Abstract: We reexamine the notions of generalized Ricci tensor and scalar curvature on a general Courant algebroid, reformulate them using objects natural w.r.t. pull-backs and reductions, and obtain them from the variation of a natural action functional. This allows us to prove, in a very general setup, the compatibility of the Poisson–Lie T-duality with the renormalization group flow and with string background equations. We thus extend the known results to a much wider class of dualities, including the cases with gauging (so called dressing cosets, or equivariant Poisson–Lie T-duality). As an illustration, we use the formalism to provide new classes of solutions of modified supergravity equations on symmetric spaces. PubDate: 2020-03-26

Abstract: In this paper lower bounds are obtained for quasi-local masses in terms of charge, angular momentum, and horizon area. In particular we treat three quasi-local masses based on a Hamiltonian approach, namely the Brown-York, Liu-Yau, and Wang-Yau masses. The geometric inequalities are motivated by analogous results for the ADM mass. They may be interpreted as localized versions of these inequalities, and are also closely tied to the conjectured Bekenstein bounds for entropy of macroscopic bodies. In addition, we give a new proof of the positivity property for the Wang-Yau mass which is used to remove the spin condition in higher dimensions. Furthermore, we generalize a recent result of Lu and Miao to obtain a localized version of the Penrose inequality for the static Wang-Yau mass. PubDate: 2020-03-25

Abstract: In this paper we study the extension of Painlevé/gauge theory correspondence to circular quivers by focusing on the special case of SU(2) \({\mathcal {N}}=2^*\) theory. We show that the Nekrasov–Okounkov partition function of this gauge theory provides an explicit combinatorial expression and a Fredholm determinant formula for the tau-function describing isomonodromic deformations of \(SL_2\) flat connections on the one-punctured torus. This is achieved by reformulating the Riemann–Hilbert problem associated to the latter in terms of chiral conformal blocks of a free-fermionic algebra. This viewpoint provides the exact solution of the renormalization group flow of the SU(2) \({\mathcal {N}}=2^*\) theory on self-dual \(\Omega \)-background and, in the Seiberg–Witten limit, an elegant relation between the IR and UV gauge couplings. PubDate: 2020-03-24

Abstract: The closed string theory minimal-area problem asks for the conformal metric of least area on a Riemann surface with the condition that all non-contractible closed curves have length greater than or equal to an arbitrary constant that can be set to one. Through every point in such a metric there is a geodesic that saturates the length condition, and the saturating geodesics in a given homotopy class form a band. The extremal metric is unknown when bands of geodesics cross, as it happens for surfaces of non-zero genus. We use recently proposed convex programs to numerically find the minimal-area metric on the square torus with a square boundary, for various sizes of the boundary. For large enough boundary the problem is equivalent to the “Swiss cross” challenge posed by Strebel. We find that the metric is positively curved in the two-band region and flat in the single-band regions. For small boundary the metric develops a third band of geodesics wrapping around it, and has both regions of positive and of negative curvature. This surface can be completed to provide the minimal-area metric on a once-punctured torus, representing a closed-string tadpole diagram. PubDate: 2020-03-24

Abstract: We study quantum chains whose Hamiltonians are perturbations by bounded interactions of short range of a Hamiltonian that does not couple the degrees of freedom located at different sites of the chain and has a strictly positive energy gap above its ground-state energy. We prove that, for small values of a coupling constant, the spectral gap of the perturbed Hamiltonian above its ground-state energy is bounded from below by a positive constant uniformly in the length of the chain. In our proof we use a novel method based on local Lie-Schwinger conjugations of the Hamiltonians associated with connected subsets of the chain. PubDate: 2020-03-20

