An explicit formula is obtained for the generalized Macdonald functions on the N-fold Fock tensor spaces, calculating a certain matrix element of a composition of several screened vertex operators. As an application, we prove the factorization property of the arbitrary matrix elements of the multi-valent intertwining operator (or refined topological vertex operator) associated with the Ding–Iohara–Miki

We derive a sharp scaling law for deviations of edge-isoperimetric sets in the lattice \({\mathbb {Z}}^d\) from the limiting Wulff shape in arbitrary dimensions. As the number n of elements diverges, we prove that the symmetric difference to the corresponding Wulff set consists of at most \(O(n^{(d-1+2^{1-d})/d})\) lattice points and that the exponent \((d-1+2^{1-d})/d\) is optimal. This extends the

We describe the exact WKB method from the point of view of abelianization, both for Schrödinger operators and for their higher-order analogues (opers). The main new example which we consider is the “\(T_3\) equation,” an order 3 equation on the thrice-punctured sphere, with regular singularities at the punctures. In this case the exact WKB analysis leads to consideration of a new sort of Darboux coordinate

We introduce super quantum Airy structures, which provide a supersymmetric generalization of quantum Airy structures. We prove that to a given super quantum Airy structure one can assign a unique set of free energies, which satisfy a supersymmetric generalization of the topological recursion. We reveal and discuss various properties of these supersymmetric structures, in particular their gauge transformations

The second author formulated quantum unique ergodicity for Eisenstein series in the prime level aspect in “Equidistribution of Eisenstein series in the level aspect”, Commun. Math. Phys. 289(3) 1150 (2009). We point out errors and correct the proofs with partially weakened claims.

We classify framed and oriented 2-1-0-extended TQFTs with values in the bicategories of Landau-Ginzburg models, whose objects and 1-morphisms are isolated singularities and (either \(\mathbb {Z}_2\)- or \((\mathbb {Z}_2 \times \mathbb {Q})\)-graded) matrix factorisations, respectively. For this we present the relevant symmetric monoidal structures and find that every object \(W\in \mathbb {k}[x_1,\dots

Suspended graphene samples are observed to be gently rippled rather than being flat. In Friedrich et al. (Z Angew Math Phys 69:70, 2018), we have checked that this nonplanarity can be rigorously described within the classical molecular-mechanical frame of configurational-energy minimization. There, we have identified all ground-state configurations with graphene topology with respect to classes of

A two-dimensional chiral conformal field theory can be viewed mathematically as the representation theory of its chiral algebra, a vertex operator algebra. Vertex operator algebras are especially well suited for studying logarithmic conformal field theory (in which correlation functions have logarithmic singularities arising from non-semisimple modules for the chiral algebra) because of the logarithmic

We define new deformable families of vertex operator algebras \(\mathfrak {A}[\mathfrak {g}, \Psi , \sigma ]\) associated to a large set of S-duality operations in four-dimensional supersymmetric gauge theory. They are defined as algebras of protected operators for two-dimensional supersymmetric junctions which interpolate between a Dirichlet boundary condition and its S-duality image. The \(\mathfrak

The magnetic Laplacian (also called the line bundle Laplacian) on a connected weighted graph is a self-adjoint operator wherein the real-valued adjacency weights are replaced by unit complex-valued weights \(\{\omega _{xy}\}_{xy\in E}\), satisfying the condition that \(\omega _{xy}=\overline{\omega _{yx}}\) for every directed edge xy. When properly interpreted, these complex weights give rise to magnetic

We determine the \(L_\infty \)-algebra that controls deformations of a relative Rota–Baxter Lie algebra and show that it is an extension of the dg Lie algebra controlling deformations of the underlying \(\mathsf {Lie}\mathsf {Rep}\) pair by the dg Lie algebra controlling deformations of the relative Rota–Baxter operator. Consequently, we define the cohomology of relative Rota–Baxter Lie algebras and

We study metastable behavior in a discrete nonlinear Schrödinger equation from the viewpoint of Hamiltonian systems theory. When there are \(n<\infty \) sites in this equation, we consider initial conditions in which almost all the energy is concentrated in one end of the system. We are interested in understanding how energy flows through the system, so we add a dissipation of size \(\gamma \) at the

