We extend authors' prior results on optimal regularity and Uhlenbeck compactness for affine connections to general connections on vector bundles. This is accomplished by deriving a vector bundle version of the RTequations, and establishing a new existence theory for these equations. These new RT-equations, non-invariant elliptic equations, provide the gauge transformations which transform the fibre component of a non-optimal connection to optimal regularity, i.e., the connection is one derivative more regular than its curvature in L p . The existence theory handles curvature regularity all the way down to, but not including, L 1 . Taken together with the affine case, our results extend optimal regularity of Kazden-DeTurck and the compactness theorem of Uhlenbeck, applicable to Riemannian geometry and compact gauge groups, to general connections on vector bundles over non-Riemannian manifolds, allowing for both compact and non-compact gauge groups. In particular, this extends optimal regularity and Uhlenbeck compactness to Yang-Mills connections on vector bundles over Lorentzian manifolds as base space, the setting of General Relativity.

We investigate the future asymptotics of spatially homogeneous space-times with a positive cosmological constant by using and further developing geometric conformal methods in General Relativity. For a large class of source fields, including fluids with anisotropic stress, we prove that the space-times are future asymptotically simple and geometrically conformally regular. We use that result in order to show the global conformal regularity of the Einstein-Maxwell system as well as the Einsteinradiation, Einstein-dust, massless Einstein-Vlasov and particular Einstein-scalar field systems for Bianchi space-times. Taking into account previous results, this implies the future non-linear stability of some of those space-times in the sense that, for small perturbations, the space-times approach locally the de Sitter solution asymptotically in time. This extends some cosmic no-hair theorems to almost spatially homogeneous space-times. However, we find that the conformal Einstein field equations preserve the Bianchi type even at conformal infinity, so the resulting asymptotic space-times have conformal hair.

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One of the main applications of resurgence in physics is the decoding of nonperturbative effects through large order relations. These relations connect perturbative asymptotic expansions of observables to expansions around other saddle points. Together, this data is unified in transseries that describe the nonperturbative structure. It is known that large order relations themselves also take the form of transseries. We study these large order transseries, uncover an interesting underlying geometry that we call the 'Borel cylinder', and show that large order transseries in turn are resurgent -that is: their nonperturbative sectors 'know about each other' through Borel residues that are essentially equal to those of the original transseries. We show that with an appropriate resummation prescription, large order relations are often exact: they can be used to exactly compute perturbative coefficients -not just their large order growth. Finally, we argue that Stokes phenomenon plays an important role for large order relations, for example if we want to extend the discrete index of the perturbative coefficients to arbitrary complex values.

We prove that the supersymmetric deformed CP 1 sigma model (the generalization of the Fateev-Onofri-Zamolodchikov model) admits an equivalent description as a generalized Gross-Neveu model. This formalism is useful for the study of renormalization properties and particularly for calculation of the one-and two-loop β-function. We show that in the UV the superdeformed model flows to the super-Thirring CFT, for which we also develop a superspace approach. It is then demonstrated that the super-Thirring model is equivalent to a sigma model with the cylinder R ˆS1 target space by an explicit computation of the correlation functions on both sides. Apart from that, we observe that the original model has another interesting conformal limit, given by the supercigar model, which as well could be described in the Gross-Neveu approach.

We consider the time evolution of the out-of-time-ordered correlator (OTOC) of two general observables A and B in a mean field chaotic quantum system described by a random Wigner matrix as its Hamiltonian. We rigorously identify three time regimes separated by the physically relevant scrambling and relaxation times. The main feature of our analysis is that we express the error terms in the optimal Schatten (tracial) norms of the observables, allowing us to track the exact dependence of the errors on their rank. In particular, for significantly overlapping observables with low rank the OTOC is shown to exhibit a significant local maximum at the scrambling time, a feature that may not have been noticed in the physics literature before. Our main tool is a novel multi-resolvent local law with Schatten norms that unifies and improves previous local laws involving either the much cruder operator norm (cf. [10]) or the ).

We study the notion of a causal time-evolution of a conserved nonlocal\nphysical quantity in a globally hyperbolic spacetime $\\mathcal{M}$. The role of\nthe `global time' is played by a chosen Cauchy temporal function $\\mathcal{T}$,\nwhereas the instantaneous configurations of the nonlocal quantity are modeled\nby probability measures $\\mu_t$ supported on the corresponding time slices\n$\\mathcal{T}^{-1}(t)$. We show that the causality of such an evolution can be\nexpressed in three equivalent ways: (i) via the causal precedence relation\n$\\preceq$ extended to probability measures, (ii) with the help of a probability\nmeasure $\\sigma$ on the space of future-directed continuous causal curves\nendowed with the compact-open topology and (iii) through a causal vector field\n$X$ of $L^\\infty_{\\textrm{loc}}$-regularity, with which the map $t \\mapsto\n\\mu_t$ satisfies the continuity equation in the distributional sense. In the\ncourse of the proof we find that the compact-open topology is sensitive to the\ndifferential properties of the causal curves, being equal to the topology\ninduced from a suitable $H^1_{\\textrm{loc}}$-Sobolev space. This enables us to\nconstruct $X$ as a vector field in a sense `tangent' to $\\sigma$. In addition,\nwe discuss the general covariance of descriptions (i)-(iii), unraveling the\ngeometrical, observer-independent notions behind them.\n

We construct a large family of quantum mechanical systems that give rise to an emergent type III 1 von Neumann algebra in the large N limit. Their partition functions are matrix integrals that appear in the study of various gauge theories. We calculate the real-time, finite temperature correlation functions in these systems and show that they are described by an emergent type III 1 von Neumann algebra at large N . The spectral density underlying this algebra is computed in closed form in terms of the eigenvalue density of a discrete matrix model. Furthermore, we explain how to systematically promote these theories to systems with a Hagedorn transition, and show that a type III 1 algebra only emerges above the Hagedorn temperature. Finally, we empirically observe in examples a correspondence between the space of states of the quantum mechanics and Calabi-Yau manifolds.

The boundary structure of 3 + 1-dimensional gravity (in the Palatini-Cartan formalism) coupled to to gauge (Yang-Mills) and matter (scalar and spinorial) fields is described through the use of the Kijowski-Tulczijew construction. In particular, the reduced phase space is obtained as the reduction of a symplectic space by some first class constraints and a cohomological description (BFV) of it is presented.