$L_{\infty}$-Algebras of Einstein-Cartan-Palatini Gravity
Marija Dimitrijević Ćirić, Grigorios Giotopoulos, Voja Radovanović, Richard J. Szabo
备注：
84 pages; v2: minor corrections and changes; v3: exposition improved and references added; Final version to be published in Journal of Mathematical Physics
We give a detailed account of the cyclic $L_\infty$-algebra formulation of general relativity with cosmological constant in the Einstein-Cartan-Palatini formalism on spacetimes of arbitrary dimension and signature, which encompasses all symmetries, field equations and Noether identities of gravity without matter fields. We present a local formulation as well as a global covariant framework, and an explicit isomorphism between the two $L_\infty$-algebras in the case of parallelizable spacetimes. By duality, we show that our $L_\infty$-algebras describe the complete BV-BRST formulation of Einstein-Cartan-Palatini gravity. We give a general description of how to extend on-shell redundant symmetries in topological gauge theories to off-shell correspondences between symmetries in terms of quasi-isomorphisms of $L_\infty$-algebras. We use this to extend the on-shell equivalence between gravity and Chern-Simons theory in three dimensions to an explicit $L_\infty$-quasi-isomorphism between differential graded Lie algebras which applies off-shell and for degenerate dynamical metrics. In contrast, we show that there is no morphism between the $L_\infty$-algebra underlying gravity and the differential graded Lie algebra governing $BF$ theory in four dimensions.
We show that the Palatini--Cartan--Holst formulation of General Relativity in tetrad variables must be complemented with additional requirements on the fields when boundaries are taken into account for the associated BV theory to induce a compatible BFV theory on the boundary.
A Palatini-type action for Einstein and Gauss-Bonnet gravity with non-trivial torsion is proposed. Three-form flux is incorporated via a deformation of the Riemann tensor, and consistency of the Palatini variational principle requires the flux to be covariantly constant and to satisfy a Jacobi identity. Studying gravity actions of third order in the curvature leads to a conjecture about general Palatini-Lovelock-Cartan gravity. We point out potential relations to string-theoretic Bianchi identities and, using the Schouten-Nijenhuis bracket, derive a set of Bianchi identities for the non-geomet…
In the framework of the Einstein-Palatini formalism, even though the projective transformation connecting the arbitrary connection with the Levi Civita connection has been floating in the literature for a long time and perhaps the result was implicitly known in the affine gravity community, yet as far as we know Julia and Silva were the first to realise its gauge character. We rederive this result by using the Rosenfeld-Dirac-Bergmann approach to constrained Hamiltonian systems and do a comprehensive self contained analysis establishing the equivalence of the Einstein-Palatini and the metric f…
We consider Riemann-Cartan gravity with minimal Palatini action, which is classically equivalent to Einstein gravity. Following the ideas of L.Lipatov \cite{LipGrav} we propose the effective action for this theory aimed at the description of the high-energy scattering of gravitating particles in the multi - Regge kinematics. For that purpose we add to the Palatini action the new terms responsible for the interaction of gravitational quanta with certain collective excitations that replace exchange by multiple gravitational excitations. We propose the heuristic explanation of its particular form…
We summarise recent perspectives on symmetries of noncommutative field theories based on homotopy algebras. We show how these viewpoints naturally lead to a new class of noncommutative field theories which possess braided gauge symmetries, and explain in detail their uses in gravity. We review how these considerations lead to a new theory of noncommutative gravity in four dimensions within the Einstein-Cartan-Palatini formalism.
We describe the cyclic $L_{\infty}$-algebra formulation of classical general relativity without matter fields in the Einstein-Cartan-Palatini formalism. Using Drinfel'd twist deformation techniques, we define a noncommutative version of the theory with braided gauge symmetries. We introduce a notion of braided $L_{\infty}$-algebra, and use it to encode the symmetries, field content, field equations and Noether identities of noncommutative gravity.