An inclusion of von Neumann factors \(M \subset \mathcal {M}\) is ergodic if it satisfies the irreducibility condition \(M'\cap \mathcal {M}=\mathbb {C}\). We investigate the relation between this and several stronger ergodicity properties, such as R-ergodicity, which requires M to admit an embedding of the hyperfinite II\(_1\) factor \(R\hookrightarrow M\) that’s ergodic in \(\mathcal {M}\). We prove

We introduce a new class of representations of the cohomological Hall algebras of Kontsevich and Soibelman, which we call cohomological Hall modules, or CoHM for short. These representations are constructed from self-dual representations of a quiver with contravariant involution \(\sigma \) and provide a mathematical model for the space of BPS states in orientifold string theory. We use the CoHM to

We study the open refined topological string amplitudes using the refined topological vertex. We determine the refinement of holonomies necessary to describe the boundary conditions of open amplitudes (which in particular satisfy the required integrality properties). We also derive the refined holonomies using the refined Chern–Simons theory.

Whenever available, refined BPS indices provide considerably more information on the spectrum of BPS states than their unrefined version. Extending earlier work on the modularity of generalized Donaldson–Thomas invariants counting D4-D2-D0 brane bound states in type IIA strings on a Calabi–Yau threefold \(\mathfrak {Y}\), we construct the modular completion of generating functions of refined BPS indices

The Hilbert space effect algebra is a fundamental mathematical structure which is used to describe unsharp quantum measurements in Ludwig’s formulation of quantum mechanics. Each effect represents a quantum (fuzzy) event. The relation of coexistence plays an important role in this theory, as it expresses when two quantum events can be measured together by applying a suitable apparatus. This paper’s

We develop a general framework for studying ergodicity of order-preserving Markov semigroups. We establish natural and in a certain sense optimal conditions for existence and uniqueness of the invariant measure and exponential convergence of transition probabilities of an order-preserving Markov process. As an application, we show exponential ergodicity and exponentially fast synchronization-by-noise

We consider the Fermi–Pasta–Ulam–Tsingou (FPUT) chain composed by \(N \gg 1\) particles and periodic boundary conditions, and endow the phase space with the Gibbs measure at small temperature \(\beta ^{-1}\). Given a fixed \({1\le m \ll N}\), we prove that the first m integrals of motion of the periodic Toda chain are adiabatic invariants of FPUT (namely they are approximately constant along the Hamiltonian

A long-standing problem in quantum gravity and cosmology is the quantization of systems in which time evolution is generated by a constraint that must vanish on solutions. Here, an algebraic formulation of this problem is presented, together with new structures and results, which show that specific conditions need to be satisfied in order for well-defined evolution to be possible.

We investigate the linear and semilinear massive Klein–Gordon equations in geometrical frameworks of type “Conformal Cyclic Cosmology” of R. Penrose, or “Singular Bouncing Scenario” as well. We give sufficient conditions on the decay of the mass to the fields be able to propagate across the Big-Bang.

In this paper, we classify non-freely acting discrete symmetries of complete intersection Calabi–Yau manifolds and their quotients by freely-acting symmetries. These non-freely acting symmetries can appear as symmetries of low-energy theories resulting from string compactifications on these Calabi–Yau manifolds, particularly in the context of the heterotic string. Hence, our results are relevant for

We ask the question whether entropy accumulates, in the sense that the operationally relevant total uncertainty about an n-partite system \(A = (A_1, \ldots A_n)\) corresponds to the sum of the entropies of its parts \(A_i\). The Asymptotic Equipartition Property implies that this is indeed the case to first order in n—under the assumption that the parts \(A_i\) are identical and independent of each

It is known that sub-extremal black hole backgrounds do not admit a (bijective) non-degenerate scattering theory in the exterior region due to the fact that the redshift effect at the event horizon acts as an unstable blueshift mechanism in the backwards direction in time. In the extremal case, however, the redshift effect degenerates and hence yields a much milder blueshift effect when viewed in the

In conformal field theory in Minkowski momentum space, the 3-point correlation functions of local operators are completely fixed by symmetry. Using Ward identities together with the existence of a Lorentzian operator product expansion (OPE), we show that the Wightman function of three scalar operators is a double hypergeometric series of the Appell \(F_4\) type. We extend this simple closed-form expression

We show that for a large class of potential functions and big coupling constant \(\lambda \) the Schrödinger cocycle over the expanding map \(x\mapsto bx ~( \text{ mod } 1)\) on \(\mathbb {T}\) has a Lyapunov exponent \(>(\log \lambda )/4\) for all energies, provided that the integer \(b\ge \lambda ^3\).

