Five-dimensional Gauge Theories Research Articles

WZW in the lightlike directions

Dimensional reduction of the M5 brane on a Lorentzian manifold along a lightlike direction results in a five-dimensional gauge theory, which can be reformulated covariantly in six dimensions, where one puts the Lie derivatives along the lightlike direction of all fields to zero up to a gauge equivalence as constraints. We find gauge and supersymmetry anomalies for certain Lorentzian six-manifolds. We cancel these anomalies by adding a certain WZW theory in the 2d space that is spanned by two lightlike directions.

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.

Spinor-helicity formalism for massive and massless amplitudes in five dimensions

Five-dimensional gauge and gravity theories are known to exhibit striking properties. D = 5 is the lowest dimension where massive tensor states appear naturally, providing a testing ground for perturbative insights into six-dimensional tensor theories. Five-dimensional supergravities are highly constrained and admit elegant geometric and algebraic formulations, with global symmetries manifest at the Lagrangian level.In this paper, we take a step towards the systematic investigation of amplitudes in five dimensions, and present a five-dimensional version of the spinor-helicity formalism, applicable to massless, massive and supersymmetric states. We give explicit representations for on-shell spinor and polarization variables such that the little-group symmetry and gauge redundancy are manifest. Massive self-dual tensor states are discussed in some detail, as well as all the on-shell supermultiplets that can appear in matter-coupled gauge and supergravity theories. As a byproduct of considering supersymmetry in the presence of central charge, we obtain massless ten-dimensional Majorana-Weyl spinors as products of five-dimensional massive spinors.We present compact expressions for superamplitudes at multiplicity three and four, including several novel superamplitudes that either do not straightforwardly uplift to six dimensions, or have not appeared in the six-dimensional literature. We discuss several examples of five-dimensional double-copy constructions in the context of gravitational theories with massive vectors and tensors, illustrating that the formalism we construct can also be used to considerably streamline the double-copy construction of mathcal{N} = 2 Maxwell-Einstein supergravities.

Five-dimensional gauge theories on spheres with negative couplings

We consider supersymmetric gauge theories on S5 with a negative Yang-Mills coupling in their large N limits. Using localization we compute the partition functions and show that the pure SU(N) gauge theory descends to an SU(N/2)+N/2× SU(N/2)−N/2× SU(2) Chern-Simons gauge theory as the inverse ’t Hooft coupling is taken to negative infinity for N even. The Yang-Mills coupling of the SU(N/2)±N/2 is positive and infinite, while that on the SU(2) goes to zero. We also show that the odd N case has somewhat different behavior. We then study the SU(N/2)N/2 pure Chern-Simons theory. While the eigenvalue density is only found numerically, we show that its width equals 1 in units of the inverse sphere radius, which allows us to find the leading correction to the free energy when turning on the Yang-Mills term. We then consider USp(2N) theories with an antisymmetric hypermultiplet and Nf< 8 fundamental hypermultiplets and carry out a similar analysis. Along the way we show that the one-instanton contribution to the partition function remains exponentially suppressed at negative coupling for the SU(N) theories in the large N limit.

Incidence geometry in a Weyl chamber I: GLn

We study the central hyperplane arrangement whose hyperplanes are the vanishing loci of the weights of the first and the second fundamental representations of gln restricted to the dual fundamental Weyl chamber. We obtain generating functions that count flats and faces of a given dimension. This counting is interpreted in physics as the enumeration of the phases of the Coulomb and mixed Coulomb-Higgs branches of a five-dimensional gauge theory with 8 supercharges in presence of hypermultiplets transforming in the fundamental and antisymmetric representation of a U(n) gauge group as described by the Intriligator-Morrison-Seiberg superpotential.

