Instanton Partition Function Research Articles

Dessins d\u2019enfants, Seiberg-Witten curves and conformal blocks

We show how to map Grothendieck’s dessins d’enfants to algebraic curves as Seiberg-Witten curves, then use the mirror map and the AGT map to obtain the corresponding 4d mathcal{N} = 2 supersymmetric instanton partition functions and 2d Virasoro conformal blocks. We explicitly demonstrate the 6 trivalent dessins with 4 punctures on the sphere. We find that the parametrizations obtained from a dessin should be related by certain duality for gauge theories. Then we will discuss that some dessins could correspond to conformal blocks satisfying certain rules in different minimal models.

Argyres-Douglas theories, S-duality and AGT correspondence

We propose a Nekrasov-type formula for the instanton partition functions of four-dimensional mathcal{N} = 2 U(2) gauge theories coupled to (A1, D2n) Argyres-Douglas theories. This is carried out by extending the generalized AGT correspondence to the case of U(2) gauge group, which requires us to define irregular states of the direct sum of Virasoro and Heisenberg algebras. Using our formula, one can evaluate the contribution of the (A1, D2n) theory at each fixed point on the U(2) instanton moduli space. As an application, we evaluate the instanton partition function of the (A3, A3) theory to find it in a peculiar relation to that of SU(2) gauge theory with four fundamental flavors. From this relation, we read off how the S-duality group acts on the UV gauge coupling of the (A3, A3) theory.

6D strings and exceptional instantons

We propose new ADHM-like methods to compute the Coulomb branch instanton partition functions of 5d and 6d supersymmetric gauge theories, with certain exceptional gauge groups or exceptional matters. We study $G_2$ theories with $n_{\bf 7}\leq 3$ matters in ${\bf 7}$, and $SO(7)$ theories with $n_{\bf 8}\leq 4$ matters in the spinor representation ${\bf 8}$. We also study the elliptic genera of self-dual instanton strings of 6d SCFTs with exceptional gauge groups or matters, including all non-Higgsable atomic SCFTs with rank $2$ or $3$ tensor branches. Some of them are tested with topological vertex calculus. We also explore a D-brane-based method to study instanton particles of 5d $SO(7)$ and $SO(8)$ gauge theories with matters in spinor representations, which further tests our ADHM-like proposals.

On the quantization of Seiberg-Witten geometry

We propose a double quantization of four-dimensional mathcal{N} = 2 Seiberg-Witten geometry, for all classical gauge groups and a wide variety of matter content. This can be understood as a set of certain non-perturbative Schwinger-Dyson identities, following the program initiated by Nekrasov [1]. The construction relies on the computation of the instanton partition function of the gauge theory on the so-called Ω-background on ℝ4, in the presence of half-BPS codimension 4 defects. The two quantization parameters are identified as the two parameters of this background. The Seiberg-Witten curve of each theory is recovered in the flat space limit. Whenever possible, we motivate our construction from type IIA string theory.

Quantum integrable systems from supergroup gauge theories

In this note, we establish several interesting connections between the super- group gauge theories and the super integrable systems, i.e. gauge theories with supergroups as their gauge groups and integrable systems defined on superalgebras. In particular, we construct the super-characteristic polynomials of super-Toda lattice and elliptic double Calogero-Moser system by considering certain orbifolded instanton partition functions of their corresponding supergroup gauge theories. We also derive an exotic generalization of \U0001d530\U0001d529(2) XXX spin chain arising from the instanton partition function of SQCD with super- gauge group, and study its Bethe ansatz equation.

Spectra of elliptic potentials and supersymmetric gauge theories

We study a relation between asymptotic spectra of the quantum mechanics problem with a four components elliptic function potential, the Darboux-Treibich-Verdier (DTV) potential, and the Omega background deformed N=2 supersymmetric SU(2) QCD models with four massive flavors in the Nekrasov-Shatashvili limit. The weak coupling spectral solution of the DTV potential is related to the instanton partition function of supersymmetric QCD with surface operator. There are two strong coupling spectral solutions of the DTV potential, they are related to the strong coupling expansions of gauge theory prepotential at the magnetic and dyonic points in the moduli space. A set of duality transformations relate the two strong coupling expansions for spectral solution, and for gauge theory prepotential.

Quiver algebras of 4D gauge theories

We construct an ϵ-deformation of W algebras, corresponding to the additive version of quiver algebras which feature prominently in the 5D version of the BPS/CFT correspondence and refined topological strings on toric Calabi–Yau’s. This new type of algebras fill in the missing intermediate level between q-deformed and ordinary W algebras. We show that ϵ-deformed W algebras are spectral duals of conventional W algebras, in particular the ϵ-deformed conformal blocks manifestly reproduce instanton partition functions of 4D quiver gauge theories in the full Ω-background and give dual integral representations of ordinary W conformal blocks.

