Exceptional theories are a group of one-parameter scalar field theories with (enhanced) vanishing soft limits in the S-matrix elements. They include the nonlinear sigma model (NLSM), Dirac-Born-Infeld scalars and the special Galileon theory. The soft behavior results from the shift symmetry underlying these theories, which leads to Ward identities generating subleading single soft theorems as well as novel Berends-Giele recursion relations. Such an approach was first applied to NLSM in 1709.08639 and 1804.08629, and here we use it to systematically study other exceptional scalar field theories. In particular, using the subleading single soft theorem for the special Galileon we identify the Feynman vertices of the corresponding extended theory, which was first discovered using the Cachazo-He-Yuan representation of scattering amplitudes. Furthermore, we present a Lagrangian for the extended theory of the special Galileon, which has a rich particle content involving biadjoint scalars, Nambu-Goldstone bosons and Galileons, as well as additional flavor structure.
The low energy structures of N=1 supersymmetric models with E_6, F_4 and E_7 gauge groups and fundamental irrep matter contents are studied herein. We identify sets of gauge invariant composites which label all flat directions in the confining/Higgs phases of these theories. The impossibility of mapping several of these primary operators rules out previously conjectured exceptional self duals reported in the literature.
We study the low-energy dynamics in three-dimensional $\mathcal{N}=2$ exceptional gauge theories with matters in a fundamental representation, especially focusing on confinement phases and on a quantum structure of the Coulomb branch in the moduli space of vacua. We argue that the confinement phases of these exceptional gauge theories have a single Coulomb branch. The 3d s-confinement phases for the exceptional gauge groups are associated with quantum-deformed moduli spaces of the corresponding 4d $\mathcal{N}=1$ exceptional gauge theories.
We begin an investigation of supersymmetric theories based on exceptional groups. The flat directions are most easily parameterized using their correspondence with gauge invariant polynomials. Symmetries and holomorphy tightly constrain the superpotentials, but due to multiple gauge invariants other techniques are needed for their full determination. We give an explicit treatment of $G_2$ and find gaugino condensation for $N_f\leq 2$, and an instanton generated superpotential for $N_f=3$. The analogy with $SU(N_c)$ gauge theories continues with modified and unmodified quantum moduli spaces for…
We consider two different versions of gauged WZW theories with the exceptional groups and gauged with any of theirs null subgroups. By constructing suitable automorphism, we establish the equivalence of these two theories. On the other hand our automorphism, relates the two dual irreducible Riemannian globally symmetric spaces with different characters based on the corresponding exceptional Lie groups.
It is expected on general grounds that the moduli space of 4d $\mathcal{N}=3$ theories is of the form $\mathbb{C}^{3r}/Γ$, with $r$ the rank and $Γ$ a crystallographic complex reflection group (CCRG). As in the case of Lie algebras, the space of CCRGs consists of several infinite families, together with some exceptionals. To date, no 4d $\mathcal{N}=3$ theory with moduli space labelled by an exceptional CCRG (excluding Weyl groups) has been identified. In this work we show that the 4d $\mathcal{N}=3$ theories proposed in \cite{Garcia-Etxebarria:2016erx}, constructed via non-geometric quotients…
We study the fermionic zero modes around 1 instanton operators for 5d supersymmetric gauge theories of type USp, SO and the exceptional groups. The major motivation is to try to understand the global symmetry enhancement pattern in these theories.
Five dimensional field theories with exceptional gauge groups are engineered from degenerations of Calabi-Yau threefolds. The structure of the Coulomb branch is analyzed in terms of relative Kähler cones. For low number of flavors, the geometric construction leads to new five dimensional fixed points.
Exceptional field theories are the manifestly duality covariant formulations of the target space theories of string/M-theory in the low-energy limit (supergravity) or for certain truncations. These theories feature a rich system of sub-theories, corresponding to different consistent truncations. We review the structure of exceptional field theory and elaborate on various potential applications. In particular, we discuss how exceptional field theories capture some of the the magic supergravity triangles, Hull's $*$-theories related to timelike dualities, as well as the generalized supergravitie…
In this paper we advance the program of using exceptional collections to understand the gauge theory description of a D-brane probing a Calabi-Yau singularity. To this end, we strengthen the connection between strong exceptional collections and fractional branes. To demonstrate our ideas, we derive a strong exceptional collection for every Y^{p,q} singularity, and also prove that this collection is simple.
We present a new construction of four dimensional $\mathcal N=3$ theories, given by M5 branes wrapping a $T^2$ in an M-theory U-fold background. The resulting setup generalizes the one used in the usual class $\mathcal S$ construction of four dimensional theories by using an extra discrete symmetry on the M5 worldvolume. Together with the M-theory U-fold description of $(0,2)$ $E$-type six-dimensional SCFTs, this allows to construct new, exceptional, $\mathcal N=3$ theories.
We discuss models with no dynamical vector fields in various dimensions which we claim might have exceptional symmetry on some loci of their parameter space. In particular we construct theories with four supercharges flowing to theories with global symmetry enhancing to $F_4$, $E_6$, and $E_7$. The main evidence for these claims is based on extracting information about the symmetry properties of the theories from their supersymmetric partition functions.
