Singular Vector of Ding-Iohara-Miki Algebra and Hall-Littlewood Limit of 5D AGT Conjecture

Abstract

In this thesis, we obtain the formula for the Kac determinant of the algebra arising from the level $N$ representation of the Ding-Iohara-Miki algebra. This formula can be proved by decomposing the level $N$ representation into the deformed $W$-algebra part and the $U(1)$ boson part, and using the screening currents of the deformed $W$-algebra. It is also discovered that singular vectors obtained by its screening currents correspond to the generalized Macdonald functions. Moreover, we investigate the $q\to 0$ limit of five-dimensional AGT correspondence. In this limit, the simplest 5D AGT conjecture is proved, that is, the inner product of the Whittaker vector of the deformed Virasoro algebra coincides with the partition function of the 5D pure gauge theory. Furthermore, the R-Matrix of the Ding-Iohara-Miki algebra is explicitly calculated, and its general expression in terms of the generalized Macdonald functions is conjectured.

PhD thesis, 67 pages with 3 figures

References in corpus (10)

• A_{N-1} conformal Toda field theory correlation functions from conformal N=2 SU(N) quiver gauge theories
• The Refined Topological Vertex
• Asymptotically free N=2 theories and irregular conformal blocks
• Whittaker vectors of the Virasoro algebra in terms of Jack symmetric polynomial
• AGT conjecture and Integrable structure of Conformal field theory for c=1
• Notes on Ding-Iohara algebra and AGT conjecture
• Kernel function and quantum algebras
• On AGT-W Conjecture and q-Deformed W-Algebra
• Quantum toroidal gl(1) and Bethe ansatz
• Norm of the Whittaker vector of the deformed Virasoro algebra