Abstract: We study the extremal process associated with the Discrete Gaussian Free Field on the square lattice and elucidate how the conformal symmetries manifest themselves in the scaling limit. Specifically, we prove that the joint process of spatial positions (x) and centered values (h) of the extreme local maxima in lattice versions of a bounded domain \(D\subset {\mathbb {C}}\) converges, as the lattice spacing tends to zero, to a Poisson point process with intensity measure \(Z^D(\mathrm{d}x)\otimes \mathrm{e}^{-\alpha h}\mathrm{d}h\), where \(\alpha \) is a constant and \(Z^D\) is a random a.s.-finite measure on D. The random measures \(\{Z^D\}\) are naturally interrelated; restrictions to subdomains are governed by a Gibbs–Markov property and images under analytic bijections f by the transformation rule \((Z^{f(D)}\circ f)(\mathrm{d}x)\,\overset{\mathrm{law}}{=}\, f'(x) ^4\, Z^D(\mathrm{d}x)\). Conditions are given that determine the laws of these measures uniquely. These identify \(Z^D\) with the critical Liouville Quantum Gravity associated with the Continuum Gaussian Free Field. PubDate: 2020-03-19

Abstract: Gauge theories can often be formulated in different but physically equivalent ways, a concept referred to as duality. Using a formalism based on graded geometry, we provide a unified treatment of all parent theories for different types of standard and exotic dualizations. Our approach is based on treating tensor fields as functions of a certain degree on graded supermanifolds equipped with a suitable number of odd coordinates. We present a universal two-parameter first order action for standard and exotic electric/magnetic dualizations and prove in full generality that it yields two dual second order theories with the desired field content and dynamics. Upon choice of parameters, the parent theory reproduces (i) the standard and exotic duals for p-forms and (ii) the standard and double duals for (p, 1) bipartite tensor fields, such as the linearized graviton and the Curtright field. Moreover, we discuss how deformations related to codimension-1 branes are included in the parent theory. PubDate: 2020-03-18

Abstract: Four quantities are fundamental in homogenization of elliptic systems in divergence form and in its applications: the field and the flux of the solution operator (applied to a general deterministic right-hand side), and the field and the flux of the corrector. Homogenization is the study of the large-scale properties of these objects. In case of random coefficients, these quantities fluctuate and their fluctuations are a priori unrelated. Depending on the law of the coefficient field, and in particular on the decay of its correlations on large scales, these fluctuations may display different scalings and different limiting laws (if any). In this contribution, we identify another crucial intrinsic quantity, motivated by H-convergence, which we refer to as the homogenization commutator and is related to variational quantities first considered by Armstrong and Smart. In the simplified setting of the random conductance model, we show what we believe to be a general principle, namely that the homogenization commutator drives at leading order the fluctuations of each of the four other quantities in a strong norm in probability, which is expressed in form of a suitable two-scale expansion and reveals the pathwise structure of fluctuations in stochastic homogenization. In addition, we show that the (rescaled) homogenization commutator converges in law to a Gaussian white noise, and we analyze to which precision the covariance tensor that characterizes the latter can be extracted from the representative volume element method. This collection of results constitutes a new theory of fluctuations in stochastic homogenization that holds in any dimension and yields optimal rates. Extensions to the (non-symmetric) continuum setting are also discussed, the details of which are postponed to forthcoming works. PubDate: 2020-03-17

Abstract: Starting from the observation that a flying saucer is a nonholonomic mechanical system whose 5-dimensional configuration space is a contact manifold, we show how to enrich this space with a number of geometric structures by imposing further nonlinear restrictions on the saucer’s velocity. These restrictions define certain ‘manœuvres’ of the saucer, which we call ‘attacking,’ ‘landing,’ or ‘\(G_2\) mode’ manœuvres, and which equip its configuration space with three kinds of flat parabolic geometry in five dimensions. The attacking manœuvre corresponds to the flat Legendrean contact structure, the landing manœuvre corresponds to the flat hypersurface type CR structure with Levi form of signature (1, 1), and the most complicated \(G_2\) manœuvre corresponds to the contact Engel structure (Engel in C R Acad Sci 116:786–788, 1893; Mano et al. in The geometry of marked contact twisted cubic structures, 2018, arXiv:1809.06455) with split real form of the exceptional Lie group \(G_2\) as its symmetries. A celebrated double fibration relating the two nonequivalent flat 5-dimensional parabolic \(G_2\) geometries is used to construct a ‘\(G_2\) joystick,’ consisting of two balls of radii in ratio \(1\!:\!3\) that transforms the difficult \(G_2\) manœuvre into the pilot’s action of rolling one of joystick’s balls on the other without slipping nor twisting. PubDate: 2020-03-17

