scholarly journals MacMahon KZ equation for Ding-Iohara-Miki algebra

2021 ◽  
Vol 2021 (4) ◽  
Author(s):  
Panupong Cheewaphutthisakun ◽  
Hiroaki Kanno
Keyword(s):  
Correlation Function ◽  
Spectral Parameter ◽  
Gauge Theories ◽  
Topological Vertex ◽  
Fock Representation ◽  
Intertwining Operator ◽  
Kz Equation ◽  
Zamolodchikov Equation

Abstract We derive a generalized Knizhnik-Zamolodchikov equation for the correlation function of the intertwiners of the vector and the MacMahon representations of Ding-Iohara-Miki algebra. These intertwiners are cousins of the refined topological vertex which is regarded as the intertwining operator of the Fock representation. The shift of the spectral parameter of the intertwiners is generated by the operator which is constructed from the universal R matrix. The solutions to the generalized KZ equation are factorized into the ratio of two point functions which are identified with generalizations of the Nekrasov factor for supersymmetric quiver gauge theories.

scholarly journals Topological vertex for 6d SCFTs with ℤ2-twist

2021 ◽  
Vol 2021 (3) ◽  
Author(s):  
Hee-Cheol Kim ◽  
Minsung Kim ◽  
Sung-Soo Kim
Keyword(s):  
Partition Function ◽  
Gauge Theories ◽  
Outer Automorphism ◽  
Elliptic Genus ◽  
Partition Functions ◽  
Topological Vertex

Abstract We compute the partition function for 6d $$ \mathcal{N} $$ N = 1 SO(2N) gauge theories compactified on a circle with ℤ2 outer automorphism twist. We perform the computation based on 5-brane webs with two O5-planes using topological vertex with two O5-planes. As representative examples, we consider 6d SO(8) and SU(3) gauge theories with ℤ2 twist. We confirm that these partition functions obtained from the topological vertex with O5-planes indeed agree with the elliptic genus computations.

scholarly journals THE QUANTUM KNIZHNIK–ZAMOLODCHIKOV EQUATION ASSOCIATED WITH $U_q(A_2^{(2)})$ FOR |q| = 1

2003 ◽  
Vol 18 (02) ◽  
pp. 225-247 ◽  
Author(s):  
TAKEO KOJIMA
Keyword(s):  
Correlation Function ◽  
Integral Representation ◽  
Critical Point ◽  
Affine Symmetry ◽  
Limiting Case ◽  
Zamolodchikov Equation

We present an integral representation to the quantum Knizhnik–Zamolodchikov equation associated with twisted affine symmetry [Formula: see text] for massless regime |q| = 1. Upon specialization, it leads to a conjectural formula for the correlation function of the Izergin–Korepin model in massless regime |q| = 1. In a limiting case q → -1, our conjectural formula reproduce the correlation function for the Izergin–Korepin model1,2 at critical point q = -1.

scholarly journals More on topological vertex formalism for 5-brane webs with O5-plane

2021 ◽  
Vol 2021 (4) ◽  
Author(s):  
Hirotaka Hayashi ◽  
Rui-Dong Zhu
Keyword(s):  
Gauge Theory ◽  
Partition Function ◽  
Vertex Operator ◽  
Vertex Function ◽  
Gauge Theories ◽  
Partition Functions ◽  
Topological Vertex ◽  
Concrete Form ◽  
Nekrasov Partition Functions ◽  
Chern Simons

Abstract We propose a concrete form of a vertex function, which we call O-vertex, for the intersection between an O5-plane and a 5-brane in the topological vertex formalism, as an extension of the work of [1]. Using the O-vertex it is possible to compute the Nekrasov partition functions of 5d theories realized on any 5-brane web diagrams with O5-planes. We apply our proposal to 5-brane webs with an O5-plane and compute the partition functions of pure SO(N) gauge theories and the pure G2 gauge theory. The obtained results agree with the results known in the literature. We also compute the partition function of the pure SU(3) gauge theory with the Chern-Simons level 9. At the end we rewrite the O-vertex in a form of a vertex operator.

