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Freund-Witten adelic formulae for Veneziano and Virasoro-Shapiro amplitudes. (English. Russian original) Zbl 0866.11068

Russ. Math. Surv. 48, No. 6, 1-39 (1993); translation from Usp. Mat. Nauk 48, No. 6(294), 3-38 (1993).
Although the title of the paper suggests that it is devoted to physical applications of the analysis on \(p\)-adic fields, it is in fact a summarization of the most essential results from analysis on \(p\)-adic fields.
This expository paper recalls some known facts on quadratic extensions of \(p\)-adic fields, their adeles and ideles, Haar measures and characters, the Mellin transform and other analysis on adelic groups. In the last chapters some integrals with respect to the Haar measure on \(\mathbb{Q}_p\) are calculated. The paper can be treated as a supplement to the book of I. M. Gelfand, M. I. Graev and I. I. Piatetski-Shapiro [Representation theory and automorphic functions (Russian), ‘Nauka’ Publishers, Moscow (1966; Zbl 0138.07201)].

MSC:

11S80 Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.)
46S10 Functional analysis over fields other than \(\mathbb{R}\) or \(\mathbb{C}\) or the quaternions; non-Archimedean functional analysis
81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
11S31 Class field theory; \(p\)-adic formal groups

Citations:

Zbl 0138.07201
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