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In this paper we propose a definition of torsion refined Gopakumar–Vafa (GV) invariants for Calabi–Yau threefolds with terminal nodal singularities that do not admit Kähler crepant resolutions. Physically, the refinement takes into account the charge of five-dimensional BPS states under a discrete gauge symmetry in M-theory. We propose a mathematical definition of the invariants in terms of the geometry of all non-Kähler crepant resolutions taken together. The invariants are encoded in the A-model topological string partition functions associated to non-commutative (nc) resolutions of the Calabi–Yau. Our main example will be a singular degeneration of the generic Calabi–Yau double cover of P 3 and leads to an enumerative interpretation of the topological string partition function of a hybrid Landau–Ginzburg model. Our results generalize a recent physical proposal made in the context of torus fibered Calabi–Yau manifolds by one of the authors and clarify the associated enumerative geometry. [ABSTRACT FROM AUTHOR]

In the framework of Atiyah's axioms of topological quantum field theory with unitarity, we give a direct proof of the fact that symmetry protected topological phases without Hall effects are classified by cobordism invariants. We first show that the partition functions of those theories are cobordism invariants after a tuning of the Euler term. Conversely, for a given cobordism invariant, we construct a unitary topological field theory whose partition function is given by the cobordism invariant, assuming that a certain bordism group is finitely generated. Two theories having the same cobordism invariant partition functions are isomorphic. [ABSTRACT FROM AUTHOR]

In the continuity of our previous paper (Crampe et al. in Commun Math Phys 349:271, 2017, arXiv:1509.05516), we define three new algebras, An(a,b,c), Bn and Cn, that are close to the braid algebra. They allow to build solutions to the Yang-Baxter equation with spectral parameters. The construction is based on a baxterisation procedure, similar to the one used in the context of Hecke or BMW algebras. The An(a,b,c) algebra depends on three arbitrary parameters, and when the parameter a is set to zero, we recover the algebra Mn(b,c) already introduced elsewhere for purpose of baxterisation. The Hecke algebra (and its baxterisation) can be recovered from a coset of the An(0,0,c) algebra. The algebra An(0,b,-b2) is a coset of the braid algebra. The two other algebras Bn and Cn do not possess any parameter, and can be also viewed as a coset of the braid algebra. [ABSTRACT FROM AUTHOR]

Skew-orthogonal polynomials (SOPs) arise in the study of the n-point distribution function for orthogonal and symplectic random matrix ensembles. Motivated by the average of characteristic polynomials of the Bures random matrix ensemble studied in Forrester and Kieburg (Commun Math Phys 342(1):151-187, 2016), we propose the concept of partial-skew-orthogonal polynomials (PSOPs) as a modification of the SOPs, and then the PSOPs with a variety of special skew-symmetric kernels and weight functions are addressed. By considering appropriate deformations of the weight functions, we derive nine integrable lattices in different dimensions. As a consequence, the tau-functions for these systems are shown to be expressed in terms of Pfaffians and the wave vectors PSOPs. In fact, the tau-functions also admit the multiple integral representations. Among these integrable lattices, some of them are known, while the others are novel to the best of our knowledge. In particular, one integrable lattice is related to the partition function of the Bures ensemble. Besides, we derive a discrete integrable lattice which can be used to compute certain vector Padé approximants. This yields the first example regarding the connection between integrable lattices and generalised inverse vector-valued Padé approximants, about which Hietarinta, Joshi, and Nijhoff pointed out that, “This field remains largely to be explored”, in the recent monograph (Hietarinta et al. in Discrete systems and integrability, vol 54. Cambridge University Press, Cambridge, 2016, [Section 4.4]). [ABSTRACT FROM AUTHOR]

A continuous groupoid homomorphism c on a locally compact second countable Hausdorff étale groupoid G gives rise to a C*-dynamical system in which every β-KMS state can be associated to a e-βc-quasi-invariant measure μ on G(0). Letting Δμ denote the set of KMS states associated to such a μ, we will prove that Δμ is a simplex for a large class of groupoids, and we will show that there is an abelian group that acts transitively and freely on the extremal points of Δμ. This abelian group can be described using the support of μ, so our theory can be used to obtain a description of all KMS states by describing the e-βc-quasi-invariant measures. To illustrate this we will describe the KMS states for the Cuntz-Krieger algebras of all finite higher rank graphs without sources and a large class of continuous one-parameter groups. [ABSTRACT FROM AUTHOR]

