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Register a new userWeb Construction of ABCDEFG and Affine Quiver Gauge Theories
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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.
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Let $mathscr{A}_q$ be the $K$-theoretic Coulomb branch of a $3d$ $mathcal{N}=4$ quiver gauge theory with quiver $Gamma$, and $mathscr{A}_q subseteq mathscr{A}_q$ be the subalgebra generated by the equivariant $K$-theory of a point together with the dressed minuscule monopole operators $M_{varpi_{i,1},f}$ and $M_{varpi^*_{i,1},f}$. In this paper, we construct an associated cluster algebra quiver $mathcal{Q}_Gamma$ and provide an embedding of the subalgebra $mathscr{A}_q$ into the quantized algebra of regular functions on the corresponding cluster variety.
Reflexive polygons have been extensively studied in a variety of contexts in mathematics and physics. We generalize this programme by looking at the 45 different lattice polygons with two interior points up to SL(2,$mathbb{Z}$) equivalence. Each corresponds to some affine toric 3-fold as a cone over a Sasaki-Einstein 5-fold. We study the quiver gauge theories of D3-branes probing these cones, which coincide with the mesonic moduli space. The minimum of the volume function of the Sasaki-Einstein base manifold plays an important role in computing the R-charges. We analyze these minimized volumes with respect to the topological quantities of the compact surfaces constructed from the polygons. Unlike reflexive polytopes, one can have two fans from the two interior points, and hence give rise to two smooth varieties after complete resolutions, leading to an interesting pair of closely related geometries and gauge theories.
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 discuss in detail the problem of counting BPS gauge invariant operators in the chiral ring of quiver gauge theories living on D-branes probing generic toric CY singularities. The computation of generating functions that include counting of baryonic operators is based on a relation between the baryonic charges in field theory and the Kaehler moduli of the CY singularities. A study of the interplay between gauge theory and geometry shows that given geometrical sectors appear more than once in the field theory, leading to a notion of multiplicities. We explain in detail how to decompose the generating function for one D-brane into different sectors and how to compute their relevant multiplicities by introducing geometric and anomalous baryonic charges. The Plethystic Exponential remains a major tool for passing from one D-brane to arbitrary number of D-branes. Explicit formulae are given for few examples, including C^3/Z_3, F_0, and dP_1.
We compute the ${cal N}=2$ supersymmetric partition function of a gauge theory on a four-dimensional compact toric manifold via equivariant localization. The result is given by a piecewise constant function of the Kahler form with jumps along the walls where the gauge symmetry gets enhanced. The partition function on such manifolds is written as a sum over the residues of a product of partition functions on $mathbb{C}^2$. The evaluation of these residues is greatly simplified by using an abstruse duality that relates the residues at the poles of the one-loop and instanton parts of the $mathbb{C}^2$ partition function. As particular cases, our formulae compute the $SU(2)$ and $SU(3)$ {it equivariant} Donaldson invariants of $mathbb{P}^2$ and $mathbb{F}_n$ and in the non-equivariant limit reproduce the results obtained via wall-crossing and blow up methods in the $SU(2)$ case. Finally, we show that the $U(1)$ self-dual connections induce an anomalous dependence on the gauge coupling, which turns out to satisfy a $mathcal{N}=2$ analog of the $mathcal{N}=4$ holomorphic anomaly equations.
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