arXiv subjects
Publications and source records attributed to Rui-Dong Zhu.
We count the Bethe states of quantum integrable models with twisted boundary conditions using the Witten index of 2d supersymmetric gauge theories. For multi-component models solvable by the nested Bethe ansatz, the result is a novel restricted occupancy problem. For the SU(3) spin chain and the t-J model, we propose formulae for the solution count on singular loci in the space of twist parameters.
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 arXiv:1709.01928. 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 $G_2$ 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.
In this article, we extend the work of arXiv:0901.4744 to a Bethe/Gauge correspondence between 2d (or resp. 3d) SO/Sp gauge theories and open XXX (resp. XXZ) spin chains with diagonal boundary conditions. The case of linear quiver gauge theories is also considered.
We interpret aspects of the Schur indices, that were identified with characters of highest weight modules in Virasoro $(p,p')=(2,2k+3)$ minimal models for $k=1,2,\dots$, in terms of paths that first appeared in exact solutions in statistical mechanics. From that, we propose closed-form fermionic sum expressions, that is, $q, t$-series with manifestly non-negative coefficients, for two infinite-series of Macdonald indices of $(A_1,A_{2k})$ Argyres-Douglas theories that correspond to $t$-refinements of Virasoro $(p,p')=(2,2k+3)$ minimal model characters, and two rank-2 Macdonald indices that correspond to $t$-refinements of $\mathcal{W}_3$ non-unitary minimal model characters. Our proposals match with computations from 4D $\mathcal{N} = 2$ gauge theories \textit{via} the TQFT picture, based on the work of J Song arXiv:1509.06730.
We test in $(A_{n-1},A_{m-1})$ Argyres-Douglas theories with $\mathrm{gcd}(n,m)=1$ the proposal of Song's in arXiv:1612.08956 that the Macdonald index gives a refined character of the dual chiral algebra. In particular, we extend the analysis to higher rank theories and Macdonald indices with surface operator, via the TQFT picture and Gaiotto-Rastelli-Razamat's Higgsing method. We establish the prescription for refined characters in higher rank minimal models from the dual $(A_{n-1},A_{m-1})$ theories in the large $m$ limit, and then provide evidence for Song's proposal to hold (at least) in some simple modules (including the vacuum module) at finite $m$. We also discuss some observed mismatch in our approach.
The topological vertex formalism for 5d $\mathcal{N}=1$ gauge theories is not only a convenient tool to compute the instanton partition function of these theories, but it is also accompanied by a nice algebraic structure that reveals various kinds of nice properties such as dualities and integrability of the underlying theories. The usual refined topological vertex formalism is derived for gauge theories with $A$-type quiver structure (and $A$-type gauge groups). In this article, we propose a construction with a web of vertex operators for all $ABCDEFG$-type and affine quivers by introducing several new vertices into the formalism, based on the reproducing of known instanton partition functions and qq-characters for these theories.
We compute the Schur index of Argyres-Douglas theories of type $(A_{N-1},A_{M-1})$ with surface operators inserted, via the Higgsing prescription proposed by D. Gaiotto, L. Rastelli and S. S. Razamat. These surface operators are obtained by turning on position-dependent vacuum expectation values of operators in a UV theory which can flow to the Argyres-Douglas theories. We focus on two series of $(A_{N-1},A_{M-1})$ theories; one with ${\rm gcd}(N,M)=1$ and the other with $M=N(k-1)$ for an integer $k\geq 2$. For these two series of Argyres-Douglas theories, our results are identical to the characters of non-vacuum modules of the associated 2d chiral algebras, which explicitly confirms a remarkable correspondence recently discovered by C. Cordova, D. Gaiotto and S.-H. Shao.
We write down a vertical representation for the elliptic Ding-Iohara-Miki algebra, and construct an elliptic version of the refined topological vertex of Awata, Feigin and Shiraishi. We show explicitly that this vertex reproduces the elliptic genus of M-strings, and that it is an intertwiner of the algebra.
We derive a one-parameter deformation of the refined topological vertex that, when used to compute non-periodic web diagrams, reproduces the six-dimensional topological string partition functions that are computed using the refined vertex and periodic web diagrams.