All Painlevé equations can be written as a time-dependent Hamiltonian system, and as such they admit a natural generalization to the case of several particles with an interaction of Calogero type (rational, trigonometric or elliptic). Recently, these systems of interacting particles have been proved to be relevant in the study of β-models. An almost two-decade-old open question by Takasaki asks whether these multi-particle systems can be understood as isomonodromic equations, thus extending the Painlevé correspondence. In this paper we answer in the affirmative by displaying explicitly suitable isomonodromic Lax pair formulations. As an application of the isomonodromic representation, we provide a construction based on discrete Schlesinger transforms, to produce solutions for these systems for special values of the coupling constants, starting from uncoupled ones; the method is illustrated for the case of the second Painlevé equation. [ABSTRACT FROM AUTHOR]

We construct non-semisimple 2 + 1-TQFTs yielding mapping class group representations in Lyubashenko’s spaces. In order to do this, we first generalize Beliakova, Blanchet and Geer’s logarithmic Hennings invariants based on quantum sl2 to the setting of finite-dimensional non-degenerate unimodular ribbon Hopf algebras. The tools used for this construction are a Hennings-augmented Reshetikhin-Turaev functor and modified traces. When the Hopf algebra is factorizable, we further show that the universal construction of Blanchet, Habegger, Masbaum and Vogel produces a 2 + 1-TQFT on a not completely rigid monoidal subcategory of cobordisms. [ABSTRACT FROM AUTHOR]

We show a sharp conformally invariant gap theorem for Yang-Mills connections in dimension 4 by exploiting an associated Yamabe-type problem. [ABSTRACT FROM AUTHOR]

Two 3-fold flops are exhibited, both of which have precisely one flopping curve. One of the two flops is new and is distinct from all known algebraic D4-flops. It is shown that the two flops are neither algebraically nor analytically isomorphic, yet their curve-counting Gopakumar-Vafa invariants are the same. We further show that the contraction algebras associated to both are not isomorphic, so the flops are distinguished at this level. This shows that the contraction algebra is a finer invariant than various curve-counting theories, and it also provides more evidence for the proposed analytic classification of 3-fold flops via contraction algebras. [ABSTRACT FROM AUTHOR]

For second countable discrete quantum groups, and more generally second countable locally compact quantum groups with trivial scaling group, we show that property (T) is equivalent to every weakly mixing unitary representation not having almost invariant vectors. This is a generalization of a theorem of Bekka and Valette from the group setting and was previously established in the case of low dual by Daws, Skalski, and Viselter. Our approach uses spectral techniques and is completely different from those of Bekka-Valette and Daws-Skalski-Viselter. By a separate argument we furthermore extend the result to second countable nonunimodular locally compact quantum groups, which are shown in particular not to have property (T), generalizing a theorem of Fima from the discrete setting. We also obtain quantum group versions of characterizations of property (T) of Kerr and Pichot in terms of the Baire category theory of weak mixing representations and of Connes and Weiss in terms of the prevalence of strongly ergodic actions. [ABSTRACT FROM AUTHOR]

A family of invariants of smooth, oriented four-dimensional manifolds is defined via handle decompositions and the Kirby calculus of framed link diagrams. The invariants are parametrised by a pivotal functor from a spherical fusion category into a ribbon fusion category. A state sum formula for the invariant is constructed via the chain-mail procedure, so a large class of topological state sum models can be expressed as link invariants. Most prominently, the Crane-Yetter state sum over an arbitrary ribbon fusion category is recovered, including the nonmodular case. It is shown that the Crane-Yetter invariant for nonmodular categories is stronger than signature and Euler invariant. A special case is the four-dimensional untwisted Dijkgraaf-Witten model. Derivations of state space dimensions of TQFTs arising from the state sum model agree with recent calculations of ground state degeneracies in Walker-Wang models. Relations to different approaches to quantum gravity such as Cartan geometry and teleparallel gravity are also discussed. [ABSTRACT FROM AUTHOR]

We propose an extension of the recently-proposed volume conjecture for closed hyperbolic 3-manifolds, to all orders in perturbative expansion. We first derive formulas for the perturbative expansion of the partition function of complex Chern-Simons theory around a hyperbolic flat connection, which produces infinitely-many perturbative invariants of the closed oriented 3-manifold. The conjecture is that this expansion coincides with the perturbative expansion of the Witten-Reshetikhin-Turaev invariants at roots of unity q=e2πi/r with r odd, in the limit r→∞. We provide numerical evidence for our conjecture. [ABSTRACT FROM AUTHOR]

In this paper, we construct global action-angle variables for a certain two-parameter family of hyperbolic van Diejen systems. Following Ruijsenaars’ ideas on the translation invariant models, the proposed action-angle variables come from a thorough analysis of the commutation relation obeyed by the Lax matrix, whereas the proof of their canonicity is based on the study of the scattering theory. As a consequence, we show that the van Diejen system of our interest is self-dual with a factorized scattering map. Also, in an appendix by S. Ruijsenaars, a novel proof of the spectral asymptotics of certain exponential type matrix flows is presented. This result is of crucial importance in our scattering-theoretical analysis. [ABSTRACT FROM AUTHOR]

