Flux Quantization on 11-dimensional Superspace (2403.16456v4)

Abstract: Flux quantization of the C-field in 11d supergravity is arguably necessary for the (UV-)completion of the theory, in that it determines the torsion charges carried by small numbers of M-branes. However, hypotheses about C-field flux-quantization ("models of the C-field") have previously been discussed only in the bosonic sector of 11d supergravity and ignoring the supergravity equations of motion. Here we highlight a duality-symmetric formulation of on-shell 11d supergravity on superspace, observe that this naturally lends itself to completion of the theory by flux quantization, and indeed that 11d super-spacetimes are put on-shell by carrying quantizable duality-symmetric super-C-field flux; the proof of which we present in detail.

• The paper derives 11D supergravity’s on-shell equations by linking super-flux Bianchi identities with flux quantization on superspace.
• It introduces a duality-symmetric method that integrates both bosonic and fermionic sectors for a comprehensive supergeometry framework.
• The work paves the way for applying these quantization insights to M-theory and exploring higher-dimensional corrections and topological effects.

11d Supergravity EoM from Super-Flux Bianchi Identity

Introduction

The paper "Flux Quantization on 11-dimensional Superspace" explores the formulation of flux quantization within the context of 11-dimensional supergravity. This contribution addresses the integration of duality-symmetric, on-shell supergravity on superspace with the quantization of the C-field, providing significant insights into the completion of 11d supergravity with respect to its conjectured formulation in M-theory. The authors present a detailed proof demonstrating that super-spacetimes satisfying the super-flux Bianchi identities correspond precisely to solutions of the 11d supergravity equations of motion. This formulation offers a pathway to quantize the C-field on superspace and extend its implications to M-theory.

Summary of Main Findings

The authors consider the problem of flux quantization in 11-dimensional supergravity, recognizing the necessity for a duality-symmetric formulation that incorporates both the bosonic and fermionic sectors of the theory. In the traditional setting, considerations of flux quantization are often restricted to the bosonic sector, ignoring contributions from the equations of motion and fermionic variables inherent to the gravitational field. This work emphasizes the requirement for supergeometry as a complete framework to appropriately address flux scenarios in 11D supergravity.

Central to the discussion is the role of the super-flux densities $G_4^s$ and $G_7^s$, built upon spinor bilinears satisfying the aforementioned duality-symmetric Bianchi identities. Through rigorous computations, the authors demonstrate that these identities imply the classical equations of motion for 11D supergravity, encompassing the gravitino and Einstein equations, thereby integrating supersymmetric effects even in on-shell conditions.

A significant implication of this proof is the absorption of Hodge duality constraints on spacetime into the superflux Bianchi identity, achieved through the super-spacetime structure, effectively alleviating the ambiguities in duality-based constraints when moving to non-superspace formulations. This revelation underscores the necessity to treat supergravity inherently within a supersymmetric framework, as classical constraints transform and simplify with the admission of fermionic contributions in superspace.

Implications and Future Directions

This completion of supergravity through super-flux quantization not only refines our understanding of the underlying super-theoretic structure but opens avenues towards its incorporation in M-theory, especially concerning small numbers of branes and subtle topological effects in solutions like super-exceptional geometric and higher-dimensional frameworks.

Future research may focus on the application of these concepts within the realms of curved superspaces, exploring their links with exceptional super-Minkowski spacetime and the consequent implications for M-theory. Another important direction involves extending the current formulation to encompass higher-order corrections and the inclusion of quantum conformal anomalies in quantized supergravity systems.

In conclusion, by solving the equations of motion through super-flux Bianchi identities, particularly leveraging the simultaneous fermionic and bosonic contributions, the paper sets novel groundwork in linking the completion of 11D supergravity with broader M-theoretic frameworks, thereby harmonizing traditional constraints and novel multi-exposed possibilities in higher-dimensional theories. The presented methodology and results serve as a cornerstone for future explorations and applications of super-flux quantization within high-energy theoretical physics.