Generalized Einstein-Scalar-Maxwell theories and locally geometric U-folds

Abstract

We give a global formulation of the coupling of four-dimensional scalar sigma models to Abelian gauge fields for the generalized situation when the "duality structure" of the Abelian gauge theory is described by a flat symplectic vector bundle (S, D, ω) defined over the scalar manifold M. The construction uses a taming of (S, ω), which encodes globally the inverse gauge couplings and theta angles of the "twisted" Abelian gauge theory in a manner that makes no use of duality frames. We show that global solutions of the equations of motion of such models give classical locally geometric U-folds. We also describe the groups of duality transformations and scalar-electromagnetic symmetries arising in such models, which involve lifting isometries of M to a particular class of flat automorphisms the bundle S and hence differ from expectations based on local analysis. The appropriate version of the Dirac quantization condition involves a discrete local system defined over M and gives rise to a smooth bundle of polarized Abelian varieties, endowed with a flat symplectic connection. This shows that a generalization of part of the mathematical structure familiar from N = 2 supergravity is already present in such purely bosonic models, without any coupling to fermions and hence without any supersymmetry. arXiv:1609.05872v1 [hep-th] 19 Sep 2016Notations and conventions. The symbols Gp, Ab and Alg denote respectively the categories of all groups, of Abelian groups and of unital R-algebras. The symbol Vect denotes the category of finitedimensional R-vector spaces and linear maps. For any category C, let C × denote its unit groupoid, i.e. the category having the same objects as C and whose morphisms are the isomorphisms of C. Given an Abelian group A, let Tors(A) denote its torsion subgroup and A tf def.= A/Tors(A) be the corresponding torsion-free group; the latter is free when it is finitely-generated.2 Indeed, the bosonic sector of the N = 2 theory should be a particular case of the bosonic sector of the N = 0 theory. Scalar nonlinear sigma models coupled to gravityWe start by recalling the global construction of four-dimensional scalar nonlinear sigma models coupled to gravity. Let M be an oriented four-manifold and Met 3,1 (M ) be the set of Lorentzian metrics defined on M . We assume that Met 3,1 (M ) is non-empty (this requires χ(M ) = 0 when M is compact).

$${\mathcal {N}}=1$$ Geometric Supergravity and Chiral Triples on Riemann Surfaces

“…A global geometric model for the generic bosonic sector of four-dimensional supergravity was constructed in [43,44]. This model involves a flat submersion π : E → M over space-time and a flat symplectic vector bundle S over E. Whereas the previous model includes the bosonic sector of any ungauged four-dimensional supergravity theory, it does not yet implement supersymmetry.…”

Section sigma models coupled to symplectic duality bundles on Lorentzian four-manifolds

“…hence that any section of such a bundle can be identified with a map from U into the fiber. Accordingly, the physics literature traditionally treats all classical fields as functions defined on U and valued in some target space, which is either a vector space or (for the scalar fields) a manifold M endowed with a Riemannian metric G. It is often also tacitly assumed that M is contractible, which implies that the duality structure [5,6] of the Abelian gauge theory coupled to the scalar fields is described by a trivial flat symplectic vector bundle S defined on M, whose data can then be encoded by a symplectic vector space S 0 (the fiber of S) [2,3]. Due to this assumption, the gauge field strengths and their Lagrangian conjugates are usually treated as two-forms defined on U and valued in S 0 .…”

“…To fully define such theories, one must decide how to interpret globally various local formulas and differential operators. Such global interpretations are generally non-unique in the sense that they depend on choices of auxiliary geometric data which are not visible in the usual local formulation [5,6,7].…”

“…In the particular case when Φ vanishes, the solutions of the equations of motion of the section sigma model are the pseudoharmonic sections studied by C. M. Wood [11,12,13,14,15,16]. The section sigma model is then coupled to Abelian gauge fields governed by a non-trivial duality structure, thereby extending the construction of [5] to this more general setting. The globally-defined theory obtained in this manner will be called generalized Einstein-Section Maxwell (GESM) theory.…”

“…When π is a topologically trivial fiber bundle and H is a trivial Ehresmann connection, the sections of π can be identified with the graphs of smooth functions from M to M and the section sigma model reduces to the ordinary scalar sigma model defined by the "scalar structure" (M, G, Φ) [5,6]. This reduction always happens when M is contractible and H is integrable, since in that case π is necessarily topologically trivial and H is necessarily a trivial Ehresmann connection.…”