A correction to this paper has been published: https://doi.org/10.1007/JHEP11(2012)010

Germany

Ioan Ghişoiu

Chile

York Schröder

Erratum to: A new method for taming tensor sum-integrals

The notion of time in general relativity must arise from an internal clock, i.e., a degree of freedom in the gravitational theory internal to the system that can serve the role of a physical clock. One such internal notion of time is the York time, corresponding to constant extrinsic curvature slicing of spacetime. We study the Hartle-Hawking wavefunction of asymptotically $\textit{AdS}$ JT gravity as a function of York time. Using both canonical quantization and the JT gravity path integral, we explicitly calculate this wavefunction and show that it satisfies a Schrodinger equation with respect to York time. We find the corresponding York Hamiltonian, which turns out to be manifestly Hermitian. Our analysis cleanly avoids operator ordering ambiguities. The dependence of the wavefunction on York time should be thought of as emerging from a unitary basis transformation of the gravitational length basis states used to compute the wavefunction, and not from time evolution of the state in the dual boundary theory.

India

Onkar Parrikar

Japan

Sunil Sake

York time in JT gravity

We relate invariants of torus knots to the counts of a class of lattice paths, which we call generalized Schröder paths. We determine generating functions of such paths, located in a region determined by a type of a torus knot under consideration, and show that they encode colored HOMFLY-PT polynomials of this knot. The generators of uncolored HOMFLY-PT homology correspond to a basic set of such paths. Invoking the knots-quivers correspondence, we express generating functions of such paths as quiver generating series, and also relate them to quadruply-graded knot homology. Furthermore, we determine corresponding A-polynomials, which provide algebraic equations and recursion relations for generating functions of generalized Schröder paths. The lattice paths of our interest explicitly enumerate BPS states associated to knots via brane constructions, as well as encode 3-dimensional <math altimg='si1.svg'><mi mathvariant='script'>N</mi><mo linebreak='goodbreak' linebreakstyle='after'>=</mo><mn>2</mn></math> theories.

Portugal

Serbia

Marko Stošić

Poland

Piotr Sułkowski

Torus knots and generalized Schröder paths

We present major progress towards the determination of the last missing piece for the pressure of a Yang-Mills plasma at high temperatures at order $\textit{g}$ in the strong coupling constant. This order is of key importance due to its role in resolving the long-standing infrared problem of finite-temperature field theory within a dimensionally reduced effective field theory setup. By systematically applying linear transformations of integration variables, or momentum shifts, we resolve equivalences between different representations of Feynman sum-integrals on the integrand level, transforming those into a canonical form. At the order $\textit{g}$ , this results in reducing a sum of $$ \mathcal{O} $$ (100000) distinct sum-integrals which are produced from all four-loop vacuum diagrams down to merely 21. Furthermore, we succeed to map 11 of those onto known lower-loop structures. This leaves only 10 genuine 4-loop sum-integrals to be evaluated, thereby bringing the finalization of three decades of theoretical efforts within reach.

Finland

Pablo Navarrete

The $\textit{g}$

General relativity (GR) as described in terms of curvature by the Einstein-Hilbert action is dynamically equivalent to theories of gravity formulated in terms of spacetime torsion or nonmetricity. This forms what is called the geometrical trinity of gravity. The theories corresponding to this trinity are, apart from GR, the teleparallel (TEGR) and symmetric teleparallel (STEGR) equivalent theories of general relativity, respectively, and their actions are equivalent to GR up to boundary terms <math><mi>B</mi></math>. We investigate how the Gibbons-Hawking-York (GHY) boundary term of GR changes under the transition to TEGR and STEGR within the framework of metric-affine gravity. We show that <math><mi>B</mi></math> is the difference between the GHY term of GR and that of metric-affine gravity. Moreover, we show that the GHY term for both TEGR and STEGR must vanish for consistency of the variational problem. Furthermore, our approach allows to extend the equivalence between GR, TEGR and STEGR beyond the Einstein-Hilbert action to any action built out of the curvature two-form, thus establishing the generalized geometrical trinity of gravity. We argue that these results will be particularly useful in view of studying gauge/gravity duality for theories with torsion and nonmetricity.

