24,406 research outputs found
Quantum Algebraic Approach to Refined Topological Vertex
We establish the equivalence between the refined topological vertex of
Iqbal-Kozcaz-Vafa and a certain representation theory of the quantum algebra of
type W_{1+infty} introduced by Miki. Our construction involves trivalent
intertwining operators Phi and Phi^* associated with triples of the bosonic
Fock modules. Resembling the topological vertex, a triple of vectors in Z^2 is
attached to each intertwining operator, which satisfy the Calabi-Yau and
smoothness conditions. It is shown that certain matrix elements of Phi and
Phi^* give the refined topological vertex C_{lambda mu nu}(t,q) of
Iqbal-Kozcaz-Vafa. With another choice of basis, we recover the refined
topological vertex C_{lambda mu}^nu(q,t) of Awata-Kanno. The gluing factors
appears correctly when we consider any compositions of Phi and Phi^*. The
spectral parameters attached to Fock spaces play the role of the K"ahler
parameters.Comment: 27 page
Consistent Histories and Quantum Reasoning
A system of quantum reasoning for a closed system is developed by treating
non-relativistic quantum mechanics as a stochastic theory. The sample space
corresponds to a decomposition, as a sum of orthogonal projectors, of the
identity operator on a Hilbert space of histories. Provided a consistency
condition is satisfied, the corresponding Boolean algebra of histories, called
a {\it framework}, can be assigned probabilities in the usual way, and within a
single framework quantum reasoning is identical to ordinary probabilistic
reasoning. A refinement rule, which allows a probability distribution to be
extended from one framework to a larger (refined) framework, incorporates the
dynamical laws of quantum theory. Two or more frameworks which are incompatible
because they possess no common refinement cannot be simultaneously employed to
describe a single physical system.Comment: Latex, 31 page
Explicit examples of DIM constraints for network matrix models
Dotsenko-Fateev and Chern-Simons matrix models, which describe Nekrasov
functions for SYM theories in different dimensions, are all incorporated into
network matrix models with the hidden Ding-Iohara-Miki (DIM) symmetry. This
lifting is especially simple for what we call balanced networks. Then, the Ward
identities (known under the names of Virasoro/W-constraints or loop equations
or regularity condition for qq-characters) are also promoted to the DIM level,
where they all become corollaries of a single identity.Comment: 46 page
Remarks on 2+1 Self-dual Chern-Simons Gravity
We study 2+1 Chern-Simons gravity at the classical action level. In
particular we rederive the linear combinations of the ``standard'' and
``exotic'' Einstein actions, from the (anti) self-duality of the ``internal''
Lorentzian indices. The relation to a genuine four-dimensional (anti)self-dual
topological theory greatly facilitates the analysis and its relation to
hyperbolic three-dimensional geometry. Finally a non-abelian vector field
``dual'' action is also obtained.Comment: 16+1 pages, LaTeX file, no figures, clarifications and comments
added, typos corrected and one reference adde
Open boundary Quantum Knizhnik-Zamolodchikov equation and the weighted enumeration of Plane Partitions with symmetries
We propose new conjectures relating sum rules for the polynomial solution of
the qKZ equation with open (reflecting) boundaries as a function of the quantum
parameter and the -enumeration of Plane Partitions with specific
symmetries, with . We also find a conjectural relation \`a la
Razumov-Stroganov between the limit of the qKZ solution and refined
numbers of Totally Symmetric Self Complementary Plane Partitions.Comment: 27 pages, uses lanlmac, epsf and hyperbasics, minor revision
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