Adjoint Matter Research Articles

Confinement and screening via holographic Wilson loops

Abstract We present the holographic dual to a family of $$ \mathcal{N} $$ N = 1 SCFTs in four dimensions, deformed by a VEV leading to a gapped system. We calculate Wilson loops in this system containing adjoint, bifundamental, and fundamental matter. We calculate the quark-antiquark energy E in terms of their separation L, finding an approximate analytic expression for E(L). This expression shows the transition between conformal, confining, and screened behaviors. Interesting phenomenology is discussed in a variety of examples. The tool used is the minimization of the F1 string action, for which the code used is made publicly available.

Remarks on QCD4 with fundamental and adjoint matter

We study 4-dimensional SU(N) gauge theory with one adjoint Weyl fermion and fundamental matter — either bosonic or fermionic. Symmetries, their ’t Hooft anomalies, and the Vafa-Witten-Weingarten theorems strongly constrain the possible bulk phases. The first part of the paper is dedicated to a single fundamental fermion or boson. As long as the adjoint Weyl fermion is massless, this theory always possesses a ℤ2Nχ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathbb{Z}}_{2N}^{\\chi } $$\\end{document} chiral symmetry, which breaks spontaneously, supporting N vacua and domain walls between them for a generic mass of the matter fields. We argue, however, that the domain walls generically undergo a phase transition, and we establish the corresponding 3d gauge theories on the walls. The rest of the paper is dedicated to studying the multi-flavor fundamental matter. Here, the phases crucially depend on the ratio of the number of colors and the number of fundamental flavors. We also discuss the limiting scenarios of heavy adjoint and fundamentals, which align neatly with our current understanding of QCD and 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} = 1 super Yang-Mills theory.

Dualities of adjoint SQCD and supersymmetry enhancement

We propose a new dual description of four-dimensional 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} = 1 SU(N) gauge theory with one adjoint (X) and Nf fundamental matters with a superpotential W = TrXp+1. The dual theory consists of the D\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{D} $$\\end{document}p[SU(N)] Argyres-Douglas theory coupled to SU(N) gauge theory and Nf fundamentals with a superpotential deformation. We study renormalization group fixed points of the Argyres-Douglas dual theories with and without superpotential deformations, and identify the conditions for them to be dual to the fixed points of adjoint SQCD. We check our proposal via matching central charges, chiral operators and superconformal indices. We find that when Nf = 2N and p = 2, the dual theory flows to 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 SU(N) superconformal QCD with 2N flavors upon suitable superpotential deformation, exhibiting supersymmetry enhancement.

Exact duality and local dynamics in SU(N) lattice gauge theory

We construct exact duality transformations in pure SU(N) Hamiltonian lattice gauge theory in ($2+1$) dimension. This duality is obtained by making a series of iterative canonical transformations on the SU(N) electric vector fields and their conjugate magnetic vector potentials on the four links around every plaquette. The resulting dual description is in terms of the magnetic scalar fields or plaquette flux loops and their conjugate electric scalar potentials. Under SU(N) gauge transformations they both transform like adjoint matter fields. The dual Hamiltonian describes the nonlocal self-interactions of these plaquette flux loops in terms of the electric scalar potentials and with inverted coupling. We show that these nonlocal loop interactions can be made local and converted into minimal couplings by introducing SU(N) auxiliary gauge fields along with new plaquette constraints. The matter fields can be included through minimal coupling. The techniques can be easily generalized to ($3+1$) dimensions.

3d [formula omitted] SO/USp adjoint SQCD: s-confinement and exact identities

We study 3d N=2 SQCD with symplectic and orthogonal gauge groups and adjoint matter. For USp(2n) with two fundamentals and SO(N) with one vector these models have been recently shown to s-confine. Here we corroborate the validity of this proposal by relating it to the confinement of USp(2n) with four fundamentals and an antisymmetric tensor, using exact mathematical results coming from the analysis of the partition function on the squashed three-sphere. Our analysis allows us to conjecture new s-confining theories for a higher number of fundamentals and vectors, in presence of linear monopole superpotentials. We then prove the new dualities through a chain of adjoint deconfinements and s-confining dualities.

