Puzzles of η-deformed AdS5 × S5

Abstract

Abstract:We derive the part of the Lagrangian for the sigma model on the η-deformed AdS 5 × S 5 space which is quadratic in fermions and has the full dependence on bosons. We then show that there exists a field redefinition which brings the corresponding Lagrangian to the standard form of type IIB Green-Schwarz superstring. Reading off the corresponding RR couplings, we observe that they fail to satisfy the supergravity equations of motion, despite the presence of κ-symmetry. However, in a special scaling limit our solution reproduces the supergravity background found by Maldacena and Russo. Further, using the fermionic Lagrangian, we compute a number of new matrix elements of the tree level world-sheet scattering matrix. We then show that after a unitary transformation on the basis of two-particle states which is not one-particle factorisable, the corresponding T-matrix factorises into two equivalent parts. Each part satisfies the classical Yang-Baxter equation and coincides with the large tension limit of the q-deformed S-matrix.

I in generalized supergravity

“…The validity of this relation has been checked for homogeneous YB deformations on a case by case basis and an outline of its derivation was given in [7]. In this letter we extend this result to the well-known η-deformations [17,19,54,55] and provide a derivation in terms of D3-branes probing the background, where I may be interpreted as a current for the center of mass motion on the D3-brane worldvolume. The conclusion that (2) holds for all YB deformations, including η-deformations, hinges on the observation that Page forms [56], special combinations of RR fluxes and NSNS two-form that appear in the D-brane action, facilitate a simple rewriting of the equations of generalized supergravity and that the Page five-form associated with the D3-brane is invariant, which we have confirmed on a case by case basis.…”

“…This is often called the η-deformation. After performing a supercoset construction the full background, including the RR sector, was worked out [19], which led to the surprise that the resulting background was not a solution of the usual type IIB supergravity but a generalized variant [12]. To date, a considerable amount of community effort has focused on the largely technical exercise of mapping out the dictionary between r -matrices and geometries, or YB deformations, including connections to (non-Abelian) T-duality [20][21][22][23][24][25][26][27][28][29][30][31][32].…”

Bubbling geometries for AdS2× S2

“…2 [21][22][23], which is analogous to a similar modification of the higher dimensional spaces [24][25][26][27][28][29]. 2 The transition between graviton, brane, and geometry regimes is governed by the scaling of energy with the string coupling constant.…”

“…To prove regularity of the metric at the location of the singular curve, we should analyze the subleading contributions to H. Let us pick a point on a curve and introduce local Cartesian coordinates (x, y, z) by choosing x direction alongḞ, z direction alongF, and y direction alongF ×Ḟ. According to (3.27), the leading contribution to the harmonic function is 29) and the next order can be written as…”

“…Any one-dimensional curve in three dimensions (x 1 , x 2 , x 3 ) can be parameterized by a profile F(v) that satisfies the normalization condition (C.3). 29 Motivated by the AdS 2 ×S 2 example, we associate such a curve with a (complex) harmonic function so vector A must be complex. Regularity of the geometry (3.5) imposes additional constraints on this vector.…”