Partition Function Of Gauge Theory Research Articles

Double Quiver Gauge Theory and BPS/CFT Correspondence

We provide a formalism using the q-Cartan matrix to compute the instanton partition function of quiver gauge theory on various manifolds. Applying this formalism to eight dimensional setups, we introduce the notion of double quiver gauge theory characterized by a pair of quivers. We also explore the BPS/CFT correspondence in eight dimensions based on the q-Cartan matrix formalism.

DE-type little strings from glued brane webs

We propose brane web configurations for D-type and E-type mathcal{N} =(1,0) little string theories based on a trivalent or quadrivalent gluing of 5-brane web diagrams. Tri-/quadri-valent gluing is a powerful way of computing 5d/6d partition functions for supersymmetric gauge theories based on the topological vertex. We generalize the gluing techniques to little string theories by introducing a new compact direction and compute their supersymmetric partition functions on Omega-deformed ℝ4×T2. As concrete examples, we consider little string theories arising from Type IIB NS5-branes probing D4 or D5 singularity. Their effective gauge theory descriptions as the affine D4 or D5 quiver gauge theory can be realized with quadrivalent or trivalent gluing, respectively. Based on these gluings of 5-brane webs, we compute their refined partition functions and compare them with the known results. We extend the computation of the partition function to little string theory engineered from IIB NS5-branes probing E6 singularity based on a trivalent gluing. We also discuss the generalization to higher rank cases and the symmetries of the partition functions.

Recurrence relation for instanton partition function in SU(N) gauge theory

We derive a residue formula and as a consequence a recurrence relation for the instanton partition function in mathcal{N} = 2 supersymmetric theory on ℂ2 with SU(N) gauge group.The particular cases of SU(2) and SU(3) gauge groups were considered in the literature before. The recurrence relation with SU(2) gauge group is long well known and was found as the Alday-Gaiotto-Tachikawa (AGT) counterpart of the Zamolodchikov relation for the Virasoro conformal blocks. In the SU(3) case a residue formula for the term with the minimal number of instantons was found and basing on it a recurrence relation was conjectured.We give a complete proof of the residue formula in all instanton orders in presence of any number of matter hypermultiplets in the adjoint and fundamental representations. The recurrence relation however describes only theories with not too much matter hypermultiplets so that the behaviour at infinity is moderate. The guideline of the proof is an algebro-geometric interpretation of the mathcal{N} = 2 supersymmetric gauge theory partition function in terms of the framed torsion-free sheaves. Lead by this interpretation we formulate a refined version of the residue formula and prove it by direct algebraic manipulations.

Instantons on Young diagrams with matters

We present the unrefined instanton partition functions of various 5d gauge theories with matter beyond the fundamental representation as sums over Young diagrams. By using these explicit expressions, we verify a range of identities among the instanton partition functions predicted by Higgsing procedures of fivebrane web diagrams and representation theory.

5d AGT correspondence of supergroup gauge theories from quantum toroidal mathfrak{gl} 1

We discuss the 5d AGT correspondence of supergroup gauge theories with A-type supergroups. We introduce two intertwiners called positive and negative intertwiners to compute the instanton partition function. The positive intertwiner is the ordinary Awata-Feigin-Shiraishi intertwiner while the negative intertwiner is an intertwiner obtained by using central charges with negative levels. We show that composition of them gives the basic Nekrasov factors appearing in supergroup partition functions. We explicitly derive the instanton partition functions of supergroup gauge theories with A and D-type quiver structures. Using the intertwiners, we briefly study the Gaiotto state, qq-characters and the relation with quiver W-algebra. Furthermore, we show that the negative intertwiner corresponds to the anti-refined topological vertex recently defined by Kimura and Sugimoto. We also discuss how superquiver theories should appear in our formalism if they exist. The existence of the AGT correspondence of the theories we study in this paper implies that there is a broader 2d/4d (5d/q-algebra) correspondence, or more generally the BPS/CFT correspondence, where new non-unitary theories play important roles.

