Short star-products for filtered quantizations
Pavel Etingof Massachusetts Institute of Technology (MIT)
7117 - 7128 of 20120 Results
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Categorification of 2d cohomological Hall algebras
Francesco Sala University of Tokyo
N=4 SYM and IIB Supergravity at 1-Loop
Francesco Aprile University of Milan
Formulations of General Relativity (Part 4 of 4)
Kirill Krasnov University of Nottingham
Algebraic structures of T[M3] and T[M4]
Sergei Gukov California Institute of Technology (Caltech) - Division of Physics Mathematics & Astronomy
Formulations of General Relativity (Part 3 of 4)
Kirill Krasnov University of Nottingham
Yangians from cohomological Hall algebras
Ben Davison University of Edinburgh
COHA of surfaces and factorization algebras
Mikhail Kapranov University of Tokyo
Networks of intertwiners, 3d theories and superalgebras
Yegor Zenkevich University of Edinburgh
Formulations of General Relativity (Part 2 of 4)
Kirill Krasnov University of Nottingham
Formulations of General Relativity (Part 1 of 4)
Kirill Krasnov University of Nottingham
Short star-products for filtered quantizations
Pavel Etingof Massachusetts Institute of Technology (MIT)
Let $A$ be a filtered Poisson algebra with Poisson bracket ${ , }$ of degree -2. A star product on $A$ is an associative product $*: A\otimes A\to A$ given by $a*b=ab+\sum_{i\ge 1}C_i(a,b)$, where $C_i$ has degree $-2i$ and $C_1(a,b)-C_1(b,a)={a, b}$. We call the product * "even", if $C_i(a,b)=(-1)^iC_i(b,a)$ for all $i$, and call it "short", if $C_i(a,b)=0$ whenever $i> min(deg(a),deg(b))$. Motivated by three-dimensional $N=4$ superconformal field theory, In 2016 Beem, Peelaers and Rastelli considered short even star-products for homogeneous symplectic singularities (more precisely, hyperK\"ahler cones) and conjectured that that they exist and depend on finitely many parameters. We prove the dependence on finitely many parameters in general and existence for a large class of examples, using the connection of this problem with zeroth Hochschild homology of quantizations suggested by Kontsevich. Beem, Peelaers and Rastelli also computed the first few terms of short quantizations for Kleinian singularities of type A, which were later computed to all orders by Dedushenko, Pufu and Yacoby. We will discuss some generalizations of these results. This is joint work with Eric Rains and Douglas Stryker.Quantum Statistical Comparison, Quantum Majorization, and Their Applications to Generalized Resource Theories
Francesco Buscemi Nagoya University
The theory of statistical comparison was formulated (chiefly by David Blackwell in the 1950s) in order to extend the theory of majorization to objects beyond probability distributions, like multivariate statistical models and stochastic transitions, and has played an important role in mathematical statistics ever since. The central concept in statistical comparison is the so-called "information ordering," according to which information need not always be a totally ordered quantity, but often takes on a multi-faceted form whose content may vary depending on its use. In this talk, after reviewing the basic ideas of statistical comparison with an emphasis on their operational character, I will discuss various generalizations to quantum theory (and beyond). I will then argue that quantum statistical comparison provides a natural framework, somehow complementary to semi-definite programming, to study quantum resource theories, with explicit examples given by the resource theories of quantum nonlocality, quantum communication, and quantum thermodynamics.