We reformulate the Palatini action for general relativity (GR) in terms of moving frames that exhibit local Galilean covariance in a large speed of light expansion. For this, we express the action in terms of variables that are adapted to a Galilean subgroup of the $GL(n,\mathbb{R})$ structure group of a general frame bundle. This leads to a novel Palatini-type formulation of GR that provides a natural starting point for a first-order non-relativistic expansion. In doing so, we show how a comparison of Lorentzian and Newton-Cartan metric-compatibility explains the appearance of torsion in the …
We study inflation driven by the Higgs field in the Einstein-Cartan formulation of gravity. In this theory, the presence of the Holst and Nieh-Yan terms with the Higgs field non-minimally coupled to them leads to three additional coupling constants. For a broad range of parameters, we find that inflation is both possible and consistent with observations. In most cases, the spectral index is given by $n_s=1-2/N_\star$ (with $N_\star$ the number of e-foldings) whereas the tensor-to-scalar ratio $r$ can vary between about $10^{-10}$ and $1$. Thus, there are scenarios of Higgs inflation in the Ein…
An abelian version of standard general relativity in the Cartan-Palatini gauge-like formulation in four dimensions has been introduced. Traditional canonical analysis utilizing similarities to the akin Husain-Kuchar SU(2) version of gravity has been performed. The model has been next quantized in the canonical path-integral Faddeev-Popov formalism yielding abelian BF theory.
We define a new homotopy algebraic structure, that we call a braided $L_\infty$-algebra, and use it to systematically construct a new class of noncommutative field theories, that we call braided field theories. Braided field theories have gauge symmetries which realize a braided Lie algebra, whose Noether identities are inhomogeneous extensions of the classical identities, and which do not act on the solutions of the field equations. We use Drinfel'd twist deformation quantization techniques to generate new noncommutative deformations of classical field theories with braided gauge symmetries, …
Symmetries and, in particular, Cartan (Noether) symmetries and conserved quantities (conservation laws) are studied for the multisymplectic formulation of first and second order Lagrangian classical field theories. Noether-type theorems are stated in this geometric framework. The concept of gauge symmetry and its geometrical meaning are also discussed in this formulation. The results are applied to study Noether and gauge symmetries for the multisymplectic description of the electromagnetic and the gravitational theory; in particular, the Einstein--Hilbert and the Einstein--Palatini approaches…
A discrete theory of gravity locally invariant under the Poincaré group is considered as in a companion paper. We define a first order theory, in the sense of Palatini, on the metric-dual Voronoi complex of a simplicial complex. We follow the same spirit of the continuum theory of General Relativity in the Cartan formalism. The field equations are carefully derived taking in account the constraints of the theory. They look very similar to first order Einstein equations in the Cartan formalism. It is shown that in the limit of {\it small deficit angles} these equations have Regge Calculus, loca…
Using the symmetry properties of two-dmensional sigma models, we introduce a notion of the Beltrami-Courant differential, so that there is a natural homotopy Gerstenhaber algebra related to it. We conjecture that the generalized Maurer-Cartan equation for the corresponding $L_{\infty}$ subalgebra gives solutions to the Einstein equations.
Higher Chern--Simons Theory in $2n+2$ Dimensions for Balanced 2-term $L_\infty$-Algebras
Beijing International Center for Mathematical Research, Peking University（北京大学数学中心）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）
Certain classical field theories admit a formal multi-particle solution, known as the perturbiner expansion, that serves as a generating function for all the tree-level scattering amplitudes and the Berends-Giele recursion relations they satisfy. In this paper it is argued that the minimal model for the $L_{\infty}$-algebra that governs a classical field theory contains enough information to determine the perturbiner expansion associated to such theory. This gives a prescription for computing the tree-level scattering amplitudes by inserting the perturbiner solution into the homotopy Maurer-Ca…
We consider finite deformations of the Hull--Strominger system. Starting from the heterotic superpotential, we identify complex coordinates on the off-shell parameter space. Expanding the superpotential around a supersymmetric vacuum leads to a third-order Maurer--Cartan equation that controls the moduli. The resulting complex effective action generalises that of both Kodaira--Spencer and holomorphic Chern--Simons theory. The supersymmetric locus of this action is described by an $L_3$ algebra.