To illustrate the more tractable properties of random dynamical systems, we consider a class of 2D maps with strong expansion on large—but non-invariant—subsets of their phase spaces. In the deterministic case, such maps are not precluded from having sinks, as derivative growth on disjoint time intervals can be cancelled when stable and unstable directions are reversed. Our main result is that when randomly perturbed, these maps possess positive Lyapunov exponents commensurate with the amount of expansion in the system. We show also that initial conditions converge exponentially fast to the stationary state, equivalently time correlations decay exponentially fast. These properties depend only on finite-time dynamics, and do not involve parameter selections, which are necessary for deterministic maps with nonuniform derivative growth. [ABSTRACT FROM AUTHOR]

We propose a model for the dS/CFT correspondence. The model is constructed in terms of a "Yang-Baxter operator" R for unitary representations of the de Sitter group SO(d, 1). This R-operator is shown to satisfy the Yang-Baxter equation, unitarity, as well as certain analyticity relations, including in particular a crossing symmetry. With the aid of this operator we construct: (a) a chiral (light-ray) conformal quantum field theory whose internal degrees of freedom transform under the given unitary representation of SO(d, 1). By analogy with the O(N) non-linear sigma model, this chiral CFT can be viewed as propagating in a de Sitter spacetime. (b) A (non-unitary) Euclidean conformal quantum field theory on Rd-1, where SO(d, 1) now acts by conformal transformations in (Euclidean) spacetime. These two theories can be viewed as dual to each other if we interpret Rd-1 as conformal infinity of de Sitter spacetime. Our constructions use semi-local generator fields defined in terms of R and abstract methods from operator algebras. [ABSTRACT FROM AUTHOR]

We want to establish the "braided action" (defined in the paper) of the DHR category on a universal environment algebra as a complete invariant for completely rational chiral conformal quantum field theories. The environment algebra can either be a single local algebra, or the quasilocal algebra, both of which are model-independent up to isomorphism. The DHR category as an abstract structure is captured by finitely many data (superselection sectors, fusion, and braiding),whereas its braided action encodes the full dynamical information that distinguishes models with isomorphic DHR categories. We show some geometric properties of the "duality pairing" between local algebras and the DHR category that are valid in general (completely rational) chiral CFTs. Under some additional assumptions whose status remains to be settled, the braided action of its DHR category completely classifies a (prime) CFT. The approach does not refer to the vacuum representation, or the knowledge of the vacuum state. [ABSTRACT FROM AUTHOR]

We introduce a notion of Rokhlin dimension for one parameter automorphism groups of $${C^*}$$ -algebras. This generalizes Kishimoto's Rokhlin property for flows, and is analogous to the notion of Rokhlin dimension for actions of the integers and other discrete groups introduced by the authors and Zacharias in previous papers. We show that finite nuclear dimension and absorption of a strongly self-absorbing $${C^*}$$ -algebra are preserved under forming crossed products by flows with finite Rokhlin dimension, and that these crossed products are stable. Furthermore, we show that a flow on a commutative $${C^*}$$ -algebra arising from a free topological flow has finite Rokhlin dimension, whenever the spectrum is a locally compact metrizable space with finite covering dimension. For flows that are both free and minimal, this has strong consequences for the associated crossed product $${C^{*}}$$ -algebras: Those containing a non-zero projection are classified by the Elliott invariant (for compact manifolds this consists of topological $${K}$$ -theory together with the space of invariant probability measures and a natural pairing given by the Ruelle-Sullivan map). [ABSTRACT FROM AUTHOR]

In this paper, we consider the classification problem of asymmetric gapped Hamiltonians, which are given as the non-degenerate part of the Hamiltonians introduced in []. We consider the C-classification, which takes into account the effect of boundaries. We show that the left and right degeneracies of edge ground states are the complete invariant. As a corollary, we consider the bulk-classification problem. We study Hamiltonians that (1) are given by translation invariant finite range interactions, (2) are gapped in the bulk, (3) are frustration-free, (4) have uniformly bounded ground state degeneracy on finite intervals, and (5) have a unique bulk ground state. We show that for the bulk-classification, any such Hamiltonians are equivalent. [ABSTRACT FROM AUTHOR]

We study one dimensional Schroedinger operators with random edge weights and their expected spectral measures $${\mu_H}$$ near zero. We prove that the measure exhibits a spike of the form $${\mu_H(-\varepsilon,\varepsilon) \sim \frac{C}{\mid{{\rm log}\varepsilon}\mid^2}}$$ (first observed by Dyson), without assuming independence or any regularity of edge weights. We also identify the limiting local eigenvalue distribution, which is different from Poisson and the usual random matrix statistics. We then use the result to compute Novikov-Shubin invariants for various groups, including lamplighter groups and lattices in the Lie group Sol. [ABSTRACT FROM AUTHOR]