Exploring 5d BPS Spectra with Exponential Networks

We develop geometric techniques for counting BPS states in five-dimensional gauge theories engineered by M theory on a toric Calabi-Yau threefold. The problem is approached by studying framed 3d-5d wall-crossing in presence of a single M5 brane wrapping a special Lagrangian submanifold $L$. The spectrum of 3d-5d BPS states is encoded by the geometry of the manifold of vacua of the 3d-5d system, which further coincides with the mirror curve describing moduli of the Lagrangian brane. Information about the BPS spectrum is extracted from the geometry of the mirror curve by construction of a nonabelianization map for exponential networks. For the simplest Calabi-Yau, $\mathbb{C}^3$ we reproduce the count of 5d BPS states encoded by the Mac Mahon function in the context of topological strings, and match predictions of 3d $tt^*$ geometry for the count of 3d-5d BPS states. We comment on applications of our construction to the study of enumerative invariants of toric Calabi-Yau threefolds.

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.

Rigid limit for hypermultiplets and five-dimensional gauge theories

We study the rigid limit of a class of hypermultiplet moduli spaces appearing in Calabi-Yau compactifications of type IIB string theory, which is induced by a local limit of the Calabi-Yau. We show that the resulting hyperkähler manifold is obtained by performing a hyperkähler quotient of the Swann bundle over the moduli space, along the isometries arising in the limit. Physically, this manifold appears as the target space of the non-linear sigma model obtained by compactification of a five-dimensional gauge theory on a torus. This allows to compute dyonic and stringy instantons of the gauge theory from the known results on D-instantons in string theory. Besides, we formulate a simple condition on the existence of a non-trivial local limit in terms of intersection numbers of the Calabi-Yau, and find an explicit form for the hypermultiplet metric including corrections from all mutually non-local D-instantons, which can be of independent interest.

Surface operators, chiral rings and localization in mathcal{N} =2 gauge theories

We study half-BPS surface operators in supersymmetric gauge theories in four and five dimensions following two different approaches. In the first approach we analyze the chiral ring equations for certain quiver theories in two and three dimensions, coupled respectively to four- and five-dimensional gauge theories. The chiral ring equations, which arise from extremizing a twisted chiral superpotential, are solved as power series in the infrared scales of the quiver theories. In the second approach we use equivariant localization and obtain the twisted chiral superpotential as a function of the Coulomb moduli of the four- and five-dimensional gauge theories, and find a perfect match with the results obtained from the chiral ring equations. In the five-dimensional case this match is achieved after solving a number of subtleties in the localization formulas which amounts to choosing a particular residue prescription in the integrals that yield the Nekrasov-like partition functions for ramified instantons. We also comment on the necessity of including Chern-Simons terms in order to match the superpotentials obtained from dual quiver descriptions of a given surface operator.

5d/6d DE instantons from trivalent gluing of web diagrams

We propose a new prescription for computing the Nekrasov partition functions of five-dimensional theories with eight supercharges realized by gauging non-perturbative flavor symmetries of three five-dimensional superconformal field theories. The topological vertex formalism gives a way to compute the partition functions of the matter theories with flavor instanton backgrounds, and the gauging is achieved by summing over Young diagrams. We apply the prescription to calculate the Nekrasov partition functions of various five-dimensional gauge theories such as SO(2N) gauge theories with or without hypermultiplets in the vector representation and also pure E6, E7, E8 gauge theories. Furthermore, the technique can be applied to computations of the Nekrasov partition functions of five-dimensional theories which arise from circle compactifications of six-dimensional minimal superconformal field theories characterized by the gauge groups SU(3), SO(8), E6, E7, E8. We exemplify our method by comparing some of the obtained partition functions with known results and find perfect agreement. We also present a prescription of extending the gluing rule to the refined topological vertex.

Emergent Yang–Mills theories from universal extra dimensions

We study emergent Yang–Mills theories which could origin from universal extra dimensions. Particularly, some vector field potential terms or polynomial vector field constraints introduced into five-dimensional (5D) non-Abelian gauge theory is shown to lead to spontaneous violation of an underlying spacetime symmetry and generate vector pseudo-Goldstone modes as conventional four-dimensional (4D) gauge boson candidates. As a special signature, apart from conventional gauge couplings, there appear an infinite number of the properly suppressed direct multi-boson (multi-photon in particular) interaction couplings in emergent Yang–Mills theories whose observation could shed light on their high-dimensional nature. Moreover, in these theories, an internal symmetry also appeared spontaneously broken to its diagonal subgroups. This breaking originates from the extra vector field components playing the role of some adjoint scalar field multiplet in the 4D spacetime. So, one naturally has the Higgs effect without a specially introduced scalar field multiplet. Remarkably, when applied to Grand Unified Theories (GUTs), this results in an automatic breakdown of emergent GUTs down to the Standard Model (SM) just at the 5D Lorentz violation scale M.

