Lax pairs on Yang-Baxter deformed backgrounds

Abstract

We explicitly derive Lax pairs for string theories on Yang-Baxter deformed backgrounds, 1) gravity duals for noncommutative gauge theories, 2) γ-deformations of S 5 , 3) Schrödinger spacetimes and 4) abelian twists of the global AdS 5 . Then we can find out a concise derivation of Lax pairs based on simple replacement rules. Furthermore, each of the above deformations can be reinterpreted as twisted boundary conditions with the undeformed background by using the rules. As another derivation, the Lax pair for gravity duals for noncommutative gauge theories is reproduced from the one for a q-deformed AdS 5 ×S 5 by taking a scaling limit.

Lax pairs for deformed Minkowski spacetimes

“…Our main purpose here is to provide the derivation as a follow-up of the previous work [40]. We first revisit a Melvin background and argue the associated Lax pair by following a simple replacement law invented in [30]. This argument enables us to deduce a general expression of Lax pair.…”

“…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]. Other deformations are associated with classical r-matrices composed of non-commuting generators and lead to deformed backgrounds obtained through a chain of dualities including S-dualities [31,32].…”

“…In section 2, we give a brief review of Yang-Baxter deformations of 4D Minkowski spacetime. In section 3, a Melvin background is revisited and the associated Lax pair is constructed by following a simple replacement rule invented in [30]. A twisted boundary condition is also discussed.…”

“…We also derive twisted boundary conditions on the undeformed coordinates by following a simple replacement rule [17,23,30]. It is easy to rewrite the twisted boundary conditions into simple forms by adopting appropriate representatives of group element.…”

“…In the first place, we have revisited a Melvin background and argued a Lax pair by adopting a simple replacement law invented in [30]. Due to this argument, one can readily deduce a general expression of Lax pair.…”