A Representation-Theoretic Approach to qq-Characters

We raise the question of whether (a slightly generalized notion of) $qq$-characters can be constructed purely representation-theoretically. In the main example of the quantum toroidal $\mathfrak{gl}_1$ algebra, geometric engineering of adjoint matter produces an explicit vertex operator $\mathsf{RR}$ which computes certain $qq$-characters, namely Hirzebruch $\chi_y$-genera, completely analogously to how the R-matrix $\mathsf{R}$ computes $q$-characters. We give a geometric proof of the independence of preferred direction for the refined vertex in this and more general non-toric settings.

Monopole deformations of 3d Seiberg-like dualities with adjoint matters

We propose new 3d mathcal{N} = 2 Seiberg-like dualities by considering various monopole superpotential deformations on 3d mathcal{N} = 2 U(Nc) SQCDs with fundamental and adjoint matter fields. We provide nontrivial evidence of these new dualities by comparing the superconformal indices, from which we analyze the change of the moduli space due to the monopole deformation. In addition, we perform the F-maximization to check the relevance of the monopole deformation for some examples, one of which is found to exhibit nontrivial symmetry enhancement in the IR. We prove such enhancement of the global symmetry using the superconformal index.

Comments on QCD3 and anomalies with fundamental and adjoint matter

’t Hooft anomaly matching is powerful for constraining the low energy phases of gauge theories. In 3d one common anomaly is the parity anomaly in a T-symmetric theory where one cannot gauge the global symmetry group without breaking the time-reversal symmetry. We find that a T-symmetric SU(N) gauge theory with either fermionic or bosonic matter in the fundamental representation of the gauge group has a parity anomaly between the flavor group and T-symmetry provided that there is also a massless Majorana fermion in the adjoint representation of the gauge group. In particular, there is always a mixed anomaly between T and U(1) baryon symmetry. We then analyze the parity anomaly in this theory, together with the more recent mod 16 time-reversal anomaly, and give some free fermion proposals as candidates for the low energy phases consistent with the anomalies. We make brief comments about the large N limit and the T-broken regimes in the conclusion as well as related anomalies in 4d.

The Hasse diagram of the moduli space of instantons

Hasse diagrams (or phase diagrams) for moduli spaces of supersymmetric field theories have been intensively studied in recent years, and many tools to compute them have been developed. The moduli space of instantons, despite being well studied, has proven difficult to deal with. In this note we explore the Hasse diagram of this moduli space from several perspectives — using the partial Higgs mechanism, using brane systems and using quiver subtraction — having to refine previously developed techniques. In particular we introduce the new concept of decorated quiver, which allows to deal with a large class of unitary quivers, including those with adjoint matter.

Off-shell color-kinematics duality for Chern-Simons

Many gauge theories possess a hidden duality between color and kinematics in their on-shell scattering amplitudes. An open problem is to formulate an off-shell realization of the duality, thus manifesting a kinematic algebra. We show that 3D Chern-Simons (CS) theory in Lorenz gauge obeys off-shell color-kinematics duality. This holds both for the gauge field and the BRST ghosts, and the duality is manifest in the Feynman rules. A kinematic algebra can be formulated through a second-order differential operator (Poisson bracket) acting on the off-shell fields, and it corresponds to 3D volume-preserving diffeomorphisms, generated by functions in Lorenz gauge. We consider several admissible double-copy constructions of CS theory with Yang-Mills theory, a higher-derivative (DF)2 gauge theory, or CS theory itself. To obtain non-vanishing amplitudes, we deform pure CS theory by including the maximum amount of adjoint matter that respects the on-shell duality. This gives a new formulation of an \U0001d4a9 = 4 CS-matter theory, with fields of unusual statistics. We argue that the color-stripped tree amplitudes of this theory are equivalent to those of the Gaiotto-Witten \U0001d4a9 = 4 CS theory with bi-fundamental matter. We further show that the double copy of the \U0001d4a9 = 4 CS theory with itself corresponds to maximally supersymmetric \U0001d4a9 = 8 Dirac-Born-Infeld theory.

