Generalized Macdonald Functions on Fock Tensor Spaces and Duality Formula for Changing Preferred Direction
- PDF / 985,666 Bytes
- 70 Pages / 439.37 x 666.142 pts Page_size
- 56 Downloads / 177 Views
Communications in
Mathematical Physics
Generalized Macdonald Functions on Fock Tensor Spaces and Duality Formula for Changing Preferred Direction Masayuki Fukuda1 , Yusuke Ohkubo2 , Jun’ichi Shiraishi2 1 Department of Physics, Faculty of Science, The University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan.
E-mail: [email protected]
2 Graduate School of Mathematical Sciences, University of Tokyo, Komaba, Tokyo 153-8914, Japan.
E-mail: [email protected]; [email protected] Received: 29 March 2019 / Accepted: 29 July 2020 Published online: 16 October 2020 – © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract: An explicit formula is obtained for the generalized Macdonald functions on the N -fold Fock tensor spaces, calculating a certain matrix element of a composition of several screened vertex operators. As an application, we prove the factorization property of the arbitrary matrix elements of the multi-valent intertwining operator (or refined topological vertex operator) associated with the Ding–Iohara–Miki algebra (DIM algebra) with respect to the generalized Macdonald functions, which was conjectured by Awata, Feigin, Hoshino, Kanai, Yanagida and one of the authors. Our proof is based on the combinatorial and analytic properties of the asymptotic eigenfunctions of the ordinary Macdonald operator of A-type, and the Euler transformation formula for Kajihara and Noumi’s multiple basic hypergeometric series. That factorization formula provides us with a reasonable algebraic description of the 5D (K-theoretic) Alday-Gaiotto– Tachikawa (AGT) correspondence, and the interpretation of the invariance under the preferred direction from the point of view of the S L(2, Z) duality of the DIM algebra. 1. Introduction Let U be the DIM algebra [DI,Mi]. As for the definition of the DIM algebra, see Definition 2.1. The central object in the present paper is the intertwining operator V(x) associated with some structure of the U modules. From the point of view of the geometric engineering, or topological vertex construction for the partition functions for the quantum supersymmetric gauge theories, we regard the V(x) as the (multi-valent) topological vertex operator. The intertwining operator V(z) is defined through a certain set of commutation relations with the U-generators. Let X (i) (z)’s (i = 1, . . . , N ) be the generating currents constructed from the standard Drinfeld current of U (Definition 2.6), N acting on the N -fold tensor space Fu = j=1 Fu j of the Fock spaces (For the definition of , see Notation 5.7).
2
M. Fukuda, Y. Ohkubo, J. Shiraishi
Definition 1.1 (Topological vertex). Let V(x) : Fu → Fv be a linear map satisfying the commutation relations x x 1− X (i) (z)V(x) = 1 − (t/q)i V(x)X (i) (z) (i = 1, . . . , N ), (1.1) z z and the normalization condition 0| V(x) |0 = 1. Here |0 (resp. 0|) is the vacuum (resp. dual vacuum) state. We refer to this operator as the Mukadé operator. Mukadé is a Japanese word which means a centipede. The operator V
Data Loading...
