In this note, we present some results on the convergence of Nekrasov partition functions as power series in the instanton counting parameter. We focus on U(N) $${\mathcal N}=2$$ gauge theories in four dimensions with matter in the adjoint and in the fundamental representations of the gauge group, respectively, and find rigorous lower bounds for the convergence radius in the two...
We prove global stability for the charged scalar field system on a background spacetime, which is close to $$1+3$$ -dimensional Minkowski space and whose outward light cones converge to those for the Schwarzs-child metric at null infinity. The key technique to this proof is the use of a modified null frame, depending only on the mass M of the metric, which captures the asymptotic...
We discuss the scaling of the effective action for the interacting scalar quantum field theory on generic spacetimes with Lorentzian signature and in a generic state (including vacuum and thermal states, if they exist). This is done constructing a flow equation, which is very close to the renown Wetterich equation, by means of techniques recently developed in the realm of...
This paper constructs in the framework of algebraic quantum field theory (AQFT) the linear Chern–Simons/Wess–Zumino–Witten system on a class of 3-manifolds M whose boundary $$\partial M$$ is endowed with a Lorentzian metric. It is proven that this AQFT is equivalent to a dimensionally reduced AQFT on a 2-dimensional manifold B, whose restriction to the 1-dimensional boundary...
In this article, we study stochastic homogenization of non-homogeneous Gaussian free fields $$\Xi ^{g,\textbf{a}} $$ and bi-Laplacian fields $$\Xi ^{b,\textbf{a}}$$ . They can be characterized as follows: for $$f=\delta $$ the solution u of $$\nabla \cdot \textbf{a} \nabla u =f$$ , $$\textbf{a}$$ is a uniformly elliptic random environment, is the covariance of $$\Xi ^{g,\textbf{a...
Motivated by the study of the Kahan–Hirota–Kimura discretisation of the Euler top, we characterise the growth and integrability properties of a collection of elements in the Cremona group of a complex projective 3-space using techniques from algebraic geometry. This collection consists of maps obtained by composing the standard Cremona transformation $${{\,\textrm{c}\,}}_3\in...
We develop a string-net construction for the (2,1)-dimensional part of a G-equivariant three-dimensional topological field theory based on a G-graded spherical fusion category. In this construction, a G-equivariant generalization of the Ptolemy groupoid enters. We compute the associated cylinder categories and show that, as expected, the model is closely related to the G...
We construct a master dynamical system on a $${\text {U}}(n)$$ quasi-Poisson manifold, $${\mathcal {M}}_d$$ , built from the double $${\text {U}}(n) \times {\text {U}}(n)$$ and $$d\ge 2$$ open balls in $$\mathbb {C}^n$$ , whose quasi-Poisson structures are obtained from $$T^* \mathbb {R}^n$$ by exponentiation. A pencil of quasi-Poisson bivectors $$P_{\underline{z}}$$ is defined...
We prove that any smooth vacuum spacetime containing a compact Cauchy horizon with surface gravity that can be normalised to a nonzero constant admits a Killing vector field. This proves a conjecture by Moncrief and Isenberg from 1983 under the assumption on the surface gravity and generalises previous results due to Moncrief–Isenberg and Friedrich–Rácz–Wald, where the generators...
As recently proved in generality by Hedenmalm and Wennman, it is a universal behavior of complex random normal matrix models that one finds a complementary error function behavior at the boundary (also called edge) of the droplet as the matrix size increases. Such behavior is seen both in the density of the eigenvalues and the correlation kernel, where the Faddeeva plasma kernel...
We use a novel physical space method to prove relatively non-degenerate integrated energy estimates for the wave equation on subextremal Schwarzschild–de Sitter spacetimes with parameters $$(M,\Lambda )$$ . These are integrated decay statements whose bulk energy density, though degenerate at highest order, is everywhere comparable to the energy density of the boundary fluxes. As...
We consider transition amplitudes in the coloured simplicial Boulatov model for three-dimensional Riemannian quantum gravity. First, we discuss aspects of the topology of coloured graphs with non-empty boundaries. Using a modification of the standard rooting procedure of coloured tensor models, we then write transition amplitudes systematically as topological expansions. We...
