Gauge Group SL Research Articles

Integrable systems and the boundary dynamics of higher spin gravity on AdS3

We introduce a new set of boundary conditions for three-dimensional higher spin gravity with gauge group SL(3, ℝ) × SL(3, ℝ), where its dynamics at the boundary is described by the members of the modified Boussinesq integrable hierarchy. In the asymptotic region the gauge fields are written in the diagonal gauge, where the excitations go along the generators of the Cartan subalgebra of sl(3, ℝ) ⊕ sl(3, ℝ). We show that the entire integrable structure of the modified Boussinesq hierarchy, i.e., the phase space, the Poisson brackets and the infinite number of commuting conserved charges, are obtained from the asymptotic structure of the higher spin theory. Furthermore, its known relation with the Boussinesq hierarchy is inherited from our analysis once the asymptotic conditions are re-expressed in the highest weight gauge. Hence, the Miura map is recovered from a purely geometric construction in the bulk. Black holes that fit within our boundary conditions, the Hamiltonian reduction at the boundary, and the generalization to higher spin gravity with gauge group SL(N, ℝ) × SL(N, ℝ) are also discussed.

Are Khovanov-Rozansky polynomials consistent with evolution in the space of knots?

R-coloured knot polynomials for m-strand torus knots Torus[m,n] are described by the Rosso-Jones formula, which is an example of evolution in n with Lyapunov exponents, labelled by Young diagrams from R⊗m. This means that they satisfy a finite-difference equation (recursion) of finite degree. For the gauge group SL(N ) only diagrams with no more than N lines can contribute and the recursion degree is reduced. We claim that these properties (evolution/recursion and reduction) persist for Khovanov-Rozansky (KR) polynomials, obtained by additional factorization modulo 1 + t, which is not yet adequately described in quantum field theory. Also preserved is some weakened version of differential expansion, which is responsible at least for a simple relation between reduced and unreduced Khovanov polynomials. However, in the KR case evolution is incompatible with the mirror symmetry under the change n −→ −n, what can signal about an ambiguity in the KR factorization even for torus knots.

On a canonical quantization of 3D Anti de Sitter pure gravity

We perform a canonical quantization of pure gravity on AdS3 using as a technical tool its equivalence at the classical level with a Chern-Simons theory with gauge group SL(2,R)xSL(2,R). We first quantize the theory canonically on an asymptotically AdS space --which is topologically the real line times a Riemann surface with one connected boundary. Using the "constrain first" approach we reduce canonical quantization to quantization of orbits of the Virasoro group and Kaehler quantization of Teichmuller space. After explicitly computing the Kaehler form for the torus with one boundary component and after extending that result to higher genus, we recover known results, such as that wave functions of SL(2,R) Chern-Simons theory are conformal blocks. We find new restrictions on the Hilbert space of pure gravity by imposing invariance under large diffeomorphisms and normalizability of the wave function. The Hilbert space of pure gravity is shown to be the target space of Conformal Field Theories with continuous spectrum and a lower bound on operator dimensions. A projection defined by topology changing amplitudes in Euclidean gravity is proposed. It defines an invariant subspace that allows for a dual interpretation in terms of a Liouville CFT. Problems and features of the CFT dual are assessed and a new definition of the Hilbert space, exempt from those problems, is proposed in the case of highly-curved AdS3.

Black holes and singularity resolution in higher spin gravity

We investigate higher spin theories of gravity in three dimensions based on the gauge group SL(N,R)*SL(N,R). In these theories the usual diffeomorphism symmetry is enhanced to include higher spin gauge transformations under which traditional geometric notions of curvature and causality are no longer invariant. This implies, for example, that apparently singular geometries can be rendered smooth by a gauge transformation, much like the resolution of orbifold singularities in string theory. The classical solutions, including the recently constructed higher spin black hole, are characterized by their holonomies around the non-contractible cycles of space-time. The black hole solutions are shown to be gauge equivalent to a BTZ black hole which is charged under a set of U(1) Chern-Simons fields. Nevertheless, depending on the choice of embedding of the gravitational gauge group, the space-time geometry may be non-trivial. We study in detail the N=3 example, where this observation allows us to find a gauge where the black hole geometry takes a simple form and the thermodynamic properties can be studied.

High spin topologically massive gravity

We propose an action to describe high spin topologically massive gravity with a negative cosmological constant. With the frame-like fields and spin connections being combined into two gauge fields, the action includes two gauge field Chern-Simons actions with different levels, and also a linear term proportional to the difference of the gauge field strengths. Such linear term play the role of imposing the torsion free conditions for high spin fields. We discuss the gauge symmetry of this action and study the fluctuations around the AdS3 vacuum. We show how to relate the the fluctuations of the gauge field to the Frondal fields, using the gauge symmetry. For the gauge group SL(n, R) × SL(n, R), we find that the fluctuations of all high spin fields up to spin n satisfy third order differential equations, and hence there generically exist massive traceless and trace mode for every spin.

