Closed Geodesics and Periods of Automorphic Forms

Sharp, Richard

doi:10.1006/aima.2001.1987

Cited by 2 publications

(2 citation statements)

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“…can be found in Refs. [94,95] and references therein. In this work we are concerned with different issues however.…”

Section: Fermat's Last Theorem Periods Of Elliptic Curves and Venezia...mentioning

confidence: 99%

New String Amplitudes From Old Fermat (Hyper)surfaces

Kholodenko

2004

Int. J. Mod. Phys. A

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The history of the discovery of bosonic string theory is well documented. This theory evolved as an attempt to find a multidimensional analogue of Euler's beta function to describe the multiparticle Veneziano amplitudes. Such an analogue had in fact been known in mathematics at least in 1922. Its mathematical meaning was studied subsequently from different angles by mathematicians such as Selberg, Weil and Deligne among others. The mathematical interpretation of this multidimensional beta function that was developed subsequently is markedly different from that described in physics literature. This work aims to bridge the gap between the mathematical and physical treatments. Using some results of recent publications (e.g. J.Geom.Phys.38 (2001) 81 ; ibid 43 (2002) 45) new topological, algebro-geometric, number-theoretic and combinatorial treatment of the multiparticle Veneziano amplitudes is developed. As a result, an entirely new physical meaning of these amplitudes is emerging: they are periods of differential forms associated with homology cycles on Fermat (hyper)surfaces. Such (hyper)surfaces are considered as complex projective varieties of Hodge type. Although the computational formalism developed in this work resembles that used in mirror symmetry calculations, many additional results from mathematics are used along with their suitable physical interpretation. For instance, the Hodge spectrum of the Fermat (hyper)surfaces is in one-to-one correspondence with the possible spectrum of particle masses. The formalism also allows us to obtain correlation functions of both conformal field theory and particle physics using the same type of the Picard-Fuchs equations whose solutions are being interpreted in terms of periods.MSC: 81T30,81T40,81T99,81U99

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“…can be found in Refs. [94,95] and references therein. In this work we are concerned with different issues however.…”

Section: Fermat's Last Theorem Periods Of Elliptic Curves and Venezia...mentioning

confidence: 99%

New String Amplitudes From Old Fermat (Hyper)surfaces

Kholodenko

2004

Int. J. Mod. Phys. A

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“…Remark 1.3. Estimating the number of closed geodesics (periods of the geodesic flow) with length less than x and certain constraints on its periods can be done using Lalley's theorem [11], see also [1], [19]. The constraints discussed in these articles restrict the periods to lie in a compact interval, say, [a, b].…”

Section: Introductionmentioning

confidence: 99%