Duality walls and defects in 5d mathcal{N}=1 theories

We propose an explicit description of “duality walls” which encode at low energy the global symmetry enhancement expected in the UV completion of certain five-dimensional gauge theories. The proposal is supported by explicit localization computations and implies that the instanton partition function of these theories satisfies novel and unexpected integral equations.

Conjugate boundary condition, hidden particles and gauge-Higgs inflation

We propose an idea that hidden particles can be separated according to gauge quantum numbers from the visible ones by the difference of boundary conditions (BCs) on extra dimensions. We formulate five-dimensional (5D) gauge theories yielding conjugate BCs besides ordinary ones on S1/Z2, and examine physical implications concerning hidden particles on an extension of the Standard Model (SM) coexisting different types of BCs. A model with conjugate boundary conditions is applied on a gauge-Higgs inflation scenario.

More on 5d descriptions of 6d SCFTs

We propose new five-dimensional gauge theory descriptions of six-dimensional $$ \mathcal{N} $$ = (1,0) superconformal field theories arising from type IIA brane configurations includ-ing an ON 0-plane. The new five-dimensional gauge theories may have SO, Sp, and SU gauge groups and further broaden the landscape of ultraviolet complete five-dimensional $$ \mathcal{N} $$ = 1 supersymmetric gauge theories. When we include an O8−-plane in addition to an ON 0-plane, T-duality yields two O7−-planes at the intersections of an ON 0-plane and two O50-planes. We propose a novel resolution of the O7−-plane with four D7-branes in such a configuration, which enables us to obtain three different types of five-dimensional gauge theories, depending on whether we resolve either none or one or two O7−-planes. Such different possibilities yield a new five-dimensional duality between a D-type SU quiver and an SO − Sp quiver theories. We also claim that a twisted circle compactification of a six-dimensional superconformal field theory may lead to a five-dimensional gauge theory different from those obtained through a simple circle compactification.

Five-dimensional Gauge-Higgs Unification: a Standard Model-like Spectrum

We study the viability of five-dimensional gauge theories as candidates for the origin of the Higgs field and its mechanism for spontaneous symmetry breaking. Within the framework of lattice field theory, we consider the simplest model of an $SU(2)$ gauge theory. We construct this theory on a five-dimensional orbifold which explicitly breaks the gauge symmetry to $U(1)$ at the fixed points of the orbifold. Using anisotropic gauge couplings, we find that this theory exhibits three distinct phases which we label as confined, Higgs and hybrid. Within the Higgs phase, close to the Higgs-hybrid phase transition, we find that the ratio of the Higgs to gauge boson masses takes Standard Model-like values. Precisely in this region of the phase diagram, we find dimensional reduction via localisation.

Defects and quantum Seiberg-Witten geometry

We study the Nekrasov partition function of the five dimensional U(N) gauge theory with maximal supersymmetry on R^4 x S^1 in the presence of codimension two defects. The codimension two defects can be described either as monodromy defects, or by coupling to a certain class of three dimensional quiver gauge theories on R^2 x S^1. We explain how these computations are connected with both classical and quantum integrable systems. We check, as an expansion in the instanton number, that the aforementioned partition functions are eigenfunctions of an elliptic integrable many-body system, which quantizes the Seiberg-Witten geometry of the five-dimensional gauge theory.

Instanton operators in five-dimensional gauge theories

We discuss instanton operators in five-dimensional gauge theories. These are defined as disorder operators which create a non-vanishing second Chern class on a foursphere surrounding their insertion point. As such they may be thought of as higherdimensional analogues of three-dimensional monopole (or ‘t Hooft) operators. We argue that they play an important role in the enhancement of the Lorentz symmetry for maximally supersymmetric Yang-Mills to SO(1, 5) at strong coupling.