NONSINGULAR COMPLEX INSTANTONS ON EUCLIDEAN SPACETIME

Building on a variation of 't Hooft's harmonic function ansatz for SU(2) instantons on ℝ4, we provide new explicit nonsingular solutions of the Yang–Mills anti-self-duality equations on Euclidean spacetime with gauge group SL(2, ℂ) and SL(3, ℝ).

Three-Dimensional Quantum Gravity, Chern-Simons Theory, and the A-Polynomial

We study three-dimensional Chern-Simons theory with complex gauge group SL(2,C), which has many interesting connections with three-dimensional quantum gravity and geometry of hyperbolic 3-manifolds. We show that, in the presence of a single knotted Wilson loop in an infinite-dimensional representation of the gauge group, the classical and quantum properties of such theory are described by an algebraic curve called the A-polynomial of a knot. Using this approach, we find some new and rather surprising relations between the A-polynomial, the colored Jones polynomial, and other invariants of hyperbolic 3-manifolds. These relations generalize the volume conjecture and the Melvin-Morton-Rozansky conjecture, and suggest an intriguing connection between the SL(2,C) partition function and the colored Jones polynomial.

Group averaging in the (p,q) oscillator representation of SL(2,R)

We investigate refined algebraic quantization with group averaging in a finite-dimensional constrained Hamiltonian system that provides a simplified model of general relativity. The classical theory has gauge group SL(2,R) and a distinguished o(p,q) observable algebra. The gauge group of the quantum theory is the double cover of SL(2,R), and its representation on the auxiliary Hilbert space is isomorphic to the (p,q) oscillator representation. When p⩾2, q⩾2 and p+q≡0 (mod 2), we obtain a physical Hilbert space with a nontrivial representation of the o(p,q) quantum observable algebra. For p=q=1, the system provides the first example known to us where group averaging converges to an indefinite sesquilinear form.

The symplectic and twistor geometry of the general isomonodromic deformation problem

Hitchin’s twistor treatment of Schlesinger’s equations is extended to the general isomonodromic deformation problem. It is shown that a generic linear system of ordinary differential equations with gauge group SL(n, C) on a Riemann surface X can be obtained by embedding X in a twistor space Z on which sl(n, C) acts. When a certain obstruction vanishes, the isomonodromic deformations are given by deforming X in Z . This is related to a description of the deformations in terms of Hamiltonian flows on a symplectic manifold constructed from affine orbits in the dual Lie algebra of a loop group.

Refined algebraic quantization in the oscillator representation of SL(2, ℝ)

We investigate refined algebraic quantization (RAQ) with group averaging in a constrained Hamiltonian system with unreduced phase space T*R4 and gauge group SL(2, ℝ). The reduced phase space ℳ is connected and contains four mutually disconnected “regular” sectors with topology R×S1, but these sectors are connected to each other through an exceptional set, where ℳ is not a manifold and where ℳ has non-Hausdorff topology. The RAQ physical Hilbert space Hphys decomposes as Hphys≃⊕Hi, where the four subspaces Hi naturally correspond to the four regular sectors of ℳ. The RAQ observable algebra Aobs, represented on Hphys, contains natural subalgebras represented on each Hi. The group averaging takes place in the oscillator representation of SL(2, ℝ) on L2(R2,2), and ensuring convergence requires a subtle choice for the test state space: the classical analog of this choice is to excise from ℳ the exceptional set while nevertheless retaining information about the connections between the regular sectors. A quantum theory with the Hilbert space Hphys and a finitely generated observable subalgebra of Aobs is recovered through both Ashtekar’s algebraic quantization and Isham’s group theoretic quantization.

Flavor symmetry of the tensor Dirac theory

Recently, the Dirac and Einstein equations were unified in a tetrad formulation of a Kaluza–Klein model with gauge group SL(2,R)×U(1). In this model, the self-adjoint modes of the tetrad describe gravity, whereas the isometric modes of the tetrad together with a scalar field describe fermions. This model gives precisely the usual Dirac–Einstein Lagrangian. In this paper we generalize the tensor Dirac theory to the larger gauge group SL(2,C)×U(1) acting on bispinors. We show that each SL(2,R)×U(1) subgroup of SL(2,C)×U(1) corresponds to a different factorization of the second-order Klein–Gordon equation into a first-order Dirac equation. Since the Noether currents are different for each factorization, the solutions describe different flavors of fermions. We show that electric charge, lepton number, and baryon number are conserved in this generalization of the Dirac theory.

Coverings and Integrability of the Gauss–Mainardi–Codazzi Equations

Using the covering theory approach (zero-curvature representations with the gauge group SL), we insert the spectral parameter into the Gauss–Mainardi–Codazzi equations in Chebyshev and geodesic coordinates. For each choice, four integrable systems are obtained.

