[Paper Review] Four-dimensional geometric supergravity and electromagnetic duality: a brief guide for mathematicians

Calin Iuliu Lazaroiu, C. S. Shahbazi|arXiv (Cornell University)|Jun 29, 2020

Black Holes and Theoretical Physics42 references4 citations

TL;DR

This paper provides a global geometric formulation of the bosonic sector of four-dimensional ungauged supergravity on an oriented four-manifold using a vertically Riemannian submersion with flat Ehresmann connection and a flat symplectic vector bundle with positive complex polarization. It characterizes the continuous electromagnetic duality (U-duality) group via a short exact sequence of automorphism groups and analyzes the Killing spinor equations, offering a mathematically rigorous framework for studying supersymmetric solutions and their topological and geometric constraints.

ABSTRACT

We give a gentle introduction to the global geometric formulation of the bosonic sector of four-dimensional supergravity on an oriented four-manifold $M$ of arbitrary topology, providing a geometric characterization of its U-duality group. The geometric formulation of four-dimensional supergravity is based on a choice of a vertically Riemannian submersion $π$ over $M$ equipped with a flat Ehresmann connection, which determines the non-linear section sigma model of the theory, and a choice of flat symplectic vector bundle $\mathcal{S}$ equipped with a positive complex polarization over the total space of $π$, which encodes the inverse gauge couplings and theta angles of the theory and determines its gauge sector. The classical fields of the theory consist of Lorentzian metrics on $M$, global sections of $π$ and two-forms valued in $\mathcal{S}$ that satisfy an algebraic relation which defines the notion of \emph{twisted} self-duality in four Lorentzian dimensions. We use this geometric formulation to investigate the group of electromagnetic duality transformations of supergravity, also known as the continuous classical U-duality group, which we characterize using a certain short exact sequence of automorphism groups of vector bundles. Moreover, we discuss the general structure of the Killing spinor equations of four-dimensional supergravity, providing several explicit examples and remarking on a few open mathematical problems. This presentation is aimed at mathematicians working in differential geometry.

Motivation & Objective

• To develop a global geometric formulation of four-dimensional ungauged supergravity accessible to mathematicians.
• To characterize the continuous electromagnetic duality (U-duality) group using bundle automorphisms and exact sequences.
• To provide a geometric and topological analysis of the Killing spinor equations in four-dimensional supergravity.
• To connect supergravity structures with differential geometry, including special holonomy, moduli spaces, and complex geometry.
• To highlight open mathematical problems in the global structure of supergravity solutions and their classification.

Proposed method

• Formalizes the bosonic sector of 4D supergravity using a vertically Riemannian submersion π over a four-manifold M with a flat Ehresmann connection.
• Introduces a flat symplectic vector bundle S over the total space of π, equipped with a positive complex polarization, to encode gauge couplings and θ-angles.
• Defines the classical fields as Lorentzian metrics on M, global sections of π, and S-valued two-forms satisfying a twisted self-duality condition.
• Uses the structure of the automorphism group of the symplectic bundle to derive a short exact sequence that characterizes the U-duality group.
• Analyzes the Killing spinor equations via the canonical lift of the Levi-Civita and Chern connections to the spinor bundle, coupling to scalar maps into the target manifold.
• Applies results from pseudo-Riemannian geometry to classify simply-connected, geodesically complete Lorentzian manifolds admitting supersymmetric solutions.

Experimental results

Research questions

• RQ1 What is the global geometric structure of the bosonic sector of four-dimensional ungauged supergravity on a four-manifold of arbitrary topology?
• RQ2 How can the electromagnetic duality (U-duality) group of supergravity be characterized geometrically using bundle automorphisms?
• RQ3 What are the necessary and sufficient geometric conditions for a Lorentzian four-manifold to admit a supersymmetric solution in N=1 chiral supergravity with vanishing superpotential?
• RQ4 How do the Killing spinor equations constrain the holonomy and global structure of spacetime in four-dimensional supergravity?
• RQ5 What are the topological and geometric obstructions to the existence of globally hyperbolic supersymmetric solutions in this framework?

Key findings

• The U-duality group is characterized as the kernel of a short exact sequence of automorphism groups of symplectic vector bundles, providing a global geometric description of electromagnetic duality.
• Supersymmetric solutions in N=1 chiral supergravity with vanishing superpotential on simply-connected, geodesically complete Lorentzian four-manifolds are classified into three types: Minkowski space, product of 2D Minkowski space and a Kähler Riemann surface, or holonomy in SO(2) ⋉ R².
• The Killing spinor equations reduce to the condition that a spinor is parallel with respect to a connection that couples the Levi-Civita and Chern connections via the scalar map.
• The scalar manifold is a complex manifold equipped with a negative Hermitian holomorphic line bundle, and the target space metric arises from the curvature of its Chern connection.
• The theory reduces to Einstein gravity coupled to a non-linear sigma model with a Kähler target space, and solutions correspond to harmonic maps with additional holomorphic constraints.
• The problem of classifying globally hyperbolic Lorentzian four-manifolds admitting supersymmetric solutions remains open, despite the classification of simply-connected cases.