Six-dimensional Theory Research Articles

Holographic Fermi surface at finite temperature in six-dimensional (2, 0) theory

We analyzed two-point Green's functions of operators in six-dimensional (2, 0) theory dual to fermions in maximal gauged supergravity in seven dimensions at finite temperature. We have considered backgrounds with both the charge parameters nonzero as well as only one charge parameter nonzero. In the former case we find all the modes admit Fermi surface(s), while in the latter, some of the modes do not have a Fermi surface. We find backgrounds corresponding to nonzero expectation value of the scalar appearing in the dual operator give rise to Fermi surface(s).

E(lementary)-strings in six-dimensional heterotic F-theory

Using E-strings, we can analyze not only six-dimensional superconformal field theories but also probe vacua of non-perturabative heterotic string. We study strings made of D3-branes wrapped on various two-cycles in the global F-theory setup. We claim that E-strings are elementary in the sense that various combinations of E-strings can form M-strings as well as heterotic strings and new kind of strings, called G-strings. Using them, we show that emissions and combinations of heterotic small instantons generate most of known six-dimensional superconformal theories, their affinizations and little string theories. Taking account of global structure of compact internal geometry, we also show that special combinations of E-strings play an important role in constructing six-dimensional theories of D- and E-types. We check global consistency conditions from anomaly cancellation conditions, both from five-branes and strings, and show that they are given in terms of elementary E-string combinations.

Holographic Fermi surfaces in the six-dimensional (2, 0) theory

We analyze the Fermi surface behavior of fermionic modes from a charged black hole background with two different charges arising in a truncated gauged supergravity theory in seven dimensions, which is dual to the six-dimensional (2, 0) theory. At zero temperature, we numerically solve equations for fermionic modes that do not couple to the gravitino. We find that each of the modes admits at least one Fermi surface. We find Fermi surfaces in both the Fermi-liquid and non-Fermi-liquid regimes.

Holography for field theory solitons

We extend a well-known D-brane construction of the AdS/dCFT correspondence to non-abelian defects. We focus on the bulk side of the correspondence and show that there exists a regime of parameters in which the low-energy description consists of two approximately decoupled sectors. The two sectors are gravity in the ambient spacetime, and a six-dimensional supersymmetric Yang-Mills theory. The Yang-Mills theory is defined on a rigid AdS4 × S2 background and admits sixteen supersymmetries. We also consider a one-parameter deformation that gives rise to a family of Yang-Mills theories on asymptotically AdS4 × S2 spacetimes, which are invariant under eight supersymmetries. With future holographic applications in mind, we analyze the vacuum structure and perturbative spectrum of the Yang-Mills theory on AdS4 × S2, as well as systems of BPS equations for finite-energy solitons. Finally, we demonstrate that the classical Yang-Mills theory has a consistent truncation on the two-sphere, resulting in maximally supersymmetric Yang-Mills on AdS4.

String-motivated one-loop amplitudes in gauge theories with half-maximal supersymmetry

We compute one-loop amplitudes in six-dimensional Yang-Mills theory with half-maximal supersymmetry from first principles: imposing gauge invariance and locality on an ansatz made from string-theory inspired kinematic building blocks yields unique expressions for the 3- and 4-point amplitudes. We check that the results are reproduced in the field-theory limit α′ → 0 of string amplitudes in K3 orbifolds, using simplifications made in a companion string-theory paper [1].

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.

4d mathcal{N}=1 from 6d mathcal{N}=left(1,0right) on a torus with fluxes

Compactifying mathcal{N}=left(1,0right) theories on a torus, with additional fluxes for global symmetries, we obtain mathcal{N}=1 supersymmetric theories in four dimensions. It is shown that for many choices of flux these models are toric quiver gauge theories with singlet fields. In particular we compare the anomalies deduced from the description of the six-dimensional theory and the anomalies of the quiver gauge theories. We also give predictions for anomalies of four-dimensional theories corresponding to general compactifications of M5-branes probing {mathrm{mathbb{C}}}^2/{mathrm{mathbb{Z}}}_k singularities.

