Barnich--Troessaert bracket as a Dirac bracket on the covariant phase space

The Barnich--Troessaert bracket is a proposal for a modified Poisson bracket on the covariant phase space for general relativity. The new bracket allows us to compute charges, which are otherwise not integrable. Yet there is a catch. There is a clear prescription for how to evaluate the new bracket for any such charge, but little is known how to extend the bracket to the entire phase space. This is a problem, because not every gravitational observable is also a charge. In this paper, we propose such an extension. The basic idea is to remove the radiative data from the covariant phase space. This requires second-class constraints. Given a few basic assumptions, we show that the resulting Dirac bracket on the constraint surface is nothing but the BT bracket. A heuristic argument is given to show that the resulting constraint surface can only contain gravitational edge modes.

Gravitational SL(2, ℝ) algebra on the light cone

In a region with a boundary, the gravitational phase space consists of radiative modes in the interior and edge modes at the boundary. Such edge modes are necessary to explain how the region couples to its environment. In this paper, we characterise the edge modes and radiative modes on a null surface for the tetradic Palatini-Holst action. Our starting point is the definition of the action and its boundary terms. We choose the least restrictive boundary conditions possible. The fixed boundary data consists of the radiative modes alone (two degrees of freedom per point). All other boundary fields are dynamical. We introduce the covariant phase space and explain how the Holst term alters the boundary symmetries. To infer the Poisson brackets among Dirac observables, we define an auxiliary phase space, where the SL(2, ℝ) symmetries of the boundary fields are manifest. We identify the gauge generators and second-class constraints that remove the auxiliary variables. All gauge generators are at most quadratic in the fundamental SL(2, ℝ) variables on phase space. We compute the Dirac bracket and identify the Dirac observables on the light cone. Finally, we discuss various truncations to quantise the system in an effective way.

$$ T\overline{T} $$-deformed fermionic theories revisited

We revisit $$ T\overline{T} $$ T T ¯ deformations of d = 2 theories with fermions with a view toward the quantization. As a simple illustration, we compute the deformed Dirac bracket for a Majorana doublet and confirm the known eigenvalue flows perturbatively. We mostly consider those $$ T\overline{T} $$ T T ¯ theories that can be reconstructed from string-like theories upon integrating out the worldsheet metric. After a quick overview of how this works when we add NSR-like or GS-like fermions, we obtain a known non-supersymmetric $$ T\overline{T} $$ T T ¯ deformation of a $$ \mathcal{N} $$ N = (1, 1) theory from the latter, based on the Noether energy-momentum. This world- sheet reconstruction implies that the latter is actually a supersymmetric subsector of a d = 3 GS-like model, implying hidden supercharges, which we do construct explicitly. This brings us to ask about different $$ T\overline{T} $$ T T ¯ deformations, such as manifestly supersymmetric $$ T\overline{T} $$ T T ¯ and also more generally via the symmetric energy-momentum. We show that, for theories with fermions, such choices often lead us to doubling of degrees of freedom, with potential unitarity issues. We show that the extra sector develops a divergent gap in the “small deformation” limit and decouples in the infrared, although it remains uncertain in what sense these can be considered a deformation.

Five-brane current algebras in type II string theories

We study the current algebras of the NS5-branes, the Kaluza-Klein (KK) five-branes and the exotic $$ {5}_2^2 $$ 5 2 2 -branes in type IIA/IIB superstring theories. Their worldvolume theories are governed by the six-dimensional $$ \mathcal{N} $$ N = (2, 0) tensor and the $$ \mathcal{N} $$ N = (1, 1) vector multiplets. We show that the current algebras are determined through the S- and T-dualities. The algebras of the $$ \mathcal{N} $$ N = (2, 0) theories are characterized by the Dirac bracket caused by the self-dual gauge field in the five-brane worldvolumes, while those of the $$ \mathcal{N} $$ N = (1, 1) theories are given by the Poisson bracket. By the use of these algebras, we examine extended spaces in terms of tensor coordinates which are the representation of ten-dimensional supersymmetry. We also examine the transition rules of the currents in the type IIA/IIB supersymmetry algebras in ten dimensions. Based on the algebras, we write down the section conditions in the extended spaces and gauge transformations of the supergravity fields.

