Gaudin models and multipoint conformal blocks. Part II. Comb channel vertices in 3D and 4D

It was recently shown that multi-point conformal blocks in higher dimensional conformal field theory can be considered as joint eigenfunctions for a system of commuting differential operators. The latter arise as Hamiltonians of a Gaudin integrable system. In this work we address the reduced fourth order differential operators that measure the choice of 3-point tensor structures for all vertices of 3- and 4-dimensional comb channel conformal blocks. These vertices come associated with a single cross ratio. Remarkably, we identify the vertex operators as Hamiltonians of a crystallographic elliptic Calogero-Moser-Sutherland model that was discovered originally by Etingof, Felder, Ma and Veselov. Our construction is based on a further development of the embedding space formalism for mixed-symmetry tensor fields. The results thereby also apply to comb channel vertices of 5- and 6-point functions in arbitrary dimension.

Extended corner symmetry, charge bracket and Einstein’s equations

We develop the covariant phase space formalism allowing for non-vanishing flux, anomalies, and field dependence in the vector field generators. We construct a charge bracket that generalizes the one introduced by Barnich and Troessaert and includes contributions from the Lagrangian and its anomaly. This bracket is uniquely determined by the choice of Lagrangian representative of the theory. We then extend the notion of corner symmetry algebra to include the surface translation symmetries and prove that the charge bracket provides a canonical representation of the extended corner symmetry algebra. This representation property is shown to be equivalent to the projection of the gravitational equations of motion on the corner, providing us with an encoding of the bulk dynamics in a locally holographic manner.

Bounded Reasoning and Higher-Order Uncertainty

A standard assumption in game theory is that players have an infinite depth of reasoning: they think about what others think and about what others think that othersthink, and so on, ad infinitum. However, in practice, players may have a finite depth of reasoning. For example, a player may reason about what other players think, but not about what others think he thinks. This paper proposes a class of type spaces that generalizes the type space formalism due to Harsanyi (1967) so that it can model players with an arbitrary depth of reasoning. I show that the type space formalism does not impose any restrictions on the belief hierarchies that can be modeled, thus generalizing the classic result of Mertens and Zamir (1985). However, there is no universal type space that contains all type spaces.

Anomalies in gravitational charge algebras of null boundaries and black hole entropy

We revisit the covariant phase space formalism applied to gravitational theories with null boundaries, utilizing the most general boundary conditions consistent with a fixed null normal. To fix the ambiguity inherent in the Wald-Zoupas definition of quasilocal charges, we propose a new principle, based on holographic reasoning, that the flux be of Dirichlet form. This also produces an expression for the analog of the Brown-York stress tensor on the null surface. Defining the algebra of charges using the Barnich-Troessaert bracket for open subsystems, we give a general formula for the central — or more generally, abelian — extensions that appear in terms of the anomalous transformation of the boundary term in the gravitational action. This anomaly arises from having fixed a frame for the null normal, and we draw parallels between it and the holographic Weyl anomaly that occurs in AdS/CFT. As an application of this formalism, we analyze the near-horizon Virasoro symmetry considered by Haco, Hawking, Perry, and Strominger, and perform a systematic derivation of the fluxes and central charges. Applying the Cardy formula to the result yields an entropy that is twice the Bekenstein-Hawking entropy of the horizon. Motivated by the extended Hilbert space construction, we interpret this in terms of a pair of entangled CFTs associated with edge modes on either side of the bifurcation surface.

Covariant phase space with boundaries

The covariant phase space method of Iyer, Lee, Wald, and Zoupas gives an elegant way to understand the Hamiltonian dynamics of Lagrangian field theories without breaking covariance. The original literature however does not systematically treat total derivatives and boundary terms, which has led to some confusion about how exactly to apply the formalism in the presence of boundaries. In particular the original construction of the canonical Hamiltonian relies on the assumed existence of a certain boundary quantity “B”, whose physical interpretation has not been clear. We here give an algorithmic procedure for applying the covariant phase space formalism to field theories with spatial boundaries, from which the term in the Hamiltonian involving B emerges naturally. Our procedure also produces an additional boundary term, which was not present in the original literature and which so far has only appeared implicitly in specific examples, and which is already nonvanishing even in general relativity with sufficiently permissive boundary conditions. The only requirement we impose is that at solutions of the equations of motion the action is stationary modulo future/past boundary terms under arbitrary variations obeying the spatial boundary conditions; from this the symplectic structure and the Hamiltonian for any diffeomorphism that preserves the theory are unambiguously constructed. We show in examples that the Hamiltonian so constructed agrees with previous results. We also show that the Poisson bracket on covariant phase space directly coincides with the Peierls bracket, without any need for non-covariant intermediate steps, and we discuss possible implications for the entropy of dynamical black hole horizons.

Kerr–Bolt black hole entropy and soft hair

Recently it has been speculated that a set of infinitesimal [Formula: see text] diffeomorphisms exist which act nontrivially on the horizon of some black holes such as Kerr and Kerr–Newman black holes.[Formula: see text] Having applied this symmetry in covariant phase space formalism, one can obtain Virasoro charges as surface integrals on the horizon. Kerr–Bolt space–time is well known for its asymptotically topology and has been studied widely in recent years. In this work, we are interested to find conserved charge associated to the Virasoro symmetry of Kerr–Bolt geometry using covariant phase space formalism. We will show right and left central charge are [Formula: see text], respectively. Our results also show good agreement with Kerr space–time in the limiting behavior.

De Sitter scalar–spinor interaction in Minkowski limit

The scalar–spinor interaction Lagrangian is presented by the Yukawa potential. In dS ambient space formalism, the interaction Lagrangian of scalar–spinor fields was obtained from a new transformation which is very similar to the gauge theory. The interaction of massless minimally coupled (mmc) scalar and spinor fields was investigated. The Minkowski limit of the mmc scalar field and massive spinor field interaction in the ambient space formalism of de Sitter spacetime is calculated. The interaction Lagrangian and mmc scalar field in the null curvature limit become zero and the local transformation in the null curvature limit become a constant phase transformation and the interaction in this limit become zero. The covariant derivative reduces to ordinary derivative too. Then, we conclude that this interaction is due to the curvature of spacetime and then the mmc scalar field may be a part of a gravitational field.

Mechanistic Modeling of Biochemical Systems Without A Priori Parameter Values Using the Design Space Toolbox v.3.0

SummaryMechanistic models of biochemical systems provide a rigorous kinetics-based description of various biological phenomena. They are indispensable to elucidate biological design principles and to devise and engineer systems with novel functionalities. To date, mathematical analysis and characterization of these models remain a challenging endeavor, the main difficulty being the lack of information for most system parameters. Here, we introduce the Design Space Toolbox v.3.0 (DST3), a software implementation of the Design Space formalism that enables mechanistic modeling of complex biological processes without requiring previous knowledge of the parameter values involved. This is achieved by making use of a phenotype-centric modeling approach, in which the system is first decomposed into a series of biochemical phenotypes. Parameter values realizing phenotypes of interest are predicted in a second step. DST3 represents the most generally applicable implementation of the Design Space formalism to date and offers unique advantages over earlier versions. By expanding the capabilities of the Design Space formalism and streamlining its distribution, DST3 represents a valuable tool for elucidating biological design principles and guiding the design and optimization of novel synthetic circuits.