A statistical physics approach to M-theory integrals

Abstract

Abstract:We explain the concepts of computational statistical physics which have proven very helpful in the study of Yang-Mills integrals, an ubiquitous new class of matrix models. Issues treated are: Absolute convergence versus Monte Carlo computability of near-singular integrals, singularity detection by Markov-chain methods, applications to asymptotic eigenvalue distributions and to numerical evaluations of multiple bosonic and supersymmetric integrals. In many cases already, it has been possible to resolve controversies between conflicting analytical results using the methods presented here.

Decay amplitudes in two-dimensional QCD

“…A more elaborate method [14] confirms that particles do not decay into states which contain the pion whereas in the massive case, the decay amplitudes are shown to be significantly large (see Figure 1). For the massless case, several decays of the type n → 1 + 1, where 1 denotes the first-excited level of 't Hooft's series, have been computed (see Figure 3).…”

“…This puts a stringent requirement on particle scattering, forbidding particle production or decay 2 . Recently, there have been suggestions [14] that the stability of the spectrum can be tested in the framework of 't Hooft's mesons decay amplitudes. The numerical computations of the amplitudes provide a detailed check of the claims presented here, namely that the infinite number of conservation laws does not imply absence of particle production or stability of the meson and zero decay amplitude.…”

“…(i). In the massless case, for a general decay k → l + p the amplitude can start from a high value (see specifically the example 14 → 3 + 3 in reference [14]) but decreases rapidly with increasing k . These results are also compatible with the earlier results of reference [3].…”

“…The numerical calculations are difficult to converge for massless fermions. A strongly convergent algorithm for very reliable numerical computations is also available in the literature [14].…”

Multidimensional Dirac strings and the Witten index of SYMCS theories with groups of higher rank

“…This question is akin to the question of the so called principal contribution to the index in maximally supersymmetric gauge matrix models. For a given simple group, the principal contribution to the index represents a bizarre fractional number (see [24], [25] and references therein) whose mathematical nature is now obscure.…”