In this paper, we obtain the transition probability formulas for the asymmetric simple exclusion process and the q-deformed totally asymmetric zero range process on the ring by applying the coordinate Bethe ansatz. We also compute the distribution function for a tagged particle with general initial condition.

We consider a family of strongly-asymmetric unimodal maps \(\{f_t\}_{t\in [0,1]}\) of the form \(f_t=t\cdot f\) where \(f:[0,1]\rightarrow [0,1]\) is unimodal, \(f(0)=f(1)=0\), \(f(c)=1\) is of the form and $$\begin{aligned} f(x)=\left\{ \begin{array}{ll} 1-K_-|x-c|+o(|x-c|)&{} \text{ for } xc, \end{array}\right. \end{aligned}$$ where we assume that \(\beta >1\). We show that such a family contains

We study the limiting absorption principle and the well-posedness of Maxwell equations with anisotropic sign-changing coefficients in the time-harmonic domain. The starting point of the analysis is to obtain Cauchy problems associated with two Maxwell systems using a change of variables. We then derive a priori estimates for these Cauchy problems using two different approaches. The Fourier approach

We investigate the long-term behavior of a random walker evolving on top of the simple symmetric exclusion process (SSEP) at equilibrium, in dimension one. At each jump, the random walker is subject to a drift that depends on whether it is sitting on top of a particle or a hole, so that its asymptotic behavior is expected to depend on the density \(\rho \in [0, 1]\) of the underlying SSEP. Our first

A set-theoretic solution of the Pentagon Equation on a non-empty set S is a map \(s:S^2\rightarrow S^2\) such that \(s_{23}s_{13}s_{12}=s_{12}s_{23}\), where \(s_{12}=s\times {{{\,\mathrm{id}\,}}}\), \(s_{23}={{{\,\mathrm{id}\,}}}\times s\) and \(s_{13}=(\tau \times {{{\,\mathrm{id}\,}}})({{{\,\mathrm{id}\,}}}\times s)(\tau \times {{{\,\mathrm{id}\,}}})\) are mappings from \(S^3\) to itself and \(\tau

We study KMS states for the C*-algebras of \(ax+b\)-semigroups of algebraic integers in which the multiplicative part is restricted to a congruence monoid, as in recent work of Bruce. We realize the extremal low-temperature KMS states as generalized Gibbs states in concrete representations induced from extremal traces of certain group C*-algebras. We use these representations to compute the type of

We construct new families of positive energy representations of affine vertex algebras together with their free field realizations by using localization technique. We introduce the twisting functor \(T_\alpha \) on the category of modules over the universal affine vertex algebra \(\mathcal {V}_\kappa (\mathfrak {g})\) of level \(\kappa \) for any positive root \(\alpha \) of \(\mathfrak {g}\), and

By developing the method of multipliers, we establish sufficient conditions on the magnetic field and the complex, matrix-valued electric potential, which guarantee that the corresponding system of Schrödinger operators has no point spectrum. In particular, this allows us to prove analogous results for Pauli operators under the same electromagnetic conditions and, in turn, as a consequence of the supersymmetric

The recently conjectured knots-quivers correspondence (Kucharski et al. in Phys Rev D 96(12):121902, 2017. arXiv:1707.02991, Adv Theor Math Phys 23(7):1849–1902, 2019. arXiv:1707.04017) relates gauge theoretic invariants of a knot K in the 3-sphere to the representation theory of a quiver \(Q_{K}\) associated to the knot. In this paper we provide geometric and physical contexts for this conjecture

