Bifundamental Matter Research Articles

3d Carrollian Chern-Simons theory & 2d Yang-Mills

Abstract With the goal of building a concrete co-dimension one holographically dual field theory for four dimensional asymptotically flat spacetimes (4d AFS) as a limit of AdS4/CFT3, we begin an investigation of 3d Chern-Simons matter (CSM) theories in the Carroll regime. We perform a Carroll (speed of light c → 0) expansion of the relativistic Chern-Simons action coupled to a massless scalar and obtain Carrollian CSM theories, which we show are invariant under the infinite dimensional 3d conformal Carroll or 4d Bondi-van der Burg-Metzner-Sachs (BMS4) symmetries, thus making them putative duals for 4d AFS. Concentrating on the leading-order electric Carroll CSM theory, we perform a null reduction of the 3d theory. Null reduction is a procedure to obtain non-relativistic theories from a higher dimensional relativistic theory. Curiously, null reduction of a Carrollian theory yields a relativistic lower-dimensional theory. We work with SU(N) × SU(M) CS theory coupled to bi-fundamental matter and show that when N = M, we obtain (rather surprisingly) a 2d Euclidean Yang-Mills theory after null reduction. We also comment on the reduction when N ≠ M and possible connections of the null-reduced Carroll theory to a candidate 2d Celestial CFT.

Wilson loop correlators at strong coupling 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} = 2 quiver gauge theories

We consider 4-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} = 2 superconformal quiver theories with SU(N)M gauge group and bi-fundamental matter and we evaluate correlation functions of n coincident Wilson loops in the planar limit of the theory. Exploiting specific untwisted/twisted combinations of these operators and using supersymmetric localization, we are able to resum the whole perturbative expansion and find exact expressions for these correlators that are valid for all values of the ’t Hooft coupling. Moreover, we analytically derive the leading strong coupling behaviour of the correlators, showing that they obey a remarkable simple rule. Our analysis is complemented by numerical checks based on a Padé resummation of the perturbative series.

't Hooft lines of ADE-type and topological quivers

We investigate 4D Chern-Simons theory with ADE gauge symmetries in the presence of interacting Wilson and ‘t Hooft line defects. We analyse the intrinsic properties of these lines’ coupling and explicate the building of oscillator-type Lax matrices verifying the RLL integrability equation. We propose gauge quiver diagrams Q_{G}^{\mu}Gμ encoding the topological data carried by the Lax operators and give several examples where Darboux coordinates are interpreted in terms of topological bi-fundamental matter. We exploit this graphical description (i) to give new results regarding solutions in representations beyond the fundamentals of sl_{N}slN, $ so_{2N}$ and e_{6,7}e6,7, and (ii) to classify the Lax operators for simply laced symmetries in a unified E_{7}7 CS theory. For quick access, a summary list of the leading topological quivers Q$ _{ADE}^{}$ is given in the conclusion section [Figures 29.(a-e), 30.(a-d) and 31.(a-d)].

A survey of family unification models with bifundamental matter

Extensions of the Standard Model have been attempted from the bottom up and from the top down, yet there remains a largely unexplored middle ground. In this paper, using the Mathematica package LieART, we exhaustively enumerate embeddings of the Standard Model within the class of theories with bifundamental fermions in product gauge group SU(a)×SU(b)×SU(c) with no more generators than E6, while achieving SM family unification rather than replication. We incorporate simple phenomenological constraints and find 151 unique models, including 9 that have only vector-like particle content beyond-the-Standard Model (BSM) particles, which we conjecture belong to 5 infinite families of such models. We describe the potentially most viable models: namely, the 9 with strictly vector-like BSM content along with the 29 models we found with no more than 30 additional BSM chiral particles. These include models with fractional electric charge color singlets, and hence magnetic monopoles with multiple Dirac charge. This latter collection of models predicts chiral particles with masses near the electroweak scale accessible to current and future collider experiments.

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.

Structure Constants in N=2 Superconformal Quiver Theories at Strong Coupling and Holography.

In a four-dimensional N=2 superconformal quiver theory with gauge group SU(N)×SU(N) and bifundamental matter, we analytically obtain the exact strong-coupling behavior of the normalized 3-point correlators of single-trace scalar operators in the large-N limit using localization techniques. We then obtain the same strong-coupling behavior from the holographic dual using the AdS/CFT correspondence at the supergravity level. This agreement confirms the validity of the analytic strong-coupling results and of the holographic correspondence in a nonmaximally supersymmetric setup in four dimensions.

Web of Seiberg-like dualities for 3D N=2 quivers

We propose a universal manipulation to obtain Seiberg-like dualities of 3d $\mathcal{N}=2$ general quiver gauge theories with unitary, symplectic and orthogonal gauge groups coupled to fundamental and bifundamental matter fields. We illustrate this with several examples of linear, circular and star-shaped quiver gauge theories. We examine the local operators in the theories by computing supersymmetric indices and also find precise matching for the proposed dualities as strong evidence. We also generalize the dualities in the presence of a boundary on which the theories obey $\mathcal{N}=(0,2)$ chiral half-BPS boundary conditions.

