Four-dimensional Gauge Theories Research Articles

Anomalies of 4d SpinG theories

We consider ’t Hooft anomalies of four-dimensional gauge theories whose fermion matter content admits SpinG(4) generalized spin structure, with G either gauged or a global symmetry. We discuss methods to directly compute w2 ∪ w3 ’t Hooft anomalies involving Stiefel-Whitney classes of gauge and flavor symmetry bundles that such theories can have on non-spin manifolds, e.g. M4 = ℂℙ2. Such anomalies have been discussed for SU(2) gauge theory with adjoint fermions, where they were shown to give an effect that was originally found in the Donaldson-Witten topological twist 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} = 2 SYM theory. We directly compute these anomalies for a variety of theories, including general G gauge theories with adjoint fermions, SU(2) gauge theory with fermions in general representations, and Spin(N) gauge theories with fundamental matter. We discuss aspects of matching these and other ’t Hooft anomalies in the IR phase where global symmetries are spontaneously broken, in particular for general Ggauge theory with Nf adjoint Weyl fermions. For example, in the case of Nf = 2 we discuss anomaly matching in the IR phase consisting of hGgauge∨\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {h}_{G_{\ extrm{gauge}}}^{\\vee } $$\\end{document} copies of a ℂℙ1 non-linear sigma model, including for the w2w3 anomalies when formulated with SpinSU2global4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\ extrm{Spin}}_{\ extrm{SU}{(2)}_{\ extrm{global}}}(4) $$\\end{document} structure.

Whittaker vectors for W\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {W}$$\\end{document}-algebras from topological recursion

We identify Whittaker vectors for Wk(g)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {W}^{\ extsf{k}}(\\mathfrak {g})$$\\end{document}-modules with partition functions of higher Airy structures. This implies that Gaiotto vectors, describing the fundamental class in the equivariant cohomology of a suitable compactification of the moduli space of G-bundles over P2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {P}^2$$\\end{document} for G a complex simple Lie group, can be computed by a non-commutative version of the Chekhov–Eynard–Orantin topological recursion. We formulate the connection to higher Airy structures for Gaiotto vectors of type A, B, C, and D, and explicitly construct the topological recursion for type A (at arbitrary level) and type B (at self-dual level). On the physics side, it means that the Nekrasov partition function for pure N=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {N} = 2$$\\end{document} four-dimensional supersymmetric gauge theories can be accessed by topological recursion methods.

Effective gravitational couplings of Kaluza-Klein gauge theories

We study the effective gravitational couplings of four-dimensional gauge theories with eight supercharges. The class of theories we analyse are arrived at via Kaluza-Klein compactification of five-dimensional gauge theories. We consider both pure SU(N) Yang-Mills theories with Chern-Simons couplings and the conformal gauge theories with 2N fundamental flavours. The resolvent of the gauge theory plays a crucial role in the calculation of these gravitational couplings. The results obtained from the Seiberg-Witten geometry are matched against independent computations using localisation.

Kac-Moody symmetry in the light front of gauge theories

We discuss the emergence of a new symmetry generator in a Hamiltonian realisation of four-dimensional gauge theories in the flat space foliated by retarded (advanced) time. It generates an asymptotic symmetry that acts on the asymptotic fields in a way different from the usual large gauge transformations. The improved canonical generators, corresponding to gauge and asymptotic symmetries, form a classical Kac-Moody charge algebra with a non-trivial central extension. In particular, we describe the case of electromagnetism, where the charge algebra is the U(1) current algebra with a level proportional to the coupling constant of the theory, κ = 4π2/e2. We construct bilinear generators yielding Virasoro algebras on the null boundary. We also provide a non-Abelian generalization of the previous symmetries by analysing the evolution of Yang-Mills theory in Bondi coordinates.

Noncommutative $$SO(2,3)_{\star }$$ gauge theory of gravity

Topological gravity (in the sense that it is metric-independent) in a 2n-dimensional spacetime can be formulated as a gauge field theory for the AdS gauge group $$SO(2,2n-1)$$ by adding a multiplet of scalar fields. These scalars can break the gauge invariance of the topological gravity action, thus making a connection with Einstein’s gravity. This review is about a noncommutative (NC) star-product deformation of the four-dimensional AdS gauge theory of gravity, including Dirac spinors and the Yang–Mills field. In general, NC actions can be expanded in powers of the canonical noncommutativity parameter $$\theta$$ using the Seiberg–Witten map. The leading-order term of the expansion is the classical action, while the higher-order $$\theta$$ -dependent terms are interpreted as new types of coupling between classical fields due to spacetime noncommutativity. We study how these perturbative NC corrections affect the field equations of motion and derive some phenomenological consequences, such as NC-deformed Landau levels of an electron. Finally, we discuss how topological gravity in four dimensions (both classical and noncommutative) appears as a low-energy sector of five-dimensional Chern–Simons gauge theory in the sense of Kaluza–Klein reduction.