We prove the conjectural relationship recently proposed in Dubrovin and Yang (Commun Number Theory Phys 11:311–336, 2017) between certain special cubic Hodge integrals of the Gopakumar–Mariño–Vafa type (Gopakumar and Vafa in Adv Theor Math Phys 5:1415–1443, 1999, Mariño and Vafa in Contemp Math 310:185–204, 2002) and GUE correlators, and the conjecture proposed in Dubrovin et al. (Adv Math 293:382–435

We prove the invariance principle for a random Lorentz-gas particle in 3 dimensions under the Boltzmann-Grad limit and simultaneous diffusive scaling. That is, for the trajectory of a point-like particle moving among infinite-mass, hard-core, spherical scatterers of radius r, placed according to a Poisson point process of density \(\varrho \), in the limit \(\varrho \rightarrow \infty \), \(r\rightarrow

Consider any Leray–Hopf weak solution of the three-dimensional Navier–Stokes equations for incompressible, viscous fluid flows. We prove that any Lagrangian trajectory associated with such a velocity field has an asymptotic expansion, as time tends to infinity, which describes its long-time behavior very precisely.

In this paper we study a generalized class of Maxwell–Boltzmann equations which in addition to the usual collision term contains a linear deformation term described by a matrix A. This class of equations arises, for instance, from the analysis of homoenergetic solutions for the Boltzmann equation considered by many authors since 1950s. Our main goal is to study a large time asymptotics of solutions

We prove the existence of a quantum isometry groups for new classes of metric spaces: (i) geodesic metrics for compact connected Riemannian manifolds (possibly with boundary) and (ii) metric spaces admitting a uniformly distributed probability measure. In the former case it also follows from recent results of the second author that the quantum isometry group is classical, i.e. the commutative \(C^*\)-algebra

We consider a scattering problem for electric potentials that have a component which is critically singular in the sense of Lebesgue spaces, and a component given by a measure supported on a compact Lipschitz hypersurface. We study direct and inverse point-source scattering under the assumptions that the potentials are real-valued and compactly supported. To solve the direct scattering problem, we

The bulk-edge correspondence predicts that interfaces between topological insulators support robust currents. We prove this principle for PDEs that are periodic away from an interface. Our approach relies on semiclassical methods. It suggests novel perspectives for the analysis of topologically protected transport.

We study the concentration properties of low-energy states for quantum systems in the semiclassical limit, in the setting of Toeplitz operators, which include quantum spin systems as a large class of examples. We establish tools proper to Toeplitz quantization to give a general subprincipal criterion for localisation. In addition, we build up symplectic normal forms in two particular settings, including

We analyse the problem of assigning sign choices to O-planes in orientifolds of type II string theory. We show that there exists a sequence of invariant p-gerbes with \(p\ge -1\), which give rise to sign choices and are related by coboundary maps. We prove that the sign choice homomorphisms stabilise with the dimension of the orientifold and we derive topological constraints on the possible sign configurations

We present a detailed account of the properties of \(\text {twister}\)s and their generalizations, \(\text {FC set}\)s, which are essential ingredients of the orbifold deconstruction procedure aimed at recognizing whether a given conformal model may be obtained as an orbifold of another one, and if so, to identify the twist group and the original model. The close analogy with the character theory of

Given a finite, directed, connected graph \(\Gamma \) equipped with a weighting \(\mu \) on its edges, we provide a construction of a von Neumann algebra equipped with a faithful, normal, positive linear functional \((\mathcal {M}(\Gamma ,\mu ),\varphi )\). When the weighting \(\mu \) is instead on the vertices of \(\Gamma \), the first author showed the isomorphism class of \((\mathcal {M}(\Gamma

We study the 2D Navier–Stokes equations linearized around the Couette flow \((y,0)^t\) in the periodic channel \({\mathbb {T}} \times [-1,1]\) with no-slip boundary conditions in the vanishing viscosity \(\nu \rightarrow 0\) limit. We split the vorticity evolution into the free evolution (without a boundary) and a boundary corrector that is exponentially localized to at most an \(O(\nu ^{1/3})\) boundary