Spectral form factor for free large N gauge theory and strings

We investigate the spectral form factor in two different systems, free large N gauge theories and highly excited string gas. In both cases, after a rapid decay of the spectral form factor at early time, new contributions come in, preventing the spectral form factor from ever becoming exponentially small. We consider U(N) gauge theories with only adjoint matter and compute the spectral form factor using a matrix integral of the thermal holonomy U. The new saddles differ from the early time saddle by preserving certain subgroups of the center symmetry. For a gas of strings, the short time decay of the spectral form factor is governed by the continuous Hagedorn density of states, which can be associated to the thermal winding mode with winding number ±1. We show that the rise of the spectral form factor comes from other winding modes that also carry momentum along the time direction. We speculate on the existence of a family of classical solutions for these string modes, similar to the Horowitz-Polchinski solution.We review a similar problem for black holes. In particular, we examine the Kontsevich-Segal criterion on complex black holes that contribute to the spectral form factor. In the canonical ensemble quantity Z(β + it), the black hole becomes unallowed at t ~ mathcal{O} (β). A way to avoid this is to consider the microcanonical ensemble, where the black hole stays allowable.

Lattice simulations of a gauge theory with mixed adjoint-fundamental matter

In this paper, we summarize our efforts in simulating Yang-Mills theories coupled to matter fields transforming under the fundamental and adjoint representations of the gauge group. In the context of composite Higgs scenarios, gauge theories with mixed representation fields have been suggested to describe the fundamental interactions well beyond the electroweak unification scale, and they are also closely related to supersymmetric QCD. In addition, they are studied as deformations of theories with pure adjoint matter in the context of adiabatic continuity. We provide some first results for bare parameter tuning and interdependence of the two representations. We also investigate how the chiral symmetry breaking or a conformal scenario can be realized and checked in such theories.

Exact results and Schur expansions in quiver Chern-Simons-matter theories

We study several quiver Chern-Simons-matter theories on the three-sphere, combining the matrix model formulation with a systematic use of Mordell’s integral, computing partition functions and checking dualities. We also consider Wilson loops in ABJ(M) theories, distinguishing between typical (long) and atypical (short) representations and focusing on the former. Using the Berele-Regev factorization of supersymmetric Schur polynomials, we express the expectation value of the Wilson loops in terms of sums of observables of two factorized copies of U(N ) pure Chern-Simons theory on the sphere. Then, we use the Cauchy identity to study the partition functions of a number of quiver Chern-Simons-matter models and the result is interpreted as a perturbative expansion in the parameters tj = −e2πmj , where mj are the masses. Through the paper, we incorporate different generalizations, such as deformations by real masses and/or Fayet-Iliopoulos parameters, the consideration of a Romans mass in the gravity dual, and adjoint matter.

On dimensional transmutation in 1 + 1D quantum hydrodynamics

Recently, a detailed correspondence was established between, on one side, four- and five-dimensional large-N supersymmetric gauge theories with N = 2 supersymmetry and adjoint matter and, on the other side, integrable 1 + 1-dimensional quantum hydrodynamics. Under this correspondence, the phenomenon of dimensional transmutation, familiar in asymptotically free quantum field theories, gets mapped to the transition from the elliptic Calogero–Moser many-body system to the closed Toda chain. In this paper, we attempt to formulate the hydrodynamical counterpart of the dimensional transmutation phenomenon inspired by the identification of the periodic intermediate long wave equation as the hydrodynamical limit of the elliptic Calogero–Moser/Ruijsenaars–Schneider system. We also conjecture that the chiral flow in the vortex fluid provides the proper framework for the microscopic description of such dimensional transmutation in 1 + 1D hydrodynamics. We provide a geometric description of this phenomenon in terms of the Atiyah Drinfeld Hitchin Manin moduli space.

Elliptic lift of the Shiraishi function as a non-stationary double-elliptic function

As a development of [1], we note that the ordinary Shiraishi functions have an insufficient number of parameters to describe generic eigenfunctions of double elliptic system (Dell). The lacking parameter can be provided by substituting elliptic instead of the ordinary Gamma functions in the coefficients of the series. These new functions (ELS-functions) are conjectured to be functions governed by compactified DIM networks which can simultaneously play the three roles: solutions to non-stationary Dell equations, Dell conformal blocks with the degenerate field (surface operator) insertion, and the corresponding instanton sums in 6d SUSY gauge theories with adjoint matter. We describe the basics of the corresponding construction and make further conjectures about the various limits and dualities which need to be checked to make a precise statement about the Dell description by double-periodic network models with DIM symmetry. We also demonstrate that the ELS-functions provide symmetric polynomials, which are an elliptic generalization of Macdonald ones, and compute the generation function of the elliptic genera of the affine Laumon spaces. In the particular U(1) case, we find an explicit plethystic formula for the 6d partition function, which is a non-trivial elliptic generalization of the (q, t) Nekrasov-Okounkov formula from 5d.

