Reviews in Mathematical Physics

The operator product expansion (OPE) on the celestial sphere of conformal primary gluons and gravitons is studied. Asymptotic symmetries imply recursion relations between products of operators whose conformal weights differ by half-integers. It is shown, for tree-level Einstein–Yang–Mills theory, that these recursion relations are so constraining that they completely fix the leading celestial OPE coefficients in terms of the Euler beta function. The poles in the beta functions are associated with conformally soft currents.

In the context of stationary bifurcation problems admitting a symmetry, this paper is focused on the key notion of Fixed Subspace (FS), and provides a review of some applications aimed at detecting bifurcating solutions in various situations. We start recalling, in its commonly used simplified version, the old Equivariant Bifurcation Lemma (EBL), where the FS is one-dimensional; then we provide a first generalization in a typical case of non-semisimple critical eigenvalues, where the presence of the symmetry produces a non-trivial situation. Next, we consider the case of FSs of dimension $>1$ in very different contexts. First, relying on the topological index theory and in particular on the Krasnosel’skii theorem, we provide a largely applicable statement of an extension of the EBL. Second, we propose a completely different and new application which combines symmetry properties with the notion of stability of bifurcating solutions. We also provide some simple examples, constructed ad hoc to illustrate the various situations.

In this work, we show convergence of the $N$-particle bosonic Schrödinger equation towards the Hartree equation. Hereby, we extend the results of [I. Anapolitanos and M. Hott, A simple proof of convergence to the Hartree dynamics in Sobolev trace norms, J. Math. Phys. 57(12) (2016) 122108; I. Anapolitanos, M. Hott and D. Hundertmark, Derivation of the Hartree equation for compound Bose gases in the mean field limit, Rev. Math. Phys. 29(07) (2017) 1750022]. We first consider the semi-relativistic Hartree equation in the defocusing and the focusing cases. We show that Pickl’s projection method [P. Pickl, Derivation of the time dependent Gross–Pitaevskii equation without positivity condition on the interaction, J. Statist. Phys. 140(1) (2010) 76–89; P. Pickl, A simple derivation of mean field limits for quantum systems, Lett. Math. Phys. 97(2) (2011) 151–164; P. Pickl, Derivation of the time dependent Gross–Pitaevskii equation with external fields, Rev. Math. Phys. 27(1) (2015) 1550003], can be adapted to this problem. Next, we extend this result to the case of fractional Hartree equations with potentials that are more singular than the Coulomb potential. Finally, in the non-relativistic case, we derive the Hartree equation assuming only $L^2$ initial data for potentials with a quantitative bound on the convergence rate.

In this paper, we extend the AKSZ formulation of the Poisson sigma model to more general target spaces, and we develop the general theory of graded geometry for poly-symplectic and poly-Poisson structures. In particular, we prove a Schwarz-type theorem and transgression for graded poly-symplectic structures, recovering the action functional and the poly-symplectic structure of the reduced phase space of the poly-Poisson sigma model, from the AKSZ construction.

We give a covariant realization of the doubled sigma-model formulation of duality-symmetric string theory within the general framework of para-Hermitian geometry. We define a notion of generalized metric on a para-Hermitian manifold and discuss its relation to Born geometry. We show that a Born geometry uniquely defines a worldsheet sigma-model with a para-Hermitian target space, and we describe its Lie algebroid gauging as a means of recovering the conventional sigma-model description of a physical string background as the leaf space of a foliated para-Hermitian manifold. Applying the Kotov–Strobl gauging leads to a generalized notion of T-duality when combined with transformations that act on Born geometries. We obtain a geometric interpretation of the self-duality constraint that halves the degrees of freedom in doubled sigma-models, and we give geometric characterizations of non-geometric string backgrounds in this setting. We illustrate our formalism with detailed worldsheet descriptions of closed string phase spaces, of doubled groups where our notion of generalized T-duality includes non-abelian T-duality, and of doubled nilmanifolds.