626 research outputs found
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 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
Macdonald operators and homological invariants of the colored Hopf link
Using a power sum (boson) realization for the Macdonald operators, we
investigate the Gukov, Iqbal, Kozcaz and Vafa (GIKV) proposal for the
homological invariants of the colored Hopf link, which include
Khovanov-Rozansky homology as a special case. We prove the polynomiality of the
invariants obtained by GIKV's proposal for arbitrary representations. We derive
a closed formula of the invariants of the colored Hopf link for antisymmetric
representations. We argue that a little amendment of GIKV's proposal is
required to make all the coefficients of the polynomial non-negative integers.Comment: 31 pages. Published version with an additional appendi
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
The MacMahon R-matrix
We introduce an -matrix acting on the tensor product of MacMahon
representations of Ding-Iohara-Miki (DIM) algebra
. This -matrix acts on pairs
of Young diagrams and retains the nice symmetry of the DIM algebra under
the permutation of three deformation parameters , and
. We construct the intertwining operator for a tensor product of
the horizontal Fock representation and the vertical MacMahon representation and
show that the intertwiners are permuted using the MacMahon -matrix.Comment: 39 page
A Quantum Deformation of the Virasoro Algebra and the Macdonald Symmetric Functions
A quantum deformation of the Virasoro algebra is defined. The Kac
determinants at arbitrary levels are conjectured. We construct a bosonic
realization of the quantum deformed Virasoro algebra. Singular vectors are
expressed by the Macdonald symmetric functions. This is proved by constructing
screening currents acting on the bosonic Fock space.Comment: 15 pages, latex fil
Selfdual strings and loop space Nahm equations
We give two independent arguments why the classical membrane fields should be
loops. The first argument comes from how we may construct selfdual strings in
the M5 brane from a loop space version of the Nahm equations. The second
argument is that there appears to be no infinite set of finite-dimensional Lie
algebras (such as for any ) that satisfies the algebraic structure
of the membrane theory.Comment: 28 pages, various additional comment
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