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Abstract: Abstract We continue the study of the ultraviolet problem for QED in \(d=3\) using Balaban’s formulation of the renormalization group. The model is defined on a fine toroidal lattice and we seek control as the lattice spacing goes to zero. Drawing on earlier papers in the series the renormalization group flow is completely controlled for weak coupling. The main result is an ultraviolet stability bound in a fixed finite volume. PubDate: 2022-06-01

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Abstract: Abstract The ergodic properties of the randomly forced Navier–Stokes system have been extensively studied in the literature during the last two decades. The problem has always been considered in bounded domains, in order to have, for example, suitable spectral properties for the Stokes operator, to ensure some compactness properties for the resolving operator of the system and the associated functional spaces, etc. In the present paper, we consider the Navier–Stokes system in an unbounded domain satisfying the Poincaré inequality. Assuming that the system is perturbed by a bounded non-degenerate noise, we establish uniqueness of stationary measure and exponential mixing in the dual-Lipschitz metric. The proof is carried out by developing the controllability approach of the papers Kuksin et al. (Geom Funct Anal 30(1):126–187, 2020) and Shirikyan (J Eur Math Soc, 2020) and using the asymptotic compactness of the dynamics. PubDate: 2022-06-01

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Abstract: Abstract This paper proposes a refinement of the usual concept of algebraic quantum field theories (AQFTs) to theories that are smooth in the sense that they assign to every smooth family of spacetimes a smooth family of observable algebras. Using stacks of categories, this proposal is realized concretely for the simplest case of 1-dimensional spacetimes, leading to a stack of smooth 1-dimensional AQFTs. Concrete examples of smooth AQFTs, of smooth families of smooth AQFTs and of equivariant smooth AQFTs are constructed. The main open problems that arise in upgrading this approach to higher dimensions and gauge theories are identified and discussed. PubDate: 2022-06-01

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Abstract: Abstract We present new analytic results on black hole perturbation theory. Our results are based on a novel relation to four-dimensional \({\mathcal {N}}=2\) supersymmetric gauge theories. We propose an exact version of Bohr-Sommerfeld quantization conditions on quasinormal mode frequencies in terms of the Nekrasov partition function in a particular phase of the \(\Omega \) -background. Our quantization conditions also enable us to find exact expressions of eigenvalues of spin-weighted spheroidal harmonics. We test the validity of our conjecture by comparing against known numerical results for Kerr black holes as well as for Schwarzschild black holes. Some extensions are also discussed. PubDate: 2022-06-01

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Abstract: Abstract We consider a two-dimensional determinantal Coulomb gas confined by a class of radial external potentials. In the limit of large number of particles, the Coulomb particles tend to accumulate on a compact set S, the support of the equilibrium measure associated with a given external potential. If the particles are forced to be completely confined in a disk \({\mathcal {D}}\) due to a hard-wall constraint on \({\partial }{\mathcal {D}}\subset {\text {Int}}S\) , then the equilibrium configuration changes and the equilibrium measure acquires a singular component at the hard wall. We study the local statistics of Coulomb particles in the vicinity of the hard wall and prove that their local correlations are expressed in terms of “Laplace-type” integrals, which appear in the context of truncated unitary matrices in the regime of weak non-unitarity. PubDate: 2022-06-01

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Abstract: Abstract We construct a quaternionic Kähler manifold from a conical special Kähler manifold with a certain type of mutually local variation of BPS structures. We give global and local explicit formulas for the quaternionic Kähler metric and specify under which conditions it is positive-definite. Locally, the metric is a deformation of the 1-loop corrected Ferrara–Sabharwal metric obtained via the supergravity c-map. The type of quaternionic Kähler metrics we obtain is related to work in the physics literature by S. Alexandrov and S. Banerjee, where they discuss the hypermultiplet moduli space metric of type IIA string theory, with mutually local D-instanton corrections. PubDate: 2022-06-01

