Representation Theory of The W1+∞W_{1+\infty} Algebra

We review the recent development in the representation theory of the $W_{1+\infty}$ algebra. The topics that we concern are, Quasifinite representation, Free field realizations, (Super) Matrix Generalization, Structure of subalgebras such as $W_\infty$ algebra, Determinant formula, Character formula. (Invited talk at ``Quantum Field Theory, Integrable Models and Beyond", YITP, 14-17 February 1994. To appear in Progress of Theoretical Physics Proceedings Supplement.)Comment: 36 pages, LaTeX, RIMS-990, YITP/K-1087, YITP/U-94-25, SULDP-1994-

Macdonald operators and homological invariants of the colored Hopf link

Using a power sum (boson) realization for the Macdonald operators, we investigate the Gukov, Iqbal, Kozcaz and Vafa (GIKV) proposal for the homological invariants of the colored Hopf link, which include Khovanov-Rozansky homology as a special case. We prove the polynomiality of the invariants obtained by GIKV's proposal for arbitrary representations. We derive a closed formula of the invariants of the colored Hopf link for antisymmetric representations. We argue that a little amendment of GIKV's proposal is required to make all the coefficients of the polynomial non-negative integers.Comment: 31 pages. Published version with an additional appendi

The MacMahon R-matrix

We introduce an $R$-matrix acting on the tensor product of MacMahon representations of Ding-Iohara-Miki (DIM) algebra $U_{q,t}(\widehat{\widehat{\mathfrak{gl}}}_1)$. This $R$-matrix acts on pairs of $3d$ Young diagrams and retains the nice symmetry of the DIM algebra under the permutation of three deformation parameters $q$, $t^{-1}$ and $\frac{t}{q}$. 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 $R$-matrix.Comment: 39 page

Irregular conformal blocks, with an application to the fifth and fourth Painlev\'e equations

We develop the theory of irregular conformal blocks of the Virasoro algebra. In previous studies, expansions of irregular conformal blocks at regular singular points were obtained as degeneration limits of regular conformal blocks; however, such expansions at irregular singular points were not clearly understood. This is because precise definitions of irregular vertex operators had not been provided previously. In this paper, we present precise definitions of irregular vertex operators of two types and we prove that one of our vertex operators exists uniquely. Then, we define irregular conformal blocks with at most two irregular singular points as expectation values of given irregular vertex operators. Our definitions provide an understanding of expansions of irregular conformal blocks and enable us to obtain expansions at irregular singular points. As an application, we propose conjectural formulas of series expansions of the tau functions of the fifth and fourth Painlev\'e equations, using expansions of irregular conformal blocks at an irregular singular point.Comment: 26 page