New quantum toroidal algebras from 5D mathcal{N} = 1 instantons on orbifolds

Quantum toroidal algebras are obtained from quantum affine algebras by a further affinization, and, like the latter, can be used to construct integrable systems. These algebras also describe the symmetries of instanton partition functions for 5D mathcal{N} = 1 supersymmetric quiver gauge theories. We consider here the gauge theories defined on an orbifold S1× ℂ2/ℤp where the action of ℤp is determined by two integer parameters (ν1, ν2). The corresponding quantum toroidal algebra is introduced as a deformation of the quantum toroidal algebra of mathfrak{gl} (p). We show that it has the structure of a Hopf algebra, and present two representations, called vertical and horizontal, obtained by deforming respectively the Fock representation and Saito’s vertex representations of the quantum toroidal algebra of mathfrak{gl} (p). We construct the vertex operator intertwining between these two types of representations. This object is identified with a (ν1, ν2)-deformation of the refined topological vertex, allowing us to reconstruct the Nekrasov partition function and the qq-characters of the quiver gauge theories.

Instanton counting in class

We compute the instanton partition functions of SCFTs in class . We obtain this result via orbifolding Dp/D(p-4) brane systems and calculating the partition function of the supersymmetric gauge theory on the worldvolume of K D(p-4) branes. Starting with D5/D1 setups probing a orbifold singularity we obtain the K instanton partition functions of 6d theories on in the presence of orbifold defects on T2 via computing the 2d superconformal index of the worldvolume theory on K D1 branes wrapping the T2. We then reduce our results to the 5d and to the 4d instanton partition functions. For k = 1 we check that we reproduce the known elliptic, trigonometric and rational Nekrasov partition functions. Finally, we show that the instanton partition functions of quivers in class can be obtained from the class mother theory partition functions with gauge factors via imposing the ‘orbifold condition’ with and , on the Coulomb moduli and the mass parameters.

Quantum elliptic Calogero-Moser systems from gauge origami

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.

The quantum DELL system

We propose quantum Hamiltonians of the double elliptic many-body integrable system (DELL) and study its spectrum. These Hamiltonians are certain elliptic functions of coordinates and momenta. Our results provide quantization of the classical DELL system which was previously found in the string theory literature. The eigenfunctions for the N-body model are instanton partition functions of 6d SU(N) gauge theory with adjoint matter compactified on a torus with a codimension two defect. As a byproduct we discover new family of symmetric orthogonal polynomials which provide an elliptic generalization to Macdonald polynomials.

On quiver W-algebras and defects from gauge origami

In this note, using Nekrasov's gauge origami framework, we study two different versions of the BPS/CFT correspondence – first, the standard AGT duality and, second, the quiver W algebra construction which has been developed recently by Kimura and Pestun. The gauge origami enables us to work with both dualities simultaneously and find exact matchings between the parameters. In our main example of an A-type quiver gauge theory, we show that the corresponding quiver qW-algebra and its representations are closely related to a large-n limit of spherical gln double affine Hecke algebra whose modules are described by instanton partition functions of a defect quiver theory.

Instantons from blow-up

We generalize Nakajima-Yoshioka blowup equations to arbitrary gauge group with hypermultiplets in arbitrary representations. Using our blowup equations, we compute the instanton partition functions for 4d mathcal{N} = 2 and 5d mathcal{N} = 1 gauge theories for arbitrary gauge theory with a large class of matter representations, without knowing explicit construction of the instanton moduli space. Our examples include exceptional gauge theories with fundamentals, SO(N ) gauge theories with spinors, and SU(6) gauge theories with rank-3 antisymmetric hypers. Remarkably, the instanton partition function is completely determined by the perturbative part.

Discrete Painlevé system for the partition function of Nf = 2 SU(2) supersymmetric gauge theory and its double scaling limit

We continue to study the matrix model of the Nf = 2 case that represents the irregular conformal block. What provides us with the Painlevé system is not the instanton partition function per se but rather a finite analog of its Fourier transform that can serve as a generating function. The system reduces to the extension of the Gross–Witten–Wadia unitary one-matrix model by the logarithmic potential while keeping the planar critical behavior intact. The double scaling limit to this critical point is a constructive way to study Argyres–Douglas type theory from IR. We elaborate upon the method of orthogonal polynomials and its relevance to these problems, developing it further for the case of a generic unitary matrix model and that of a special case with the logarithmic potential.