We define the analytic torsion associated with a Riemann surface endowed with a metric having Poincaré-type singularities in the neighborhood of a finite number of points and a Hermitian vector bundle with at most logarithmic singularities at those points, coming from the metric on the negative power of the canonical line bundle twisted by the divisor of the points. Then we provide a relation between

For each partition \(\underline{p}\) of an integer \(N\ge 2\), consisting of r parts, an integrable hierarchy of Lax type Hamiltonian PDE has been constructed recently by some of us. In the present paper we show that any tau-function of the \(\underline{p}\)-reduced r-component KP hierarchy produces a solution of this integrable hierarchy. Along the way we provide an algorithm for the explicit construction

We explain how dispersionless integrable hierarchy in 2d topological field theory arises from the Kodaira–Spencer gravity (BCOV theory). The infinitely many commuting Hamiltonians are given by the current observables associated to the infinite abelian symmetries of the Kodaira–Spencer gravity. We describe a BV framework of effective field theories that leads to the B-model interpretation of dispersionless

The process of quantum measurement is considered in the algebraic framework of quantum field theory on curved spacetimes. Measurements are carried out on one quantum field theory, the “system”, using another, the “probe”. The measurement process involves a dynamical coupling of “system” and “probe” within a bounded spacetime region. The resulting “coupled theory” determines a scattering map on the

We study the eigenvalues of the Toeplitz quantization of complex-valued functions on the torus subject to small random perturbations given by a complex-valued random matrix whose entries are independent copies of a random variable with mean 0, variance 1 and bounded fourth moment. We prove that the eigenvalues of the perturbed operator satisfy a Weyl law with probability close to one, which proves

Given a closed orientable hyperbolic manifold of dimension \(\ne 3\) we prove that the multiplicity of the Pollicott-Ruelle resonance of the geodesic flow on perpendicular one-forms at zero agrees with the first Betti number of the manifold. Additionally, we prove that this equality is stable under small perturbations of the Riemannian metric and simultaneous small perturbations of the geodesic vector

The gauged sigma model with target \({\mathbb {P}}^1\), defined on a Riemann surface \(\Sigma \), supports static solutions in which \(k_{+}\) vortices coexist in stable equilibrium with \(k_{-}\) antivortices. Their moduli space is a noncompact complex manifold \({\textsf {M}}_{(k_{+},k_{-})}(\Sigma )\) of dimension \(k_{+}+k_{-}\) which inherits a natural Kähler metric \(g_{L^2}\) governing the model’s

In Maulik and Thomas (in preparation) the Vafa–Witten theory of complex projective surfaces is lifted to oriented \({\mathbb {C}}^*\)-equivariant cohomology theories. Here we study the K-theoretic refinement. It gives rational functions in \(t^{1/2}\) invariant under \(t^{1/2}\leftrightarrow t^{-1/2}\) which specialise to numerical Vafa–Witten invariants at \(t=1\). On the “instanton branch” the invariants

The goal of the paper is to apply the theory of integrable systems to construct explicit sections of line bundles over the combinatorial model of the moduli space of pointed Riemann surfaces based on Strebel differentials. These line bundles are tensor products of the determinants of the Hodge or Prym vector bundles with the standard tautological line bundles \(\mathcal {L}_j\), and the sections are

We study n non-intersecting Brownian motions corresponding to initial configurations which have a vanishing density in the large n limit at an interior point of the support. It is understood that the point of vanishing can propagate up to a critical time, and we investigate the nature of the microscopic space-time correlations near the critical point and critical time. We show that they are described

Inspired by a formula of Stern that relates scalar curvature to harmonic functions, we evaluate the mass of an asymptotically flat 3-manifold along faces and edges of a large coordinate cube. In terms of the mean curvature and dihedral angle, the resulting mass formula relates to Gromov’s scalar curvature comparison theory for cubic Riemannian polyhedra. In terms of the geodesic curvature and turning

We establish a generalisation of the fundamental state convertibility theorem in quantum information to the context of bipartite quantum systems modelled by commuting semi-finite von Neumann algebras. Namely, we establish a generalisation to this setting of Nielsen’s theorem on the convertibility of quantum states under local operations and classical communication (LOCC) schemes. Along the way, we