In this paper we construct new solutions of the Kähler-Yang-Mills equations, by applying dimensional reduction methods to the product of the complex projective line with a compact Riemann surface. The resulting equations, which we call gravitating vortex equations, describe abelian vortices on the Riemann surface with back reaction of the metric. As a particular case of these gravitating vortices on the Riemann sphere we find solutions of the Einstein-Bogomol'nyi equations, which physically correspond to Nielsen-Olesen cosmic strings in the Bogomol'nyi phase. We use this to provide a Geometric Invariant Theory interpretation of an existence result by Y. Yang for the Einstein-Bogomol'nyi equations, applying a criterion due to G. Székelyhidi. [ABSTRACT FROM AUTHOR]

Recently, it was understood that modified concepts of locality played an important role in the study of extended quantum systems out of equilibrium, in particular in so-called generalized Gibbs ensembles. In this paper, we rigorously study pseudolocal charges and their involvement in time evolutions and in the thermalization process of arbitrary states with strong enough clustering properties. We show that the densities of pseudolocal charges form a Hilbert space, with inner product determined by thermodynamic susceptibilities. Using this, we define the family of pseudolocal states, which are determined by pseudolocal charges. This family includes thermal Gibbs states at high enough temperatures, as well as (a precise definition of) generalized Gibbs ensembles. We prove that the family of pseudolocal states is preserved by finite time evolution, and that, under certain conditions, the stationary state emerging at infinite time is a generalized Gibbs ensemble with respect to the evolution dynamics. If the evolution dynamics does not admit any conserved pseudolocal charges other than the evolution Hamiltonian, we show that any stationary pseudolocal state with respect to these dynamics is a thermal Gibbs state, and that Gibbs thermalization occurs. The framework is that of translation-invariant states on hypercubic quantum lattices of any dimensionality (including quantum chains) and finite-range Hamiltonians, and does not involve integrability. [ABSTRACT FROM AUTHOR]

We prove that generically, both in a topological and measure-theoretical sense, an invariant Lagrangian Diophantine torus of a Hamiltonian system is superexponentially stable in the sense that nearby solutions remain close to the torus for an interval of time which dominate any time that is exponentially large with respect to the inverse of the distance to the torus. More specifically, we prove stability over times that are doubly exponentially large with respect to the inverse of the distance to the torus. We also prove that for an arbitrary small perturbation of a generic integrable Hamiltonian system, there exists a set of almost full positive Lebesgue measure of KAM tori which are superexponentially stable with the previous estimates. Our results hold true for real-analytic but more generally for Gevrey smooth systems. [ABSTRACT FROM AUTHOR]

We give an elementary and explicit proof of the first fundamental theorem of invariant theory for the orthosymplectic supergroup by generalising the geometric method of Atiyah, Bott and Patodi to the supergroup context. We use methods from super-algebraic geometry to convert invariants of the orthosymplectic supergroup into invariants of the corresponding general linear supergroup on a different space. In this way, super Schur-Weyl-Brauer duality is established between the orthosymplectic supergroup of superdimension ( m|2 n) and the Brauer algebra with parameter m − 2 n. The result may be interpreted either in terms of the group scheme OSp( V) over $${{\mathbb C}}$$ , where V is a finite dimensional super space, or as a statement about the orthosymplectic Lie supergroup over the infinite dimensional Grassmann algebra $${\Lambda}$$ . We take the latter point of view here, and also state a corresponding theorem for the orthosymplectic Lie superalgebra, which involves an extra invariant generator, the super-Pfaffian. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a variation of the notion of topological phase reflecting metric structure of the position space. This framework contains not only periodic and non-periodic systems with symmetries in Kitaev's periodic table but also topological crystalline insulators. We also define the bulk and edge indices as invariants taking values in the twisted equivariant K-groups of Roe algebras as generalizations of existing invariants such as the Hall conductance or the Kane-Mele $${\mathbb{Z}_2}$$ -invariant. As a consequence, we obtain a new mathematical proof of the bulk-edge correspondence by using the coarse Mayer-Vietoris exact sequence. As a new example, we study the index of reflection-invariant systems. [ABSTRACT FROM AUTHOR]

In this paper, we study holomorphic discs in K3 surfaces and defined the open Gromov-Witten invariants. Using this new invariant, we can establish a version of correspondence between tropical discs and holomorphic discs with non-trivial invariants. We give an example of wall-crossing phenomenon of the invariant and expect it satisfies Kontsevich-Soibelman wall-crossing formula. [ABSTRACT FROM AUTHOR]

