Kontsevich–Witten model from 2+1 gravity: new exact combinatorial solution

2004

Int. J. Mod. Phys. A

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

“…This makes calculation of the Veneziano amplitudes similar to our earlier calculations of the Witten-Kontsevich averages, Ref. [64].…”

Section: Veneziano Amplitudes From Fermat Hypersurfacessupporting

confidence: 69%

New String Amplitudes From Old Fermat (Hyper)surfaces

2004

Int. J. Mod. Phys. A

“…Using this observation, perhaps superimposed with our earlier treatment of the Witten-Konsevich model, Ref. [7], one can develop, in principle, some quantum mechanical model whose partition function will coincide with the Poincare ′ polynomial discussed earlier. There is much faster way however to arrive at the final destination.…”

Section: Connections With Kp Hierarchymentioning

confidence: 66%

“…We begin with observation (taken from our earlier study of the WittenKontsevich model, Ref. [7]) that there is one-to-one correspondence between the Young tableaux and directed random walks. Let us recall details of this correspondence now.…”

Section: Motivationmentioning

confidence: 99%

New strings for old Veneziano amplitudes

Journal of Geometry and Physics

2006

In this part of our four parts work we use theory of polynomial invariants of finite pseudo-reflection groups in order to reconstruct both the Veneziano and Veneziano-like (tachyon-free) amplitudes and the generating function reproducing these amplitudes. We demonstrate that such generating function and amplitudes associated with it can be recovered with help of finite dimensional exactly solvable N=2 supersymmetric quantum mechanical model known earlier from works of Witten, Stone and others. Using the Lefschetz isomorphism theorem we replace traditional supersymmetric calculations by the group-theoretic thus solving the Veneziano model exactly using standard methods of representation theory. Mathematical correctness of our arguments relies on important theorems by Shepard and Todd, Serre and Solomon proven respectively in the early fifties and sixties and documented in the monograph by Bourbaki. Based on these theorems, we explain why the developed formalism leaves all known results of conformal field theories unchanged. We also explain why these theorems impose stringent requirements connecting analytical properties of scattering amplitudes with symmetries of space-time in which such amplitudes act.

“…with coefficients K λ,n known as Kostka numbers [16], f λ being the number of standard Young tableaux of shape λ and the notation λ ⊢ k meaning that λ is partition of k. Through such connection with Schur polynomials one can develop connections with Kadomtsev-Petviashvili (KP) hierarchy of nonlinear exactly integrable systems on one hand and with the theory of Schubert varieties on another [15]. We shall provide more details on such a connection in Part 4.…”

Section: Brief Review Of the Veneziano Amplitudesmentioning

confidence: 99%

“…[33] or our earlier work, Ref. [15], and/or Part 2 for rigorous definitions and further discussion) via …”

Section: Analytical Properties Of the Multiparticle Veneziano And Venmentioning

confidence: 99%

New strings for old Veneziano amplitudes

Journal of Geometry and Physics

2005

The bosonic string theory evolved as an attempt to find a physical/quantum mechanical model capable of reproducing Euler's beta function (Veneziano amplitude) and its multidimensional analogue. The multidimensional analogue of beta function was studied mathematically for some time from different angles by mathematicians such as Selberg,Weil and Deligne among many others. The results of their studies apparently were not taken into account in physics literature on string theory. In a recent publication [IJMPA 19 (2004) 1655] an attempt was made to restore the missing links. The results of this publication are incomplete, however, since no attempts were made at reproduction of known spectra of both open an closed bosonic strings or at restoration of the underlying model(s) reproducing such spectra. Nevertheless, discussed in this publication the existing mathematical interpretation of the multidimensional analogue of Euler's beta function as one of the periods associated with the corresponding differential form "living" on the Fermat-type (hyper) surfaces, happens to be crucial for restoration of the quantum/statistical mechanical model reproducing such generalized beta function. Unlike the traditional formulations, this model is supersymmetric. Details leading to restoration of this model will be presented in the forthcoming Parts 2-4 of our work. They are devoted respectively to the group-theoretic, symplectic and combinatorial treatments of this model. In this paper the discussion is restricted mainly to study of analytical properties of the multiparticle Veneziano and Veneziano-like (tachyon-free) amplitudes. In the last case, we demonstrate that the Veneziano-like amplitudes alone (with parameters adjusted accordingly) are capable of reproducing known spectra of both open and closed bosonic strings. The choice of parameters is subject to some constraints dictated by the mathematical interpretation of these amplitudes as periods of Fermat-type (hyper)surfaces considered as complex manifolds of Hodge type.