Adjoint Chiral Multiplet Research Articles

Non-invertible symmetries along 4d RG flows

We explore novel examples of RG flows preserving a non-invertible self-duality symmetry. Our main focus is on 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 quadratic superpotential deformations of 4d 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 super-Yang-Mills theory with gauge algebra suN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathfrak{su}(N) $$\\end{document}. A theory that can be obtained in this way is the so-called 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* SYM where all adjoint chiral multiplets have a mass. Such IR theory exhibits a rich structure of vacua which we thoroughly examine. Our analysis elucidates the physics of spontaneous breaking of self-duality symmetry occurring in the degenerate gapped vacua. The construction can be generalized, taking as UV starting point a theory of class S\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{S} $$\\end{document}, to demonstrate how non-invertible self-duality symmetries exist in a variety of 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 SCFTs. We finally apply this understanding to prove that the conifold theory has a non-invertible self-duality symmetry.

Operator spectroscopy for 4D SCFTs with a=c

We study a rich set of four-dimensional superconformal field theories with both central charges identical: $a=c$. These are constructed via the diagonal $\mathcal{N}=2$ or $\mathcal{N}=1$ gauging of the flavor symmetry $G$ of a collection of $\mathcal{N}=2$ Argyres-Douglas theories of type ${\mathcal{D}}_{p}(G)$, with or without adjoint chiral multiplets, in 2106.12579 and 2111.12092. We compute superconformal indices of some theories where the rank of $G$ is low, performing a refined test for unitarity, and further determine the relevant and marginal operator content in detail. We find that most of these theories flow to interacting superconformal field theories with $a=c$ in the infrared.

Infinitely many 4D N=1 SCFTs with a=c

We study a rich set of four-dimensional $\mathcal{N}=1$ superconformal field theories (SCFTs) with both central charges identical: $a = c$. We construct them via the diagonal $\mathcal{N}=1$ gauging of the flavor symmetry $G$ of a collection of $\mathcal{N}=2$ Argyres--Douglas theories of type $\mathcal{D}_p(G)$, with or without additional adjoint chiral multiplets. In this way, we construct infinitely-many theories that flow to interacting SCFTs with $a = c$ in the infrared. Finally, we briefly highlight the features of the SCFTs without $a = c$ that arise from generalizing this construction.

Subleading corrections to the free energy in a theory with N5/3 scaling

We numerically investigate the sphere partition function of a Chern-Simons-matter theory with SU(N) gauge group at level k coupled to three adjoint chiral multiplets that is dual to massive IIA theory. Beyond the leading order N5/3 behavior of the free energy, we find numerical evidence for a term of the form (2/9) log N. We conjecture that this term may be universal in theories with N5/3 scaling in the large-N limit with the Chern-Simons level k held fixed.

Equivariant U(N) Verlinde algebra from Bethe/gauge correspondence

We compute the topological partition function (twisted index) of mathcal{N} = 2 U(N) Chern-Simons theory with an adjoint chiral multiplet on Σg × S1. The localization technique shows that the underlying Frobenius algebra is the equivariant Verlinde algebra which is obtained from the canonical quantization of the complex Chern-Simons theory regularized by U(1) equivariant parameter t. Our computation relies on a Bethe/Gauge correspondence which allows us to represent the equivariant Verlinde algebra in terms of the Hall-Littlewood polynomials Pλ(xB, t) with a specialization by Bethe roots xB of the q-boson model. We confirm a proposed duality to the Coulomb branch limit of the lens space superconformal index of four dimensional mathcal{N} = 2 theories for SU(2) and SU(3) with lower levels. In SU(2) case we also present more direct computation based on Jeffrey-Kirwan residue operation.

Topologically twisted SUSY gauge theory, gauge-Bethe correspondence and quantum cohomology

We calculate the partition function and correlation functions in A-twisted 2d mathcal{N} = (2, 2) U(N) gauge theories and topologically twisted 3d mathcal{N} = 2 U(N) gauge theories containing an adjoint chiral multiplet with particular choices of R-charges and the magnetic fluxes for flavor symmetries. According to the Gauge-Bethe correspondence, they correspond to the Heisenberg XXX1/2 and XXZ1/2 spin chain models, respectively. We identify the partition function with the inverse of the norm of the Bethe eigenstate. Correlation functions are identified to coefficients of the expectation value of Baxter Q-operator. In addition, we consider correlation functions of 2d mathcal{N} = (2, 2)* theories and their relations to the equivariant integration of the equivariant quantum cohomology classes of the cotangent bundle of Grassmann manifolds and the equivariant quantum cohomology ring. Also, we study the twisted chiral ring relations of supersymmetric Wilson loops in 3d mathcal{N} = 2* theories and the Bethe subalgebra of the XXZ1/2 spin chain models.