We consider the weakly asymmetric simple exclusion process in the presence of a slow bond and starting from the invariant state, namely the Bernoulli product measure of parameter $${\rho \in (0,1)}$$ . The rate of passage of particles to the right (resp. left) is $${\frac{1}{2} + \frac{a}{2n^{\gamma}}}$$ (resp. $${\frac{1}{2} - \frac{a}{2n^{\gamma}}}$$ ) except at the bond of vertices $${\{-1,0\}}$$ where the rate to the right (resp. left) is given by $${\frac{\alpha}{2n^\beta} + \frac{a}{2n^{\gamma}}}$$ (resp. $${\frac{\alpha}{2n^\beta}-\frac{a}{2n^{\gamma}}}$$ ). Above, $${\alpha > 0}$$ , $${\gamma \geq \beta \geq 0}$$ , $${a\geq 0}$$ . For $${\beta < 1}$$ , we show that the limit density fluctuation field is an Ornstein-Uhlenbeck process defined on the Schwartz space if $${\gamma > \frac{1}{2}}$$ , while for $${\gamma = \frac{1}{2}}$$ it is an energy solution of the stochastic Burgers equation. For $${\gamma \geq \beta =1}$$ , it is an Ornstein-Uhlenbeck process associated to the heat equation with Robin's boundary conditions. For $${\gamma \geq \beta > 1}$$ , the limit density fluctuation field is an Ornstein-Uhlenbeck process associated to the heat equation with Neumann's boundary conditions. [ABSTRACT FROM AUTHOR]

The Lie algebra $${\mathcal{D}}$$ of regular differential operators on the circle has a universal central extension $${\hat{\mathcal{D}}}$$ . The invariant subalgebra $${\hat{\mathcal{D}}^+}$$ under an involution preserving the principal gradation was introduced by Kac, Wang, and Yan. The vacuum $${\hat{\mathcal{D}}^+}$$ -module with central charge $${c \in \mathbb{C}}$$ , and its irreducible quotient $${\mathcal{V}_c}$$ , possess vertex algebra structures, and $${\mathcal{V}_c}$$ has a nontrivial structure if and only if $${c \in \frac{1}{2}\mathbb{Z}}$$ . We show that for each integer $${n > 0}$$ , $${\mathcal{V}_{n/2}}$$ and $${\mathcal{V}_{-n}}$$ are $${\mathcal{W}}$$ -algebras of types $${\mathcal{W}(2, 4,\dots,2n)}$$ and $${\mathcal{W}(2, 4,\dots, 2n^2 + 4n)}$$ , respectively. These results are formal consequences of Weyl's first and second fundamental theorems of invariant theory for the orthogonal group $${{\rm O}(n)}$$ and the symplectic group $${{\rm Sp}(2n)}$$ , respectively. Based on Sergeev's theorems on the invariant theory of $${{\rm Osp}(1, 2n)}$$ we conjecture that $${\mathcal{V}_{-n+1/2}}$$ is of type $${\mathcal{W}(2, 4,\dots, 4n^2 + 8n + 2)}$$ , and we prove this for $${n = 1}$$ . As an application, we show that invariant subalgebras of $${\beta\gamma}$$ -systems and free fermion algebras under arbitrary reductive group actions are strongly finitely generated. [ABSTRACT FROM AUTHOR]

We consider a gapped periodic quantum system with time-reversal symmetry of fermionic (or odd) type, i.e. the time-reversal operator squares to $${-\mathbb{1}}$$ . We investigate the existence of periodic and time-reversal invariant Bloch frames in dimensions 2 and 3. In 2 d, the obstruction to the existence of such a frame is shown to be encoded in a $${\mathbb{Z}_2}$$ -valued topological invariant, which can be computed by a simple algorithm. We prove that the latter agrees with the Fu-Kane index. In 3 d, instead, four $${\mathbb{Z}_2}$$ invariants emerge from the construction, again related to the Fu-Kane-Mele indices. When no topological obstruction is present, we provide a constructive algorithm yielding explicitly a periodic and time-reversal invariant Bloch frame. The result is formulated in an abstract setting, so that it applies both to discrete models and to continuous ones. [ABSTRACT FROM AUTHOR]

