scholarly journals Virasoro Action on Imaginary Verma Modules and the Operator Form of the KZ-Equation

2012 ◽  
Vol 102 (2) ◽  
pp. 125-148
Author(s):  
Ben L. Cox ◽  
Vyacheslav Futorny ◽  
Renato A. Martins
Keyword(s):  
Verma Modules ◽  
Operator Form ◽  
Kz Equation

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 MATRIX ELEMENTS OF Uq[su(2)k] VERTEX OPERATORS VIA BOSONIZATION

1994 ◽  
Vol 09 (25) ◽  
pp. 4431-4447 ◽  
Author(s):  
A. H. BOUGOURZI ◽  
ROBERT A. WESTON
Keyword(s):  
Vertex Operators ◽  
Matrix Elements ◽  
Point Function ◽  
Single Integral ◽  
Kz Equation ◽  
Arbitrary Level

We construct bosonized vertex operators (VO's) and conjugate vertex operators (CVO's) of Uq[ su (2)k] for arbitrary level k and representation j ≤ k/2. Both are obtained directly as two solutions of the defining condition of vertex operators — namely that they intertwine Uq[ su (2)k] modules. We construct the screening charge and present a formula for the n-point function. Specializing to j = 1/2 we construct all VO's and CVO's explicitly. The existence of the CVO allows us to place the calculation of the two-point function on the same footing as k = 1; that is, it is obtained without screening currents and involves only a single integral from the CVO. This integral is evaluated and the resulting function is shown to obey the q-KZ equation and to reduce simply to both the expected k = 1 and q = 1 limits.

scholarly journals 8-Vertex Correlation Functions and Twist Covariance of q-KZ Equation

1998 ◽  
Vol 10 (08) ◽  
pp. 1027-1059
Author(s):  
C. Frønsdal ◽  
A. Galindo
Keyword(s):  
Hopf Algebras ◽  
Casimir Operator ◽  
Intertwining Operators ◽  
Vertex Operators ◽  
Vertex Model ◽  
Standard Quantum ◽  
R Matrix ◽  
Point Correlation ◽  
Point Correlation Function ◽  
Kz Equation

We study the vertex operators Φ(z) associated with standard quantum groups. The element Z=RR t is a "Casimir operator" for quantized Kac–Moody algebras and the quantum Knizhnik–Zamolodchikov (q-KZ) equation is interpreted as the statement :ZΦ(z):=Φ(z). We study the covariance of the q-KZ equation under twisting, first within the category of Hopf algebras, and then in the wider context of quasi Hopf algebras. We obtain the intertwining operators associated with the elliptic R-matrix and calculate the two-point correlation function for the eight-vertex 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.

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