Vacuum misalignment in presence of four-Fermi operators

We consider the issue of vacuum misalignment induced by four-Fermi couplings in a generic strongly coupled four-dimensional gauge theory. After briefly reviewing the general formalism, we focus on the case of partial compositness-like operators at leading order, which is relevant in applications to phenomenology. We show that the interactions between an elementary fermion and composite spin-1/2 operators in various representations contribute to the effective potential with relative sign differences. Thus the correct sign required to misalign the vacuum is guaranteed to occur for some representations but not all of them. The overall sign dictating the specific representations responsible for misalignment can in principle be determined on the lattice. We also comment on the likely sign for some simple cases.

Celestial holography meets twisted holography: 4d amplitudes from chiral correlators

We propose a new program for computing a certain integrand of scattering amplitudes of four-dimensional gauge theories which we call the form factor integrand, starting from 6d holomorphic theories on twistor space. We show that the form factor integrands can be expressed as sums of products of 1.) correlators of a 2d chiral algebra, related to the algebra of asymptotic symmetries uncovered recently in the celestial holography program, and 2.) OPE coefficients of a 4d non-unitary CFT. We prove that conformal blocks of the chiral algebras are in one-to-one correspondence with local operators in 4d. We use this bijection to recover the Parke-Taylor formula, the CSW formula, and certain one-loop scattering amplitudes. Along the way, we explain and derive various aspects of celestial holography, incorporating techniques from the twisted holography program such as Koszul duality. This perspective allows us to easily and efficiently recover the infinite-dimensional chiral algebras of asymptotic symmetries recently extracted from scattering amplitudes of massless gluons and gravitons in the celestial basis. We also compute some simple one-loop corrections to the chiral algebras and derive the three-dimensional bulk theories for which these 2d algebras furnish an algebra of boundary local operators.

Techni-composite Higgs models with symmetric and asymmetric dark matter candidates

We propose a novel class of composite models that feature both a technicolor and a composite Higgs vacuum limit, resulting in an asymmetric dark matter candidate. These techni-composite Higgs models are based on an extended left-right electroweak symmetry with a pseudo-Nambu Goldstone boson Higgs and stable dark matter candidates charged under a global $\mathrm{U}(1{)}_{X}$ symmetry, connected to the baryon asymmetry at high temperatures via the $\mathrm{SU}(2{)}_{\mathrm{R}}$ sphaleron. We consider, as explicit examples, four-dimensional gauge theories with fermions charged under a new confining gauge group ${G}_{\mathrm{HC}}$.

Holographic bubbles with Jecco: expanding, collapsing and critical

Cosmological phase transitions can proceed via the nucleation of bubbles that subsequently expand and collide. The resulting gravitational wave spectrum depends crucially on the properties of these bubbles. We extend our previous holographic work on planar bubbles to cylindrical bubbles in a strongly-coupled, non-Abelian, four-dimensional gauge theory. This extension brings about two new physical properties. First, the existence of a critical bubble, which we determine. Second, the bubble profile at late times exhibits a richer self-similar structure, which we verify. These results require a new 3+1 evolution code called Jecco that solves the Einstein equations in the characteristic formulation in asymptotically AdS spaces. Jecco is written in the Julia programming language and is freely available. We present an outline of the code and the tests performed to assess its robustness and performance.

Gauge Theory and the Analytic Form of the Geometric Langlands Program

We present a gauge-theoretic interpretation of the “analytic” version of the geometric Langlands program, in which Hitchin Hamiltonians and Hecke operators are viewed as concrete operators acting on a Hilbert space of quantum states. The gauge theory ingredients required to understand this construction—such as electric–magnetic duality between Wilson and ’t Hooft line operators in four-dimensional gauge theory—are the same ones that enter in understanding via gauge theory the more familiar formulation of geometric Langlands, but now these ingredients are organized and applied in a novel fashion.

Composite self-interacting dark matter and Higgs

We propose a novel mechanism in composite models that provides self-interacting dark matter along with the Higgs itself as composite particles, alleviating the Standard Model naturalness problem and explaining small-scale discrepancies such as the core-cusp and “too big to fail” problems. These dark matter candidates are stable due to global U(1) symmetries of the composite dynamics and their strong self-interactions are created by the novel mechanism based on top-quark partial compositeness. The relic density of the dark matter candidates is particle anti-particle symmetric and due to thermal freeze-out. We implement this mechanism in a four-dimensional gauge theory with a minimal number of fermions charged under a new confining gauge group GHC.

Tensor renormalization group study of (3+1)-dimensional ℤ2 gauge-Higgs model at finite density

We investigate the critical endpoints of the (3+1)-dimensional ℤ2 gauge-Higgs model at finite density together with the (2+1)-dimensional one at zero density as a benchmark using the tensor renormalization group method. We focus on the phase transition between the Higgs phase and the confinement phase at finite chemical potential along the critical end line. In the (2+1)-dimensional model, the resulting endpoint is consistent with a recent numerical estimate by the Monte Carlo simulation. In the (3+1)-dimensional case, however, the location of the critical endpoint shows disagreement with the known estimates by the mean-field approximation and the Monte Carlo studies. This is the first application of the tensor renormalization group method to a four-dimensional lattice gauge theory and a key stepping stone toward the future investigation of the phase structure of the finite density QCD.

