Superconformal Gauge Theories Research Articles

A matrix-model approach to integrated correlators in a N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 SYM theory

In a N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 superconformal gauge theory with matter hypermultiplets transforming in the symmetric and anti-symmetric representations of SU(N), we study the integrated correlators of two Coulomb-branch operators and two moment-map operators using localization. In the corresponding matrix model we identify the operator associated with the integrated insertions of moment-map operators and provide for it an exact expression valid for all values of the coupling constant in the planar limit. This allows us to study the integrated correlators at strong-coupling where we show that they behave as the 2-point functions of the Coulomb-branch operators, up to an overall constant dependent only on the conformal dimensions of the latter. The strong-coupling relation between integrated correlators and 2-point functions turns out to be the same as in N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 SYM at large N, despite the reduced amount of supersymmetry in our theory.

Supersymmetric Cardy formula and the Weak Gravity Conjecture in AdS/CFT

The Weak Gravity Conjecture (WGC) in anti-de Sitter spacetime (AdS) asserts the existence of an operator in the boundary conformal field theory (CFT) whose scaling dimension-to-charge ratio satisfies a certain upper bound. This bound is specified by the ratio of the conformal central charge c and the flavor central charge kF. We propose a modified bound in AdS5/CFT4, determined by a combination of two central charges 3c − 2a instead of c. This combination arises in the Cardy-like limit of the 4d superconformal index, which captures the Bekenstein-Hawking entropy of large BPS black holes in AdS5. Using the new bound, we find that certain superconformal field theories (SCFTs) that are previously thought to violate the AdS WGC, including SQCDs in the conformal window, do satisfy the WGC. We check this version of the WGC against all possible superconformal gauge theories with SU(N) gauge group admitting a large N limit when the superpotential is absent. We conjecture the modified version of the WGC is a generic property of any 4d SCFT, regardless of the existence of a weakly coupled gravity dual or a large N limit.

Seiberg-Witten Geometry of Four-Dimensional $\mathcal N=2$ Quiver Gauge Theories

Seiberg-Witten geometry of mass deformed $\mathcal N=2$ superconformal ADE quiver gauge theories in four dimensions is determined. We solve the limit shape equations derived from the gauge theory and identify the space $\mathfrak M$ of vacua of the theory with the moduli space of the genus zero holomorphic (quasi)maps to the moduli space ${\rm Bun}_{\mathbf G} (\mathcal E)$ of holomorphic $G^{\mathbb C}$-bundles on a (possibly degenerate) elliptic curve $\mathcal E$ defined in terms of the microscopic gauge couplings, for the corresponding simple ADE Lie group $G$. The integrable systems $\mathfrak P$ underlying the special geometry of $\mathfrak M$ are identified. The moduli spaces of framed $G$-instantons on ${\mathbb R}^{2} \times {\mathbb T}^{2}$, of $G$-monopoles with singularities on ${\mathbb R}^{2} \times {\mathbb S}^{1}$, the Hitchin systems on curves with punctures, as well as various spin chains play an important role in our story. We also comment on the higher-dimensional theories.

Exact strong coupling results in mathcal{N} = 2 Sp(2N) superconformal gauge theory from localization

We apply the localization technique to compute the free energy on four-sphere and the circular BPS Wilson loop in the four-dimensional mathcal{N} = ∈ superconformal Sp(2N) gauge theory containing vector multiplet coupled to four hypermultiplets in fundamental representation and one hypermultiplet in rank-2 antisymmetric representation. This theory is unique among similar mathcal{N} = ∈ superconformal models that are planar-equivalent to mathcal{N} = ∆ SYM in that the corresponding localization matrix model has the interaction potential containing single-trace terms only. We exploit this property to show that, to any order in large N expansion and an arbitrary ’t Hooft coupling λ, the free energy and the Wilson loop satisfy Toda lattice equations. Solving these equations at strong coupling, we find remarkably simple expressions for these observables which include all corrections in 1/N and 1/ sqrt{lambda } . We also compute the leading exponentially suppressed mathcal{O}left({e}^{-sqrt{lambda }}right) corrections and consider a generalization to the case when the fundamental hypermultiplets have a non-zero mass. The string theory dual of this mathcal{N} = ∈ gauge theory should be a particular orientifold of AdS5 × S5 type IIB string and we discuss the string theory interpretation of the obtained strong-coupling results.

