The non-Abelian gauge theory of matrix big bangs

Abstract

We study at the classical and quantum mechanical level the time-dependent Yang-Mills theory that one obtains via the generalisation of discrete light-cone quantisation to singular homogeneous plane waves. The non-Abelian nature of this theory is known to be important for physics near the singularity, at least as far as the number of degrees of freedom is concerned. We will show that the quartic interaction is always subleading as one approaches the singularity and that close enough to t=0 the evolution is driven by the diverging tachyonic mass term. The evolution towards asymptotically flat space-time also reveals some surprising features.Comment: 29 pages, 8 eps figures, v2: minor changes, references added: v3 small typographical changes

Aspects of plane wave (matrix) string dynamics

“…When one considers perturbative string theory, the relevant dynamics of closed strings (in the lightcone gauge) is that of the transverse string modes on the cylinder S 1 × R, and it is well known that in Brinkmann variables this reduces to the standard free action of massive scalar fields with time-dependent mass matrix −A ab (t) (the frequency/profile of the Brinkmann plane wave). The same is true for non-perturbative matrix string theory: in [24] it was shown that the near-singularity t → 0 evolution of the plane wave matrix big bang models of [13] is dominated by the same time-dependent mass terms, while the quartic interaction and the accompanying non-Abelian nature of the dynamics are subleading. In the isotropic case we have focussed on one is thus in both cases confronted with the problem of the quantisation of a single scalar field on S 1 × R with time-dependent mass term m 2 (t) = ω 2 (t) (or with an infinite number of Fourier modes with masses m n (t) 2 = m(t) 2 + n 2 , n ∈ Z).…”

“…also the assessment and discussion in [27]). For instance, as argued in [24], for the plane wave matrix big bang models of [13] near the singularity the usual quartic interaction term for the non-Abelian matrix string coordinates becomes irrelevant, due to the characteristic and inevitable presence of tachyonic mass terms in these models (for this reason, formally this argument applies to all the modes of the matrix string, not just the quantum mechanical zero mode). As a consequence, it does not appear that (in the spirit of the hope expressed in the context of the original matrix big bang model of [17]) the extra non-geometric degrees of freedom at the singularity arising from the weakly-coupled non-Abelian matrix string (which are actually also far from massless in the present case) can lead to a resolution of the singularity in this class of models.…”

“…We investigated the way that the dilaton depends on the regularisation, and found that the strongest constraints from dilaton regularity actually come from the elimination of diverging linear dilatons at large times t. This would be relevant if one thinks of this as a model for big bang and emergent spacetime, in the spirit of [17]. However, in that case one needs to study the late-time behaviour of the non-Abelian theory [24,44,45]. On the other hand if, in the spirit of the Penrose limit, one considers the plane wave only as a near-singularity approximation of the full metric, then the large-t behaviour is irrelevant simply because it is also an artefact of the approximation.…”

“…We recall and reiterate here that it was argued in [24], on the basis of numerical and analytical investigations of various toy-models, that both the classical and the quantum evolution of the matrix (string) system near strong string coupling (and hence weak gauge coupling) singularities are generically driven entirely by the divergent tachyonic mass terms. In particular, the quartic interaction potential, which comes with a time-dependent coefficient g 2 Y M ∼ t 2q with q > 0, can be shown to be self-consistently small in quantum-mechanical perturbation theory in that regime and appears to (perhaps somewhat disappointingly) play no role there.…”

“…Curiously exactly the same (decoupled, Abelian) string mode equations arise from the analysis of strong string coupling singularities in the context of the (non-Abelian) matrix string models [13] for singular plane waves, in which the strong string coupling implies weak (time-dependent) Yang-Mills gauge coupling. This comes about because, as shown in [24], near the singularity the usual quartic interaction term for the non-Abelian matrix string coordinates becomes irrelevant. 2 Thus the near-singularity behaviour of the matrix string coordinates is governed only by the decoupled (non-self-interacting) kinetic and harmonic oscillator (mass) terms ∼ ω n (t) 2 , as above, and one formally encounters the same problems and issues as in the perturbative string theory model of [19], now however at strong string coupling.…”