512 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
Five-dimensional AGT Conjecture and the Deformed Virasoro Algebra
We study an analog of the AGT relation in five dimensions. We conjecture that
the instanton partition function of 5D N=1 pure SU(2) gauge theory coincides
with the inner product of the Gaiotto-like state in the deformed Virasoro
algebra. In four dimensional case, a relation between the Gaiotto construction
and the theory of Braverman and Etingof is also discussed.Comment: 12 pages, reference added, minor corrections (typos, notation
changes, etc
Quantum Algebras and Macdonald Polynomials
We derive a quantum deformation of the algebra and its quantum Miura
transformation, whose singular vectors realize the Macdonald polynomials.Comment: LaTeX file, 17-pages, no-figures, a reference adde
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