Mink4 × S2 solutions of 10 and 11 dimensional supergravity

Abstract

We complete the classification of Mink 4 solutions preserving N = 2 supersymmetry and SU(2) Rsymmetry parameterised by a round S 2 factor. We consider eleven-dimensional supergravity and relax the assumptions of earlier works in type II theories. We show that, using chains of dualities, all solutions of this type can be generated from one of two master classes: an SU(2)-structure in M-theory and a conformal Calabi-Yau in type IIB. Finally, using our results, we recover AdS 5 × S 2 solutions in M-theory and construct a compact Minkowski solution with Atiyah-Hitchin singularity. IntroductionSome of the most physically relevant solutions in supergravity are those exhibiting a warped Minkowski factor. From early on, the main reason for this is that Minkowski vacua of string and M-theory are required to make contact with known particle physics phenomena in four dimensions, where one should arrange for the co-dimensions to be compact. Another reason, which clearly gained considerable traction with the advent of the AdS/CFT correspondence, is that all AdS solutions admit a description in terms of a foliation of Minkowski over a non compact interval -namely the Poincaré patch.The most simple way to realise a Minkowski vacua from ten or eleven dimensions is to assume that the compact internal space accommodates some holonomy group, specifically SU(3) or G 2 for compactifications of string theory or M-theory down to four dimensions respectively. Such solutions preserve (at least) N = 2 supersymmetry and have been well studied in the literature [1][2][3][4][5], with a resurgence of interest in the G 2 case in recent years [6-9] (see also [10][11][12][13] for G 2 arising in a heterotic context). However, such manifolds support neither fluxes nor a warping of the Minkowski directions, so it is reasonable to expect that they represent a rather small region of the space of possible solutions. The inclusion of fluxes requires one to generalise the notion of holonomy group to structure group (or G-structure) [14][15][16][17]. It is well known that no compact regular solutions of this type exist [18-20] -indeed a necessary element of such constructions are localised singularities, namely O-planes and their generalisations through string dualities, or lifts to M-theory (e.g. Atiyah-Hitchin singularities). Most progress constructing compact solutions with fluxes has happened within another restrictive ansatz, with the internal space assumed to be conformally a holonomy manifold [21][22][23], allowing one to broadly use the same mathematical tools as before. At this point all Mink d solutions for d > 1 (with the exception of Mink 2 in eleven dimensions) have been classified (see [24][25][26][27][28] for eleven dimensions, [29][30][31][32][33][34][35][36] for ten dimensions), however finding solutions beyond the ansatz of warped holonomy has proved challenging -see [38,39] for success in this direction. The issue appears to be one of tractability, so it would be helpful to have some additional guiding principle.AdS solutions necessita...

Electrostatic Description of Five-dimensional SCFTs

“…Operating with these equations, sometimes one is able to write the f i and the fluxes in terms of a single function V ( y) and its derivatives where the function V solves a linear PDE. Some situations are known for which the V -function solves a non-linear PDE [30,60,61].…”

Type II solutions on AdS3 × S3 × S3 with large superconformal symmetry

“…Having such spinors we shall then find every solution with an S 3 × S 3 factor consistent with them. This follows the line of reasoning of the earlier works [33][34][35][36][37], where many of the technical details exploited here were originally worked out. Here it will be possible to give the explicit local form of every type II solution consistent with the SO(4) spinor.…”

Breaking supersymmetry with pure spinors

“…This reduction occurs when we impose that one of the complex directions which define j, ω in (2.8) is actually part of the external space R 1,5 , as in (2.11). This is not the only way to get an identity structure assuming a six-dimensional Minkowski space; however it was found in [18] that these cases give rise to parametric deformations of the usual Imamura solution which can be generated by chains of dualities. Since none of these dualities requires the presence of the SO(3) symmetry in the co-dimensions of the D6-brane, we don't expect the discussion is much different dropping this assumption.…”

“…Natural candidates are the backgrounds contained in the R 1,3 classification of [7,20], where the internal space is taken to be a warped product of a two-dimensional sphere with an unconstrained four-dimensional manifold, M 6 = S 2 × M 4 . In [18] it was proven that the complete list of possible classes of this type in type II theories can be obtained, through a web of duality, starting from two master classes: a conformal Calabi-Yau case in type IIB, which we analyze in the next section, and the IIA solution discussed in [7, appendix C].…”

“…In particular, from the fact that the Imamura solution with spherical symmetry can be obtained from this class after two T-dualities[18].…”