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Abstract: Abstract Two decades ago, Lieb and Loss (Self-energy of electrons in non-perturbative QED. Preprint arXiv:math-ph/9908020 and mp-arc #99–305, 1999) approximated the ground state energy of a free, nonrelativistic electron coupled to the quantized radiation field by the infimum \(E_{\alpha , \Lambda }\) of all expectation values \(\langle \phi _{el} \otimes \psi _{ph} H_{\alpha , \Lambda } (\phi _{el} \otimes \psi _{ph}) \rangle \) , where \(H_{\alpha , \Lambda }\) is the corresponding Hamiltonian with fine structure constant \(\alpha >0\) and ultraviolet cutoff \(\Lambda < \infty \) , and \(\phi _{el}\) and \(\psi _{ph}\) are normalized electron and photon wave functions, respectively. Lieb and Loss showed that \(c \alpha ^{1/2} \Lambda ^{3/2} \le E_{\alpha , \Lambda } \le c^{-1} \alpha ^{2/7} \Lambda ^{12/7}\) for some constant \(c >0\) . In the present paper, we prove the existence of a constant \(C < \infty \) , such that $$\begin{aligned} \bigg \frac{E_{\alpha , \Lambda }}{F_1 \, \alpha ^{2/7} \, \Lambda ^{12/7}} - 1 \bigg \ \le \ C \, \alpha ^{4/105} \, \Lambda ^{-4/105} \end{aligned}$$ holds true, where \(F_1 >0\) is an explicit universal number. This result shows that Lieb and Loss’ upper bound is actually sharp and gives the asymptotics of \(E_{\alpha , \Lambda }\) uniformly in the limit \(\alpha \rightarrow 0\) and in the ultraviolet limit \(\Lambda \rightarrow \infty \) . PubDate: 2022-06-01

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Abstract: Abstract We revisit the Riemann–Hilbert problem determined by Donaldson–Thomas invariants for the resolved conifold and for other small crepant resolutions. While this problem can be recast as a system of TBA-type equations in the conformal limit, solutions are ill-defined due to divergences in the sum over infinite trajectories in the spectrum of D2–D0-brane bound states. We explore various prescriptions to make the sum well defined, show that one of them reproduces the existing solution in the literature, and identify an alternative solution which is better behaved in a certain limit. Furthermore, we show that a suitable asymptotic expansion of the \(\tau \) function reproduces the genus expansion of the topological string partition function for any small crepant resolution. As a by-product, we conjecture new integral representations for the triple sine function, similar to Woronowicz integral representation for Faddeev’s quantum dilogarithm. PubDate: 2022-06-01

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Abstract: Abstract We prove that the quasi-periodic Schrödinger operator with a finitely differentiable potential has purely absolutely continuous spectrum for all phases if the frequency is Diophantine and the potential is sufficiently small in the corresponding \(C^k\) topology. This extends the work of Eliasson [19] and Avila–Jitomirskaya [5] from the analytic topology to the finitely differentiable one which is much broader, revealing the interesting phenomenon that small oscillation of the potential leads to both zero Lyapunov exponent in the whole spectrum and purely absolutely continuous spectrum. Our result is based on a refined quantitative \(C^{k,k_0}\) almost reducibility theorem which only requires a quite low initial regularity “ \(k>14\tau +2\) ” and much of the regularity “ \(k_0\le k-2\tau -2\) ” is conserved in the end, where \(\tau \) is the Diophantine constant of the frequency. PubDate: 2022-05-23

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Abstract: Abstract The main result of this paper is a complete proof of a new Lieb–Thirring-type inequality for Jacobi matrices originally conjectured by Hundertmark and Simon. In particular, it is proved that the estimate on the sum of eigenvalues does not depend on the off-diagonal terms as long as they are smaller than their asymptotic value. An interesting feature of the proof is that it employs a technique originally used by Hundertmark–Laptev–Weidl concerning sums of singular values for compact operators. This technique seems to be novel in the context of Jacobi matrices. PubDate: 2022-05-21

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Abstract: Abstract We discuss a set of heterotic and type II string theory compactifications to \(1+1\) dimensions that are characterized by factorized internal worldsheet CFTs of the form \(V_1\otimes \bar{V}_2\) , where \(V_1, V_2\) are self-dual (super) vertex operator algebras. In the cases with spacetime supersymmetry, we show that the BPS states form a module for a Borcherds–Kac–Moody (BKM) (super)algebra, and we prove that for each model the BKM (super)algebra is a symmetry of genus zero BPS string amplitudes. We compute the supersymmetric indices of these models using both Hamiltonian and path integral formalisms. The path integrals are manifestly automorphic forms closely related to the Borcherds–Weyl–Kac denominator. Along the way, we comment on various subtleties inherent to these low-dimensional string compactifications. PubDate: 2022-05-16

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Abstract: Abstract We derive a precise asymptotic formula for the density of the small singular values of the real Ginibre matrix ensemble shifted by a complex parameter z as the dimension tends to infinity. For z away from the real axis the formula coincides with that for the complex Ginibre ensemble we derived earlier in Cipolloni et al. (Prob Math Phys 1:101–146, 2020). On the level of the one-point function of the low lying singular values we thus confirm the transition from real to complex Ginibre ensembles as the shift parameter z becomes genuinely complex; the analogous phenomenon has been well known for eigenvalues. We use the superbosonization formula (Littelmann et al. in Comm Math Phys 283:343–395, 2008) in a regime where the main contribution comes from a three dimensional saddle manifold. PubDate: 2022-05-16

