Twists of global AdS 5 × S 5 and their Non - Commutative Field Theory Dual

We consider the Melvin Twist of AdS5×S5 under U(1)×U(1) isometry of the boundary S3 of the global AdS5 geometry and identify its field theory dual. We also study the thermodynamics of the Melvin deformed theory. Melvin twist, also known as the T-s-T transformation, is a powerful solution generating technique in supergravity and string theories [1–6]. The procedure relies on having a U(1)× U(1) compact isometry along which one performs a sequence of T-duality, twist, and a Tduality. The twist is an SL(2, R) transformation on the complex structure of the T-dual torus. As such, the Melvin twist can simply be thought of as an SL(2, R) transformation acting on the Kähler structure of the torus parameterized by U(1)× U(1). Interesting closed string backgrounds, such as Melvin universes, null branes, pp-waves, and Gödel universes can be constructed by applying the Melvin Twist procedure to the Minkowski background. The construction reveals the hidden simplicity of these closed string backgrounds: they are dual to flat spaces. As a result, world sheet sigma model for strings in these backgrounds are exactly solvable and have been studied extensively [7–12]. The same procedure can be applied to black p-brane backgrounds to construct various asymptotically non-trivial space-time geometries [13]. Melvin twist applied to the Dp-brane background and the subsequent near horizon limit gives rise to supergravity duals for a variety of decoupled field theories depending on the orientation of the brane and the Melvin twist. If both of the U(1) isometries are along the brane, one generally obtains a non-commutative field theory, typically with non-constant non-commutativity parameter [14–19]. If one of the U(1) is transverse to the brane, then one obtains a dipole field theory [20–22]. Taking both of the U(1)’s to be transverse to the brane gives rise to the construction of Lunin and Maldacena [23]. The list of models constructed along these lines is summarized in table 1. These theories are S-dual to NCOS theories [24, 25]. They are also closely related to “Puff Field Theory” which was studied recently in [26, 27]. The hidden simplicity of Melvin twists in the context of gauge theory duals manifests itself as preservation of integrability. The fact that q/β-deformed N = 4 SYM remains integrable was pointed out in [28,29]. A broader class of integrable twists were studied in [30, 31]. In this article, we consider the effect of twisting along the U(1)×U(1) ∈ SO(4) isometry of the S3. More specifically, we consider AdS5 × S5 solution of type IIB theory ds = R [ − cosh ρdτ 2 + dρ + sinh ρ(dθ + sin θdφ1 + cos θdφ2) + dΩ5 ] B = 0 e = λ 4πN (1) where λ is the ’t Hooft coupling λ = 2g YMN = 4πgsN = R α′2 , (2) An earlier discussion of a construction of this type is [6]. 1 Type of Twist Model Melvin Twist Hashimoto-Thomas model Melvin Shift Twist Seiberg-Witten Model Null Melvin Shift Twist Aharony-Gomis-Mehen model Null Melvin Twist Dolan-Nappi model Melvin Null Twist Hashimoto-Sethi model Melvin R Twist Bergman-Ganor model Null Melvin R Twist Ganor-Varadarajan model R Melvin R Twist Lunin-Maldacena model Table 1: Catalog of non-commutative gauge theories viewed as a world volume theory of D-branes in a “X” Melvin “Y” twist background. This table originally appeared in [18]. and perform a Melvin twist on the torus parameterized by the coordinates (φ1, φ2). This is equivalent to acting on the Kahler structure ρ = 1 α ( Bφ1φ2 + i √ gφ1φ1gφ2φ2 ) (3) by an SL(2, R) transformation ρ → ρ′ = ρ χρ+ 1 (4) giving rise to a background ds = α′ √ λ [ − cosh ρdτ 2 + dρ + sinh ρ ( dθ + sin θdφ1 + cos 2 θdφ2 1 + χ2λ cos2 θ sin θ sinh ρ )