Gauge Theory Perspective Research Articles

Post Newtonian emission of gravitational waves from binary systems: a gauge theory perspective

Using the AGT correspondence and localization, we derive a combinatorial formula for the Post-Newtonian expansion of the wave form describing the gravitational emission from binary systems made of objects of extremely different masses. The results are written as a double instanton series describing the expansion of the gravitational wave at large distances and small velocities, and are tested against previous formulae in the literature for Schwarschild and Kerr black holes at the 5th and 3rd Post Newtonian order respectively beyond the quadrupole approximation. Tidal effects show up in the wave form at the 5th PN order, providing a quantitative measure of sizes and reflectivity properties of the gravity solution.

Encoding a one-dimensional topological gauge theory in a Raman-coupled Bose-Einstein condensate

Topological gauge theories provide powerful effective descriptions of certain strongly correlated systems, a prime example being the Chern-Simons gauge theory of fractional quantum Hall states. Engineering topological gauge theories in controlled quantum systems is of both conceptual and practical importance, as it would provide access to systems with exotic excitations such as anyons without the need for strong correlations. Here, we discuss a scheme to engineer the chiral BF theory, a minimal model of a topological gauge theory corresponding to a one-dimensional reduction of the Chern-Simons theory, with ultracold atoms. Using the local conservation laws of the theory, we encode its quantum Hamitonian into an ultracold quantum gas with chiral interactions. Building on a seminal proposal by Edmonds et al. (Phys. Rev. Lett. 110, 085301 (2013)), we show how to implement it in a Raman-coupled Bose-Einstein condensate with imbalanced scattering lengths, as we have recently realized experimentally (Fr\"olian et al., Nature 608, 293 (2022)). We discuss the properties of the chiral condensate from a gauge theory perspective, and assess the validity of the effective quantum description for accessible experimental parameters via numerical simulations. Our approach lays the foundation for realizing topological gauge theories in higher dimensions with Bose-Einstein condensates.

Einstein gravity as a gauge theory for the conformal group

General relativity in dimension n = p + q can be formulated as a gauge theory for the conformal group , along with an additional field reducing the structure group down to the Poincaré group . In this paper, we propose a new variational principle for Einstein geometry which realizes this fact. Importantly, as opposed to previous treatments in the literature, our action functional gives first order field equations and does not require supplementary constraints on gauge fields, such as torsion-freeness. Our approach is based on the ‘first order formulation’ of conformal tractor geometry. Accordingly, it provides a straightforward variational derivation of the tractor version of the Einstein equation. To achieve this, we review the standard theory of tractor geometry with a gauge theory perspective, defining the tractor bundle a priori in terms of an abstract principal bundle and providing an identification with the standard conformal tractor bundle via a dynamical soldering form. This can also be seen as a generalization of the so called Cartan–Palatini formulation of general relativity in which the ‘internal’ orthogonal group is extended to an appropriate parabolic subgroup of the conformal group.

Gravitational Dynamics—A Novel Shift in the Hamiltonian Paradigm

It is well known that Einstein’s equations assume a simple polynomial form in the Hamiltonian framework based on a Yang-Mills phase space. We re-examine the gravitational dynamics in this framework and show that time evolution of the gravitational field can be re-expressed as (a gauge covariant generalization of) the Lie derivative along a novel shift vector field in spatial directions. Thus, the canonical transformation generated by the Hamiltonian constraint acquires a geometrical interpretation on the Yang-Mills phase space, similar to that generated by the diffeomorphism constraint. In classical general relativity this geometrical interpretation significantly simplifies calculations and also illuminates the relation between dynamics in the ‘integrable’ (anti)self-dual sector and in the full theory. For quantum gravity, it provides a point of departure to complete the Dirac quantization program for general relativity in a more satisfactory fashion. This gauge theory perspective may also be helpful in extending the ‘double copy’ ideas relating the Einstein and Yang-Mills dynamics to a non-perturbative regime. Finally, the notion of generalized, gauge covariant Lie derivative may also be of interest to the mathematical physics community as it hints at some potentially rich structures that have not been explored.

Calabi-Yau Manifolds, Hermitian Yang-Mills Instantons, and Mirror Symmetry

We address the issue of why Calabi-Yau manifolds exist with a mirror pair. We observe that the irreducible spinor representation of the Lorentz group Spin(6) requires us to consider the vector spaces of two forms and four forms on an equal footing. The doubling of the two-form vector space due to the Hodge duality doubles the variety of six-dimensional spin manifolds. We explore how the doubling is related to the mirror symmetry of Calabi-Yau manifolds. Via the gauge theory formulation of six-dimensional Riemannian manifolds, we show that the curvature tensor of a Calabi-Yau manifold satisfies the Hermitian Yang-Mills equations on the Calabi-Yau manifold. Therefore, the mirror symmetry of Calabi-Yau manifolds can be recast as the mirror pair of Hermitian Yang-Mills instantons. We discuss the mirror symmetry from the gauge theory perspective.