Johanna Erdmenger

Bastian Heß

René Meyer

Italy

Ioannis Matthaiakakis

Gibbons-Hawking-York boundary terms and the generalized geometrical trinity of gravity

We derive analytic results for scalar massless bosonic vacuum sum-integrals at two loops. Building upon a recent factorization proof of massive two-loop vacuum integrals, we are able to solve the corresponding Matsubara sums and map the result onto one-loop structures, thereby proving factorization also in the sum-integral setting. Analytic results are provided for generic integer-valued propagator- and numerator-powers of the class of sum-integrals under consideration, allowing to eliminate them from any perturbative expansion, dramatically simplifying the evaluation of some observables encountered e.g. in hot QCD.

Canada

Andrei Davydychev

Factorizing two-loop vacuum sum-integrals

We investigate the structure of a particular class of massive vacuum Feynman integrals at two loops. This class enjoys the linear relation m1 + m2 = m3 between its three propagator masses, corresponding to zeros of the associated Källén function. Apart from having applications in thermal field theory, the integrals can be mapped onto one-loop three-point functions with collinear external momenta, suggesting the term “collinear” masses. We present a closed-form solution for these integrals, proving that they can always be factorized into products of one-loop cases, for all integer-valued propagator powers.

Recursion-free solution for two-loop vacuum integrals with “collinear” masses

The Brown-York stress tensor provides a means for defining quasilocal gravitational charges in subregions bounded by a timelike hypersurface. We consider the generalization of this stress tensor to null hypersurfaces. Such a stress tensor can be derived from the on-shell subregion action of general relativity associated with a Dirichlet variational principle, which fixes an induced Carroll structure on the null boundary. The formula for the mixed-index tensor T  i  j takes a remarkably simple form that is manifestly independent of the choice of auxiliary null vector at the null surface, and we compare this expression to previous proposals for null Brown-York stress tensors. The stress tensor we obtain satisfies a covariant conservation equation with respect to any connection induced from a rigging vector at the hypersurface, as a result of the null constraint equations. For transformations that act covariantly on the boundary structures, the Brown-York charges coincide with canonical charges constructed from a version of the Wald-Zoupas procedure. For anomalous transformations, the charges differ by an intrinsic functional of the boundary geometry, which we explicity verify for a set of symmetries associated with finite null hyper-surfaces. Applications of the null Brown-York stress tensor to symmetries of asymptotically flat spacetimes and celestial holography are discussed.

United States

Venkatesa Chandrasekaran

Éanna Flanagan

Ibrahim Shehzad

Antony Speranza

Brown-York charges at null boundaries

We compute the Hamiltonian surface charges of gravity for a family of conservative boundary conditions, that include Dirichlet, Neumann, and York’s mixed boundary conditions defined by holding fixed the conformal induced metric and the trace of the extrinsic curvature. We show that for all boundary conditions considered, canonical methods give the same answer as covariant phase space methods improved by a boundary Lagrangian, a prescription recently developed in the literature and thus supported by our results. The procedure also suggests a new integrable charge for the Einstein-Hilbert Lagrangian, different from the Komar charge for non-Killing and non-tangential diffeomorphisms. We study how the energy depends on the choice of boundary conditions, showing that both the quasi-local and the asymptotic expressions are affected. Finally, we generalize the analysis to non-orthogonal corners, confirm the matching between covariant and canonical results without any change in the prescription, and discuss the subtleties associated with this case.

France

Gloria Odak

Simone Speziale

Brown-York charges with mixed boundary conditions

Initial value problem in general relativity is often solved numerically; with only a few exceptions one of which is the “model” solution of Bowen and York where an analytical form of the solution is available. The solution describes a dynamical, time-asymmetric, gravitating system with mass and linear momentum. Here we revisit this solution and correct an error which turns out to be important for identifying the energy-content of the solution. Depending on the linear momentum, the ratio of the non-stationary part of the initial energy to the total ADM energy takes values between [0, 0.592). This non-stationary part is expected to be turned into gravitational waves during the evolution of the system to possibly settle down to a black hole with mass and linear momentum. In the ultra-relativistic case (the high momentum limit), the maximum amount of gravitational wave energy is 59.2% of the total ADM energy. We also give a detailed account of the general solution of the Hamiltonian constraint.

Türkiye

Emel Altas

Bayram Tekin

Bowen–York model solution redux