Localization vs holography in 4d mathcal{N} = 2 quiver theories

We study 4-dimensional mathcal{N} = 2 superconformal quiver gauge theories obtained with an orbifold projection from mathcal{N} = 4 SYM, and compute the 2- and 3-point correlation functions among chiral/anti-chiral single-trace scalar operators and the corresponding structure constants. Exploiting localization, we map the computation to an interacting matrix model and obtain expressions for the correlators and the structure constants that are valid for any value of the ’t Hooft coupling in the planar limit of the theory. At strong coupling, these expressions simplify and allow us to extract the leading behavior in an analytic way. Finally, using the AdS/CFT correspondence, we compute the structure constants from the dual supergravity theory and obtain results that perfectly match the strong-coupling predictions from localization.

Three-point functions in a mathcal{N} = 2 superconformal gauge theory and their strong-coupling limit

We study the 3-point functions of single-trace scalar operators in a four-dimensional mathcal{N} = 2 SYM theory with gauge group SU(N) and matter in the symmetric plus anti-symmetric representation, which has a vanishing β-function. By mapping this computation to the matrix model arising from localization on a 4-sphere we are able to resum the perturbative expansion in the large-N ’t Hooft limit and derive the behavior of the correlators at strong coupling. Finally, by combining our results on the 3-point functions with those on the 2-point functions that have been recently found, we obtain the normalized 3-point coefficients of this conformal field theory at strong coupling and find that they depend in a simple way on the conformal dimensions of the single-trace operators.

3d mathcal{N} = 3 generalized Giveon-Kutasov duality

We generalize the Giveon-Kutasov duality for the 3d mathcal{N} = 3 U(N)k,k+nN Chern-Simons matter gauge theory with F fundamental hypermultiplets by introducing SU(N) and U(1) Chern-Simons levels differently. We study the supersymmetric partition functions and the superconformal indices of the duality, which supports the validity of the duality proposal. From the duality, we can map out the low-energy phases: for example, confinement appears for F + k − N = −n = 1 or N = 2F = k = −n = 2. For F + k − N < 0, supersymmetry is spontaneously broken, which is in accord with the fact that the partition function vanishes. In some cases, the theory shows supersymmetry enhancement to 3d mathcal{N} = 4. For k = 0, we comment on the magnetic description dual to the so-called “ugly” theory, where the usual decoupled sector is still interacting with others for n ≠ 0. We argue that the SU(N)0 “ugly-good” duality (which corresponds to the n → ∞ limit in our setup) is closely related to the S-duality of the 4d mathcal{N} = 2 SU(N) superconformal gauge theory with 2N fundamental hypermultiplets. By reducing the number of flavors via real masses, we suggest possible ways to flow to the “bad” theories.

Emergent Sasaki-Einstein geometry and AdS/CFT

A central problem in any quantum theory of gravity is to explain the emergence of the classical spacetime geometry in some limit of a more fundamental, microscopic description of nature. The gauge/gravity-correspondence provides a framework in which this problem can, in principle, be addressed. This is a holographic correspondence which relates a supergravity theory in five-dimensional Anti-deSitter space to a strongly coupled superconformal gauge theory on its 4-dimensional flat Minkowski boundary. In particular, the classical geometry should therefore emerge from some quantum state of the dual gauge theory. Here we confirm this by showing how the classical metric emerges from a canonical state in the dual gauge theory. In particular, we obtain approximations to the Sasaki-Einstein metric underlying the supergravity geometry, in terms of an explicit integral formula involving the canonical quantum state in question. In the special case of toric quiver gauge theories we show that our results can be computationally simplified through a process of tropicalization.

Extended superconformal higher-spin gauge theories in four dimensions

Using the off-shell formulation for mathcal{N} = 2 conformal supergravity in four dimensions, we describe superconformal higher-spin multiplets of conserved currents in a curved background and present their associated unconstrained gauge prepotentials. The latter are used to construct locally superconformal chiral actions, which are demonstrated to be gauge invariant in arbitrary conformally flat backgrounds. The main mathcal{N} = 2 results are then generalised to the mathcal{N} -extended case. We also present the gauge-invariant field strengths for on-shell massless higher-spin mathcal{N} = 2 supermultiplets in anti-de Sitter space. These field strengths prove to furnish representations of the mathcal{N} = 2 superconformal group.

Classification of large N superconformal gauge theories with a dense spectrum

We classify the large N limits of four-dimensional supersymmetric gauge theories with simple gauge groups that flow to superconformal fixed points. We restrict ourselves to the ones without a superpotential and with a fixed flavor symmetry. We find 35 classes in total, with 8 having a dense spectrum of chiral gauge-invariant operators. The central charges a and c for the dense theories grow linearly in N in contrast to the N2 growth for the theories with a sparse spectrum. The difference between the central charges a − c can have both signs, and it does not vanish in the large N limit for the dense theories. We find that there can be multiple bands separated by a gap, or a discrete spectrum above the band. We also find a criterion on the matter content for the fixed point theory to possess either a dense or sparse spectrum. We discover a few curious aspects regarding supersymmetric RG flows and a-maximization along the way. For all the theories with the dense spectrum, the AdS version of the Weak Gravity Conjecture (including the convex hull condition for the cases with multiple U(1)’s) holds for large enough N even though they do not have weakly-coupled gravity duals.