Evidence for the holographic dual ofN=3solution in massive type IIA supergravity

We calculate the Kaluza-Klein spectrum of spin-2 fluctuations around the ${\cal N}=3$ warped ${\rm AdS}_4\times M_6$ solution in massive IIA supergravity. This solution was conjectured to be dual to the $D=3$ ${\cal N}=3$ superconformal ${\rm SU}(N)$ Chern-Simons matter theory with level $k$ and 2 adjoint chiral multiplets. The ${\rm SO}(3)_R\times{\rm SO}(3)_D$ isometry of the ${\cal N}=3$ solution is identified with the ${\rm SU}(2)_F\times {\rm SU}(2)_{\cal R}$ global symmetry of the dual ${\cal N}=3$ SCFT. We show that the ${\rm SO}(3)_R\times{\rm SO}(3)_D$ quantum numbers and the ${\rm AdS}$ energies carried by the BPS spin-2 modes match precisely with those of the spin-2 gauge invariant operators in the short multiplets of operators in the ${\cal N}=3$ SCFT. We also compute the Euclidean action of the ${\cal N}=3$ solution and the free energy of the ${\cal N}=3$ SCFT on $S^3$, in the limit $N\gg k$. Remarkably, the results show a complete agreement.

Factorization of the 3d superconformal index with an adjoint matter

We work out the factorization of the 3d superconformal index for $$ \mathcal{N}=2 $$ U(N c ) gauge theory with one adjoint chiral multiplet as well as N f fundamental, N a anti-fundamental chiral multiplets. Using the factorization, one can prove the Seiberg-like duality for $$ \mathcal{N}=4 $$ U(N c ) theory with N f hypermultiplets at the index level. We explicitly show that monopole operators violating unitarity bound in a bad theory are mapped to free hypermultiplets in the dual side. For $$ \mathcal{N}=2 $$ U(N c ) theory with one adjoint matter X, N f fundamental, N a anti-fundamental chiral multiplets with superpotential W = trX n+1, we work out Seiberg-like duality for this theory. The index computation provides combinatorial identities for a dual pair, which we carry out intensive numerical checks.

D-TERM DYNAMICAL SUSY BREAKING

We consider an [Formula: see text] supersymmetric U(N) gauge theory with an adjoint chiral multiplet. By developing a self-consistent Hartree-Fock approximation to the leading order which is reminiscent of that of the BCS/NJL in the superconductivity/chiral symmetry, we show that the [Formula: see text] supersymmetry is spontaneously broken, giving a mixed Majorana-Dirac mass term for gaugino due to the nonvanishing D-term VEV and F-term one induced by D-term.

Dirac gauginos and unification in F-theory

Supersymmetric models in which the gauginos acquire Dirac masses, rather than Majorana masses, offer an appealing alternative to the minimal supersymmetric standard model, especially in the light of the bounds set on superpartner masses by the 2011 LHC data. Dirac gauginos require the presence of chiral multiplets in the adjoint representation of the gauge group, and the realisation of such scenarios in F-theory is the subject of this paper. The chiral adjoints drastically alter the usual picture of gauge coupling unification, but this is disturbed anyway in F-theory models with non-trivial hypercharge flux. The interplay between these two factors is explored, and it is found for example that viable F-theory unification can be achieved at around the reduced Planck scale, if there is an extra vector-like pair of singlet leptons with TeV-scale mass. I then discuss the conditions which must be satisfied by the geometry and hypercharge flux of an F-theory model with Dirac gauginos. One nice possibility is for the visible sector to be localised on a K3 surface, and this is discussed in some detail. Finally, I describe how to achieve an unbroken discrete R-symmetry in such compactifications, which is an important ingredient in many models with Dirac gauginos, and write down a simple example which has adjoint chiral multiplets, an appropriate R-symmetry, and allows for viable breaking of SU(5) by hypercharge flux.

Localization and large N reduction on [formula omitted] for the planar and M-theory limit

We show a large N reduction on S3 in a BPS sector for a broad class of theories: N⩾2 supersymmetric Chern–Simons theory with any number of adjoint and bi-fundamental chiral multiplets. We show that a localization method can be applied to the reduced model and the path integral can be written by a multi-contour integral. By taking a particular localization configuration, we also show that the large N equivalence between the original theory on S3 and the reduced model holds for the free energy and the expectation value of BPS Wilson loops. It turns out that the large N reduction on S3 holds also for the M-theory limit.

