Non-integrability in non-relativistic theories

Abstract

Generic non-relativistic theories giving rise to non-integrable string solutions are classified. Our analysis boils down to a simple algebraic condition for the scaling parameters of the metric. Particular cases are the Lifshitz and the anisotropic Lifshitz spacetimes, for which we find that for trivial dilaton dependence the only integrable physical theory is that for z = 1. For the hyperscaling violation theories we conclude that the vast majority of theories are non-integrable, while only for a small class of physical theories, where the Fermi surfaces belong to, integrability is not excluded. Schrödinger theories are also analyzed and a necessary condition for non-integrability is found. Our analysis is also applied to cases where the exponential of the dilaton is a monomial of the holographic coordinate.

Lunin-Maldacena backgrounds from the classical Yang-Baxter equation — towards the gravity/CYBE correspondence

“…It also contains the three-dimensional Schrödinger spacetime as a subspace and, for the subsector analysis, one can use the results obtained in a series of works [50][51][52][53]. All of the results are consistent to the recent analysis [54].…”

Chaos in the BMN matrix model

“…Recently, chaotic string solutions have been well studied in various ways [2][3][4][5][6][7][8][9][10][11][12]. Motivation lies on potential applications to the AdS/CFT correspondence [13][14][15] beyond integrability [16], but it has not been clarified yet what corresponds to chaotic strings in the gauge-theory side.…”

Chaotic strings in a near Penrose limit of AdS5 × T1,1

“…Apart from this integrable example, there are many non-integrable AdS/CFT dualities in which chaotic string solutions appear. For example, when the internal space is given by a Sasaki-Einstein manifold like T 1,1 [6] and Y p,q [7], the string world-sheet theory exhibits the chaotic behavior (For other examples, see [8][9][10][11][12][13][14][15][16]). …”