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Abstract: Abstract We show that one-dimensional Schrödinger operators whose potentials arise by randomly concatenating words from an underlying set exhibit exponential dynamical localization (EDL) on any compact set which trivially intersects a finite set of critical energies. We do so by first giving a new proof of spectral localization for such operators and then showing that once one has the existence of a complete orthonormal basis of eigenfunctions (with probability one), the same estimates used to prove it naturally lead to a proof of the aforementioned EDL result. The EDL statements provide new localization results for several classes of random Schrödinger operators including random polymer models and generalized Anderson models. PubDate: 2022-05-13

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Abstract: Abstract Using methods of microlocal analysis, we prove that the regularization of divergent amplitudes stays a pure ultraviolet problem in string-localized field theories, despite the weaker localization. Thus, power counting does not lose its significance as an indicator for renormalizability. It also follows that standard techniques can be used to regularize divergent amplitudes in string-localized field theories. PubDate: 2022-05-11

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Abstract: Abstract We establish Lieb–Thirring type estimates for the Schrödinger operator with a singular measure serving as potential. PubDate: 2022-05-08

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Abstract: Abstract For suitable charges of the constituents, the phase space of multi-centered BPS black holes in \(\mathcal {N}=2\) four-dimensional supergravity famously exhibits scaling regions where the distances between the centers can be made arbitrarily small, so that the bound state becomes indistinguishable from a single-centered black hole. In this note, we establish necessary conditions on the Dirac product of charges for the existence of such regions for any number of centers, generalizing the standard triangular inequalities in the three-center case. Furthermore, we show the same conditions are necessary for the existence of multi-centered solutions at the attractor point. We prove that similar conditions are also necessary for the existence of self-stable Abelian representations of the corresponding quiver, as suggested by the duality between the Coulomb and Higgs branches of supersymmetric quantum mechanics. PubDate: 2022-05-08

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Abstract: Abstract We introduce and study a class of entanglement criteria based on the idea of applying local contractions to an input multipartite state, and then computing the projective tensor norm of the output. More precisely, we apply to a mixed quantum state a tensor product of contractions from the Schatten class \(S_1\) to the Euclidean space \(\ell _2\) , which we call entanglement testers. We analyze the performance of this type of criteria on bipartite and multipartite systems, for general pure and mixed quantum states, as well as on some important classes of symmetric quantum states. We also show that previously studied entanglement criteria, such as the realignment and the SIC POVM criteria, can be viewed inside this framework. This allows us to answer in the positive two conjectures of Shang, Asadian, Zhu, and Gühne by deriving systematic relations between the performance of these two criteria. PubDate: 2022-05-07

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Abstract: Abstract The Fokker–Planck equation is a partial differential equation which is a key ingredient in many models in physics. This paper aims to obtain a quantum counterpart of Fokker–Planck dynamics, as a means to describing quantum Fokker–Planck dynamics. Given that relevant models relate to the description of large systems, the quantization of the Fokker–Planck equation should be done in a manner that respects this fact and is therefore carried out within the setting of non-commutative analysis based on general von Neumann algebras. Within this framework, we present a quantization of the generalized Laplace operator and then go on to incorporate a potential term conditioned to non-commutative analysis. In closing, we then construct and examine the asymptotic behaviour of the corresponding Markov semigroups. We also present a non-commutative Csiszar–Kullback inequality formulated in terms of a notion of relative entropy and show that for more general systems, good behaviour with respect to this notion of entropy ensures similar asymptotic behaviour of the relevant dynamics. PubDate: 2022-05-01

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Abstract: Abstract Let \([\lambda ,\mu ]\) be an interval contained in a spectral gap of a periodic Schrödinger operator H. Consider \(H(\alpha )=H-\alpha V\) where V is a fast decaying positive function. We study the asymptotic behavior of the number of eigenvalues of \(H(\alpha )\) in \([\lambda ,\mu ]\) as \(\alpha \rightarrow \infty \) . PubDate: 2022-05-01

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Abstract: Abstract We continue the study of fuzzy geometries inside Connes’ spectral formalism and their relation to multimatrix models. In this companion paper to Pérez-Sánchez (Ann Henri Poincaré 22:3095–3148, 2021, arXiv:2007.10914), we propose a gauge theory setting based on noncommutative geometry, which—just as the traditional formulation in terms of almost-commutative manifolds—has the ability to also accommodate a Higgs field. However, in contrast to ‘almost-commutative manifolds’, the present framework, which we call gauge matrix spectral triples, employs only finite-dimensional algebras. In a path-integral quantization approach to the Spectral Action, this allows to state Yang–Mills–Higgs theory (on four-dimensional Euclidean fuzzy space) as an explicit random multimatrix model obtained here, whose matrix fields exactly mirror those of the Yang–Mills–Higgs theory on a smooth manifold. PubDate: 2022-04-23