Small vacuum energy from small equivalence violation in scalar gravity

The theory of scalar gravity proposed by Nordström, and refined by Einstein and Fokker, provides a striking analogy to general relativity. In its modern form, scalar gravity appears as the low-energy effective field theory of the spontaneous breaking of conformal symmetry within a CFT, and is AdS/CFT dual to the original Randall-Sundrum I model, but without a UV brane. Scalar gravity faithfully exhibits several qualitative features of the cosmological constant problem of standard gravity coupled to quantum matter, and the Weinberg no-go theorem can be extended to this case as well. Remarkably, a solution to the scalar gravity cosmological constant problem has been proposed, where the key is a very small violation of the scalar equivalence principle, which can be elegantly formulated as a particular type of deformation of the CFT. In the dual AdS picture this involves implementing Goldberger-Wise radion stabilization where the Goldberger-Wise field is a pseudo-Nambu Goldstone boson. In quantum gravity however, global symmetries protecting pNGBs are not expected to be fundamental. We provide a natural six-dimensional gauge theory origin for this global symmetry and show that the violation of the equivalence principle and the size of the vacuum energy seen by scalar gravity can naturally be exponentially small. Our solution may be of interest for study of non-supersymmetric CFTs in the spontaneously broken phase.

Supersymmetric R\xe9nyi entropy and Anomalies in 6d (1,0) SCFTs

A closed formula of the universal part of supersymmetric Rényi entropy Sq for six-dimensional (1, 0) superconformal theories is proposed. Within our arguments, Sq across a spherical entangling surface is a cubic polynomial of ν = 1/q, with 4 coefficients expressed as linear combinations of the ’t Hooft anomaly coefficients for the R-symmetry and gravitational anomalies. As an application, we establish linear relations between the c-type Weyl anomalies and the ’t Hooft anomaly coefficients. We make a conjecture relating the supersymmetric Rényi entropy to an equivariant integral of the anomaly polynomial in even dimensions and check it against known data in 4d and 6d.

Effective field theory for magnetic compactifications

Magnetic flux plays an important role in compactifications of field and string theories in two ways, it generates a multiplicity of chiral fermion zero modes and it can break supersymmetry. We derive the complete four-dimensional effective action for mathcal{N} = 1 supersymmetric Abelian and non-Abelian gauge theories in six dimensions compactified on a torus with flux. The effective action contains the tower of charged states and it accounts for the mass spectrum of bosonic and fermionic fields as well as their level-dependent interactions. This allows us to compute quantum corrections to the mass and couplings of Wilson lines. We find that the one-loop corrections vanish, contrary to the case without flux. This can be traced back to the spontaneous breaking of symmetries of the six-dimensional theory by the background gauge field, with the Wilson lines as Goldstone bosons.

Holography, brane intersections and six-dimensional SCFTs

We study supersymmetric intersections of NS5-, D6- and D8-branes in type IIA string theory. We focus on the supergravity description of this system and identify a “near horizon” limit in which we recover the recently classified supersymmetric seven-dimensional AdS solutions of massive type IIA supergravity. Using a consistent truncation to seven-dimensional gauged supergravity we construct a universal supersymmetric deformation of these AdS vacua. In the holographic dual six-dimensional (1,0) superconformal field theory this deformation describes a universal RG flow on the tensor branch of the vacuum moduli space triggered by a vacuum expectation value for a protected scalar operator of dimension four.

Spin-2 spectrum of six-dimensional field theories

We analyze the mass spectrum of spin-2 excitations around the gravity duals of "linear quiver" supersymmetric conformal field theories (SCFT's) in six dimensions. We show that for the entire family of gravity solutions it satisfies a bound which corresponds to a unitarity bound for the scaling dimension of the dual field theory operators. We determine the masses of excitations which belong to short multiplets, and for certain gravity solutions we obtain the Kaluza-Klein modes and the corresponding mass spectrum, fully and explicitly. Finally, we discuss an intuitive picture of the dual operators in terms of the effective descriptions of the SCFT's.

Line operators from M-branes on compact Riemann surfaces

In this paper, we determine the charge lattice of mutually local Wilson and 't Hooft line operators for class S theories living on M5-branes wrapped on compact Riemann surfaces. The main ingredients of our analysis are the fundamental group of the N-cover of the Riemann surface, and a quantum constraint on the six-dimensional theory. The latter plays a central role in excluding some of the possible lattices and imposing consistency conditions on the charges. This construction gives a geometric explanation for the mutual locality among the lines, fixing their charge lattice and the structure of the four-dimensional gauge group.