Strong-coupling results for mathcal{N} = 2 superconformal quivers and holography

We consider mathcal{N} = 2 superconformal quiver gauge theories in four dimensions and evaluate the chiral/anti-chiral correlators of single-trace operators. We show that it is convenient to form particular twisted and untwisted combinations of these operators suggested by the dual holographic description of the theory. The various twisted sectors are orthogonal and the correlators in each sector have always the same structure, as we show at the lowest orders in perturbation theory with Feynman diagrams. Using localization we then map the computation to a matrix model. In this way we are able to obtain formal expressions for the twisted correlators in the planar limit that are valid for all values of the ‘t Hooft coupling λ, and find that they are proportional to 1/λ at strong coupling. We successfully test the correctness of our extrapolation against a direct numerical evaluation of the matrix model and argue that the 1/λ behavior qualitatively agrees with the holographic description.

Exact results in a mathcal{N} = 2 superconformal gauge theory at strong coupling

We consider the mathcal{N} = 2 SYM theory with gauge group SU(N) and a matter content consisting of one multiplet in the symmetric and one in the anti-symmetric representation. This conformal theory admits a large-N ’t Hooft expansion and is dual to a particular orientifold of AdS5 × S5. We analyze this gauge theory relying on the matrix model provided by localization à la Pestun. Even though this matrix model has very non-trivial interactions, by exploiting the full Lie algebra approach to the matrix integration, we show that a large class of observables can be expressed in a closed form in terms of an infinite matrix depending on the ’t Hooft coupling λ. These exact expressions can be used to generate the perturbative expansions at high orders in a very efficient way, and also to study analytically the leading behavior at strong coupling. We successfully compare these predictions to a direct Monte Carlo numerical evaluation of the matrix integral and to the Padé resummations derived from very long perturbative series, that turn out to be extremely stable beyond the convergence disk |λ| < π2 of the latter.

Superconformal index with surface defects for class [formula omitted

We study surface defects in 4d N=1SU(N) superconformal gauge theories of class Sk obtained from the 6d (1,0) theories of type AN−1, which are worldvolume theories on N M5-branes at C2/Zk singularities, compactified on Riemann surfaces with punctures. We evaluate the superconformal indices with surface defects for the orbifold of linear quiver gauge theories by two methods. First we apply a method based on Riemann surface description and obtain the superconformal index of the theories in the presence of surface defects labelled by arbitrary symmetric representations of su(N). Then we propose another description for the same surface defects, which involves 4d-2d coupled systems, by identifying which 2d N=(0,2) theories should be coupled. We compute the index of the 4d-2d systems and reproduce the results obtained from the first method. Finally we study the 2d TQFT structure of the index for class Sk theories by obtaining several eigenfunctions and eigenvalues of the difference operators that capture the surface defects and checking their relation.

(Symplectic) leaves and (5d Higgs) branches in the Poly(go)nesian Tropical Rain Forest

We derive the structure of the Higgs branch of 5d superconformal field theories or gauge theories from their realization as a generalized toric polygon (or dot diagram). This approach is motivated by a dual, tropical curve decomposition of the (p, q) 5-brane-web system. We define an edge coloring, which provides a decomposition of the generalized toric polygon into a refined Minkowski sum of sub-polygons, from which we compute the magnetic quiver. The Coulomb branch of the magnetic quiver is then conjecturally identified with the 5d Higgs branch. Furthermore, from partial resolutions, we identify the symplectic leaves of the Higgs branch and thereby the entire foliation structure. In the case of strictly toric polygons, this approach reduces to the description of deformations of the Calabi-Yau singularities in terms of Minkowski sums.

Mass deformations of unoriented quiver theories

We study the interplay between mass deformations and unoriented projections of super-conformal quiver gauge theories resulting from D3-branes at (toric) Calabi-Yau singularities. We focus on simple orbifold cases (ℂ3/ℤ3 and ℂ3/ℤ4) and their non-orbifold descendants. This allows us to generalize the construction rules and clarify points that have been previously overlooked. In particular we spell out the conditions of anomaly cancellations as well as super-conformal invariance that typically require the introduction of flavour branes, which in turn may spoil toric symmetry. Finally, we discuss duality cascades in this context and the interplay between Seiberg/toric duality and unoriented projection with (or without) mass deformations.