Abstract: We identify various structures on the configuration space C of a flying saucer, moving in a three-dimensional smooth manifold M. Always C is a five-dimensional contact manifold. If M has a projective structure, then C is its twistor space and is equipped with an almost contact Legendrean structure. Instead, if M has a conformal structure, then the saucer moves according to a CR structure on C. With yet another structure on M, the contact distribution in C is equipped with a cone over a twisted cubic. This defines a certain type of Cartan geometry on C (more specifically, a type of ‘parabolic geometry’) and we provide examples when this geometry is ‘flat,’ meaning that its symmetries comprise the split form of the exceptional Lie algebra \({\mathfrak {g}}_2\). PubDate: 2020-03-16

Abstract: In this paper, we study the linear inviscid damping for the linearized \(\beta \)-plane equation around shear flows. We develop a new method to give the explicit decay rate of the velocity for a class of monotone shear flows. This method is based on the space-time estimate and the vector field method in the sprit of the wave equation. For general shear flows including the Sinus flow, we also prove the linear damping by establishing the limiting absorption principle, which is based on the compactness method introduced by Wei et al. (Ann PDE 5:3, 2019). The main difficulty is that the Rayleigh–Kuo equation has more singular points due to the Coriolis effects so that the compactness argument becomes more involved and delicate. PubDate: 2020-03-14

Abstract: We consider the weakly coupled \(\phi ^4 \) theory on \(\mathbb {Z}^4 \), in a weak magnetic field h, and at the chemical potential \(\nu _c \) for which the theory is critical if \(h=0\). We prove that, as \(h\searrow 0\), the magnetization of the model behaves as \((h\log h^{-1})^{\frac{1}{3}} \), and so exhibits a logarithmic correction to mean field scaling behavior. This result is well known to physicists, but had never been proven rigorously. Our proof uses the classic construction of the critical theory by Gawedzki and Kupiainen (Commun Math Phys 99(2):197–252, 1985), and a cluster expansion with large blocks. PubDate: 2020-03-12

Abstract: We consider canonical Gibbsian ensembles of Euler point vortices on the 2-dimensional torus or in a bounded domain of \(\mathbb {R}^2\). We prove that under the Central Limit scaling of vortices intensities, and provided that the system has zero global space average in the bounded domain case (neutrality condition), the ensemble converges to the so-called energy–enstrophy Gaussian random distributions. This can be interpreted as describing Gaussian fluctuations around the mean field limit of vortices ensembles of Caglioti et al. (Commun Math Phys 143(3):501–525, 1992) and Kiessling and Wang (J Stat Phys 148(5):896–932, 2012), and it generalises the result on fluctuations of Bodineau and Guionnet (Ann Inst H Poincaré Probab Stat 35(2):205–237, 1999). The main argument consists in proving convergence of partition functions of vortices. PubDate: 2020-03-12

Abstract: For a dilute system of non-relativistic bosons interacting through a positive, radial potential v with scattering length a we prove that the ground state energy density satisfies the bound \(e(\rho ) \ge 4\pi a \rho ^2 (1- C \sqrt{\rho a^3} \,)\). PubDate: 2020-03-12

Abstract: In many contexts one encounters Hermitian operators M on a Hilbert space whose dimension is so large that it is impossible to write down all matrix entries in an orthonormal basis. How does one determine whether such M is positive semidefinite' Here we approach this problem by deriving asymptotically optimal bounds to the distance to the positive semidefinite cone in Schatten p-norm for all integer \(p\in [1,\infty )\), assuming that we know the moments \(\mathbf {tr}(M^k)\) up to a certain order \(k=1,\ldots , m\). We then provide three methods to compute these bounds and relaxations thereof: the sos polynomial method (a semidefinite program), the Handelman method (a linear program relaxation), and the Chebyshev method (a relaxation not involving any optimization). We investigate the analytical and numerical performance of these methods and present a number of example computations, partly motivated by applications to tensor networks and to the theory of free spectrahedra. PubDate: 2020-03-11

