Abstract: By enforcing invariance under S-duality in type IIB string theory compactified on a Calabi–Yau threefold, we derive modular properties of the generating function of BPS degeneracies of D4–D2–D0 black holes in type IIA string theory compactified on the same space. Mathematically, these BPS degeneracies are the generalized Donaldson-Thomas invariants counting coherent sheaves with support on a divisor \({\mathcal {D}}\), at the large volume attractor point. For \({\mathcal {D}}\) irreducible, this function is closely related to the elliptic genus of the superconformal field theory obtained by wrapping M5-brane on \({\mathcal {D}}\) and is therefore known to be modular. Instead, when \({\mathcal {D}}\) is the sum of n irreducible divisors \({\mathcal {D}}_i\), we show that the generating function acquires a modular anomaly. We characterize this anomaly for arbitrary n by providing an explicit expression for a non-holomorphic modular completion in terms of generalized error functions. As a result, the generating function turns out to be a (mixed) mock modular form of depth \(n-1\). PubDate: 2019-11-16

Abstract: The first example of a quantum group was introduced by P. Kulish and N. Reshetikhin. In the paper Kulish et al. (J Soviet Math 23:2435–2441, 1983), they found a new algebra which was later called \(U_q (\mathfrak {sl}(2))\). Their example was developed independently by V. Drinfeld and M. Jimbo, which resulted in the general notion of quantum group. Later, a complimentary approach to quantum groups was developed by L. Faddeev, N. Reshetikhin, and L. Takhtajan in Faddeev et al. (Leningr Math J 1:193–225, 1990). Recently, the so-called Belavin–Drinfeld cohomology (twisted and non-twisted) have been introduced in the literature to study and classify certain families of quantum groups and Lie bialgebras. Later, the last two authors interpreted non-twisted Belavin–Drinfeld cohomology in terms of non-abelian Galois cohomology \(H^1({\mathbb {F}}, {\mathbf {H}})\) for a suitable algebraic \({\mathbb {F}}\)-group \({\mathbf {H}}\). Here \({\mathbb {F}}\) is an arbitrary field of zero characteristic. The non-twisted case is thus fully understood in terms of Galois cohomology. The twisted case has only been studied using Galois cohomology for the so-called (“standard”) Drinfeld–Jimbo structure. The aim of the present paper is to extend these results to all twisted Belavin–Drinfeld cohomology and thus, to present classification of quantum groups in terms of Galois cohomology and the so-called orders. Low dimensional cases \(\mathfrak {sl}(2)\) and \(\mathfrak {sl}(3)\) are considered in more details using a theory of cubic rings developed by B. N. Delone and D. K. Faddeev in Delone and Faddeev (The theory of irrationalities of the third degree. Translations of mathematical monographs, vol 10. AMS, Providence, 1964). Our results show that there exist yet unknown quantum groups for Lie algebras of the types \(A_n, D_{2n+1}, E_6\), not mentioned in Etingof et al. (J Am Math Soc 13:595–609, 2000). PubDate: 2019-11-14

Abstract: We study the qualitative behavior of nonlinear Dirac equations arising in quantum field theory on complete Riemannian manifolds. In particular, we derive monotonicity formulas and Liouville theorems for solutions of these equations. Finally, we extend our analysis to Dirac-harmonic maps with curvature term. PubDate: 2019-11-13

Abstract: The isospectral deformations of the Frobenius–Stickelberger–Thiele (FST) polynomials introduced in Spiridonov et al. (Commun Math Phys 272:139–165, 2007) are studied. For a specific choice of the deformation of the spectral measure, one is led to an integrable lattice (FST lattice), which is indeed an isospectral flow connected with a generalized eigenvalue problem. In the second part of the paper the spectral problem used previously in the study of the modified Camassa–Holm (mCH) peakon lattice is interpreted in terms of the FST polynomials together with the associated FST polynomials, resulting in a map from the mCH peakon lattice to a negative flow of the finite FST lattice. Furthermore, it is pointed out that the degenerate case of the finite FST lattice unexpectedly maps to the interlacing peakon ODE system associated with the two-component mCH equation studied in Chang et al. (Adv Math 299:1–35, 2016). PubDate: 2019-11-13