Entropic destruction of heavy quarkonium in heavy quark cloud

Previous research has shown that the peak of the quarkonium entropy at the deconfinement transition would be related to the entropic force which induces the melting of quarkonium. In this article, we study the effect of backreaction on the entropic force in a strongly coupled plasma of adjoint matter. The backreaction covered here comes from the presence of static heavy quarks evenly distributed over such a plasma. It is found that the inclusion of backreaction increases the entropic force thus enhancing the quarkonium dissociation, in accord with the findings of the imaginary potential.

Flowing to mathcal{N} = 3 Chern-Simons-matter theory

New renormalisation group flows of three-dimensional Chern-Simons theories with a single gauge group SU(N ) and adjoint matter are found holographically. These RG flows have an infrared fixed point given by a CFT with mathcal{N} = 3 supersymmetry and SU(2) flavour symmetry. The ultraviolet fixed point is again described by a CFT with either mathcal{N} = 2 and SU(3) symmetry or mathcal{N} = 1 and G2 symmetry. The gauge/gravity duals of these RG flows are constructed as domain-wall solutions of a gauged supergravity model in four dimensions that enjoys an embedding into massive IIA supergravity. A concrete RG flow that brings a mass deformation of the mathcal{N} = 2 CFT into the mathcal{N} = 3 CFT at low energies is described in detail.

Dynamics of non-Abelian strings in the theory interpolating from N=2 to N=1 supersymmetric QCD

We study the dynamics of non-Abelian vortex strings supported in ${\mathcal N}=2$ supersymmetric QCD with the U$(N)$ gauge group and $N_f=N$ quark flavors deformed by the mass $\mu$ of the adjoint matter. In the limit of large $\mu$ the bulk four-dimensional theory flows to ${\mathcal N}=1$ supersymmetric QCD. The dynamics of orientational zero modes of the non-Abelian string is described by the world sheet CP$(N-1)$ model. At $\mu=0$ this model has ${\mathcal N}= \left(2,2\right) $ supersymmetry while at large $\mu$ it flows to the non-supersymmetric CP$(N-1)$ model. We solve the world sheet model in the large $N$ approximation and find a rich phase structure with respect to the deformation parameter $\mu$ and quark mass differences. The phases include two strong coupling phases and two Higgs phases. In particular, the Higgs phase at small $\mu$ supports CP$(N-1)$ model kinks representing confined monopoles of the bulk QCD, while in the large-$\mu$ Higgs phase monopoles disappear.

3d “chiral” Kutasov-Schwimmer duality

We propose a “chiral” version of Kutasov-Schwimmer duality in a 3d N=2SU(N) gauge theory with F fundamental matters, F¯ anti-fundamental matters and an adjoint matter X with a tree-level superpotential W=trXk+1. The theory exhibits a rich structure of the baryonic and (dressed) Coulomb branch operators. At first sight, the duality seems incorrect due to a mismatch of the anti-baryonic branch of the moduli space under the duality transformation. The duality well works by realizing that the anti-baryonic operators are identified with some of the dressed Coulomb branch corresponding to non-abelian monopoles. This generalizes the SU(N) “chiral” duality with (anti-)fundamental matters, which we previously proposed.

New and old fermionic dualities from 3d bosonization

We construct novel fermion-fermion dualities in 2 + 1-dimensions using 3d bosonization dualities. This is achieved by relating two-node quiver theories using both the flavor-bounded and flavor-violated 3d bosonization dualities. Such quivers can be viewed as a generalization of the fermionic particle-vortex duality. A special case of these quivers exhibits a ℤ2 symmetry under interchange of the two nodes. Using orbifold techniques, we show that such dualities provide a novel way of deriving known 3d bosonization dualities with adjoint matter, thus unifying the non-Abelian bosonization dualities in an even larger duality web. We then use this construction to derive new dualities involving adjoint matter.