We present a theory of Galilean-invariant conventional and chiral $p_x \pm ip_y$ fermionic superfluids at zero temperature in two spatial dimensions in terms of a dual gauge theory. Our formulation is general coordinate invariant. The parity-violating effects are encoded in the Wen-Zee term that gives rise to the Hall viscosity and edge current. We show that the relativistic superfluid with the Euler current reduces to the chiral superfluid in the limit $c\to\infty$. Using Newton-Cartan geometry we construct the covariant formulation of the effective theory and calculate the energy current.
We show that the vertex algebra W{1+ \infty} with central charge -1 is isomorphic to a tensor product of the simple W_3 algebra with central charge -2 and a Heisenberg vertex algebra generated by a free bosonic field. We construct a family of irreducible modules of the W_3 algebra with central charge -2 in terms of free fields and calculate the full character formulas of these modules with respect to the full Cartan subalgebra of the W_3 algebra.
We study the emergent scenario, which is proposed to avoid the big bang singularity, in the Einstein-Cartan (EC) theory with a positive cosmological constant and a perfect fluid by analyzing the existence and stability of the Einstein static (ES) solutions. We find that there is no stable ES solution for a spatially flat or open universe. However, for a spatially closed universe, the stable ES solution does exist, and in the same existence parameter regions, there also exists an unstable one. With the slow decrease of the equation of state $w$ of the perfect fluid, the stable and unstable crit…
We review various classical unified theories of gravity and other interactions that have appeared in the literature, paying special attention to scenarios in which spacetime remains four-dimensional, while an "internal" space is enlarged. The starting point for each such unification scenario is a particular formalism for General Relativity. We thus start by reviewing, besides the usual Einstein-Hilbert and Palatini formulations, the Einstein-Cartan, MacDowell-Mansouri and BF (both non-chiral and chiral) formulations. Each of these introduces some version of "internal" bundle and a dynamical va…
General Relativity in dimension $n = p + q$ can be formulated as a gauge theory for the conformal group $SO(p+1,q+1)$, along with an additional field reducing the structure group down to the Poincaré group $ISO(p,q)$. 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 …
We summarise some of our recent works on $L_\infty$-algebras and quasi-groups with regard to higher principal bundles and their applications in twistor theory and gauge theory. In particular, after a lightning review of $L_\infty$-algebras, we discuss their Maurer-Cartan theory and explain that any classical field theory admitting an action can be reformulated in this context with the help of the Batalin-Vilkovisky formalism. As examples, we explore higher Chern-Simons theory and Yang-Mills theory. We also explain how these ideas can be combined with those of twistor theory to formulate maxima…
In contrast to the Einstein-Hilbert action, the action for self-dual gravity contains vierbeins. They are eleminated at the level of observables by an $SL(2,\mathbb{C})$ gauge condition implied by the action. We argue that despite this condition, new "edge" or superselected state vectors corresponding to maps of the spheres $S^2_{\infty}$ at infinity to $SL(2, \mathbb{C})$ arise. They are characterised by new quantum numbers and they lead to mixed states. For black holes, they arise both at the horizon and the spatial infinity and may be relevant for the black hole information paradox. Similar…
We show that the low-lying excitations of the one-dimensional Bose gas are described, at all orders in a 1/N expansion and at the first order in the inverse of the coupling constant, by an effective hamiltonian written in terms of an extended conformal algebra, namely the Cartan subalgebra of the $W_{1+\infty}\times \bar{W}_{1+\infty}$ algebra. This enables us to construct the first interaction term which corrects the hamiltonian of free fermions equivalent to a hard-core boson system.
