30 research outputs found
On elementary proof of AGT relations from six dimensions
The actual definition of the Nekrasov functions participating in the AGT
relations implies a peculiar choice of contours in the LMNS and Dotsenko-Fateev
integrals. Once made explicit and applied to the original triply-deformed
(6-dimensional) version of these integrals, this approach reduces the AGT
relations to symmetry in q_{1,2,3}, which is just an elementary identity for an
appropriate choice of the integration contour (which is, however, a little
non-traditional). We illustrate this idea with the simplest example of N=(1,1)
U(1) SYM in six dimensions, however, all other cases can be evidently
considered in a completely similar way.Comment: 5 page
Ding-Iohara-Miki symmetry of network matrix models
Ward identities in the most general "network matrix model" can be described
in terms of the Ding-Iohara-Miki algebras (DIM). This confirms an expectation
that such algebras and their various limits/reductions are the relevant
substitutes/deformations of the Virasoro/W-algebra for (q, t) and (q_1, q_2,
q_3) deformed network matrix models. Exhaustive for these purposes should be
the Pagoda triple-affine elliptic DIM, which corresponds to networks associated
with 6d gauge theories with adjoint matter (double elliptic systems). We
provide some details on elliptic qq-characters.Comment: 20 pages, 2 figure
Duality in elliptic Ruijsenaars system and elliptic symmetric functions
We demonstrate that the symmetric elliptic polynomials
originally discovered in the study of generalized Noumi-Shiraishi functions are
eigenfunctions of the elliptic Ruijsenaars-Schneider (eRS) Hamiltonians that
act on the mother function variable (substitute of the Young-diagram
variable ). This means they are eigenfunctions of the dual eRS system.
At the same time, their orthogonal complements in the Schur scalar product,
are eigenfunctions of the elliptic reduction of the Koroteev-Shakirov
(KS) Hamiltonians. This means that these latter are related to the dual eRS
Hamiltonians by a somewhat mysterious orthogonality transformation, which is
well defined only on the full space of time variables, while the coordinates
appear only after the Miwa transform. This observation explains the
difficulties with getting the apparently self-dual Hamiltonians from the double
elliptic version of the KS Hamiltonians.Comment: 15 page
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
Irreducible representations of simple Lie algebras by differential operators
We describe a systematic method to construct arbitrary highest-weight
modules, including arbitrary finite-dimensional representations, for any finite
dimensional simple Lie algebra . The Lie algebra generators are
represented as first order differential operators in variables. All rising
generators are universal in the sense that they do not depend on
representation, the weights enter (in a very simple way) only in the
expressions for the lowering operators . We present explicit formulas
of this kind for the simple root generators of all classical Lie algebras
Spectral Duality Between Heisenberg Chain and Gaudin Model
In our recent paper we described relationships between integrable systems
inspired by the AGT conjecture. On the gauge theory side an integrable spin
chain naturally emerges while on the conformal field theory side one obtains
some special reduced Gaudin model. Two types of integrable systems were shown
to be related by the spectral duality. In this paper we extend the spectral
duality to the case of higher spin chains. It is proved that the N-site GL(k)
Heisenberg chain is dual to the special reduced k+2-points gl(N) Gaudin model.
Moreover, we construct an explicit Poisson map between the models at the
classical level by performing the Dirac reduction procedure and applying the
AHH duality transformation.Comment: 36 page
Spectral Duality in Integrable Systems from AGT Conjecture
We describe relationships between integrable systems with N degrees of
freedom arising from the AGT conjecture. Namely, we prove the equivalence
(spectral duality) between the N-cite Heisenberg spin chain and a reduced gl(N)
Gaudin model both at classical and quantum level. The former one appears on the
gauge theory side of the AGT relation in the Nekrasov-Shatashvili (and further
the Seiberg-Witten) limit while the latter one is natural on the CFT side. At
the classical level, the duality transformation relates the Seiberg-Witten
differentials and spectral curves via a bispectral involution. The quantum
duality extends this to the equivalence of the corresponding Baxter-Schrodinger
equations (quantum spectral curves). This equivalence generalizes both the
spectral self-duality between the 2x2 and NxN representations of the Toda chain
and the famous AHH duality