Rigid holography and six-dimensional N = 2 0 $$ \mathcal{N}=\left(2,0\right) $$ theories on AdS5 × S $$ \mathbb{S} $$ 1

Field theories on anti-de Sitter (AdS) space can be studied by realizing them as low-energy limits of AdS vacua of string/M theory. In an appropriate limit, the field theories decouple from the rest of string/M theory. Since these vacua are dual to conformal field theories, this relates some of the observables of these field theories on AdS space to a subsector of the dual conformal field theories. We exemplify this ‘rigid holography’ by studying in detail the six-dimensional $$ \mathcal{N}=\left(2,0\right) $$ A K − 1 superconformal field theory (SCFT) on AdS5 × $$ \mathbb{S} $$ 1, with equal radii for AdS5 and for $$ \mathbb{S} $$ 1. We choose specific boundary conditions preserving sixteen supercharges that arise when this theory is embedded into Type IIB string theory on AdS5 × $$ \mathbb{S} $$ 5/ℤ K . On ℝ4,1 × $$ \mathbb{S} $$ 1, this six-dimensional theory has a 5(K − 1)-dimensional moduli space, with unbroken five-dimensional SU(K) gauge symmetry at (and only at) the origin. On AdS5 × $$ \mathbb{S} $$ 1, the theory has a 2(K − 1)-dimensional ‘moduli space’ of supersymmetric configurations. We argue that in this case the SU(K) gauge symmetry is unbroken everywhere in the ‘moduli space’ and that this five-dimensional gauge theory is coupled to a four-dimensional theory on the boundary of AdS5 whose coupling constants depend on the ‘moduli’. This involves non-standard boundary conditions for the gauge fields on AdS5. Near the origin of the ‘moduli space’, the theory on the boundary contains a weakly coupled four-dimensional $$ \mathcal{N}=2 $$ supersymmetric SU(K) gauge theory. We show that this implies large corrections to the metric on the ‘moduli space’. The embedding in string theory implies that the six-dimensional $$ \mathcal{N}=\left(2,0\right) $$ theory on AdS5 × $$ \mathbb{S} $$ 1 with sources on the boundary is a subsector of the large N limit of various four-dimensional $$ \mathcal{N}=2 $$ quiver SCFTs that remains non-trivial in the large N limit. The same subsector appears universally in many different four-dimensional $$ \mathcal{N}=2 $$ SCFTs. We also discuss a decoupling limit that leads to $$ \mathcal{N}=\left(2,0\right) $$ ‘little string theories’ on AdS5 × $$ \mathbb{S} $$ 1.

CPphase from twisted Higgs vacuum expectation value in extra dimension

We propose a new mechanism for generating a CP phase via Higgs a vacuum expectation value originating from geometry of an extra dimension. A twisted boundary condition is the key to produce an extra-dimension coordinate-dependent vacuum expectation value, which contains a CP phase degree of freedom and can be a new source of a CP phase in higher-dimensional gauge theories. As an illustrative example, we apply our mechanism to a five-dimensional gauge theory with point interactions and show that our mechanism can dynamically produce a nontrivial CP-violating phase with electroweak symmetry breaking, even though the five-dimensional model does not include any CP-violating phases of Yukawa couplings in the five-dimensional Lagrangian because of a single generation of five-dimensional fermions. We apply our mechanism to a model with point interactions, which has no source of CP-violating phases in the couplings of the higher-dimensional action, and show that a nontrivial CP phase dynamically appears.

M5-BRANES, TORIC DIAGRAMS AND GAUGE THEORY DUALITY

We explore a duality between five-dimensional supersymmetric linear quiver gauge theories compactified on a circle, which are the five-dimensional uplifts of four-dimensional superconformal linear quiver theories. We find a correspondence between the gauge theory parameters of two dual theories, under which identical infrared effective coupling constants are obtained on the Coulomb branch. Two independent approaches using M-theory and the topological string theory give a consistent result.