A landscape of field theories

Studying a quantum field theory involves a choice of space-time manifold and a choice of background for any global symmetries of the theory. We argue that many more choices are possible when specifying the background. In the context of branes in string theory, the additional data corresponds to a choice of supergravity tensor fluxes. We propose the existence of a landscape of field theory backgrounds, characterized by the space-time metric, global symmetry background and a choice of tensor fluxes. As evidence for this landscape, we study the supersymmetric six-dimensional (2,0) theory compactified to two dimensions. Different choices of metric and flux give rise to distinct two-dimensional theories, which can preserve differing amounts of supersymmetry.

Aspects of superconformal multiplets in D > 4

We explicitly construct and list all unitary superconformal multiplets, along with their index contributions, in five and six dimensions. From this data, we uncover various unifying themes in the representation theory of five- and six-dimensional superconformal field theories. At the same time, we provide a detailed argument for the complete classification of unitary irreducible representations in five dimensions using a combination of physical and mathematical techniques.

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.

Anomalies, renormalization group flows, and the a-theorem in six-dimensional (1, 0) theories

We establish a linear relation between the $a$-type Weyl anomaly and the 't Hooft anomaly coefficients for the $R$-symmetry and gravitational anomalies in six-dimensional $(1,0)$ superconformal field theories. For RG flows onto the tensor branch, where conformal symmetry is spontaneously broken, supersymmetry relates the anomaly mismatch $\Delta a$ to the square of a four-derivative interaction for the dilaton. This establishes the $a$-theorem for all such flows. The four-derivative dilaton interaction is in turn related to the Green-Schwarz-like terms that are needed to match the 't Hooft anomalies on the tensor branch, thus fixing their relation to $\Delta a$. We use our formula to obtain exact expressions for the $a$-anomaly of $N$ small $E_8$ instantons, as well as $N$ M5-branes probing an orbifold singularity, and verify the $a$-theorem for RG flows onto their Higgs branches. We also discuss aspects of supersymmetric RG flows that terminate in scale but not conformally invariant theories with massless gauge fields.

Spectrum in the presence of brane-localized mass on torus extra dimensions

The lightest mass eigenvalue of a six-dimensional theory compactified on a torus is numerically evaluated in the presence of the brane-localized mass term. The dependence on the cutoff scale $\Lambda$ is non-negligible even when $\Lambda$ is two orders of magnitude above the compactification scale, which indicates that the mass eigenvalue is sensitive to the size of the brane, in contrast to five-dimensional theories. We obtain an approximate expression of the lightest mass in the thin brane limit, which well fits the numerical calculations, and clarifies its dependence on the torus moduli parameter $\tau$. We find that the lightest mass is typically much lighter than the compactification scale by an order of magnitude even in the limit of a large brane mass.

Symmetry enhancements via 5d instantons, q W $$ q\mathcal{W} $$ -algebrae and (1, 0) superconformal index

We explore $$ \mathcal{N}=\left(1,0\right) $$ superconformal six-dimensional theories arising from M5 branes probing a transverse A k singularity. Upon circle compactification to 5 dimensions, we describe this system with a dual pq-web of five-branes and propose the spectrum of basic five-dimensional instanton operators driving global symmetry enhancement. For a single M5 brane, we find that the exact partition function of the 5d quiver gauge theory matches the 6d (1, 0) index, which we compute by letter counting. We finally show that S-duality of the pq-web implies new relations among vertex correlators of $$ q\mathcal{W} $$ algebrae.

Higher order QCD predictions for associated Higgs production with anomalous couplings to gauge bosons

We present predictions for the associated production of a Higgs boson at NLO+PS accuracy, including the effect of anomalous interactions between the Higgs and gauge bosons. We present our results in different frameworks, one in which the interaction vertex between the Higgs boson and Standard Model $W$ and $Z$ bosons is parameterized in terms of general Lorentz structures, and one in which Electroweak symmetry breaking is manifestly linear and the resulting operators arise through a six-dimensional effective field theory framework. We present analytic calculations of the Standard Model and Beyond the Standard Model contributions, and discuss the phenomenological impact of the higher order pieces. Our results are implemented in the NLO Monte Carlo program MCFM, and interfaced to shower Monte Carlos through the {\sc Powheg} box framework.