The Λ-BMS4 charge algebra

The surface charge algebra of generic asymptotically locally (A)dS4 spacetimes without matter is derived without assuming any boundary conditions. Surface charges associated with Weyl rescalings are vanishing while the boundary diffeomorphism charge algebra is non-trivially represented without central extension. The Λ-BMS4 charge algebra is obtained after specifying a boundary foliation and a boundary measure. The existence of the flat limit requires the addition of corner terms in the action and symplectic structure that are defined from the boundary foliation and measure. The flat limit then reproduces the BMS4 charge algebra of supertranslations and super-Lorentz transformations acting on asymptotically locally flat spacetimes. The BMS4 surface charges represent the BMS4 algebra without central extension at the corners of null infinity under the standard Dirac bracket, which implies that the BMS4 flux algebra admits no non-trivial central extension.

The Hamiltonian dynamics of Hořava gravity

We consider the Hamiltonian formulation of Hořava gravity in arbitrary dimensions, which has been proposed as a renormalizable gravity model for quantum gravity without the ghost problem. We study the full constraint analysis of the non-projectable Hořava gravity whose potential, $$\mathcal{V}(R)$$V(R), is an arbitrary function of the (intrinsic) Ricci scalar R but without the extension terms which depend on the proper acceleration $$a_i$$ai. We find that there exist generally three distinct cases of this theory, A, B, and C, depending on (i) whether the Hamiltonian constraint generates new (second-class) constraints or just fixes the associated Lagrange multipliers, or (ii) whether the IR Lorentz-deformation parameter $${\lambda }$$λ is at the conformal point or not. It is found that, for Cases A and C, the dynamical degrees of freedom are the same as in general relativity, while, for Case B, there is one additional phase-space degree of freedom, representing an extra (odd) scalar graviton mode. This would achieve the dynamical consistency of a restricted model at the fully non-linear level and be a positive result in resolving the long-standing debates about the extra graviton modes of the Hořava gravity. Several exact solutions are also studied as some explicit examples of the new constraints. The structure of the newly obtained, “extended” constraint algebra seems to be generic to Hořava gravity and its general proof would be a challenging problem. Some other challenging problems, which include the path integral quantization and the Dirac bracket quantization are discussed also.

Singular Lagrangians and precontact Hamiltonian systems

In this paper, we discuss the singular Lagrangian systems on the framework of contact geometry. These systems exhibit a dissipative behavior in contrast with the symplectic scenario. We develop a constraint algorithm similar to the presymplectic one studied by Gotay and Nester (the geometrization of the well-known Dirac–Bergmann algorithm). We also construct the Hamiltonian counterpart and prove the equivalence with the Lagrangian side. A Dirac–Jacobi bracket is constructed similar to the Dirac bracket.

Liouville’s Theorem & Classical Statistical Mechanics

This chapter returns to the discussion of constrained Hamiltonian dynamics, now in the canonical setting, including topics such as regular Lagrangians, constraint surfaces, Hessian conditions and the constrained action principle. The standard approach to Hamiltonian mechanics is to treat all the variables as being independent; in the constrained case, a constraint function links the variables so they are no longer independent. In this chapter, the Dirac–Bergmann theory for singular Lagrangians is developed, using an action-based approach. The chapter then investigates consistency conditions and Dirac’s different types of constraints (i.e. first-class constraints, second-class constraints, primary constraints and secondary constraints) before deriving the Dirac bracket from simple arguments. The Jackiw–Fadeev constraint formulation is then discussed before the chapter closes with the Güler formulation for a constrained Hamilton–Jacobi theory.

The canonical structure of the superstring action

We consider the canonical structure of the Green–Schwarz superstring in 9 + 1 dimensions using the Dirac constraint formalism; it is shown that its structure is similar to that of the superparticle in 2 + 1 and 3 + 1 dimensions. A key feature of this structure is that the primary fermionic constraints can be divided into two groups using field-independent projection operators; if one of these groups is eliminated through use of a Dirac bracket then the second group of primary fermionic constraints becomes first class. (This is what also happens with the superparticle action.) These primary fermionic first-class constraints can be used to find the generator of a local fermionic gauge symmetry of the action. We also consider the superstring action in other dimensions of space–time to see if the fermionic gauge symmetry can be made simpler than it is in 2 + 1, 3 + 1, and 9 + 1 dimensions. With a 3 + 3 dimensional target space, we find that such a simplification occurs. We finally show how in five dimensions there is no first-class fermionic constraint.