Conformal S-dualities from O-planes

We study 4d SCFTs obtained by orientifold projections on necklace quivers with fractional branes. The models obtained by this procedure are \U0001d4a9 = 1 linear quivers with unitary, symplectic and orthogonal gauge groups, bifundamental and tensorial matter. Remarkably, models that are not dual in the unoriented case can have the same central charges and superconformal index after the projection. The reason for this behavior rests upon the ubiquitous presence of adjoint fields with R-charge one. We claim that the presence of such fields is at the origin of the notion of inherited S-duality on the models’ conformal manifold.

Notes on hyperloops in mathcal{N} = 4 Chern-Simons-matter theories

We present new circular Wilson loops in three-dimensional mathcal{N} = 4 quiver Chern-Simons-matter theory on S3. At any given node of the quiver, a two-parameter family of operators can be obtained by opportunely deforming the 1/4 BPS Gaiotto-Yin loop. Including then adjacent nodes, the coupling to the bifundamental matter fields allows to enlarge this family and to construct loop operators based on superconnections. We discuss their classification, which depends on both discrete data and continuous parameters subject to an identification. The resulting moduli spaces are conical manifolds, similar to the conifold of the 1/6 BPS loops of the ABJ(M) theory.

Holography for 2D N=(0,4) quantum field theory

We study the correspondence between ${\mathrm{AdS}}_{3}$ massive IIA supergravity vacua and two-dimensional $\mathcal{N}=(0,4)$ quiver quantum field theories. After categorizing all kinds of gravity solutions, we demystify the ones that seem to reflect anomalous gauge theories. In particular, we prove that there are bound states of D-branes on the boundary of the space that provide the dual quiver theory with exactly the correct amount of flavor symmetry in order to cancel its gauge anomalies. Then we propose that the structure of the field theory should be complemented with additional bifundamental matter, which we argue may only be $\mathcal{N}=(4,4)$ hypermultiplets. Finally, we construct a Bogomol'nyi-Prasad-Sommerfield (BPS) string configuration and use the old and new supersymmetric matter to build its dual ultraviolet operator. During this holographic synthesis, we uncover some interesting features of the quiver superpotential and associate the proposed operator with the same classical mass of its dual BPS string.

Five-dimensional gauge theories from shifted web diagrams

In previous works (arXiv:1610.07916, arXiv:1711.07921, arXiv:1807.00186) we studied a class of toric Calabi-Yau threefolds which engineer six-dimensional supersymmetric gauge theories with gauge group $U(N)$ and adjoint matter. The K\"ahler moduli space of these manifolds can be extended through flop transformations to include regions which are described by so-called shifted toric web diagrams. In this paper we analyse gauge theories that are engineered by these shifted toric web diagrams and argue that in specific limits, some of the them engineer five-dimensional quiver gauge theories with gauge group $G\subset U(N)$ and with fundamental and bi-fundamental matter. We discuss several examples in detail and describe how the matter sector is obtained from the six-dimensional parent theory.

Counting trees in supersymmetric quantum mechanics

We study the supersymmetric ground states of the Kronecker model of quiver quantum mechanics. This is the simplest quiver with two gauge groups and bifundamental matter fields, and appears universally in four-dimensional \mathcal N = 2 systems. The ground state degeneracy may be written as a multi-dimensional contour integral, and the enumeration of poles can be simply phrased as counting bipartite trees. We solve this combinatorics problem, thereby obtaining exact formulas for the degeneracies of an infinite class of models. We also develop an algorithm to compute the angular momentum of the ground states, and present explicit expressions for the refined indices of theories where one rank is small.

Bosonizing three-dimensional quiver gauge theories

We start with the recently conjectured 3d bosonization dualities and gauge global symmetries to generate an infinite sequence of new dualities. These equate theories with non-Abelian product gauge groups and bifundamental matter. We uncover examples of Bose/Bose and Fermi/Fermi dualities, as well as a sequence of dualities between theories with scalar matter in two-index representations. Our conjectures are consistent with level/rank duality in massive phases.

Vacuum structure of bifundamental gauge theories at finite topological angles

We discuss possible vacuum structures of SU(n) × SU(n) gauge theories with bifundamental matters at finite θ angles. In order to give a precise constraint, a mixed ’t Hooft anomaly is studied in detail by gauging the center ℤn one-form symmetry of the bifundamental gauge theory. We propose phase diagrams that are consistent with the con-straints, and also give a heuristic explanation of the result based on the dual superconductor scenario of confinement.

Asymptotics of ground state degeneracies in quiver quantum mechanics

We study the growth of the ground state degeneracy in the Kronecker model of quiver quantum mechanics. This is the simplest quiver with two gauge groups and bifundamental matter elds, and appears universally in the context of BPS state counting in fourdimensionalN = 2 systems. For large ranks, the ground state degeneracy is exponential with slope a modular function that we are able to compute at integral values of its argument. We also observe that the exponential of the slope is an algebraic number and determine its associated algebraic equation explicitly in several examples. The speed of growth of the degeneracies, together with various physical features of the bound states, suggests a dual string interpretation.