String theory of the Omega deformation

In this article, we construct a supersymmetric real mass deformation for the adjoint chiral multiplets in the gauge theory describing the dynamics of a stack of D2-branes in type II string theory. We do so by placing the D2-branes into the T-dual of a supersymmetric NS fluxbrane background. We furthermore note that this background is the string theoretic realization of an Omega-deformation of flat space in the directions transverse to the branes where the deformation parameters satisfy \epsilon_1 = - \epsilon_2. This \Omega-deformation therefore serves to give supersymmetric real masses to the chiral multiplets of the 3D gauge theory. To illustrate the physical effect of the real mass term, we derive BPS-saturated classical solutions for the branes rotating in this background. Finally, we reproduce all the same structure in the presence of NS fivebranes and comment on the relationship to the gauge theory/spin-chain correspondence of Nekrasov and Shatashvili.

Probing non-toric geometry with rotating membranes

Recently Martelli and Sparks presented the first non-toric AdS 4/CFT 3 duality relation between M-theory on AdS 4 × V 5 , 2 / Z k and a class of three-dimensional N = 2 quiver Chern–Simons-matter theories. V 5 , 2 is a seven-dimensional homogeneous Sasaki–Einstein manifold with isometry group SO ( 5 ) × U ( 1 ) R , which is in general broken to SU ( 2 ) × U ( 1 ) × U ( 1 ) R by the orbifold projection if k > 1 . The dual field theory is described by the A 1 quiver, U ( N ) k × U ( N ) − k gauge group, four bifundamentals, two adjoint chiral multiplets interacting via a cubic superpotential. We explore this proposal by studying various classical membrane solutions moving in V 5 , 2 . Rotating membrane solutions of folded, wrapped, spike, and giant magnon types are presented with their dispersion relations. We also discuss their dual operators in the Chern–Simons-matter theory.

K-VORTEX DYNAMICS IN $\mathcal{N}\, = \,1^*$ THEORY AND IN ITS GRAVITY DUAL

We study magnetic flux tubes in the Higgs vacuum of the [Formula: see text] mass deformation of SU(Nc), [Formula: see text] SYM and its large Nc string dual, the Polchinski-Strassler geometry. Choosing equal masses for the three adjoint chiral multiplets, for all Nc we identify a "colour-flavour locked" symmetry, SO(3)C+F which leaves the Higgs vacuum invariant. At weak coupling, we find explicit non-Abelian k-vortex solutions carrying a ℤNc-valued magnetic flux, with topological winding 0 < k < Nc. These k-strings spontaneously break SO(3)C+F to U(1)C+F resulting in an S2 moduli space of solutions. The world-sheet sigma model is a nonsupersymmetric ℂℙ1 model with a theta angle θ1+1 = k(Nc - k)θ3+1 where θ3+1 is the Yang-Mills vacuum angle. We find numerically that k-vortex tensions follow the Casimir scaling law Tk ∝ k(Nc - k) for large Nc. In the large Nc IIB string dual, the SO(3)C+F symmetry is manifest in the geometry interpolating between AdS5 × S5 and the interior metric due to a single D5 -brane carrying D3 -brane charge. We identify candidate k-vortices as expanded probe D3 -branes formed from a collection of k D -strings. The resulting k-vortex tension exhibits precise Casimir scaling, and the effective world-sheet theta angle matches the semiclassical result. S -duality maps the Higgs to the confining phase so that confining string tensions at strong 't Hooft coupling also exhibit Casimir scaling in [Formula: see text] theory in the large Nc limit.

Flavors in the microscopic approach to 𝒩 = 1 gauge theories

In this note, we solve an extended version of the N=1 super Yang-Mills theory with gauge group U(N), an adjoint chiral multiplet and Nf flavors of quarks, by using the N=1 microscopic formalism based on Nekrasov's sums over colored partitions. Our main new result is the computation of the general mesonic operators. We prove that the generalized Konishi anomaly equations with flavors are satisfied at the non-perturbative level. This yields in particular a microscopic, first principle derivation of the matrix model disk diagram contributions that must be included in the Dijkgraaf-Vafa approach.