Topological strings and Wilson loops

We propose the refined topological string correspondence to the expectation values of half-BPS Wilson loop operators in 5d mathcal{N} = 1 gauge theory partition function on the Omega-deformed background {mathbb{R}}_{upepsilon_{1,2}}^4 × S1. We provide the refined topological vertex method and the refined holomorphic anomaly equation method in the topological string theory, from which we have exact computations on the 5d Wilson loops partition functions in both A- and B-models. Finally, with the exact results we have in B-model, we recover the quantum periods of local ℙ1 × ℙ1 model and local ℙ2 model in the study of quantum geometry and we further give a refined generalization of A-period.

Symmetry defects and orbifolds of two-dimensional Yang–Mills theory

We describe discrete symmetries of two-dimensional Yang–Mills theory with gauge group G associated with outer automorphisms of G, and their corresponding defects. We show that the gauge theory partition function with defects can be computed as a path integral over the space of twisted G-bundles and calculate it exactly. We argue that its weak-coupling limit computes the symplectic volume of the moduli space of flat twisted G-bundles on a surface. Using the defect network approach to generalised orbifolds, we gauge the discrete symmetry and construct the corresponding orbifold theory, which is again two-dimensional Yang–Mills theory but with gauge group given by an extension of G by outer automorphisms. With the help of the orbifold completion of the topological defect bicategory of two-dimensional Yang–Mills theory, we describe the reverse orbifold using a Wilson line defect for the discrete gauge symmetry. We present our results using two complementary approaches: in the lattice regularisation of the path integral, and in the functorial approach to area-dependent quantum field theories with defects via regularised Frobenius algebras.

Decomposition of BPS moduli spaces and asymptotics of supersymmetric partition functions

We present a prototype for Wilsonian analysis of asymptotics of supersymmetric partition functions of non-abelian gauge theories. Localization allows expressing such partition functions as an integral over a BPS moduli space. When the limit of interest introduces a scale hierarchy in the problem, asymptotics of the partition function is obtained in the Wilsonian approach by i) decomposing (in some suitable scheme) the BPS moduli space into various patches according to the set of light fields (lighter than the scheme dependent cut-off Λ) they support, ii) localizing the partition function of the effective field theory on each patch (with cut-offs set by the scheme), and iii) summing up the contributions of all patches to obtain the final asymptotic result (which is scheme-independent and accurate as Λ → ∞). Our prototype concerns the Cardy-like asymptotics of the 4d superconformal index, which has been of interest recently for its application to black hole microstate counting in AdS5/CFT4. As a byproduct of our analysis we obtain the most general asymptotic expression for the index of gauge theories in the Cardy-like limit, encompassing and extending all previous results.

Circular quiver gauge theories, isomonodromic deformations and W_N fermions on the torus

We study the relation between class mathcal {S} theories on punctured tori and isomonodromic deformations of flat SL(N) connections on the two-dimensional torus with punctures. Turning on the self-dual Omega -background corresponds to a deautonomization of the Seiberg–Witten integrable system which implies a specific time dependence in its Hamiltonians. We show that the corresponding tau -function is proportional to the dual gauge theory partition function, the proportionality factor being a nontrivial function of the solution of the deautonomized Seiberg–Witten integrable system. This is obtained by mapping the isomonodromic deformation problem to W_N free fermion correlators on the torus.

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.

Irregular conformal blocks, Painlev\xe9 III and the blow-up equations

We study the relation of irregular conformal blocks with the Painlevé III3 equation. The functional representation for the quasiclassical irregular block is shown to be consistent with the BPZ equations of conformal field theory and the Hamilton-Jacobi approach to Painlevé III3. It leads immediately to a limiting case of the blow-up equations for dual Nekrasov partition function of 4d pure supersymmetric gauge theory, which can be even treated as a defining system of equations for both c = 1 and c → ∞ conformal blocks. We extend this analysis to the domain of strong-coupling regime where original definition of conformal blocks and Nekrasov functions is not known and apply the results to spectral problem of the Mathieu equations. Finally, we propose a construction of irregular conformal blocks in the strong coupling region by quantization of Painlevé III3 equation, and obtain in this way a general expression, reproducing c = 1 and quasiclassical c → ∞ results as its particular cases. We have also found explicit integral representations for c = 1 and c = −2 irregular blocks at infinity for some special points.

