Isometric embeddings and noncommutative branes in homogeneous gravitational waves

Abstract

We characterize the worldvolume theories on symmetric D-branes in a six-dimensional Cahen-Wallach pp-wave supported by a constant Neveu-Schwarz three-form flux. We find a class of flat noncommutative euclidean D3-branes analogous to branes in a constant magnetic field, as well as curved noncommutative lorentzian D3-branes analogous to branes in an electric background. In the former case the noncommutative field theory on the branes is constructed from first principles, related to dynamics of fuzzy spheres in the worldvolumes, and used to analyse the flat space limits of the string theory. The worldvolume theories on all other symmetric branes in the background are local field theories. The physical origins of all these theories are described through the interplay between isometric embeddings of branes in the spacetime and the Penrose-Güven limit of AdS 3 × S 3 with Neveu-Schwarz three-form flux. The noncommutative field theory of a non-symmetric spacetime-filling D-brane is also constructed, giving a spatially varying but time-independent noncommutativity analogous to that of the Dolan-Nappi model.

Symmetry, gravity and noncommutativity

“…For brevity we will only consider translation generators. Using the twist element (6.38) we arrive at the twisted coproducts [63] above to obtain the usual quantization of coadjoint orbits in nw(4) ∨ [62]. In exactly the same way that the noncommutative space Ê 3 ∞ of Section 6.1 above can be viewed as a collection of all fuzzy spheres, we can regard the noncommutative geometry of NW(4) as a foliation by all noncommutative S1-branes.…”

“…Let us now introduce the one-formΛ := − i ϑ −1 x − 0 + ϑ x + z dz − z dz (6.34)on the null hypersurfaces of constant light-cone position x − = x − 0 , and compute the corresponding two-form gauge transformation of the B-field in (6.33) to getB −→ B + dΛ = − i ϑ dx + ∧ z dz − z dz + 2 i ϑ x − 0 dz ∧ dz . (6.35)Applying the Seiberg-Witten formula (5.3) to the closed string background fields (6.32) and (6.35) we compute[62]…”

Noncommutative field theory on homogeneous gravitational waves

“…The background NW 6 arises as the Penrose-Güven limit [53,37] of an AdS 3 × S 3 background [11]. While this limit is a useful tool for understanding various aspects of string dy-namics, it is not in general suitable for describing the quantum geometry of embedded Dsubmanifolds [38]. In the following we will resort to a more direct quantization of the spacetime NW 6 and its D-submanifolds.…”

“…The symmetric D-branes wrapping twisted conjugacy classes of the Lie group N were classified in [61]. Their quantization was analysed in [38] and it was found that, in the semi-classical regime, only the untwisted euclidean D3-branes support a noncommutative worldvolume geometry. We study these D3-branes as a special case of our more general constructions and find exact agreement with the predictions of the boundary conformal field theory analysis [28].…”

“…In Section 3 we construct star-products which are equivalent to the Kontsevich product for the pertinent Poisson geometry. These products are much simpler and more tractable than the star-product on NW 6 which was constructed in [38] through the noncommutative foliation of NW 6 by D3-branes corresponding to quantized coadjoint orbits. Throughout this paper we will work with three natural star-products which we construct explicitly in closed form.…”