Framed sheaves on root stacks and supersymmetric gauge theories on ALE spaces

We develop a new approach to the study of supersymmetric gauge theories on ALE spaces using the theory of framed sheaves on root toric stacks, which illuminates relations with gauge theories on R4 and with two-dimensional conformal field theory. We construct a stacky compactification of the minimal resolution Xk of the Ak−1 toric singularity C2/Zk, which is a projective toric orbifold Xk such that Xk∖Xk is a Zk-gerbe. We construct moduli spaces of torsion free sheaves on Xk which are framed along the compactification gerbe. We prove that this moduli space is a smooth quasi-projective variety, compute its dimension, and classify its fixed points under the natural induced toric action. We use this construction to compute the partition functions and correlators of chiral BPS operators for N=2 quiver gauge theories on Xk with nontrivial holonomies at infinity. The partition functions are computed with and without couplings to bifundamental matter hypermultiplets and expressed in terms of toric blowup formulas, which relate them to the corresponding Nekrasov partition functions on the affine toric open subsets of Xk. We compare our new partition functions with previous computations, explore their connections to the representation theory of affine Lie algebras, and find new constraints on fractional instanton charges in the coupling to fundamental matter. We show that the partition functions in the low energy limit are characterized by the Seiberg–Witten curves, and in some cases also by suitable blowup equations involving Riemann theta-functions on the Seiberg–Witten curve with characteristics related to the nontrivial holonomies.

3d mirror symmetry as a canonical transformation

We generalize the free Fermi-gas formulation of certain 3d ${\cal N}=3$ supersymmetric Chern-Simons-matter theories by allowing Fayet-Iliopoulos couplings as well as mass terms for bifundamental matter fields. The resulting partition functions are given by simple modifications of the argument of the Airy function found previously. With these extra parameters it is easy to see that mirror-symmetry corresponds to linear canonical transformations on the phase space (or operator algebra) of the 1-dimensional fermions.

Quivers, words and fundamentals

A systematic study of holomorphic gauge invariant operators in general $$ \mathcal{N} $$ = 1 quiver gauge theories, with unitary gauge groups and bifundamental matter fields, was recently presented in [1]. For large ranks a simple counting formula in terms of an infinite product was given. We extend this study to quiver gauge theories with fundamental matter fields, deriving an infinite product form for the refined counting in these cases. The infinite products are found to be obtained from substitutions in a simple building block expressed in terms of the weighted adjacency matrix of the quiver. In the case without fundamentals, it is a determinant which itself is found to have a counting interpretation in terms of words formed from partially commuting letters associated with simple closed loops in the quiver. This is a new relation between counting problems in gauge theory and the Cartier-Foatamonoid. For finite ranks of the unitary gauge groups, the refined counting is given in terms of expressions involving Littlewood-Richardson coefficients.

6d Conformal matter

A single M5-brane probing G, an ADE-type singularity, leads to a system which has G x G global symmetry and can be viewed as "bifundamental" (G,G) matter. For the A_N series, this leads to the usual notion of bifundamental matter. For the other cases it corresponds to a strongly interacting (1,0) superconformal system in six dimensions. Similarly, an ADE singularity intersecting the Horava-Witten wall leads to a superconformal matter system with E_8 x G global symmetry. Using the F-theory realization of these theories, we elucidate the Coulomb/tensor branch of (G,G') conformal matter. This leads to the notion of fractionalization of an M5-brane on an ADE singularity as well as fractionalization of the intersection point of the ADE singularity with the Horava-Witten wall. Partial Higgsing of these theories leads to new 6d SCFTs in the infrared, which we also characterize. This generalizes the class of (1,0) theories which can be perturbatively realized by suspended branes in IIA string theory. By reducing on a circle, we arrive at novel duals for 5d affine quiver theories. Introducing many M5-branes leads to large N gravity duals.

Anomalous dimensions in non-supersymmetric bifundamental Chern-Simons theories

Non-abelian Chern-Simons theories coupled to fermions are known to provide an interesting class of non-supersymmetric conformal fixed points \cite{Giombi:2011kc}. These theories, particularly those based on bifundamental matter, are important because they may provide simple non-supersymmetric examples of the AdS/CFT correspondence. For instance, it seems natural to conjecture that $O(N)_{-k}\times O(N)_k$ Chern-Simons theory coupled to Majorana fermions transforming in a bi-vector representation may be dual to pure Einstein gravity with a small negative cosmological constant in the "M-theory" limit where $k=1$ and $N$ is large. While it is extremely difficult to directly study such bifundamental theories when $k=1$ or even at strong 't Hooft coupling $\lambda=\frac{N}{k}$, it is possible to calculate physical quantities to all orders in $\lambda$ in a $U(M)_{k_M} \times U(N)_{k_N}$ theory, in the limit $M \ll N$, in an $M/N$ expansion. To illustrate this, we calculate the anomalous dimension of the primary operator $\bar{\psi}{\psi}$, to first order in $M/N$, to all orders in $\lambda_M=\frac{N}{k_M}$, but with $\lambda_N=\frac{N}{k_N}=0$. We also comment on possible bosonization dualities for bifundamental Chern-Simons theories.