This paper deals with several issues concerning the algebraic quantization of the real Proca field in a globally hyperbolic spacetime and the definition and existence of Hadamard states for that field. In particular, extending previous work, we construct the so-called Møller $$*$$ -isomorphism between the algebras of Proca observables on paracausally related spacetimes, proving...
Following Nag–Sullivan, we study the representation of the group $$\textrm{Diff}^+(S^1)$$ of diffeomorphisms of the circle on the Hilbert space of holomorphic functions. Conformal welding provides triangular decompositions for the corresponding symplectic transformations. We apply Berezin formalism and lift this decomposition to operators acting on the Fock space. This lift...
The classical Buscher rules d escribe T-duality for metrics and B-fields in a topologically trivial setting. On the other hand, topological T-duality addresses aspects of non-trivial topology while neglecting metrics and B-fields. In this article, we develop a new unifying framework for both aspects.
The relative entropy of entanglement $$E_R$$ is defined as the distance of a multipartite quantum state from the set of separable states as measured by the quantum relative entropy. We show that this optimisation is always achieved, i.e. any state admits a closest separable state, even in infinite dimensions; also, $$E_R$$ is everywhere lower semi-continuous. We use this to...
Exponentially expanding space–times play a central role in contemporary cosmology, most importantly in the theory of inflation and in the dark energy driven expansion in the late universe. In this work, we give a complete list of de Sitter solutions of the semiclassical Einstein equation (SCE), where classical gravity is coupled to the expected value of a renormalized stress...
In the TQFT formalism of Moore–Tachikawa for describing Higgs branches of theories of class $${\mathcal {S}}$$ , the space associated to the unpunctured sphere in type $${{\mathfrak {g}}}$$ is the universal centraliser $${\mathfrak {Z}}_G$$ , where $${{\mathfrak {g}}}=Lie(G)$$ . In more physical terms, this space arises as the Coulomb branch of pure $${\mathcal {N}}=4$$ gauge...
Obtaining rigorous and general results about the non-equilibrium dynamics of extended many-body systems is a difficult task. In quantum lattice models with short-range interactions, the Lieb–Robinson bound tells us that the spatial extent of operators grows at most linearly in time. But what happens within this light-cone? We discuss rigorous results on ergodicity and the...
We present a numerical approximation scheme for the Tomita–Takesaki modular operator of local subalgebras in linear quantum fields, working at one-particle level. This is applied to the local subspaces for double cones in the vacuum sector of a massive scalar free field in $$(1+1)$$ - and $$(3+1)$$ -dimensional Minkowski spacetime, using a discretization of time-0 data in...
We present a general construction of integrable degenerate $$\mathcal {E}$$ -models on a 2d manifold $$\Sigma $$ using the formalism of Costello and Yamazaki based on 4d Chern–Simons theory on $$\Sigma \times {\mathbb {C}}{P}^1$$ . We begin with a physically motivated review of the mathematical results of Benini et al. (Commun Math Phys 389(3):1417–1443, 2022. https://doi.org/10...
It was conjectured by Herman that an analytic Lagrangian Diophantine quasi-periodic torus $${\mathcal {T}}_0$$ , invariant by a real-analytic Hamiltonian system, is always accumulated by a set of positive Lebesgue measure of other Lagrangian Diophantine quasi-periodic invariant tori. While the conjecture is still open, we will prove the following weaker statement: there exists an...
We study the fractional Hardy inequality on the integers. We prove the optimality of the Hardy weight and hence affirmatively answer the question of sharpness of the constant.
Consider a quantum cat map M associated with a matrix $$A\in {{\,\textrm{Sp}\,}}(2n,{\mathbb {Z}})$$ , which is a common toy model in quantum chaos. We show that the mass of eigenfunctions of M on any nonempty open set in the position–frequency space satisfies a lower bound which is uniform in the semiclassical limit, under two assumptions: (1) there is a unique simple...
We address a recent conjecture stated by Z. Van Herstraeten and N. J. Cerf. They claim that the Shannon entropy for positive Wigner functions is bounded below by a positive constant, which can be attained only by Gaussian pure states. We introduce an alternative definition of entropy for all absolutely integrable Wigner functions, which is the Shannon entropy for positive Wigner...