Supersymmetric Yang–Mills theories:not quite the usual perspective

In this paper, we take up an old thread of development concerning the characterization of supersymmetric theories without any use of anticommuting variables that goes back to one of the authors’ very early work (Nicolai H 1980 Nucl. Phys. B 176 419). Our special focus here will be on the formulation of supersymmetric Yang–Mills theories, extending previous results beyond D = 4 dimensions. This perspective is likely to provide new insights into these theories, and in particular the maximally extended N = 4 theory. As a new result we re-derive the admissible dimensions for interacting (pure) super-Yang–Mills theories to exist. This article is dedicated to the memory of Peter Freund, amongst many other things an early contributor to supersymmetry, and an author of one of the very first papers on superconformal gauge theories (Crispim Romao J et al 1977 Nucl. Phys. B 122 170). The final section contains some personal reminiscences of H N’s encounters with Peter Freund.

A note on chiral trace relations from qq-characters

We study chiral operators in four-dimensional mathcal{N} = 2 supersymmetric gauge theories. We provide a general procedure to derive the exact relations among the vacuum expectation values of chiral operators in the Ω-background using the non-perturbative Dyson-Schwinger equations of qq-characters. We demonstrate our procedure using several examples. For pure SU(N ) gauge theory and SU(N ) gauge theory with 2N fundamental hypermultiplets, we verify the relations conjectured previously in the literature. We also briefly discuss the relations in SU(N) linear superconformal quiver gauge theories.

Crepant resolutions of [formula omitted] and the generalized Kronheimer construction (in view of the gauge/gravity correspondence)

As a continuation of a general program started in two previous publications, in the present paper we study the Kähler quotient resolution of the orbifold ℂ3∕Z4, comparing with the results of a toric description of the same. In this way we determine the algebraic structure of the exceptional divisor, whose compact component is the second Hirzebruch surface F2. We determine the explicit Kähler geometry of the smooth resolved manifold Y, which is the total space of the canonical bundle of F2. We study in detail the chamber structure of the space of stability parameters (corresponding in gauge theory to the Fayet–Iliopoulos parameters) that are involved in the construction of the desingularizations either by generalized Kronheimer quotient, or as algebro-geometric quotients. The walls of the chambers correspond to two degenerations; one is a partial desingularization of the quotient, which is the total space of the canonical bundle of the weighted projective space P[1,1,2], while the other is the product of the ALE space A1 by a line, and is related to the full resolution in a subtler way. These geometrical results will be used to look for exact supergravity brane solutions and dual superconformal gauge theories.

‐Algebras, the BV Formalism, and Classical Fields

AbstractWe summarise some of our recent works on ‐algebras and quasi‐groups with regard to higher principal bundles and their applications in twistor theory and gauge theory. In particular, after a lightning review of ‐algebras, we discuss their Maurer–Cartan theory and explain that any classical field theory admitting an action can be reformulated in this context with the help of the Batalin–Vilkovisky formalism. As examples, we explore higher Chern–Simons theory and Yang–Mills theory. We also explain how these ideas can be combined with those of twistor theory to formulate maximally superconformal gauge theories in four and six dimensions by means of ‐quasi‐isomorphisms, and we propose a twistor space action.

A Family of GL&lt;sub&gt;r&lt;/sub&gt; Multiplicative Higgs Bundles on Rational Base

In this paper we study a restricted family of holomorphic symplectic leaves in the Poisson-Lie group ${\rm GL}_r(\mathcal{K}_{\mathbb{P}^1_x})$ with rational quadratic Sklyanin brackets induced by a one-form with a single quadratic pole at $\infty \in \mathbb{P}_{1}$. The restriction of the family is that the matrix elements in the defining representation are linear functions of $x$. We study how the symplectic leaves in this family are obtained by the fusion of certain fundamental symplectic leaves. These symplectic leaves arise as minimal examples of (i) moduli spaces of multiplicative Higgs bundles on $\mathbb{P}^{1}$ with prescribed singularities, (ii) moduli spaces of $U(r)$ monopoles on $\mathbb{R}^2 \times S^1$ with Dirac singularities, (iii) Coulomb branches of the moduli space of vacua of 4d $\mathcal{N}=2$ supersymmetric $A_{r-1}$ quiver gauge theories compactified on a circle. While degree 1 symplectic leaves regular at $\infty \in \mathbb{P}^1$ (Coulomb branches of the superconformal quiver gauge theories) are isomorphic to co-adjoint orbits in $\mathfrak{gl}_{r}$ and their Darboux parametrization and quantization is well known, the case irregular at infinity (asymptotically free quiver gauge theories) is novel. We also explicitly quantize the algebra of functions on these moduli spaces by presenting the corresponding solutions to the quantum Yang-Baxter equation valued in Heisenberg algebra (free field realization).