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.

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.

Topological vertex/anti-vertex and supergroup gauge theory

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.

Integrating over quiver variety and BPS/CFT correspondence

We show the vertex operator formalism for the quiver gauge theory partition function and the qq-character of the highest weight module on quiver, both associated with the integral over the quiver variety.

Partition functions for supersymmetric gauge theories on spheres

In this paper we briefly review the main idea of the localization technique and its extension suitable in supersymmetric gauge field theory. We analyze the partition function of the vector multiplets with supercharges and its blocks on the even- and odd-dimensional spheres and squashed spheres. We exploit the so-called Faà di Bruno’s formula and show that multipartite partition functions can be written in the form of expansion series of the Bell polynomials. Applying the restricted specialization argument we show that q-infinite-product representation of partition functions admits presentation in terms of the Patterson–Selberg (or the Ruelle-type) spectral functions.

3d expansions of 5d instanton partition functions

We propose a set of novel expansions of Nekrasov’s instanton partition functions. Focusing on 5d supersymmetric pure Yang-Mills theory with unitary gauge group on {mathrm{mathbb{C}}}_{q,{t}^{-1}}^2times {mathbb{S}}^1 , we show that the instanton partition function admits expansions in terms of partition functions of unitary gauge theories living on the 3d subspaces {mathrm{mathbb{C}}}_qtimes {mathbb{S}}^1 , {mathrm{mathbb{C}}}_{t^{-1}}times {mathbb{S}}^1 and their intersection along {mathbb{S}}^1 . These new expansions are natural from the BPS/CFT viewpoint, as they can be matched with Wq,t correlators involving an arbitrary number of screening charges of two kinds. Our constructions generalize and interpolate existing results in the literature.

5-brane webs for 5d mathcal{N} = 1 G2 gauge theories

We propose 5-brane webs for 5d mathcal{N} = 1 G2 gauge theories. From a Higgsing of the SO(7) gauge theory with a hypermultiplet in the spinor representation, we construct two types of 5-brane web configurations for the pure G2 gauge theory using an O5-plane or an tilde{mathrm{O}5} -plane. Adding flavors to the 5-brane web for the pure G2 gauge theory is also discussed. Based on the obtained 5-brane webs, we compute the partition functions for the 5d G2 gauge theories using the recently suggested topological vertex formulation with an O5-plane, and we find agreement with known results.

Analytic continuation of dimensions in supersymmetric localization

We compute the perturbative partition functions for gauge theories with eight supersymmetries on spheres of dimension d ≤ 5, proving a conjecture by the second author. We apply similar methods to gauge theories with four supersymmetries on spheres with d ≤ 3. The results are valid for non-integer d as well. We further propose an analytic continuation from d = 3 to d = 4 that gives the perturbative partition function for an mathcal{N} =1 gauge theory. The results are consistent with the free multiplets and the one-loop β-functions for general mathcal{N} = 1 gauge theories. We also consider the analytic continuation of an mathcal{N} = 1 preserving mass deformation of the maximally supersymmetric gauge theory and compare to recent holographic results for mathcal{N} = 1∗ super Yang-Mills. We find that the general structure for the real part of the free energy coming from the analytic continuation is consistent with the holographic results.

More large N limits of 3d gauge theories

In this paper we study the large N solution to matrix models describing the partition functions of 3d supersymmetric gauge theories on . The model we focus on has a single gauge group and fundamental fields, whose number scales with N, also known as the Veneziano limit. The novel point in our model is our choice of masses for the fundamental fields. Instead of vanishing or fixed masses, we consider a linear distribution. We show that the model can still be solved by standard large N techniques and explore the different phases of the model. We also comment about other natural mass distributions.