THE ZERO CURVATURE FORMULATION OF TB, sTB HIERARCHY AND TOPOLOGICAL ALGEBRAS

A particular dispersive generalization of long water wave equation in (1+1) dimensions, which is important in the study of matrix models without scaling limit, known as two-Boson (TB) equation, as well as the associated hierarchy has been derived from the zero curvature condition on the gauge group SL (2, R) ⊗ U (1). The supersymmetric extension of the two-Boson (sTB) hierarchy has similarly been derived from the zero curvature condition associated with the gauge supergroup OSp (2|2). Topological algebras arise naturally as the second Hamiltonian structure of these classical integrable systems, indicating a close relationship of these models with 2-D topological field theories.

SELF-DUAL SUPERGRAVITY AND SUPERSYMMETRIC YANG-MILLS THEORY COUPLED TO GREEN-SCHWARZ SUPERSTRING

We present the canonical set of superspace constraints for self-dual supergravity, a “self-dual” tensor multiplet and a self-dual Yang-Mills multiplet with N=1 supersymmetry in the space-time with signature (+,+, −, −). For this set of constraints, the consistency of the self-duality conditions on these multiplets with supersymmetry is manifest. The energy-momentum tensors of all the self-dual “matter” multiplets vanish, to be consistent with the self-duality of the Riemann tensor. In particular, the special significance of the “self-dual” tensor multiplet is noted. This result fills the gap left over in our previous series of papers, with respect to the consistent couplings among the self-dual matter multiplets. We also couple these nontrivial backgrounds to a Green-Schwarz superstring σ model, under the requirement of invariance under fermionic (kappa) symmetry. The finiteness of the self-dual supergravity is discussed, based on its “off-shell” structure. A set of exact solutions for the “self-dual” tensor and self-dual Yang-Mills multiplets for the gauge group SL(2) on a self-dual gravitational instanton background is given, and its consistency with the Green-Schwarz string σ model is demonstrated.

QQCD2 AND G/G MODEL

The 2D lattice gauge theory with a quantum gauge group SL q(2) is considered. When q=ei2π/(k+2), its weak coupling partition function coincides with the one of the G/G coset model (i.e. equals the Verlinde numbers). However, despite such a remarkable coincidence, these models are not equivalent but, in certain sense, dual to each other.

W-gravity from Chern-Simons theory

Starting with three-dimensional Chern-Simons theory with gauge group Sl(N, R) , we derive an action S cov invariant under both left and right W N transformations. We give an interpretation of S cov in terms of anomalies, and discuss its relation with Toda theory.

On N = 2 strings and classical scattering solutions of self-dual Yang-Mills in (2,2) space-time

Ooguri and Vafa have shown that the open N = 2 string corresponds to self-dual Yang-Mills (SDYM) and also that, in perturbation theory, it has a vanishing four-particle scattering amplitude. We discuss how the dynamics of the three-particle scattering implies that on-shell states can only scatter if their momenta lie in the same self-dual plane and then investigate classical SDYM with the aim of compairing exact solutions with the tree-level perturbation theory predictions. In particular for the gauge group SL(2, C ) with a plane wave Hirota ansatz SDYM reduces to a complicated set of algebraic relations due to de Vega. Here we solve these conditions and the solutions are shown to correspond to collisions of plane wave kinks. The main result is that for a class of kinks the resulting phase shifts are non-zero, the solution as a whole is not pure gauge and so the scattering seems non-trivial. However the stress-energy and lagrangian density are confined to string-like regions in the space-time and in particular are zero for the incoming/outgoing kinks so the solution does not correspond to physical four-point scattering.

The zero curvature formulation of the KP and the sKP equations

The Kadomtsev-Petviashvili equation is derived from the zero curvature condition associated with the gauge group SL (2, R ) in 2 + 1 dimensions. A fermionic extension of the KP equation is also obtained using the zero curvature condition of the supergroup OSp(2∥1), which reduces upon appropriate restriction to the Kupershmidt equation.

SELF-DUALITY AND GENERALIZED KdV FLOWS

We obtain the (N+1)-th flow of the generalized (N–1)-KdV hierarchy from self-dual Yang-Mills equations with gauge group SL(N) and space-time signature (2, 2). The dimensional reduction is performed by using a pair of orthogonal Killing vector fields (one time-like and one null) and we generalize previous results by Mason and Sparling to N≥2. We illustrate our method with explicit examples and determine the form of the self-dual solutions for N=2, 3, 4. Applications of this formalism and its possible generalizations are also discussed briefly.

Complex static solutions and wave solutions to (2+1)-dimensional topologically massive gauge theories

Complex static solutions with zero action and zero energy-momentum tensor are constructed for the (2+1)-dimensional topologically massive SU(2) gauge theory by generalizing the vortex-like solutions. These solutions can be treated as real gauge potentials for the noncompact gauge group SL(2, C ). Wave solutions are also exhibited.