Given a group $${G}$$ , and an abelian $${C^*}$$ -algebra $${\mathfrak{A}}$$ , the antihomomorphisms $${\Theta\colon G\rightarrow {\mathrm{Aut}}(\mathfrak{A})}$$ are in one-to-one with those left actions $${\Phi\colon G\times {\mathrm{Spec}}(\mathfrak{A})\rightarrow {\mathrm{Spec}}(\mathfrak{A})}$$ whose translation maps $${\Phi_g}$$ are continuous; whereby continuities of $${\Theta}$$ and $${\Phi}$$ turn out to be equivalent if $${\mathfrak{A}}$$ is unital. In particular, a left action $${\phi\colon G \times X\rightarrow X}$$ can be uniquely extended to the spectrum of a $${C^*}$$ -subalgebra $${\mathfrak{A}}$$ of the bounded functions on $${X}$$ if $${\phi_g^*(\mathfrak{A})\subseteq \mathfrak{A}}$$ holds for each $${g\in G}$$ . In the present paper, we apply this to the framework of loop quantum gravity. We show that, on the level of the configuration spaces, quantization and reduction in general do not commute, i.e., that the symmetry-reduced quantum configuration space is (strictly) larger than the quantized configuration space of the reduced classical theory. Here, the quantum-reduced space has the advantage to be completely characterized by a simple algebraic relation, whereby the quantized reduced classical space is usually hard to compute. [ABSTRACT FROM AUTHOR]

For an anyon model in two spatial dimensions described by a modular tensor category, the topological S-matrix encodes the mutual braiding statistics, the quantum dimensions, and the fusion rules of anyons. It is nontrivial whether one can compute the S-matrix from a single ground state wave function. Here, we define a class of Hamiltonians consisting of local commuting projectors and an associated matrix that is invariant under local unitary transformations. We argue that the invariant is equivalent to the topological S-matrix. The definition does not require degeneracy of the ground state. We prove that the invariant depends on the state only, in the sense that it can be computed by any Hamiltonian in the class of which the state is a ground state. As a corollary, we prove that any local quantum circuit that connects two ground states of quantum double models (discrete gauge theories) with non-isomorphic abelian groups must have depth that is at least linear in the system's diameter. As a tool for the proof, a manifestly Hamiltonian-independent notion of locally invisible operators is introduced. This gives a sufficient condition for a many-body state not to be generated from a product state by any small depth quantum circuit; this is a many-body entanglement witness. [ABSTRACT FROM AUTHOR]

We construct the first nontrivial examples of Calabi-Yau monopoles. Our main interest in these comes from Donaldson and Segal's suggestion (Geometry of Special Holonomy and Related Topics. Surveys in Differential Geometry, vol 16, pp 1-41, ) that it may be possible to define an invariant of certain noncompact Calabi-Yau manifolds from these gauge theoretical equations. We focus on the Stenzel metric on the cotangent bundle of the 3-sphere $${T^* \mathbb{S}^3}$$ and study monopoles under a symmetry assumption. Our main result constructs the moduli of these symmetric monopoles and shows that these are parametrized by a positive real number known as the mass of the monopole. In other words, for each fixed mass we show that there is a unique monopole that is invariant in a precise sense. Moreover, we also study the large mass limit under which we give precise results on the bubbling behavior of our monopoles. Towards the end, an irreducible SU(2) Hermitian-Yang-Mills connection on the Stenzel metric is constructed explicitly. [ABSTRACT FROM AUTHOR]

We develop a representation theory for $${\lambda}$$ -lattices, arising as standard invariants of subfactors, and for rigid C*-tensor categories, including a definition of their universal C*-algebra. We use this to give a systematic account of approximation and rigidity properties for subfactors and tensor categories, like (weak) amenability, the Haagerup property and property (T). We determine all unitary representations of the Temperley-Lieb-Jones $${\lambda}$$ -lattices and prove that they have the Haagerup property and the complete metric approximation property. We also present the first subfactors with property (T) standard invariant and that are not constructed from property (T) groups. [ABSTRACT FROM AUTHOR]

It has previously been suggested that small subsystems of closed quantum systems thermalize under some assumptions; however, this has been rigorously shown so far only for systems with very weak interaction between subsystems. In this work, we give rigorous analytic results on thermalization for translation-invariant quantum lattice systems with finite-range interaction of arbitrary strength, in all cases where there is a unique equilibrium state at the corresponding temperature. We clarify the physical picture by showing that subsystems relax towards the reduction of the global Gibbs state, not the local Gibbs state, if the initial state has close to maximal population entropy and certain non-degeneracy conditions on the spectrumare satisfied.Moreover,we showthat almost all pure states with support on a small energy window are locally thermal in the sense of canonical typicality. We derive our results from a statement on equivalence of ensembles, generalizing earlier results by Lima, and give numerical and analytic finite size bounds, relating the Ising model to the finite de Finetti theorem. Furthermore, we prove that global energy eigenstates are locally close to diagonal in the local energy eigenbasis, which constitutes a part of the eigenstate thermalization hypothesis that is valid regardless of the integrability of the model. [ABSTRACT FROM AUTHOR]

