Effective action for non-geometric fluxes duality covariant actions

Published for SISSA by Springer Received: February 10, 2017 Revised: June 14, 2017 Accepted: July 5, 2017 Published: July 14, 2017 Kanghoon Lee,a Soo-Jong Reya,b and Yuho Sakatania,c a Fields, Gravity & Strings, CTPU, Institute for Basic Sciences, Daejeon 34047, Korea b School of Physics & Astronomy and Center for Theoretical Physics, Seoul National University, Seoul 08826, Korea c Department of Physics, Kyoto Prefectural University of Medicine, Kyoto 606-0823, Japan E-mail: , , Abstract: The (heterotic) double field theories and the exceptional field theories are manifestly duality covariant formulations, describing low-energy limit of various superstring and M-theory compactifications. These field theories are known to be reduced to the standard descriptions by introducing appropriately parameterized generalized metric and by applying suitably chosen section conditions. In this paper, we apply these formulations to non-geometric backgrounds. We introduce different parameterizations for the generalized metric in terms of the dual fields which are pertinent to non-geometric fluxes. Under certain simplifying assumptions, we construct new effective action for non-geometric backgrounds. We then study the non-geometric backgrounds sourced by exotic branes and find their U -duality monodromy matrices. The charge of exotic branes obtained from these monodromy matrices agrees with the charge obtained from the non-geometric flux integral. Keywords: Flux compactifications, String Duality ArXiv ePrint: 1612.08738 Open Access, c The Authors. Article funded by SCOAP3 . https://doi.org/10.1007/JHEP07(2017)075 JHEP07(2017)075 Effective action for non-geometric fluxes duality covariant actions Contents 1 2 General framework 2.1 Parameterization of Lie algebra 2.2 The generalized metric 2.3 Example: Double Field Theory 2.4 Example: Einstein gravity 2.5 Effective action for non-geometric fluxes 3 3 6 8 10 11 3 Non-geometric fluxes in EFT: M-theory 3.1 Parameterization of the generalized vielbein 3.1.1 n = 7: G = SL(5) 3.1.2 n = 6: G = SO(5, 5) 3.1.3 n = 5: G = E6(6) 3.1.4 n = 4: G = E7(7) 3.2 Eleven-dimensional effective action 3.3 Reduction to the type IIA theory 11 13 13 16 16 17 18 19 4 Non-geometric fluxes in EFT: type IIB section 4.1 Parameterizations of the generalized vielbein 4.1.1 n = 7: G = SL(5) 4.1.2 n = 6: G = SO(5, 5) 4.1.3 n = 5: G = E6(6) 4.1.4 n = 4: G = E7(7) 4.2 Ten-dimensional effective action 20 21 22 22 23 24 25 5 Non-geometric fluxes in heterotic DFT 5.1 Parameterization of generalized vielbein 5.1.1 Parameterization from coset representative 5.2 Non-geometric fluxes and action 27 28 30 33 6 Exotic branes and non-geometric fluxes 6.1 Exotic branes in the heterotic DFT 6.1.1 Symmetric dual five-brane 6.1.2 Neutral and gauge branes 6.1.3 Generalized metric and monodromy 6.2 Exotic branes in the M-theory 6.2.1 53 -brane 6.2.2 26 -brane 6.3 Exotic branes in the type IIB theory 6.3.1 522 -brane 35 36 37 39 40 42 42 43 44 44 –i– JHEP07(2017)075 1 Introduction 6.3.2 6.3.3 p37−p -brane 164 -brane 45 46 47 A Notations A.1 Ed(d) algebras: M-theory section A.2 Ed(d) algebras: type IIB section 48 49 50 B Calculation of the EFT action B.1 Redefinitions of coordinates B.2 External part B.3 Internal (potential) part B.4 Summary 53 53 61 64 68 C Double-vielbein formalism for gauged DFT C.1 Parameterization from defining properties of double-vielbein C.2 Connection and curvature C.3 Nongeometric fluxes and action 68 68 72 73 D Exotic branes 74 I do not wish, at this stage, to examine the logical justification of this form of argumentation; for the present, I am considering it as a practice, which we can observe in the habits of men and animals. Bertrand Russell, ‘Philosophy’. 1 Introduction Recently, a significant progress has been achieved for novel formulations of supergravity in which duality symmetries in string and M-theory compactification are manifest. They include the double field theory (DFT) [1–7], the exceptional field theory (EFT) [8–26] (see also [27–34] for closely related attempts) as well as the generalized geometry [35– 40]. One important advantage of these formulations is that they can treat wide variety of spacetimes, such as non-geometric backgrounds [41–44], that are not globally describable in the conventional formulation of supergravity. As pointed out in [45, 46], non-geometric backgrounds arise quite naturally in superstring theories. Backgrounds sourced by exotic branes [47–53] are concrete examples. As an application of DFT and related formulations such as the β-supergravity [54–61], a background of a particular exotic brane, so-called a 522 -brane, was studied in [45, 46, 62–72] and the exotic 522 -brane was identified with a magnetic source of the non-geometric Q-flux [64, 70, 72]. –1– JHEP07(2017)075 7 Discussion 1 Note that the section condition or the strong constraint in DFT/EFT can be relaxed through the generalized Scherk-Schwarz reduction [82], which provides all the fluxes in the maximal and half-maximal gauged supergravity [80]. In this paper we will restrict ourselves to the usual section condition, and the non-geometric fluxes considered in this paper are included in the same duality orbit with geometric fluxes. However, extension of the non-geometric fluxes to the gauged DFT/EFT would be straightforward via generalized Scherk-Schwarz reduction. –2– JHEP07(2017)075 One reason why the exotic 522 -brane received special attention is that the non-geometric Q-flux, which is intrinsic to the 522 -brane background, is related to a T -duality monodromy, and the much developed DFTs efficiently describe such background. It is known that backgrounds of other exotic branes possess other non-geometric fluxes that are related to the Q-flux via U -duality transformations [51, 73]. In order to describe such non-geometric backgrounds, variants of the β-supergravity, which can describe the background of an exotic p-brane (called a p37−p -brane) or a 164 -brane, was proposed in [74]. There, each of these exotic branes was identified as the magnetic sources of a non-geometric P -flux [75–77] or a non-geometric Q-flux associated with a 6-vector, β m1 ···m6 [74]. However, the reformulation of [74] is applicable only to a limited situation; coexistence of different non-geometric fluxes are not allowed and existence of isometries are assumed. In fact, EFT, a manifestly Ed(d) U -duality covariant formulation of the supergravity, is a more suitable formulation, and indeed, backgrounds of the exotic 53 -brane, 522 -brane, and the 523 -brane were studied in SL(5) EFT [78, 79]. One of the main purposes of this paper is to systematically identify the non-geometric fluxes in Ed(d) EFT for the cases of 4 ≤ d ≤ 7. The goal of this paper is to develop effective actions for a certain class of non-geometric flux backgrounds in Type II string and M-theories. Our starting point is the duality covariant action in an extended fie (...truncated)