Mathematics

We introduce the notion of an oscillatory formal distribution supported at a point. We prove that a formal distribution is given by a formal oscillatory integral if and only if it is an oscillatory distribution that has a certain nondegeneracy property. We give an algorithm that recovers the jet of infinite order of the integral kernel of a formal oscillatory integral at the critical point from the corresponding formal distribution. We also prove that a star product ⋆ on a Poisson manifold M is natural in the sense of Gutt and Rawnsley if and only if the formal distribution f⊗g↦(f⋆g)(x) is oscillatory for every x∈M .

The formality morphism \boldsymbol{\mathcal{F}}=\{\mathcal{F}_n , n\geqslant1\} in Kontsevich's deformation quantization is a collection of maps from tensor powers of the differential graded Lie algebra (dgLa) of multivector fields to the dgLa of polydifferential operators on finite-dimensional affine manifolds. Not a Lie algebra morphism by its term \mathcal{F}_1 alone, the entire set \boldsymbol{\mathcal{F}} is an L_\infty -morphism instead. It induces a map of the Maurer-Cartan elements, taking Poisson bi-vectors to deformations \mu_A\mapsto\star_{A[[\hbar]]} of the usual multiplication of functions into associative noncommutative \star -products of power series in \hbar . The associativity of \star -products is then realized, in terms of the Kontsevich graphs which encode polydifferential operators, by differential consequences of the Jacobi identity. The aim of this paper is to illustrate the work of this algebraic mechanism for the Kontsevich \star -products (in particular, with harmonic propagators). We inspect how the Kontsevich weights are correlated for the orgraphs which occur in the associator for \star and in its expansion using Leibniz graphs with the Jacobi identity at a vertex.

We construct a categorification of the modular data associated with every family of unipotent characters of the spetsial complex reflection group G(d,1,n) . The construction of the category follows the decomposition of the Fourier matrix as a Kronecker tensor product of exterior powers of the character table S of the cyclic group of order d . The representation of the quantum universal enveloping algebra of the general linear Lie algebra gl m , with quantum parameter an even root of unity of order 2d , provides a categorical interpretation of the matrix ⋀ m S . We also prove some positivity conjectures of Cuntz at the decategorified level.

The Macdonald process is a stochastic process on the collection of partitions that is a (q,t) -deformed generalization of the Schur process. In this paper, we approach the Macdonald process identifying the space of symmetric functions with a Fock representation of a Heisenberg algebra. By using the free field realization of operators diagonalized by the Macdonald symmetric functions, we propose a method of computing several correlation functions with respect to the Macdonald process. It is well-known that expectation value of several observables for the Macdonald process admit determinantal expression. We find that this determinantal structure is apparent in free field realization of the corresponding operators and, furthermore, it has a natural interpretation in the language of free fermions at the Schur limit. We also propose a generalized Macdonald measure motivated by recent studies on generalized Macdonald functions whose existence relies on the Hopf algebra structure of the Ding--Iohara--Miki algebra.

We define the quantum group D + 4 -- a free quantum version of the demihyperoctahedral group D 4 (the smallest representative of the Coxeter series D ). In order to do so, we construct a free analogue of the property that a 4×4 matrix has determinant one. Such analogues of determinants are usually very hard to define for free quantum groups in general and our result only holds for the matrix size N=4 . The free D + 4 is then defined by imposing this generalized determinant condition on the free hyperoctahedral group H + 4 . Moreover, we give a detailed combinatorial description of the representation category of D + 4 .

Using M-theory in physics, Cho, Gang, and Kim (JHEP 2020, 115 (2020) ) recently outlined a program that connects two parallel subjects of three dimensional manifolds, namely, geometric topology and quantum topology. They suggest that classical topological invariants such as Chern-Simons invariants of SL(2,C) -flat connections and adjoint Reidemeister torsions of a three manifold can be packaged together to produce a (2+1) -topological quantum field theory, which is essentially equivalent to a modular tensor category. It is further conjectured that every modular tensor category can be obtained from a three manifold and a semi-simple Lie group. In this paper, we study this program mathematically, and provide strong support for the feasibility of such a program. The program produces an algorithm to generate the potential modular T -matrix and the quantum dimensions of a candidate modular data. The modular S -matrix follows from essentially a trial-and-error procedure. We find modular tensor categories that realize candidate modular data constructed from Seifert fibered spaces and torus bundles over the circle that reveal many subtleties in the program. We make a number of improvements to the program based on our computations. Our main result is a mathematical construction of a premodular category from each Seifert fibered space with three singular fibers and a family of torus bundles over the circle with Thurston SOL geometry. The premodular categories from Seifert fibered spaces are related to Temperley-Lieb-Jones categories and the ones from torus bundles over the circle are related to metaplectic categories. We conjecture that a resulting premodular category is modular if and only if the three manifold is a Z 2 -homology sphere and condensation of bosons in premodular categories leads to either modular or super-modular categories.

We introduce analogues of the Fubini-Study metrics and the corresponding Levi-Civita connections on quantum projective spaces, following the approach of Beggs and Majid. We define the quantum metrics as two-tensors, symmetric in the appropriate sense, in terms of the differential calculi introduced by Heckenberger and Kolb. We define connections on these calculi and show that they are torsion free and cotorsion free, where the latter condition uses the quantum metric and is a weaker notion of metric compatibility.

In physics, it is believed that the consistency of two dimensional conformal field theory follows from the bootstrap equation. In this paper, we introduce the notion of a full vertex algebra by analyzing the bootstrap equation, which is a "real analytic" generalization of a Z -graded vertex algebra. We also give a mathematical formulation of the consistency of four point correlation functions in two dimensional conformal field theory and prove it for a full vertex algebra with additional assumptions on the conformal symmetry. In particular, we show that the bootstrap equation together with the conformal symmetry implies the consistency of four point correlation functions. As an application, a deformable family of full vertex algebras parametrized by the Grassmanian is constructed, which appears in the toroidal compactification of string theory. This give us examples satisfying the above assumptions.

The purpose of this work is to present a complet categorical point of view of the association between finite dimmensional representations of a compact quantum group and quantum vector bundles with quantum linear connections using M. Durdevichs theory.

We show that if C is a fusion 2 -category in which the endomorphism category of the unit object is Vec or SVec , then the indecomposable objects of C form a finite group.