We provide a classification of type AII topological quantum systems in dimension d = 1, 2, 3, 4. Our analysis is based on the construction of a topological invariant, the FKMM-invariant, which completely classifies 'Quaternionic" vector bundles ( a.k.a. 'symplectic" vector bundles) in dimension $${d\leqslant 3}$$ . This invariant takes value in a proper equivariant cohomology theory and, in the case of examples of physical interest, it reproduces the familiar Fu-Kane-Mele index. In the case d = 4 the classification requires a combined use of the FKMM-invariant and the second Chern class. Among the other things, we prove that the FKMM-invariant is a bona fide characteristic class for the category of 'Quaternionic" vector bundles in the sense that it can be realized as the pullback of a universal topological invariant. [ABSTRACT FROM AUTHOR]

Using canonical 1-parameter family of Hermitian connections on the tangent bundle, we provide invariant solutions to the Strominger system on certain complex Lie groups and their quotients. Both flat and non-flat cases are discussed in detail. This paper answers a question proposed by Andreas and Garcia-Fernandez in Comm Math Phys 332(3):1381-1383, . [ABSTRACT FROM AUTHOR]

We consider the C-classification of gapped Hamiltonians introduced in Fannes et al. (Commun Math Phys 144:443-490, ) and Nachtergaele (Commun Math Phys 175:565-606, ) as parent Hamiltonians of translation invariant finitely correlated states. Within this family, we show that the number of edge modes, which is equal at the left and right edge, is the complete invariant. The construction proves that translation invariance of the 'bulk' ground state does not need to be broken to establish C-equivalence, namely that the spin chain does not need to be blocked. [ABSTRACT FROM AUTHOR]

We prove the sharp $${L^1-L^\infty}$$ time-decay estimate for the 2 D -Schrödinger equation with a general family of scaling critical electromagnetic potentials. [ABSTRACT FROM AUTHOR]

Spectral measures provide invariants for braided subfactors via fusion modules. In this paper we study joint spectral measures associated to the rank two Lie group G, including the McKay graphs for the irreducible representations of G and its maximal torus, and fusion modules associated to all known G modular invariants. [ABSTRACT FROM AUTHOR]

In this article, we study complex Gaussian multiplicative chaos. More precisely, we study the renormalization theory and the limit of the exponential of a complex log-correlated Gaussian field in all dimensions (including Gaussian Free Fields in dimension 2). Our main working assumption is that the real part and the imaginary part are independent. We also discuss applications in 2 D string theory; in particular we give a rigorous mathematical definition of the so-called Tachyon fields, the conformally invariant operators in critical Liouville Quantum Gravity with a c = 1 central charge, and derive the original KPZ formula for these fields. [ABSTRACT FROM AUTHOR]

We establish reflection positivity for Gibbs trace states for a class of gauge-invariant, reflection-invariant Hamiltonians describing parafermion interactions on a lattice. We relate these results to recent work in the condensed-matter physics literature. [ABSTRACT FROM AUTHOR]

We introduce symplectic structures on 'Lie pairs' of (real or complex) Lie algebroids as studied by Chen et al. (From Atiyah classes to homotopy Leibniz algebras. , ), encompassing homogeneous symplectic spaces, symplectic manifolds with a $${\mathfrak{g}}$$ -action, and holomorphic symplectic manifolds. We show that to each such symplectic Lie pair are associated Rozansky-Witten-type invariants of three-manifolds and knots, given respectively by weight systems on trivalent and chord diagrams. [ABSTRACT FROM AUTHOR]

We prove the Chern-Gauss-Bonnet Theorem using sigma models whose source supermanifolds have super dimension 0|2. Along the way we develop machinery for understanding manifold invariants encoded by families of 0| δ-dimensional Euclidean field theories and their quantization. [ABSTRACT FROM AUTHOR]

A string theoretic derivation is given for the conjecture of Hausel, Letellier and Rodriguez-Villegas on the cohomology of character varieties with marked points. Their formula is identified with a refined BPS expansion in the stable pair theory of a local root stack, generalizing previous work of the first two authors in collaboration with Pan. Haiman's geometric construction for Macdonald polynomials is shown to emerge naturally in this context via geometric engineering. In particular this yields a new conjectural relation between Macdonald polynomials and refined local orbifold curve counting invariants. The string theoretic approach also leads to a new spectral cover construction for parabolic Higgs bundles in terms of holomorphic symplectic orbifolds. [ABSTRACT FROM AUTHOR]

