118 research outputs found
Superalgebraic methods in the classical theory of representations. Capelli's identity, the Koszul map and the center of the enveloping algebra U(gl(n))
We show how the use of superalgebraic methods sheds new light on some
classical themes of representation theory and it leads to significant
simplifications of traditional proofs
Chern-Simons theory on spherical Seifert manifolds, topological strings and integrable systems
We consider the Gopakumar-Ooguri-Vafa correspondence, relating
Chern-Simons theory at large to topological strings, in the context of
spherical Seifert 3-manifolds. These are quotients of the three-sphere by the free action of a
finite isometry group. Guided by string theory dualities, we propose a large
dual description in terms of both A- and B-twisted topological strings on
(in general non-toric) local Calabi-Yau threefolds. The target space of the
B-model theory is obtained from the spectral curve of Toda-type integrable
systems constructed on the double Bruhat cells of the simply-laced group
identified by the ADE label of . Its mirror A-model theory is realized
as the local Gromov-Witten theory of suitable ALE fibrations on ,
generalizing the results known for lens spaces. We propose an explicit
construction of the family of target manifolds relevant for the correspondence,
which we verify through a large analysis of the matrix model that expresses
the contribution of the trivial flat connection to the Chern-Simons partition
function. Mathematically, our results put forward an identification between the
expansion of the LMO invariant of
and a suitably restricted Gromov-Witten/Donaldson-Thomas
partition function on the A-model dual Calabi-Yau. This expansion, as
well as that of suitable generating series of perturbative quantum invariants
of fiber knots in , is computed by the Eynard-Orantin
topological recursion.Comment: 65 page
Young-Capelli bitableaux, Capelli immanants in U(gl(n)) and the Okounkov quantum immanants
We propose a new approach to a unified study of determinants, permanents,
immanants, (determinantal) bitableaux and symmetrized bitableaux in the
polynomial algebra as well as of their Lie analogues in the
enveloping algebra . This leads to new relevant classes of elements
in : Capelli bitableaux, right Young-Capelli bitableaux and Capelli
immanants. The set of standard Capelli bitableaux and the set of standard right
Young-Capelli bitableaux are bases of , whose action on the
Gordan-Capelli basis of have remarkable properties. Capelli
immanants can be efficiently computed and provide a system of generators of
. The Okounkov quantum immanants are proved to be simple linear
combinations of Capelli immanants. Several examples are provided throughout the
paper.Comment: arXiv admin note: text overlap with arXiv:1608.0678
Whitney algebras and Grassmann's regressive products
Geometric products on tensor powers of an exterior
algebra and on Whitney algebras \cite{crasch} provide a rigorous version of
Grassmann's {\it regressive products} of 1844 \cite{gra1}. We study geometric
products and their relations with other classical operators on exterior
algebras, such as the Hodge operators and the {\it join} and {\it meet}
products in Cayley-Grassmann algebras \cite{BBR, Stew}. We establish encodings
of tensor powers and of Whitney algebras in
terms of letterplace algebras and of their geometric products in terms of
divided powers of polarization operators. We use these encodings to provide
simple proofs of the Crapo and Schmitt exchange relations in Whitney algebras
and of two typical classes of identities in Cayley-Grassmann algebras
Open Topological Strings and Integrable Hierarchies: Remodeling the A-Model
We set up, purely in A-model terms, a novel formalism for the global solution of the open and closed topological A-model on toric Calabi-Yau threefolds. The starting point is to build on recent progress in the mathematical theory of open Gromov-Witten invariants of orbifolds; we interpret the localization formulae as relating D-brane amplitudes to closed string amplitudes perturbed with twisted masses through an analogue of the "loop insertion operatorâ of matrix models. We first generalize this form of open/closed string duality to general toric backgrounds in all chambers of the stringy KĂ€hler moduli space; secondly, we display a neat connection of the (gauged) closed string side to tau functions of 1+1 Hamiltonian integrable hierarchies, and exploit it to provide an effective computation of open string amplitudes. In doing so, we also provide a systematic treatment of the change of flat open moduli induced by a phase transition in the closed moduli space. We test our proposal in detail by providing an extensive number of checks. We also use our formalism to give a localization-based derivation of the Hori-Vafa spectral curves as coming from a resummation of A-model disc instanton
Integrable hierarchies and the mirror model of local CP1
We study structural aspects of the Ablowitz-Ladik (AL) hierarchy in the light
of its realization as a two-component reduction of the two-dimensional Toda
hierarchy, and establish new results on its connection to the Gromov-Witten
theory of local CP1. We first of all elaborate on the relation to the Toeplitz
lattice and obtain a neat description of the Lax formulation of the AL system.
We then study the dispersionless limit and rephrase it in terms of a conformal
semisimple Frobenius manifold with non-constant unit, whose properties we
thoroughly analyze. We build on this connection along two main strands. First
of all, we exhibit a manifestly local bi-Hamiltonian structure of the
Ablowitz-Ladik system in the zero-dispersion limit. Secondarily, we make
precise the relation between this canonical Frobenius structure and the one
that underlies the Gromov-Witten theory of the resolved conifold in the
equivariantly Calabi-Yau case; a key role is played by Dubrovin's notion of
"almost duality" of Frobenius manifolds. As a consequence, we obtain a
derivation of genus zero mirror symmetry for local CP1 in terms of a dual
logarithmic Landau-Ginzburg model.Comment: 27 pages, 1 figur
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