Abstract: We find a relation between Lagrangian Floer pairing of a symplectic manifold and Kapustin–Li pairing of the mirror Landau–Ginzburg model under localized mirror functor. They are conformally equivalent with an interesting conformal factor\((vol^{Floer}/vol)^2\), which can be described as a ratio of Lagrangian Floer volume class and classical volume class. For this purpose, we introduce B-invariant of Lagrangian Floer cohomology with values in Jacobian ring of the mirror potential function. And we prove what we call a multi-crescent Cardy identity under certain conditions, which is a generalized form of Cardy identity. As an application, we discuss the case of general toric manifold, and the relation to the work of Fukaya–Oh–Ohta–Ono and their Z-invariant. Also, we compute the conformal factor \((vol^{Floer}/vol)^2\) for the elliptic curve quotient \(\mathbb {P}^1_{3,3,3}\), which gives a modular form. PubDate: 2019-11-12

Abstract: In this paper, we investigate the small scale equidistribution properties of randomised sums of Laplacian eigenfunctions (i.e. random waves) on a compact manifold. We prove small scale expectation and variance results for random waves on all compact manifolds. Here, “small scale” refers to balls of radius \(r(\lambda )\rightarrow 0\) such that \(r/r_{\text {Planck}}\rightarrow \infty \), where \(r_{\text {Planck}}\) is the Planck scale. For balls at a larger scale (although still \(r(\lambda )\rightarrow 0\)) we also obtain estimates showing that the probability that a random wave fails to equidistribute decays exponentially with the eigenvalue. PubDate: 2019-11-11

Abstract: We describe a covariant framework to construct a globalized version for the perturbative quantization of nonlinear split AKSZ Sigma Models on manifolds with and without boundary, and show that it captures the change of the quantum state as one changes the constant map around which one perturbs. This is done by using concepts of formal geometry. Moreover, we show that the globalized quantum state can be interpreted as a closed section with respect to an operator that squares to zero. This condition is a generalization of the modified Quantum Master Equation as in the BV-BFV formalism, which we call the modified “differential” Quantum Master Equation. PubDate: 2019-11-08

Abstract: This paper investigates the relations between the Toda conformal field theories, quantum group theory and the quantisation of moduli spaces of flat connections. We use the free field representation of the \({{\mathcal {W}}}\)-algebras to define natural bases for spaces of conformal blocks of the Toda conformal field theory associated to the Lie algebra \({\mathfrak {s}}{\mathfrak {l}}_3\) on the three-punctured sphere with representations of generic type associated to the three punctures. The operator-valued monodromies of degenerate fields can be used to describe the quantisation of the moduli spaces of flat \(\mathrm {SL}(3)\)-connections. It is shown that the matrix elements of the monodromies can be expressed as Laurent polynomials of more elementary operators which have a simple definition in the free field representation. These operators are identified as quantised counterparts of natural higher rank analogs of the Fenchel–Nielsen coordinates from Teichmüller theory. Possible applications to the study of the non-Lagrangian SUSY field theories are briefly outlined. PubDate: 2019-11-07

Abstract: We consider the Wulff problem arising from the study of the Heitmann–Radin energy of N atoms sitting on a periodic lattice. Combining the sharp quantitative Wulff inequality in the continuum setting with a notion of quantitative closeness between discrete and continuum energies, we provide very short proofs of fluctuation estimates of Voronoi-type sets associated with almost minimizers of the discrete problem about the continuum limit Wulff shape. In the particular case of exact energy minimizers, we recover the well-known, sharp \(N^{3/4}\) scaling law for all considered planar lattices, as well as a sub-optimal scaling law for the cubic lattice in dimension \(d\ge 3\). PubDate: 2019-11-06