An invariant state of a quantum Markov semigroup is an equilibrium state if it satisfies a quantum detailed balance condition. In this paper, we introduce a notion of entropy production for faithful normal invariant states of a quantum Markov semigroup on $${\mathcal{B}{\mathsf{h}}}$$ as a numerical index measuring 'how far' they are from equilibrium. The entropy production rate is defined as the derivative of the relative entropy of the one-step forward and backward evolution, in analogy with the classical probabilistic concept. We prove an explicit trace formula expressing the entropy production rate in terms of the completely positive part of the generator of a norm continuous quantum Markov semigroup, showing that it turns out to be zero if and only if a standard quantum detailed balance condition holds. [ABSTRACT FROM AUTHOR]

An irreducible II-subfactor $${A\subset B}$$ is exactly 1-supertransitive if $${B\ominus A}$$ is reducible as an A − A bimodule. We classify exactly 1-supertransitive subfactors with index at most $${6\frac{1}{5}}$$ , leaving aside the composite subfactors at index exactly 6 where there are severe difficulties. Previously, such subfactors were only known up to index $${3+\sqrt{5} \approx 5.23}$$ . Our work is a significant extension, and also shows that index 6 is not an insurmountable barrier. There are exactly three such subfactors with index in $${(3+\sqrt{5}, 6 \frac{1}{5}]}$$ , all with index $${3+2\sqrt{2}}$$ . One of these comes from SO(3) at a root of unity, while the other two appear to be closely related, and are 'braided up to a sign'. This is the published version of . [ABSTRACT FROM AUTHOR]

We show local and global well-posedness results for the Hartree equation where γ is a bounded self-adjoint operator on $${L^2(\mathbb{R}^d)}$$ , ρ( x) = γ( x, x) and w is a smooth short-range interaction potential. The initial datum γ(0) is assumed to be a perturbation of a translation-invariant state γ = f(−Δ) which describes a quantum system with an infinite number of particles, such as the Fermi sea at zero temperature, or the Fermi-Dirac and Bose-Einstein gases at positive temperature. Global well-posedness follows from the conservation of the relative (free) energy of the state γ( t), counted relatively to the stationary state γ. We indeed use a general notion of relative entropy, which allows us to treat a wide class of stationary states f(−Δ). Our results are based on a Lieb-Thirring inequality at positive density and on a recent Strichartz inequality for orthonormal functions, which are both due to Frank, Lieb, Seiringer and the first author of this article. [ABSTRACT FROM AUTHOR]

We formulate a refinement of SU( N) Chern-Simons theory on a three-manifold M via an index in the (2, 0) theory on N M5 branes. The refined Chern-Simons theory is defined on any M with a semi-free circle action. We give an explicit solution of the theory, in terms of a one-parameter refinement of the S and T matrices of Chern-Simons theory, related to the theory of Macdonald polynomials. The ordinary and refined Chern-Simons theory are similar in many ways; for example, the Verlinde formula holds in both. Refined Chern-Simons theory gives rise to new topological invariants of Seifert three-manifolds and torus knots inside them. We conjecture that the invariants are certain indices on knot homology groups. For torus knots in S colored by fundamental representation, the index equals the Poincaré polynomials of the knot homology theory categorifying the HOMFLY polynomial. As a byproduct, we show that our theory on S has a large- N dual which is the refined topological string on $${X=\mathcal{O}(-1) \oplus \mathcal{O}(-1) \rightarrow {\rm I\!P}^1}$$ ; this supports the conjecture by Gukov, Schwarz and Vafa relating the spectrum of BPS states on X to SL knot homology. We also provide a matrix model description of some amplitudes of the refined Chern-Simons theory on S. [ABSTRACT FROM AUTHOR]

In this paper we consider the total space of the canonical bundle of $${\mathbb{P}^2}$$ and we use a proposal by Hosono, together with results of Seidel and Auroux-Katzarkov-Orlov, to deduce the physical mirror equivalence between $${D^b_{\mathbb{P}^2}(K_{\mathbb{P}2})}$$ and the derived Fukaya category of its mirror which assigns the expected central charge to BPS states. By construction, our equivalence is compatible with the mirror map relating the complex and the Kähler moduli spaces and with the computation of Gromov-Witten invariants. [ABSTRACT FROM AUTHOR]

We define one-point disk invariants of a smooth projective Calabi-Yau complete intersection in the presence of an anti-holomorphic involution via localization. We show that these invariants are rational numbers and obtain a formula for them which confirms, in particular, a conjecture by Jinzenji-Shimizu [(Int J Geom Method M 11(1):1456005, ), Conjecture 1]. [ABSTRACT FROM AUTHOR]

We prove an asymptotic bound on the eta invariant of a family of coupled Dirac operators on an odd dimensional manifold. In the case when the manifold is the unit circle bundle of a positive line bundle over a complex manifold, we obtain precise formulas for the eta invariant. [ABSTRACT FROM AUTHOR]