BRAIDING MATRICES OF CONFORMAL BLOCKS AND COSET MODELS

1990 ◽  
Vol 05 (01) ◽  
pp. 211-222 ◽  
Author(s):  
H. ITOYAMA ◽  
A. SEVRIN
Keyword(s):  
Ising Model ◽  
Spectral Parameter ◽  
Field Theories ◽  
Conformal Field Theories ◽  
Conformal Blocks ◽  
Toda System ◽  
Conformal Field ◽  
Coset Models ◽  
Zamolodchikov Equation ◽  
Current Algebras

We explain, from the Knizhnik-Zamolodchikov equation, the coincidence of the braiding matrices of conformal field theories having current algebras with face Boltzmann weights (at infinite spectral parameter) of the corresponding generalized Toda system (GTS). A vertex-height correspondence is introduced in the WZW theory. Braiding matrices of coset models are found to factorize into those of the WZW theories and, as an example, we evaluate those of the Ising model.

scholarly journals Quasi-Hopf twist and elliptic Nekrasov factor

2021 ◽  
Vol 2021 (12) ◽  
Author(s):  
Panupong Cheewaphutthisakun ◽  
Hiroaki Kanno
Keyword(s):  
Commutation Relation ◽  
Gamma Function ◽  
Gauge Theories ◽  
Toroidal Algebra ◽  
Elliptic Gamma Function ◽  
Kz Equation ◽  
Elliptic Quantum

Abstract We investigate the quasi-Hopf twist of the quantum toroidal algebra of $$ {\mathfrak{gl}}_1 $$ gl 1 as an elliptic deformation. Under the quasi-Hopf twist the underlying algebra remains the same, but the coproduct is deformed, where the twist parameter p is identified as the elliptic modulus. Computing the quasi-Hopf twist of the R matrix, we uncover the relation to the elliptic lift of the Nekrasov factor for instanton counting of the quiver gauge theories on ℝ4× T2. The same R matrix also appears in the commutation relation of the intertwiners, which implies an elliptic quantum KZ equation for the trace of intertwiners. We also show that it allows a solution which is factorized into the elliptic Nekrasov factors and the triple elliptic gamma function.

scholarly journals ON THE EQUIVALENCE BETWEEN TOPOLOGICALLY AND NON-TOPOLOGICALLY MASSIVE ABELIAN GAUGE THEORIES

2000 ◽  
Vol 15 (02) ◽  
pp. 121-131 ◽  
Author(s):  
E. HARIKUMAR ◽  
M. SIVAKUMAR
Keyword(s):  
Correlation Function ◽  
Gauge Theory ◽  
Phase Space ◽  
Gauge Theories ◽  
Abelian Gauge ◽  
First Order ◽  
Space Extension ◽  
Canonical Variables ◽  
Canonical Approach ◽  
Total Divergence

We analyze the equivalence between topologically massive gauge theory (TMGT) and different formulations of non-topologically massive gauge theories (NTMGTs) in the canonical approach. The different NTMGTs studied are Stückelberg formulation of (a) a first-order formulation involving one- and two-form fields, (b) Proca theory, and (c) massive Kalb–Ramond theory. We first quantize these reducible gauge systems by using the phase space extension procedure and using it, identify the phase space variables of NTMGTs which are equivalent to the canonical variables of TMGT and show that under this the Hamiltonian also get mapped. Interestingly it is found that the different NTMGTs are equivalent to different formulations of TMGTs which differ only by a total divergence term. We also provide covariant mappings between the fields in TMGT to NTMGTs at the level of correlation function.

scholarly journals 6d/5d exceptional gauge theories from web diagrams

2021 ◽  
Vol 2021 (7) ◽  
Author(s):  
Hirotaka Hayashi ◽  
Hee-Cheol Kim ◽  
Kantaro Ohmori
Keyword(s):  
Gauge Theory ◽  
Gauge Theories ◽  
Partition Functions ◽  
Topological Vertex ◽  
Nekrasov Partition Functions ◽  
Chern Simons ◽  
The Web ◽  
Kaluza Klein

Abstract We construct novel web diagrams with a trivalent or quadrivalent gluing for various 6d/5d theories from certain Higgsings of 6d conformal matter theories on a circle. The theories realized on the web diagrams include 5d Kaluza-Klein theories from circle compactifications of the 6d G2 gauge theory with 4 flavors, the 6d F4 gauge theory with 3 flavors, the 6d E6 gauge theory with 4 flavors and the 6d E7 gauge theory with 3 flavors. The Higgsings also give rise to 5d Kaluza-Klein theories from twisted compactifications of 6d theories including the 5d pure SU(3) gauge theory with the Chern-Simons level 9 and the 5d pure SU(4) gauge theory with the Chern-Simons level 8. We also compute the Nekrasov partition functions of the theories by applying the topological vertex formalism to the newly obtained web diagrams.

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