Abstract: Using ideas from Jones, lattice gauge theory and loop quantum gravity, we construct \(1+1\) -dimensional gauge theories on a spacetime cylinder. Given a separable compact group G, we construct localized time-zero fields on the spatial torus as a net of C*-algebras together with an action of the gauge group that is an infinite product of G over the dyadic rationals and, using a recent machinery of Jones, an action of Thompson’s group T as a replacement of the spatial diffeomorphism group. Adding a family of probability measures on the unitary dual of G we construct a state and obtain a net of von Neumann algebras carrying a state-preserving gauge group action. For abelian G, we provide a very explicit description of our algebras. For a single measure on the dual of G, we have a state-preserving action of Thompson’s group and semi-finite von Neumann algebras. For \(G={\mathbf {S}}\) the circle group together with a certain family of heat-kernel states providing the measures, we obtain hyperfinite type III factors with a normal faithful state providing a nontrivial time evolution via Tomita–Takesaki theory (KMS condition). In the latter case, we additionally have a non-singular action of the group of rotations with dyadic angles, as a subgroup of Thompson’s group T, for geometrically motivated choices of families of heat-kernel states. PubDate: 2019-11-04

Abstract: Recent works have shown that an instance of a Brownian surface (such as the Brownian map or Brownian disk) a.s. has a canonical conformal structure under which it is equivalent to a \(\sqrt{8/3}\) -Liouville quantum gravity (LQG) surface. In particular, Brownian motion on a Brownian surface is well-defined. The construction in these works is indirect, however, and leaves open a basic question: is Brownian motion on a Brownian surface the limit of simple random walk on increasingly fine discretizations of that surface, the way Brownian motion on \(\mathbb {R}^2\) is the \(\epsilon \rightarrow 0\) limit of simple random walk on \(\epsilon \mathbb {Z}^2\) ' We answer this question affirmatively by showing that Brownian motion on a Brownian surface is (up to time change) the \(\lambda \rightarrow \infty \) limit of simple random walk on the Voronoi tessellation induced by a Poisson point process whose intensity is \(\lambda \) times the associated area measure. Among other things, this implies that as \(\lambda \rightarrow \infty \) the Tutte embedding (a.k.a. harmonic embedding) of the discretized Brownian disk converges to the canonical conformal embedding of the continuum Brownian disk, which in turn corresponds to \(\sqrt{8/3}\) -LQG. Along the way, we obtain other independently interesting facts about conformal embeddings of Brownian surfaces, including information about the Euclidean shapes of embedded metric balls and Voronoi cells. For example, we derive moment estimates that imply, in a certain precise sense, that these shapes are unlikely to be very long and thin. PubDate: 2019-11-04

Abstract: We study the global dynamics of the wave equation, Maxwell’s equation and the linearized Bianchi equations on a fixed anti-de Sitter (AdS) background. Provided dissipative boundary conditions are imposed on the dynamical fields we prove uniform boundedness of the natural energy as well as both degenerate (near the AdS boundary) and non-degenerate integrated decay estimates. Remarkably, the non-degenerate estimates “lose a derivative”. We relate this loss to a trapping phenomenon near the AdS boundary, which itself originates from the properties of (approximately) gliding rays near the boundary. Using the Gaussian beam approximation we prove that non-degenerate energy decay without loss of derivatives does not hold. As a consequence of the non-degenerate integrated decay estimates, we also obtain pointwise-in-time decay estimates for the energy. Our paper provides the key estimates for a proof of the non-linear stability of the anti-de Sitter spacetime under dissipative boundary conditions. Finally, we contrast our results with the case of reflecting boundary conditions. PubDate: 2019-11-04

Abstract: In this paper, we investigate topological aspects of indices of twisted geometric operators on manifolds equipped with fibered boundaries. We define K-groups relative to the pushforward for boundary fibration, and show that indices of twisted geometric operators, defined by complete \(\Phi \) or edge metrics, can be regarded as the index pairing over these K-groups. We also prove various properties of these indices using groupoid deformation techniques. Using these properties, we give an application to the localization problem of signature operators for singular fiber bundles. PubDate: 2019-11-03

Abstract: We consider a one-dimensional infinite chain of coupled charged harmonic oscillators in a magnetic field with a small stochastic perturbation of order \(\epsilon \). We prove that for a space–time scale of order \(\epsilon ^{-1}\) the density of energy distribution (Wigner distribution) evolves according to a linear phonon Boltzmann equation. We also prove that an appropriately scaled limit of solutions of the linear phonon Boltzmann equation is a solution of the fractional diffusion equation with exponent 5 / 6. PubDate: 2019-11-01