Categorification of 2d cohomological Hall algebras
Francesco Sala University of Tokyo
Let $\mathcal{M}$ denote the moduli stack of either coherent sheaves on a smooth projective surface or Higgs sheaves on a smooth projective curve $X$. The convolution algebra structure on the Borel-Moore homology of $\mathcal{M}$ is an instance of two-dimensional cohomological Hall algebras. In the present talk, I will describe a full categorication of the cohomological Hall algebra of $\mathcal{M}$. This is achieved by exhibiting a derived enhancement of $\mathcal{M}$. Furthermore, this method applies also to several other moduli stacks, such as the moduli stack of vector bundles with flat connections on $X$ and the moduli stack of finite-dimensional representations of the fundamental group of $X$. In the second part of the talk, I will focus on the case of curves and discuss some relations between the Betti, de Rham, and Dolbeaut categorified cohomological Hall algebras. This is based on a work in progress with Mauro Porta.N=4 SYM and IIB Supergravity at 1-Loop
Francesco Aprile University of Milan
Using knowledge about the spectrum of operators in N=4 SYM, consistency of OPE, and analytic bootstrap techniques, I will obtain 1- loop corrections for 4-pt functions of single particle half-BPS operators of IIB supergravity on AdS_5\timesS^5. Along the way, I will discuss a general formula for the leading anomalous dimension of all double-trace operators in the supergravity regime.
Formulations of General Relativity (Part 4 of 4)
Kirill Krasnov University of Nottingham
The goal of this series is to collect various different formulations of General Relativity, with emphasis on four spacetime dimensions and formulations that use differential forms. The (non-exhaustive) list of formulations to be covered is per this plan:
Lecture 1): Motivations, followed by the usual Einstein-Hilbert to start with, first order Palatini, second order pure affine connection Eddington-Schroedinger.
Lecture 2) Cartan's geometry of soldering. First order Einstein-Cartan tetrad formulation, second order pure spin connection formulation, MacDowell-Mansouri formulation.
Lecture 3) Non-chiral BF-type formulations. Explicit pure spin connection Lagrangian, field redefinitions, BF plus potential for the 2-form field formulation.
Lecture 4) Chiral formulations in four dimensions. Chiral Einstein-Cartan, Plebanski formulation, pure SU(2) connection formulation, SU(2) BF plus potential for the 2-form field. Self-dual gravity. Concluding remarks.
Algebraic structures of T[M3] and T[M4]
Sergei Gukov California Institute of Technology (Caltech) - Division of Physics Mathematics & Astronomy
The talk will focus on tensor categories associated with 3d N=2 theories and chiral algebras associated with 2d N=(0,2) theories, as well as their combinations that involve 3d N=2 theories "sandwiched" by half-BPS boundary conditions and interfaces. Such situations, originally studied in a joint work with A.Gadde and P.Putrov, have a variety of applications, including applications to topology of 3-manifolds and 4-manifolds where Kirby moves translate into novel dualities of 3d N=2 and 2d N=(0,2) theories and where the corresponding algebraic structures can be related to COHAs. After reviewing some elements of that story going back to 2013, I will focus on the latest developments in the area of "3d Modularity" where mock Jacobi forms, SL(2,Z) Weil representations, quantum modular forms, non-semisimple modular tensor categories, and chiral algebras of logarithmic CFTs make a surprising appearance (based on recent and ongoing work with M.Cheng, S.Chun, F.Ferrari, S.Harrison and B.Feigin).Formulations of General Relativity (Part 3 of 4)
Kirill Krasnov University of Nottingham
The goal of this series is to collect various different formulations of General Relativity, with emphasis on four spacetime dimensions and formulations that use differential forms. The (non-exhaustive) list of formulations to be covered is per this plan:
Lecture 1): Motivations, followed by the usual Einstein-Hilbert to start with, first order Palatini, second order pure affine connection Eddington-Schroedinger.
Lecture 2) Cartan's geometry of soldering. First order Einstein-Cartan tetrad formulation, second order pure spin connection formulation, MacDowell-Mansouri formulation.
Lecture 3) Non-chiral BF-type formulations. Explicit pure spin connection Lagrangian, field redefinitions, BF plus potential for the 2-form field formulation.
Lecture 4) Chiral formulations in four dimensions. Chiral Einstein-Cartan, Plebanski formulation, pure SU(2) connection formulation, SU(2) BF plus potential for the 2-form field. Self-dual gravity. Concluding remarks.
