Explicit examples of DIM constraints for network matrix models
Received: May
Explicit examples of DIM constraints for network matrix models
Hidetoshi Awata 1 2 6 7 8 9 10 11
Hiroaki Kanno 1 2 4 6 7 8 9 10 11
Takuya Matsumoto 1 2 6 7 8 9 10 11
Andrei Mironov 0 1 2 3 5 7 8 9 10 11
Alexei Morozov 0 1 2 3 7 8 9 10 11
Andrey Morozov 0 1 2 3 7 8 9 10 11
g Yusuke Ohkubo 1 2 6 7 8 9 10 11
Yegor Zenkevich 1 2 7 8 9 10 11
Field Theories, Topological Strings
0 National Research Nuclear University MEPhI
1 Leninsky pr. , 53, Moscow 119991 , Russia
2 Nagoya , 464-8602 , Japan
3 Institute for Information Transmission Problems
4 KMI, Nagoya University
5 Theory Department, Lebedev Physics Institute
6 Graduate School of Mathematics, Nagoya University
7 60-letiya Oktyabrya pr. , 7a, Moscow 117312 , Russia
8 Bratiev Kashirinyh , 129, Chelyabinsk 454001 , Russia
9 Kashirskoe sh. , 31, Moscow 115409 , Russia
10 Bol.Karetny , 19 (1), Moscow 127994 , Russia
11 Bol.Cheremushkinskaya , 25, Moscow 117218 , Russia
Dotsenko-Fateev and Chern-Simons matrix models, which describe Nekrasov functions for SYM theories in di erent 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.
Conformal and W Symmetry; Supersymmetric gauge theory; Topological
1 Introduction 2
Basic example: theme with variations
The main theme
Variation I: matrix elements in the free- eld theory
Variation II: generating functions
Variation III: DF model
Variation IV: multi- eld case
Variation V: Chern-Simons (CS) model
Variation VI: correlators with vertex operators
Variation VII: Nekrasov functions
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3.1
3.2
3.3
3.4
3.5
3.6
3.7
Variation VIII: network model level. Network as a Feynman diagram
Variation IX: balanced network model
2.10 Variation X: q-deformation 2.11 Variation XI: q; t; : : :-deformations 2.12 Variation XII: -deformation to non-unit Virasoro central charge
3
DIM calculus for balanced network model
DIM algebra
Bosonization in the case of special slopes
Relation to topological vertex
Building screening charges and vertex operators
Network partition function
Examples of conformal blocks Compacti ed network and the a ne screening operator
4
5
6
The action of Virasoro and DIM(gl1)
Vertical action of DIM
Conclusion
A Properties of the DIM algebras and their limits
A.1 Constructing DIM(gl1) from W1+1 algebra
A.2 Elliptic DIM(gl1) algebra
A.3 Rank > 1: DIM(gln) = quantum toroidal algebra of type gln
A.4 A ne Yangian of gl1 [139]
{ 1 {
u sitting on the horizontal leg of the topological
vertex (denoted by the dashed line) is the same as its action on the product of two representations
| the \vertical" jv and \diagonal"
=
uv. b) Appropriate contraction of two intertwiners is also
an intertwiner. This gives the vertex operator of the corresponding conformal eld theory with
deformed Virasoro symmetry, corresponding to a single vertical brane in gure 2.
1
Introduction
Nekrasov functions, describing instanton corrections in supersymmetric Yang-Mills
theories [1]{[11], and AGT related conformal blocks [12{16] possess rich symmetries that can
be separated into large and in nitesimal. The former describe dualities between di
erent models, while the latter de ne equations on the partition functions in each particular
case. They are also known as \Virasoro constraints" [17, 18] for associated conformal or
Dotsenko-Fateev (DF) matrix models [19{30], which are further promoted to network
matrix models [31{34], looking like convolutions of re ned topological vertices [35{37] and
possessing direct topological string interpretation.
As conjectured in a number of papers throughout recent years [38]{[49] and
recently summarized in [50], in full generality the symmetry underlying the AGT
correspondence [51{53], is the Ding-Iohara-Miki algebra (DIM) [54]{[69], in particular, the
innitesimal Ward identities are controlled by DIM from which the (deformed) Virasoro and
WK emerge as subalgebras in particular representations. In other words, the full symmetry
of the Seiberg-Witten theory seems to be the Pagoda triple-a ne elliptic DIM algebra (not
yet fully studied and even de ned), and particular models (brane patterns or Calabi-Yau
toric varieties labeled by integrable systems a la [3, 4]) are associated with its particular
representations. The ordinary DF matrix models arise when one speci es \vertical" and
\horizontal" directions, then convolutions of topological vertices can be split into vertex
operators and screening charges, and the DIM algebra constraints can be attributed in the
usual way [70{79] to commutativity of screening charges with the action of the algebra in
the given representation. Dualities are associated wit (...truncated)