Abstract: We show that every (graded) derivation on the algebra of free quantum fields and their Wick powers in curved spacetimes gives rise to a set of anomalous Ward identities for time-ordered products, with an explicit formula for their classical limit. We study these identities for the Koszul–Tate and the full BRST differential in the BV–BRST formulation of perturbatively interacting quantum gauge theories, and clarify the relation to previous results. In particular, we show that the quantum BRST differential, the quantum antibracket and the higher-order anomalies form an \(L_\infty \) algebra. The defining relations of this algebra ensure that the gauge structure is well-defined on cohomology classes of the quantum BRST operator, i.e., observables. Furthermore, we show that one can determine contact terms such that also the interacting time-ordered products of multiple interacting fields are well defined on cohomology classes. An important technical improvement over previous treatments is the fact that all our relations hold off-shell and are independent of the concrete form of the Lagrangian, including the case of open gauge algebras. PubDate: 2019-11-01

Abstract: It has been known for a while that the effective geometrical description of compactified strings on d-dimensional target spaces implies a generalization of geometry with a doubling of the sets of tangent space directions. This generalized geometry involves an O(d,d) pairing \({\eta}\) and an O(2d) generalized metric \({\mathcal{H}}\). More recently it has been shown that in order to include T-duality as an effective symmetry, the generalized geometry also needs to carry a phase space structure or more generally a para-Hermitian structure encoded into a skew-symmetric pairing \({\omega}\). The consistency of string dynamics requires this geometry to satisfy a set of compatibility relations that form what we call a Born geometry. In this work we prove an analogue of the fundamental theorem of Riemannian geometry for Born geometry. We show that there exists a unique connection which preserves the Born structure \({(\eta,\omega,\mathcal{H})}\) and which is torsionless in a generalized sense. This resolves a fundamental ambiguity that is present in the double field theory formulation of effective string dynamics. PubDate: 2019-11-01

Abstract: We study the exchange of energy between the modes of the optical branch and those of the acoustic one in a diatomic chain of particles, with masses m1 and m2. We prove that, at small temperature and provided \({m_1\gg m_2}\), for the majority of the initial data the energy of each branch is approximately constant for times of order \({\beta^{S/2}}\), where \({S=\lfloor \sqrt{m_1/m_2}/2\rfloor}\) and \({\beta}\) is the inverse temperature. The result is uniform in the thermodynamic limit. PubDate: 2019-11-01

Abstract: Let \({\mathfrak {g}}\) be a complex simple Lie algebra. We study the family of Bethe subalgebras in the Yangian \(Y({\mathfrak {g}})\) parameterized by the corresponding adjoint Lie group G. We describe their classical limits as subalgebras in the algebra of polynomial functions on the formal Lie group \(G_1[[t^{-1}]]\). In particular we show that, for regular values of the parameter, these subalgebras are free polynomial algebras with the same Poincaré series as the Cartan subalgebra of the Yangian. Next, we extend the family of Bethe subalgebras to the De Concini–Procesi wonderful compactification \(\overline{G}\supset G\) and describe the subalgebras corresponding to generic points of any stratum in \(\overline{G}\) as Bethe subalgebras in the Yangian of the corresponding Levi subalgebra in \({\mathfrak {g}}\). In particular, we describe explicitly all Bethe subalgebras corresponding to the closure of the maximal torus in the wonderful compactification. PubDate: 2019-11-01

Abstract: In this paper, we prove the existence of \(H^2\)-regular coordinates on Riemannian 3-manifolds with boundary, assuming only \(L^2\)-bounds on the Ricci curvature, \(L^4\)-bounds on the second fundamental form of the boundary, and a positive lower bound on the volume radius. The proof follows by extending the theory of Cheeger–Gromov convergence to include manifolds with boundary in the above low regularity setting. The main tools are boundary harmonic coordinates together with elliptic estimates and a geometric trace estimate, and a rigidity argument using manifold doubling. Assuming higher regularity of the Ricci curvature, we also prove corresponding higher regularity estimates for the coordinates. PubDate: 2019-11-01

Abstract: We provide a full characterisation of quantum differentiability (in the sense of Connes) on quantum tori. We also prove a quantum integration formula which differs substantially from the commutative case. PubDate: 2019-11-01