

MSC:

14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
81T45 Topological field theories in quantum mechanics
14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
33B15 Gamma, beta and polygamma functions
11S80 Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.)
53D45 Gromov-Witten invariants, quantum cohomology, Frobenius manifolds

Citations:

Zbl 1251.14012
PDFBibTeX XMLCite
Full Text: DOI arXiv