Gravity: A gauge theory perspective

The evolution of a generally covariant theory is under-determined. One hundred years ago such dynamics had never before been considered; its ramifications were perplexing, its future important role for all the fundamental interactions under the name gauge principle could not be foreseen. We recount some history regarding Einstein, Hilbert, Klein and Noether and the novel features of gravitational energy that led to Noether’s two theorems. Under-determined evolution is best revealed in the Hamiltonian formulation. We developed a covariant Hamiltonian formulation. The Hamiltonian boundary term gives covariant expressions for the quasi-local energy, momentum and angular momentum. Gravity can be considered as a gauge theory of the local Poincaré group. The dynamical potentials of the Poincaré gauge theory of gravity are the frame and the connection. The spacetime geometry has in general both curvature and torsion. Torsion naturally couples to spin; it could have a significant magnitude and yet not be noticed, except on a cosmological scale where it could have significant effects.

Holographic Kondo model in various dimensions

We study the addition of localised impurities to U(N) Supersymmetric Yang-Mills theories in (p + 1)-dimensions by using the gauge/gravity correspondence. From the gravity side, the impurities are introduced by considering probe D(8 − p)-branes extending along the time and radial directions and wrapping an (7 − p)-dimensional submanifold of the internal (8 − p)-sphere, so that the degrees of freedom are point-like from the gauge theory perspective. We analyse both the configuration in which the branes generate straight flux tubes — corresponding to actual single impurities — and the one in which connected flux tubes are created — corresponding to dimers. We discuss the thermodynamics of both the configurations and the related phase transition. In particular, the specific heat of the straight flux-tube configuration is negative for p < 3, while it is never the case for the connected one. We study the stability of the system by looking at the impurity fluctuations. Finally, we characterise the theory by computing one- and two-point correlators of the gauge theory operators dual to the impurity fluctuations. Because of the underlying generalised conformal structure, such correlators can be expressed in terms of an effective coupling constant (which runs because of its dimensionality) and a generalised conformal dimension.

Dimer models, integrable systems and quantum Teichmüller space

We introduce a correspondence between dimer models (and hence superconformal quivers) and the quantum Teichmüller space of the Riemann surfaces associated to them by mirror symmetry. Via the untwisting map, every brane tiling gives rise to a tiling of the Riemann surface with faces surrounding punctures. We explain how to obtain an ideal triangulation by dualizing this tiling. In order to do so, tiling nodes of valence greater than 3 (equivalently superpotential terms of order greater than 3 in the corresponding quiver gauge theories) must be decomposed by the introduction of 2-valent nodes. From a quiver gauge theory perspective, this operation corresponds to integrating-in massive fields. Fock coordinates in Teichmüller space are in one-to-one correspondence with chiral fields in the quiver. We present multiple explicit examples, including infinite families of theories, illustrating how the right number of Fock coordinates is generated by this procedure. Finally, we explain how Chekhov and Fock commutation relations between coordinates give rise to the commutators associated to dimer models by Goncharov and Kenyon in the context of quantum integrable systems. For generic dimer models (i.e. those containing nodes that are not 3-valent), this matching requires the introduction of a natural generalization of Chekhov and Fock rules. We also explain how urban renewal in the original brane tiling (Seiberg duality for the quivers) is mapped to flips of the ideal triangulation.

Loop quantization from a lattice gauge theory perspective

Loop quantization from a lattice gauge theory perspective

New tests of the pp-wave correspondence

The pp-wave/SYM correspondence is an equivalence relation, H_{string}= Delta -J, between the Hamiltonian H_{string} of string field theory in the pp-wave background and the dilatation operator Delta in N=4 Super Yang-Mills in the double scaling limit. We calculate matrix elements of these operators in string field theory and in gauge theory.In the string theory Hilbert space we use the natural string basis,and in the gauge theory we use the basis which is isomorphic to it. States in this basis are specific linear combinations of the original BMN operators, and were constructed previously for the case of two scalar impurities. We extend this construction to incorporate BMN operators with vector and mixed impurities. This enables us to verify from the gauge theory perspective two key properties of the three-string interaction vertex of Spradlin and Volovich: (1) the vanishing of the three-string amplitude for string states with one vector and one scalar impurity; and (2) the relative minus sign in the string amplitude involving states with two vector impurities compared to that with two scalar impurities. This implies a spontaneous breaking of the Z_2 symmetry of the string field theory in the pp-wave background. Furthermore, we calculate the gauge theory matrix elements of Delta -J for states with an arbitrary number of scalar impurities. In all cases we find perfect agreement with the corresponding string amplitudes derived from the three-string vertex.