For a quiver with weighted arrows we define gauge-theory K-theoretic Walgebra generalizing the definition of Shiraishi et al., and Frenkel and Reshetikhin. In particular, we show that the qq-character construction of gauge theory presented by Nekrasov is isomorphic to the definition of the W-algebra in the operator formalism as a commutant of screening charges in the free field representation. Besides, we allow arbitrary quiver and expect interesting applications to representation theory of generalized Borcherds-Kac-Moody Lie algebras, their quantum affinizations and associated W-algebras.

We study an exotic state which is localized only at an intersection of edges of a topological material. This "edge-of-edge" state is shown to exist generically. We construct explicitly generic edge-of-edge states in 5-dimensional Weyl semimetals and their dimensional reductions, such as 4-dimensional topological insulators of class A and 3-dimensional chiral topological insulators of class AIII. The existence of the edge-of-edge state is due to a topological charge of the edge states. The notion of the Berry connection is generalized to include the space of all possible boundary conditions, where Chern-Simons forms are shown to be nontrivial.

We introduce quiver gauge theory associated with the non-simply-laced type fractional quiver, and define fractional quiver W-algebras by using construction of [1, 2] with representation of fractional quivers.1 The M-theory brane picture for A-series is rotated by 90 degrees.

We define elliptic generalization of W-algebras associated with arbitrary quiver using our construction [1] with six-dimensional gauge theory.

We systematically study the interesting relations between the quantum elliptic Calogero-Moser system (eCM) and its generalization, and their corresponding supersymmetric gauge theories. In particular, we construct the suitable characteristic polynomial for the eCM system by considering certain orbifolded instanton partition function of the corresponding gauge theory. This is equivalent to the introduction of certain co-dimension two defects. We next generalize our construction to the folded instanton partition function obtained through the so-called "gauge origami" construction and precisely obtain the corresponding characteristic polynomial for the doubled version, named the elliptic double Calogero-Moser (edCM) system. X-function in this case. Finally we demonstrate the validity of X-function by recovering the correct commuting Hamiltonians which are expressed in terms of the Dunkl operators generalized to the edCM system. We should comment here that the connection between edCM systems and the so-called "folded instanton" configuration derived from gauge origami construction was noticed in [7], in this work we firmly established this connection by working out the relevant details in steps.We discuss various future directions in Section 6. We relegate our various definitions of functions and some of the computational details in a series of Appendices.1 This X-function itself is also known as the fundamental q-character of A 0 quiver constructed in [14]. See also [15] for another construction through the quantum toroidal algebra of gl 1 . As mentioned in this paper, we need to consider the orbifolded version of the X-function in order to extract the commuting Hamiltonians of the eCM system. 2 The trigonometric version is studied, e.g., in [20,21]. 3 ゲージ折紙 (日本語); 規範摺紙 (中文 繁體字); 规范折纸 (中文 简体字).(2.12)where the symbol PV means the principal value integral. In 2 → 0 limit, the integration should be dominated by saddle point configurations, which yield:dyG(x αi − y)ρ(y) + log(qR(x αi )) = 0.(2.13)

We investigate the parity-broken phase structure for staggered and naive fermions in the Gross-Neveu model as a toy model of QCD. We consider a generalized staggered Gross-Neveu model including two types of four-point interactions. We use generalized mass terms to split the doublers for both staggered and naive fermions.The phase boundaries derived from the gap equations show that the mass splitting of tastes results in an Aoki phase both in the staggered and naive cases. We also discuss the continuum limit of these models and explore taking the chirally-symmetric limit by fine-tuning a mass parameter and two coupling constants. This supports the idea that in lattice QCD we can derive one-or two-flavor staggered fermions by tuning the mass parameter, which are likely to be less expensive than Wilson fermions in

The orbifold generalization of the partition function, which would describe the gauge theory on the ALE space, is investigated from the combinatorial perspective. It is shown that the root of unity limit q → exp(2πi/k) of the q-deformed partition function plays a crucial role in the orbifold projection while the limit q → 1 applies to R 4 . Then starting from the combinatorial representation of the partition function, a new type of multimatrix model is derived by considering its asymptotic behavior. It is also shown that Seiberg-Witten curve for the corresponding gauge theory arises from the spectral curve of this multi-matrix model. *

We propose a new vertex formalism, called anti-refined topological vertex (anti-vertex for short), to compute the generalized topological string amplitude, which gives rise to the supergroup gauge theory partition function. We show the one-to-many correspondence between the gauge theory and the Calabi-Yau geometry, which is peculiar to the supergroup theory, and the relation between the ordinary vertex formalism and the vertex/anti-vertex formalism through the analytic continuation.