Non-Abeliank-vortex dynamics in 𝒩 = 1* theory and its gravity dual

We study magnetic flux tubes in the Higgs vacuum of the N=1^* mass deformation of SU(N_c), N=4 SYM and its large N_c string dual, the Polchinski-Strassler geometry. Choosing equal masses for the three adjoint chiral multiplets, for all N_c we identify a "colour-flavour locked" symmetry, SO(3)_{C+F} which leaves the Higgs vacuum invariant. At weak coupling, we find explicit non-Abelian k-vortex solutions carrying a Z_{N_c}-valued magnetic flux, with winding, 0 < k < N_c. These k-strings spontaneously break SO(3)_{C+F} to U(1)_{C+F} resulting in an S^2 moduli space of solutions. The world-sheet sigma model is a nonsupersymmetric CP^1 model with a theta angle \theta_{1+1} = k(N_c-k)\theta_{3+1} where \theta_{3+1} is the Yang-Mills vacuum angle. We find numerically that k-vortex tensions follow the Casimir scaling law T_k \propto k (N_c-k) for large N_c. In the large N_c IIB string dual, the SO(3)_{C+F} symmetry is manifest in the geometry interpolating between AdS_5 x S^5 and the interior metric due to a single D5-brane carrying D3-brane charge. We identify candidate k-vortices as expanded probe D3-branes formed from a collection of k D-strings. The resulting k-vortex tension exhibits precise Casimir scaling, and the effective world-sheet theta angle matches the semiclassical result. S-duality maps the Higgs to the confining phase so that confining string tensions at strong 't Hooft coupling also exhibit Casimir scaling in N=1^* theory in the large N_c limit.

Extended 𝒩 = 1 super Yang-Mills theory

We solve a generalization of ordinary N=1 super Yang-Mills theory with gauge group U(N) and an adjoint chiral multiplet X for which we turn on both an arbitrary tree-level superpotential term \int d^{2}\theta Tr W(X) and an arbitrary field-dependent gauge kinetic term \int d^{2}\theta Tr V(X)W^{\alpha}W_{\alpha}. When W=0, the model reduces to the extended Seiberg-Witten theory recently studied by Marshakov and Nekrasov. We use two different points of view: a ''macroscopic'' approach, using generalized anomaly equations, the Dijkgraaf-Vafa matrix model and the glueball superpotential; and the recently proposed ''microscopic'' approach, using Nekrasov's sum over colored partitions and the quantum microscopic superpotential. The two formalisms are based on completely different sets of variables and statistical ensembles. Yet it is shown that they yield precisely the same gauge theory correlators. This beautiful mathematical equivalence is a facet of the open/closed string duality. A full microscopic derivation of the non-perturbative N=1 gauge dynamics follows.

Glueball operators and the microscopic approach to 𝒩 = 1 gauge theories

We explain how to generalize Nekrasov's microscopic approach to = 2 gauge theories to the = 1 case, focusing on the typical example of the U(N) theory with one adjoint chiral multiplet X and an arbitrary polynomial tree-level superpotential Tr W(X). We provide a detailed analysis of the generalized glueball operators and a non-perturbative discussion of the Dijkgraaf-Vafa matrix model and of the generalized Konishi anomaly equations. We compute in particular the non-trivial quantum corrections to the Virasoro operators and algebra that generate these equations. We have performed explicit calculations up to two instantons, that involve the next-to-leading order corrections in Nekrasov's Ω-background.

Microscopic quantum superpotential in 𝒩 = 1 gauge theories

We consider the N=1 super Yang-Mills theory with gauge group U(N), adjoint chiral multiplet X and tree-level superpotential Tr W(X). We compute the quantum effective superpotential W_mic as a function of arbitrary off-shell boundary conditions at infinity for the scalar field X. This effective superpotential has a remarkable property: its critical points are in one-to-one correspondence with the full set of quantum vacua of the theory, providing in particular a unified picture of solutions with different ranks for the low energy gauge group. In this sense, W_mic is a good microscopic effective quantum superpotential for the theory. This property is not shared by other quantum effective superpotentials commonly used in the literature, like in the strong coupling approach or the glueball superpotentials. The result of this paper is a first step in extending Nekrasov's microscopic derivation of the Seiberg-Witten solution of N=2 super Yang-Mills theories to the realm of N=1 gauge theories.

Adding flavour to the Polchinski-Strassler background

As an extension of holography with flavour, we analyze in detail the embedding of a D7-brane probe into the Polchinski-Strassler gravity background, in which the breaking of conformal symmetry is induced by a 3-form flux G_3. This corresponds to giving masses to the adjoint chiral multiplets. We consider the N=2 supersymmetric case in which one of the adjoint chiral multiplets is kept massless while the masses of the other two are equal. This setup requires a generalization of the known expressions for the backreaction of G_3 in the case of three equal masses to generic mass values. We work to second order in the masses to obtain the embedding of D7-brane probes in the background. At this order, the 2-form potentials corresponding to the background flux induce an 8-form potential which couples to the worldvolume of the D7-branes. We show that the embeddings preserve an SU(2) x SU(2) symmetry. We study possible embeddings both analytically in a particular approximation, as well as numerically. The embeddings preserve supersymmetry, as we investigate using the approach of holographic renormalization. The meson spectrum associated to one of the embeddings found reflects the presence of the adjoint masses by displaying a mass gap.