Schrödinger geometries arising from Yang-Baxter deformations

Abstract

We present further examples of the correspondence between solutions of type IIB supergravity and classical r-matrices satisfying the classical Yang-Baxter equation (CYBE). In the previous works, classical r-matrices have been composed of generators of only one of either so(2, 4) or so(6) . In this paper, we consider some examples of r-matrices with both of them. The r-matrices of this kind contain (generalized) Schrödinger spacetimes and gravity duals of dipole theories. It is known that the generalized Schrödinger spacetimes can also be obtained via a certain class of TsT transformations called null Melvin twists. The metric and NS-NS two-form are reproduced by following the Yang-Baxter sigma-model description.

Lax pairs for deformed Minkowski spacetimes

“…The deformations include TsT transformations of AdS 5 ×S 5 [13][14][15] such as γ-deformations of S 5 [16,17], gravity duals of noncommutative gauge theories [18,19], and Schrödinger geometries [20][21][22]. These backgrounds can also be realized as the undeformed AdS 5 ×S 5 with twisted boundary conditions [17,[23][24][25][26][27][28][29][30].…”

Yang-Baxter deformations of Minkowski spacetime

“…Jordanian deformations based on the cybe have been proposed in [38]. In the latter case, there are a lot of classical r-matrices satisfying the cybe and some of them are associated with well-known gravitational backgrounds, such as Lunin-Maldacena-Frolov backgrounds [39,40], gravity duals for non-commutative gauge theories [41,42], Schrödinger spacetimes [43][44][45][46][47] and gravity duals for dipole theories [48][49][50][51][52], as shown in a series of works [53][54][55][56][57]. Very recently, the reality of the classical action has been revisited in [58] and a unified picture of deformed integrable sigma models has been provided in [59].…”

Lax pairs on Yang-Baxter deformed backgrounds

“…Furthermore, one can find many solutions of the CYBE. In fact, in a series of papers [22][23][24][25][26][27][28][29], many examples of (skew-symmetric) classical r-matrices have been identified with the wellknown backgrounds such as γ-deformations of S 5 [30,31], gravity duals for noncommutative (NC) gauge theories [32,33] and Schrödinger spacetimes [34][35][36], in addition to new backgrounds [22]. This identification may be called the gravity/CYBE correspondence [23] (For a short summary, see [37]) and indicate that the moduli space of (a certain class of) solutions of type IIB supergravity can be described by the CYBE.…”

“…Explicit forms of Lax pairs are computed, while the associated classical r-matrices have already been obtained in [23,24,27]. …”

“…The resulting Lax pairs agree with Frolov's results [31] up to gauge transformations. Section 5 argues Lax pairs for Schrödinger spacetimes by evaluating the abstract forms given in [27]. In section 6, we derive Lax pairs for abelian twists of the global AdS 5 from the results of [23].…”