Surface defects in gauge theory and KZ equation

We study the regular surface defect in the \(\Omega \)-deformed four-dimensional supersymmetric gauge theory with gauge group SU(N) with 2N hypermultiplets in fundamental representation. We prove its vacuum expectation value obeys the Knizhnik–Zamolodchikov equation for the 4-point conformal block of the \(\widehat{{\mathfrak {sl}}}_{N}\)-current algebra, originally introduced in the context of two-dimensional conformal field theory. The level and the vertex operators are determined by the parameters of the \(\Omega \)-background and the masses of the hypermultiplets; the cross-ratio of the 4 points is determined by the complexified gauge coupling. We clarify that in a somewhat subtle way the branching rule is parametrized by the Coulomb moduli. This is an example of the BPS/CFT relation.

Bubble wall velocity from holography

Cosmological phase transitions proceed via the nucleation of bubbles that subsequently expand and collide. The resulting gravitational wave spectrum depends crucially on the bubble wall velocity. We use holography to compute the wall velocity from first principles in a strongly coupled, non-Abelian, four-dimensional gauge theory. The wall velocity is determined dynamically in terms of the nucleation temperature. We verify that ideal hydrodynamics provides a good description of the system everywhere except near the wall.

Symmetric Mass Generation in Lattice Gauge Theory

We construct a four-dimensional lattice gauge theory in which fermions acquire mass without breaking symmetries as a result of gauge interactions. Our model consists of reduced staggered fermions transforming in the bifundamental representation of an SU(2)×SU(2) gauge symmetry. This fermion representation ensures that single-site bilinear mass terms vanish identically. A symmetric four-fermion operator is however allowed, and we give numerical results that show that a condensate of this operator develops in the vacuum.

Conformal windows beyond asymptotic freedom

We study four-dimensional gauge theories coupled to fermions in the fundamental and mesonlike scalars. All requisite beta functions are provided for general gauge group and fermion representation. In the regime where asymptotic freedom is absent, we determine all interacting fixed points using perturbation theory up to three loop in the gauge and two loop in the Yukawa and quartic couplings. We find that the conformal window of ultraviolet fixed points is narrowed down by finite-$N$ corrections beyond the Veneziano limit. We also find a new infrared fixed point whose main features, such as scaling exponents, UV-IR connecting trajectories, and phase diagram, are provided. Both fixed points collide upon varying the number of fermion flavors ${N}_{\mathrm{f}}$, and conformality is lost through a saddle-node bifurcation. We further revisit the prospect for ultraviolet fixed points in the large-${N}_{\mathrm{f}}$ limit where matter field fluctuations dominate. Unlike at weak coupling, we do not find clear evidence for new scaling solutions even in the presence of scalar and Yukawa couplings.

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.

The mixed 0-form/1-form anomaly in Hilbert space: pouring the new wine into old bottles

We study four-dimensional gauge theories with arbitrary simple gauge group with 1-form global center symmetry and 0-form parity or discrete chiral symmetry. We canonically quantize on \U0001d54b3, in a fixed background field gauging the 1-form symmetry. We show that the mixed 0-form/1-form ’t Hooft anomaly results in a central extension of the global-symmetry operator algebra. We determine this algebra in each case and show that the anomaly implies degeneracies in the spectrum of the Hamiltonian at any finite- size torus. We discuss the consistency of these constraints with both older and recent semiclassical calculations in SU(N) theories, with or without adjoint fermions, as well as with their conjectured infrared phases.

Infinite-dimensional fermionic symmetry in supersymmetric gauge theories

We establish the existence of an infinite-dimensional fermionic symmetry in four-dimensional supersymmetric gauge theories by analyzing semiclassical photino dynamics in abelian mathcal{N} = 1 theories with charged matter. The symmetry is parametrized by a spinor-valued function on an asymptotic S2 at null infinity. It is not manifest at the level of the Lagrangian, but acts non-trivially on physical states, and its Ward identity is the soft photino theorem. The infinite-dimensional fermionic symmetry resides in the same mathcal{N} = 1 supermultiplet as the physically non-trivial large gauge symmetries associated with the soft photon theorem.

Defects, nested instantons and comet-shaped quivers

We introduce and study a surface defect in four-dimensional gauge theories supporting nested instantons with respect to the parabolic reduction of the gauge group at the defect. This is engineered from a mathrm{{D3/D7}}-branes system on a non-compact Calabi–Yau threefold X. For X=T^2times T^*{{mathcal {C}}}_{g,k}, the product of a two torus T^2 times the cotangent bundle over a Riemann surface {{mathcal {C}}}_{g,k} with marked points, we propose an effective theory in the limit of small volume of {mathcal C}_{g,k} given as a comet-shaped quiver gauge theory on T^2, the tail of the comet being made of a flag quiver for each marked point and the head describing the degrees of freedom due to the genus g. Mathematically, we obtain for a single mathrm{{D7}}-brane conjectural explicit formulae for the virtual equivariant elliptic genus of a certain bundle over the moduli space of the nested Hilbert scheme of points on the affine plane. A connection with elliptic cohomology of character varieties and an elliptic version of modified Macdonald polynomials naturally arises.