Membranes in string theory, trees, the Weil conjectures, and the Ramanujan numbers. (English) Zbl 0671.22010
It has become recently a fashion to try to construct the ultimate theories of matter on the microscopic level (below the Planck scale) by considering some more exotic mathematical structures than those used usually by physicists. In particular, there are attempts to use the p- adic numbers to construct the higher-dimensional membrane theories with scattering amplitudes generalizing the Virasoro-Shapiro amplitudes of dual models. In the paper the previous description of membranes by p-adic algebraic extensions, due to J. L. Gervais, is generalized, using p-adic group theory, to obtain, by algebraic extensions, an arbitrarily high- dimensional cohomology (as determined by the factors of Artin-Mazur zeta function). As a second result the factorization of the inverse partition function for 26-dimensional bosonic string is interpreted as the existence of 11-dimensional membrane. Finally, it has become possible recently to prove the index theorems by constructing some supersymmetric sigma models; also recently discovered invariants for low-dimensional manifolds can be interpreted in terms of supersymmetric quantum models. According to this line of thought the author suggests the possibility of interpreting also the cohomological content of Weyl-Deligne theorem in terms of supersymmetric quantum mechanics (in its p-adic form). However, no model has been explicitly constructed, only some indications are given.
Reviewer: P.Maslanka
MSC:
22E70 | Applications of Lie groups to the sciences; explicit representations |
83E15 | Kaluza-Klein and other higher-dimensional theories |
11E95 | \(p\)-adic theory |
22E50 | Representations of Lie and linear algebraic groups over local fields |
58J50 | Spectral problems; spectral geometry; scattering theory on manifolds |
53C80 | Applications of global differential geometry to the sciences |
74K05 | Strings |
11S80 | Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.) |
22E41 | Continuous cohomology of Lie groups |
Keywords:
p-adic numbers; higher-dimensional membrane theories; high-dimensional cohomology; Artin-Mazur zeta function; inverse partition function; 26- dimensional bosonic string; index theorems; supersymmetric sigma models; supersymmetric quantum modelsReferences:
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