Abstract: For \(0<\gamma <2\) and \(\delta >0\), we consider the Liouville graph distance, which is the minimal number of Euclidean balls of \(\gamma \)-Liouville quantum gravity measure at most \(\delta \) whose union contains a continuous path between two endpoints. In this paper, we show that the renormalized distance is tight and thus has subsequential scaling limits at \(\delta \rightarrow 0\). In particular, we show that for all \(\delta >0\) the diameter with respect to the Liouville graph distance has the same order as the typical distance between two endpoints. PubDate: 2020-03-11

Abstract: The affine motion of two-dimensional (2d) incompressible fluids surrounded by vacuum can be reduced to a completely integrable and globally solvable Hamiltonian system of ordinary differential equations for the deformation gradient in \(\mathrm{SL}(2,\mathbb {R})\). In the case of perfect fluids, the motion is given by geodesic flow in \(\mathrm{SL}(2,\mathbb {R})\) with the Euclidean metric, while for magnetically conducting fluids (MHD), the motion is governed by a harmonic oscillator in \(\mathrm{SL}(2,\mathbb {R})\). A complete classification of the dynamics is given including rigid motions, rotating eddies with stable and unstable manifolds, and solutions with vanishing pressure. For perfect fluids, the displacement generically becomes unbounded, as \(t\rightarrow \pm \infty \). For MHD, solutions are bounded and generically quasi-periodic and recurrent. PubDate: 2020-03-11

Abstract: We study, using mean curvature flow methods, \(2+1\) dimensional cosmologies with a positive cosmological constant and matter satisfying the dominant and the strong energy conditions. If the spatial slices are compact with non-positive Euler characteristic and are initially expanding everywhere, then we prove that the spatial slices reach infinite volume, asymptotically converge on average to de Sitter and they become, almost everywhere, physically indistinguishable from de Sitter. This holds true notwithstanding the presence of initial arbitrarily-large density fluctuations and the formation of black holes. PubDate: 2020-03-11

Abstract: We construct a formal global quantization of the Poisson Sigma Model in the BV-BFV formalism using the perturbative quantization of AKSZ theories on manifolds with boundary and analyze the properties of the boundary BFV operator. Moreover, we consider mixed boundary conditions and show that they lead to quantum anomalies, i.e. to a failure of the (modified differential) Quantum Master Equation. We show that it can be restored by adding boundary terms to the action, at the price of introducing corner terms in the boundary operator. We also show that the quantum Grothendieck BFV operator on the total space of states is a differential, i.e. squares to zero, which is necessary for a well-defined BV cohomology. PubDate: 2020-03-10

Abstract: In a previous article, we introduced the first passage set (FPS) of constant level \(-a\) of the two-dimensional continuum Gaussian free field (GFF) on finitely connected domains. Informally, it is the set of points in the domain that can be connected to the boundary by a path along which the GFF is greater than or equal to \(-a\). This description can be taken as a definition of the FPS for the metric graph GFF, and in the current article, we prove that the metric graph FPS converges towards the continuum FPS in the Hausdorff distance. We also draw numerous consequences; in particular, we obtain a relatively simple proof of the fact that certain natural interfaces of the metric graph GFF converge to \(\hbox {SLE}_4\) level lines. These results improve our understanding of the continuum GFF, by strengthening its relationship with the critical Brownian loop-soup. Indeed, a new construction of the FPS using clusters of Brownian loops and excursions helps to strengthen the known GFF isomorphism theorems, and allows us to use Brownian loop-soup techniques to prove technical results on the geometry of the GFF. We also obtain a new representation of Brownian loop-soup clusters, and as a consequence, we prove that the clusters of a critical Brownian loop-soup admit a non-trivial Minkowski content in the gauge \(r\mapsto \log r ^{1/2}r^2\). PubDate: 2020-03-09