Web construction of ABCDEFG and affine quiver gauge theories

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 ormalism 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 ABCDEF G-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.

Fiber-base duality from the algebraic perspective

Quiver 5D mathcal{N}=1 gauge theories describe the low-energy dynamics on webs of (p, q)-branes in type IIB string theory. S-duality exchanges NS5 and D5 branes, mapping (p, q)-branes to branes of charge (−q, p), and, in this way, induces several dualities between 5D gauge theories. On the other hand, these theories can also be obtained from the compactification of topological strings on a Calabi-Yau manifold, for which the S-duality is realized as a fiber-base duality. Recently, a third point of view has emerged in which 5D gauge theories are engineered using algebraic objects from the Ding-Iohara-Miki (DIM) algebra. Specifically, the instanton partition function is obtained as the vacuum expectation value of an operator mathcal{T} constructed by gluing the algebra’s intertwiners (the equivalent of topological vertices) following the rules of the toric diagram/brane web. Intertwiners and mathcal{T} -operators are deeply connected to the co-algebraic structure of the DIM algebra. We show here that S-duality can be realized as the twist of this structure by Miki’s automorphism.

A note on the algebraic engineering of 4D [formula omitted] super Yang–Mills theories

Some BPS quantities of N=1 5D quiver gauge theories, like instanton partition functions or qq-characters, can be constructed as algebraic objects of the Ding–Iohara–Miki (DIM) algebra. This construction is applied here to N=2 super Yang–Mills theories in four dimensions using a degenerate version of the DIM algebra. We build up the equivalent of horizontal and vertical representations, the first one being defined using vertex operators acting on a free boson's Fock space, while the second one is essentially equivalent to the action of Vasserot–Shiffmann's Spherical Hecke central algebra. Using intertwiners, the algebraic equivalent of the topological vertex, we construct a set of T-operators acting on the tensor product of horizontal modules, and the vacuum expectation values of which reproduce the instanton partition functions of linear quivers. Analysing the action of the degenerate DIM algebra on the T-operator in the case of a pure U(m) gauge theory, we further identify the degenerate version of Kimura–Pestun's quiver W-algebra as a certain limit of q-Virasoro algebra. Remarkably, as previously noticed by Lukyanov, this particular limit reproduces the Zamolodchikov–Faddeev algebra of the sine-Gordon model.

Chiral rings for surface operators in 4d and 5d SQCD

Chiral rings of two-dimensional (2,2) theories coupled to 4d mathcal{N} = 2 theories with matter hypermultiplets are studied. Specifically, the vacua of the twisted superpotential of the 2d theories with vanishing sum of matter charges are computed by considering the resolvent of the bulk theory. The solutions to the chiral ring equations are also obtained from the instanton partition function via Higgsing for simple surface operators and via the orbifold description for full surface operators. These 2d/4d coupled theories are lifted to 3d/5d theories and vacua are found similarly in two different methods: by solving the 3d chiral ring equations taking into account the effect of 5d resolvent and by computing the 5d instanton partition function in the presence of a surface operator. We also check the Seiberg-like duality for both 2d/4d and 3d/5d coupled systems with a specific Chern-Simons coefficient for the latter.

Beyond triality: dual quiver gauge theories and little string theories

The web of dual gauge theories engineered from a class of toric Calabi-Yau threefolds is explored. In previous work, we have argued for a triality structure by compiling evidence for the fact that every such manifold XN,M (for given (N, M)) engineers three a priori different, weakly coupled quiver gauge theories in five dimensions. The strong coupling regime of the latter is in general described by Little String Theories. Furthermore, we also conjectured that the manifold XN,M is dual to XN ′,M ′ if NM = N′M′ and gcd(N, M) = gcd(N′, M′). Combining this result with the triality structure, we currently argue for a large number of dual quiver gauge theories, whose instanton partition functions can be computed explicitly as specific expansions of the topological partition function {mathcal{Z}}_{N,M} of XN,M. We illustrate this web of dual theories by studying explicit examples in detail. We also undertake first steps in further analysing the extended moduli space of XN,M with the goal of finding other dual gauge theories.

Analyticity of Nekrasov Partition Functions

We prove that the K-theoretic Nekrasov instanton partition functions have a positive radius of convergence in the instanton counting parameter and are holomorphic functions of the Coulomb parameters in a suitable domain. We discuss the implications for the AGT correspondence and the analyticity of the norm of Gaiotto states for the deformed Virasoro algebra. The proof is based on random matrix techniques and relies on an integral representation of the partition function, due to Moore, Nekrasov, and Shatashvili, which we also prove.