Reflection states are introduced in the vertical and horizontal modules of the Ding-Iohara-Miki (DIM) algebra (quantum toroidal $\mathfrak{gl}_1$). Webs of DIM representations are in correspondence with $(p,q)$-web diagrams of type IIB string theory, under the identification of the algebraic intertwiner of Awata, Feigin and Shiraishi with the refined topological vertex. Extending the correspondence to the vertical reflection states, it is possible to engineer the $\mathcal{N}=1$ quiver gauge theory of D-type (with unitary gauge groups). In this way, the Nekrasov instanton partition function is reproduced from the evaluation of expectation values of intertwiners. This computation leads to the identification of the vertical reflection state with the orientifold plane of string theory. We also provide a translation of this construction in the Iqbal-Kozcaz-Vafa refined topological vertex formalism.
The integrability of 4d $\mathcal{N}=2$ gauge theories has been explored in various contexts, for example the Seiberg-Witten curve and its quantization. Recently, Maulik and Okounkov proposed that an integrable lattice model is associated with the gauge theory, through an R-matrix, to which we refer as MO's R-matrix in this paper, constructed in the instanton moduli space. In this paper, we study the R-matrix using the Ding-Iohara-Miki (DIM) algebra. We provide a concrete boson realization of the universal R-matrix in DIM and show that the defining conditions for MO's R-matrix can be derived from this free boson oscillator expression. Several consistency checks for the oscillator expression are also performed.
Instanton partition functions of $\mathcal{N}=1$ 5d Super Yang-Mills reduced on $S^1$ can be engineered in type IIB string theory from the $(p,q)$-branes web diagram. To this diagram is superimposed a web of representations of the Ding-Iohara-Miki (DIM) algebra that acts on the partition function. In this correspondence, each segment is associated to a representation, and the (topological string) vertex is identified with the intertwiner operator constructed by Awata, Feigin and Shiraishi. We define a new intertwiner acting on the representation spaces of levels $(1,n)\otimes(0,m)\to(1,n+m)$, thereby generalizing to higher rank $m$ the original construction. It allows us to use a folded version of the usual $(p,q)$-web diagram, bringing great simplifications to actual computations. As a result, the characterization of Gaiotto states and vertical intertwiners, previously obtained by some of the authors, is uplifted to operator relations acting in the Fock space of horizontal representations. We further develop a method to build qq-characters of linear quivers based on the horizontal action of DIM elements. While fundamental qq-characters can be built using the coproduct, higher ones require the introduction of a (quantum) Weyl reflection acting on tensor products of DIM generators.
The instanton partition functions of $\mathcal{N}=1$ 5d super Yang-Mills are built using elements of the representation theory of quantum $\mathcal{W}_{1+\infty}$ algebra: Gaiotto state, intertwiner, vertex operator. This algebra is also known under the names of Ding-Iohara-Miki and quantum toroidal $\widehat{\mathfrak{gl}}(1)$ algebra. Exploiting the explicit action of the algebra on the partition function, we prove the regularity of the 5d qq-characters. These characters provide a solution to the Schwinger-Dyson equations, and they can also be interpreted as a quantum version of the Seiberg-Witten curve.
Recently an orthogonal basis of $\mathcal{W}_N$-algebra (AFLT basis) labeled by $N$-tuple Young diagrams was found in the context of 4D/2D duality. Recursion relations among the basis are summarized in the form of an algebra SH$^c$ which is universal for any $N$. We show that it has an $\mathfrak{S}_3$ automorphism which is referred to as triality. We study the level-rank duality between minimal models, which is a special example of the automorphism. It is shown that the nonvanishing states in both systems are described by $N$ or $M$ Young diagrams with the rows of boxes appropriately shuffled. The reshuffling of rows implies there exists partial ordering of the set which labels them. For the simplest example, one can compute the partition functions for the partially ordered set (poset) explicitly, which reproduces the Rogers-Ramanujan identities. We also study the description of minimal models by SH$^c$. Simple analysis reproduces some known properties of minimal models, the structure of singular vectors and the $N$-Burge condition in the Hilbert space.
Recently, Maulik and Okounkov proposed an integrable lattice model where the degree of freedom at each site is identical to the Hilbert space of free boson in two dimensions. We give a brief review of their construction and explain the relation with $\mathcal{W}_n$ algebra and Calogero-Sutherland model. As a generalization, we examine the Yangian associated with $\mathcal{N}=1$ superconformal algebra which describes a supersymmetric extension of Calogero-Sutherland model and compare it with the literature.