[Paper Review] Transgression forms as source for topological gravity and Chern-Simons-Higgs theories

Omar Valdivia|arXiv (Cornell University)|Nov 6, 2014

Black Holes and Theoretical Physics6 references3 citations

TL;DR

This thesis proposes a novel construction of topological gravity and Chern-Simons-Higgs theories using transgression forms as Lagrangians in higher-dimensional gauge theories. By formulating Poincaré-invariant topological gravity via a gauged Wess–Zumino–Witten model based on coset space $\frac{ISO(d-1,1)}{SO(d-1,1)}$, and extending dimensional reduction of Chern–Simons theories via equivariant principal bundles, the work derives new Chern–Simons–Higgs models with higher-order gauge and diffeomorphism-invariant Higgs couplings, revealing geometric moduli spaces isomorphic to quiver varieties.

ABSTRACT

Two main gauge invariant off-shell models are studied in this Thesis. I) Poincare-invariant topological gravity in even dimensions is formulated as a transgression field theory whose gauge connections are associated to linear and nonlinear realizations of the Poincare group ISO(d-1,1). The resulting theory is a gauged Wess-Zumino-Witten model whereby the transition functions relating gauge fields belong to the coset ISO(d-1,1)/SO(d-1,1). The supersymmetric extension leads to topological supergravity in two dimensions starting from a transgression field theory for the super-Poincare group in three dimensions. The construction is extended to a three-dimensional Chern-Simons theory of gravity invariant under the Maxwell algebra, where the corresponding Maxwell gauged Wess-Zumino-Witten model is obtained. II) dimensional reduction of Chern-Simons theories with arbitrary gauge group in a formalism based on equivariant principal bundles is considered. For the classical gauge groups the relations between equivariant principal bundles and quiver bundles is clarified, and show that the reduced quiver gauge theories are all generically built on the same universal symmetry breaking pattern. The reduced model is a novel Chern-Simons-Higgs theory consisting of a Chern-Simons term valued in the residual gauge group plus a higher order gauge and diffeomorphism invariant coupling of Higgs fields with the gauge fields. The moduli spaces of solutions provide in some instances geometric representations of certain quiver varieties as moduli spaces of flat invariant connections. In the context of dimensional reductions involving non-compact gauge groups, the reduction of five-dimensional supergravity induce novel couplings between gravity and matter. The resulting model is regarded as to a quiver gauge theory of AdS(3)xU(1) gravity involving a non-minimal coupling to scalar Higgs fermion fields.

Motivation & Objective

• To develop a gauge-invariant, off-shell formulation of topological gravity in even dimensions using transgression forms as Lagrangians.
• To generalize Chern–Simons theories via dimensional reduction using equivariant principal bundles, leading to novel Chern–Simons–Higgs models.
• To establish a correspondence between reduced quiver gauge theories and universal symmetry breaking patterns in classical gauge groups.
• To explore supersymmetric extensions, including topological supergravity in two dimensions and Maxwell algebra-invariant gravity.
• To demonstrate that moduli spaces of flat invariant connections geometrically realize quiver varieties in specific cases.

Proposed method

• Constructs topological gravity in even dimensions as a transgression field theory in one higher dimension, with gauge connections realizing linear and nonlinear realizations of the Poincaré group $ISO(d-1,1)$.
• Identifies the coset space $\frac{ISO(d-1,1)}{SO(d-1,1)}$ as the parametrization of a scalar field in the fundamental representation of the gauge group.
• Applies the $S$-expansion procedure to derive the Maxwell algebra and its invariant tensors, enabling construction of a Maxwell gauged Wess–Zumino–Witten model.
• Uses equivariant principal bundles to formalize dimensional reduction of Chern–Simons theories, preserving gauge and diffeomorphism invariance.
• Derives a Chern–Simons–Higgs theory with a residual gauge group and higher-order couplings between Higgs fields and gauge fields.
• Relates the moduli spaces of flat invariant connections to quiver varieties via the structure of reduced quiver gauge theories.

Experimental results

Research questions

• RQ1 How can transgression forms be used as Lagrangians to construct topological gravity theories in even dimensions?
• RQ2 What is the role of the coset space $\frac{ISO(d-1,1)}{SO(d-1,1)}$ in realizing scalar fields and gauge symmetry in topological gravity?
• RQ3 How does dimensional reduction of Chern–Simons theories via equivariant bundles lead to new Chern–Simons–Higgs models with higher-order couplings?
• RQ4 Can the moduli spaces of flat invariant connections in reduced theories be geometrically interpreted as quiver varieties?
• RQ5 What are the implications of non-compact gauge groups in dimensional reduction, particularly in five-dimensional supergravity?

Key findings

• The Poincaré-invariant topological gravity in even dimensions is realized as a gauged Wess–Zumino–Witten model with transition functions in $\frac{ISO(d-1,1)}{SO(d-1,1)}$, where the coset scalar field transforms in the fundamental representation of the gauge group.
• The supersymmetric extension yields topological supergravity in two dimensions from a transgression theory invariant under the supersymmetric Poincaré group in three dimensions.
• A three-dimensional Chern–Simons theory of gravity invariant under the Maxwell algebra is constructed, with its gauged Wess–Zumino–Witten model derived via $S$-expansion.
• Dimensional reduction of Chern–Simons theories with classical gauge groups leads to quiver gauge theories built on a universal symmetry breaking pattern, with a residual Chern–Simons term and higher-order Higgs couplings.
• The moduli spaces of flat invariant connections in the reduced models are shown to geometrically represent certain quiver varieties, establishing a direct link between gauge theory and algebraic geometry.
• Reduction of five-dimensional supergravity yields a novel quiver gauge theory of AdS${}_{3}\times\mathrm{U}(1)$ gravity with non-minimal couplings to scalar and fermionic Higgs fields.