Yangians from cohomological Hall algebras
Ben Davison University of Edinburgh
I will explain various features of the preprojective CoHA, a kind of universal algebra of correspondences generalising the algebras of endomorphisms of cohomology of quiver varieties considered by Nakajima. In particular I will focus on features of this algebra that become visible after viewing it as a dimensional reduction of the Kontsevich-Soibelman) critical CoHA associated to a related 3-dimensional Calabi-Yau category. Many nice features emerge from this view, e.g. an embedding in a related shuffle algebra, a formula for the graded dimension of the algebra, a flat non cocommutative deformation, a cocommutative coproduct, a geometric doubling procedure, the PBW theorem, an isomorphism with the Yangian considered by Maulik and Okounkov... I will focus on the perverse filtration, which is the key feature of my joint work with Sven Meinhardt, and gives rise to most of the aboveCOHA of surfaces and factorization algebras
Mikhail Kapranov University of Tokyo
This is a report on joint work with E. Vasserot. For a smooth quasiprojective surface S we consider the stack Coh(S) of coherent sheaves on S with compact support and make the Borel-Moore homology of Coh(S) into an associative algebra using a derived version of the Hall multiplication diagram. Subcategories in Coh(S) given by various conditions on dimension of support give rise to various types of Hecke operators. In particular, we consider the COHA R(S) of sheaves with 0-dimensional support and show that its chain level lift is a factorization coalgebra on S with values in homotopy associative algebras. This allows us to find the size (graded dimension) of R(S).Networks of intertwiners, 3d theories and superalgebras
Yegor Zenkevich University of Edinburgh
Refined topological vertex formalism allows one to conveniently compute partition functions of topological strings on toric CY backgrounds. These partition functions reproduce instanton partition functions of 5d N=1 gauge theories, obtained from the CY by the geometric engineering procedure. In the algebraic language the vertices can be described as intertwiners of Fock representations of a quantum toroidal algebra. I will present a «Higgsed» version of refined topological vertex formalism which computes vortex partition functions of certain N=2* 3d theories, and show how it naturally arises in the algebraic approach. The new formalism gives a streamlined way to write down the screening charges of a general class of q-deformed W-algebras, including those associated with superalgebras. The obtained partition functions are automatically eigenfunctions of Ruijsenaars-Schneider Hamiltonians or their supersymmetric generalizations.Formulations of General Relativity (Part 2 of 4)
Kirill Krasnov University of Nottingham
The goal of this series is to collect various different formulations of General Relativity, with emphasis on four spacetime dimensions and formulations that use differential forms. The (non-exhaustive) list of formulations to be covered is per this plan:
Lecture 1): Motivations, followed by the usual Einstein-Hilbert to start with, first order Palatini, second order pure affine connection Eddington-Schroedinger.
Lecture 2) Cartan's geometry of soldering. First order Einstein-Cartan tetrad formulation, second order pure spin connection formulation, MacDowell-Mansouri formulation.
Lecture 3) Non-chiral BF-type formulations. Explicit pure spin connection Lagrangian, field redefinitions, BF plus potential for the 2-form field formulation.
Lecture 4) Chiral formulations in four dimensions. Chiral Einstein-Cartan, Plebanski formulation, pure SU(2) connection formulation, SU(2) BF plus potential for the 2-form field. Self-dual gravity. Concluding remarks.
Formulations of General Relativity (Part 1 of 4)
Kirill Krasnov University of Nottingham
The goal of this series is to collect various different formulations of General Relativity, with emphasis on four spacetime dimensions and formulations that use differential forms. The (non-exhaustive) list of formulations to be covered is per this plan:
Lecture 1): Motivations, followed by the usual Einstein-Hilbert to start with, first order Palatini, second order pure affine connection Eddington-Schroedinger.
Lecture 2) Cartan's geometry of soldering. First order Einstein-Cartan tetrad formulation, second order pure spin connection formulation, MacDowell-Mansouri formulation.
Lecture 3) Non-chiral BF-type formulations. Explicit pure spin connection Lagrangian, field redefinitions, BF plus potential for the 2-form field formulation.
Lecture 4) Chiral formulations in four dimensions. Chiral Einstein-Cartan, Plebanski formulation, pure SU(2) connection formulation, SU(2) BF plus potential for the 2-form field. Self-dual gravity. Concluding remarks.