[Paper Review] Gauging as constraining: the universal generalised geometry action in two dimensions

Athanasios Chatzistavrakidis, Andreas Deser+2 more|arXiv (Cornell University)|May 14, 2017

Homotopy and Cohomology in Algebraic Topology2 references8 citations

TL;DR

This paper presents a generalized framework for gauging two-dimensional sigma models by treating foliations—rather than global symmetries—as the fundamental geometric structure. By reformulating the ungauged theory using auxiliary 1-forms in the generalized tangent bundle and restricting them to Dirac structures, the method systematically identifies consistent gauge theories without requiring isometries or global symmetries, with the key result being that only specific small Dirac structures allow consistent gauging, leading to target spaces like tori or circles without additional freezing of degrees of freedom.

ABSTRACT

One of the central concepts in modern theoretical physics, gauge symmetry, is typically realised by lifting a finite-dimensional global symmetry group of a given functional to an infinite-dimensional local one by extending the functional to include gauge fields. In this contribution we review the construction of gauged actions for two-dimensional sigma models, considering a more general notion to be gauged, namely that of a (possibly singular) foliation. In particular, the original action does not need to have any global symmetry for this purpose. Moreover, reformulating the ungauged theory by means of auxiliary 1-form fields taking values in the generalised tangent bundle over the target, all possible such gauge theories result from restriction of these fields to take values in (possibly small) Dirac structures. This turns all the remaining 1-form fields into gauge fields and leads to the presence of a local symmetry. We recall all needed mathematical notions, those of (higher) Lie algebroids, Courant algebroids, and Dirac structures.

Motivation & Objective

• To develop a general framework for gauging 2D sigma models that does not rely on global symmetries or isometries, but instead on the geometry of foliations.
• To extend the concept of gauging beyond traditional Lie group actions by using generalized geometry and Courant algebroids.
• To identify conditions under which consistent gauging is possible, particularly avoiding unphysical freezing of degrees of freedom in the target space.
• To demonstrate that consistent gauging corresponds precisely to restriction of auxiliary 1-form fields to small Dirac structures in the generalized tangent bundle.
• To clarify the role of minimal vs. non-minimal coupling in enabling gauging of non-isometric directions, especially in singular foliations.

Proposed method

• Reformulate the ungauged 2D sigma model using auxiliary 1-form fields valued in the generalized tangent bundle of the target manifold.
• Identify all possible gauge theories as arising from restricting these 1-form fields to small Dirac subbundles of the Courant algebroid.
• Use the invariance conditions (24), (31), and (32) to determine consistent gauge couplings via the connection coefficients $\omega^{a}_{b}$, $\theta_{a}$, and $\phi_{b}$.
• Apply the universal action (30) to allow non-minimal coupling, introducing additional tunable parameters to solve invariance conditions beyond minimal coupling.
• Construct explicit solutions for $\omega^{a}_{b}$ and $\phi_{b}$ in specific examples, such as $\omega^{3}_{4} = -\mathrm{d}X^{2}$ for the $D_{(\rho_{3},\rho_{4})}$ structure.
• Verify that the resulting quotient spaces (e.g., 2-tori, circles) correspond to the physical target spaces of the gauged theory, with no extra freezing of propagation.

Experimental results

Research questions

• RQ1 Can consistent gauge theories be constructed for 2D sigma models without assuming any underlying global symmetry or isometry?
• RQ2 Under what geometric conditions does a foliation admit a consistent gauging via generalized geometry?
• RQ3 Can the target space of the gauged theory be made to coincide with the leaf space of the foliation, avoiding unphysical freezing of degrees of freedom?
• RQ4 How does the inclusion of non-minimal coupling extend the set of possible gaugings, particularly for non-isometric directions?
• RQ5 What role do Dirac structures in Courant algebroids play in classifying consistent gaugings in generalized geometry?

Key findings

• Only specific small Dirac structures, such as $D_{(\rho_{3},\rho_{4})}$, allow consistent gauging, while others like $D_{(\rho_{1})}$ or $D_{(\rho_{2})}$ do not admit solutions to the invariance conditions.
• For the $D_{(\rho_{3},\rho_{4})}$ structure, the solution $\omega^{3}_{4} = -\mathrm{d}X^{2}$ leads to a gauged theory with a 2-torus as the target space, preserving all physical degrees of freedom.
• The $D_{(\rho_{1},\rho_{3},\rho_{4})}$ structure yields a circle as the target space via $\omega^{1}_{4} = -\mathrm{d}X^{2}$ and $\omega^{4}_{3} = -\mathrm{d}X^{2}$, demonstrating consistent gauging of a 3-torus foliation.
• For the $D_{(\rho_{2},\rho_{3},\rho_{4})}$ structure, the solution $\omega^{2}_{4} = \mathrm{d}X^{1} - (X^{4} + X^{1}X^{2})\mathrm{d}X^{2} + \mathrm{d}X^{3}$ results in a circle target space, confirming consistent gauging of a 3-nilmanifold foliation.
• In the non-minimal coupling framework, a new Dirac structure $D'_{(\rho_{4})}$ allows gauging of the $\rho_{4}$-foliation in Euclidean worldsheet signature, with $\theta = (x^{1}x^{2} + x^{4})\mathrm{d}x^{2} - \mathrm{d}x^{3}$ and $\phi = \mathrm{d}x^{2}$, which was not possible under minimal coupling.
• The $\rho_{1}$-foliation remains ungaugable even in the non-minimal framework, indicating that not all foliations admit consistent gauging, even with extended parameters.