We investigate the patching of double and exceptional field theories. In double field theory the patching conditions imposed on the spacetime after solving the strong section condition imply that the 3-form field strength $H$ is exact. A similar conclusion can be reached for the form field strengths of exceptional field theories after some plausive assumptions are made on the relation between the transition functions of the additional coordinates and the patching data of the form field strengths. We illustrate the issues that arise, and explore several alternative options which include the int…
We find the Seiberg-Witten geometry for four dimensional N=2 supersymmetric E_6 gauge theories with massless fundamental hypermultiplets, by geometrically embedding them in type II string theories compactified on Calabi-Yau threefolds. The resulting geometry completely agrees with that of recent works, which are based on the technique of N=1 confining phase superpotentials. We also derive the Seiberg-Witten geometry for E_7 gauge theories with massive fundamental hypermultiplets.
We derive four dimensional gauge theories with exceptional groups $F_4$, $E_8$, $E_7$, and $E_7$ with matter, by starting from the duality between the heterotic string on $K3$ and F-theory on a elliptically fibered Calabi-Yau 3-fold. This configuration is compactified to four dimensions on a torus, and by employing toric geometry, we compute the type IIB mirrors of the Calabi-Yaus of the type IIA string theory. We identify the Seiberg-Witten curves describing the gauge theories as ALE spaces fibered over a $P^1$ base.
We apply the recent proposal for mirrors of nonabelian (2,2) supersymmetric two-dimensional gauge theories to make predictions for two-dimensional supersymmetric gauge theories with exceptional gauge groups G2, F4, E6, E7, and E8. We compute the mirror Landau-Ginzburg models and predict excluded Coulomb loci and Coulomb branch relations (quantum cohomology). We also discuss the relationship between weight lattice normalizations and theta angle periodicities in the proposal, and explore different conventions for the mirrors. Finally, we discuss the behavior of pure gauge theories with exception…
We examine exceptional field theory through the lens of the generalised double copy formalism. This allows us to construct classical solutions in M-theory using a generalised Kerr-Schild ansatz and along the way indicates hints towards a single copy of M-theory. Based on a talk at the Nankai Symposium given by DSB.
Stacks of D3-branes placed at the tip of a cone over a del Pezzo surface provide a way of geometrically engineering a small but rich class of gauge/gravity dualities. We develop tools for understanding the resulting quiver gauge theories using exceptional collections. We prove two important results for a general quiver gauge theory: 1) we show the ordering of the nodes can be determined up to cyclic permutation and 2) we derive a simple formula for the ranks of the gauge groups (at the conformal point) in terms of the numbers of bifundamentals. We also provide a detailed analysis of four node …
Higher-form symmetries have proved useful in constraining the dynamics of a number of quantum field theories. In the context of the Argyres-Douglas (AD) theories of the $(G,G')$ type, we find that the 1-form symmetries are invariant under the Higgs branch flow, and that they are captured by the non-Higgsable sector at a generic point on the Higgs branch of the AD theory in question. As a consequence, dimensional reduction of an AD theory with a non-trivial 1-form symmetry to 3d leads to a free sector. We utilize these observations, along with other results, to propose systematically the mirror…
We study the four-dimensional superconformal N=2 gauge theories engineered by the Type IIB superstring on Arnold's 14 exceptional unimodal singularities (a.k.a. Arnold's strange duality list), thus extending the methods of 1006.3435 to singularities which are not the direct sum of minimal ones. In particular, we compute their BPS spectra in several `strongly coupled' chambers. From the TBA side, we construct ten new periodic Y-systems, providing additional evidence for the existence of a periodic Y-system for each isolated quasi-homogeneous singularity with $\hat c<2$ (more generally, for each…
We use conformal embeddings involving exceptional affine Kac-Moody algebras to derive new dualities of three-dimensional topological field theories. These generalize the familiar level-rank duality of Chern-Simons theories based on classical gauge groups to the setting of exceptional gauge groups. For instance, one duality sequence we discuss is $(E_{N})_{1}\leftrightarrow SU(9-N)_{-1}$. Others such as $SO(3)_{8}\leftrightarrow PSU(3)_{-6},$ are dualities among theories with classical gauge groups that arise due to their embedding into an exceptional chiral algebra. We apply these equivalences…
We construct novel web diagrams with a trivalent or quadrivalent gluing for various 6d/5d theories from certain Higgsings of 6d conformal matter theories on a circle. The theories realized on the web diagrams include 5d Kaluza-Klein theories from circle compactifications of the 6d $G_2$ gauge theory with 4 flavors, the 6d $F_4$ gauge theory with 3 flavors, the 6d $E_6$ gauge theory with 4 flavors and the 6d $E_7$ gauge theory with 3 flavors. The Higgsings also give rise to 5d Kaluza-Klein theories from twisted compactifications of 6d theories including the 5d pure SU(3) gauge theory with the C…
We treat the topic of the closures of the nilpotent orbits of the Lie algebras of Exceptional groups through their descriptions as moduli spaces, in terms of Hilbert series and the highest weight generating functions for their representation content. We extend the set of known Coulomb branch quiver theory constructions for Exceptional group minimal nilpotent orbits, or reduced single instanton moduli spaces, to include all orbits of Characteristic Height 2, drawing on extended Dynkin diagrams and the unitary monopole formula. We also present a representation theoretic formula, based on localis…
We introduce an infrared regulator in Yang--Mills theories under the form of a mass term for the nonabelian fields. We show that the resulting action, built in a covariant linear gauge, is multiplicatively renormalizable by proving the validity at all orders of the Slavnov identity defining the theory.
Double Field Theory (DFT) and Exceptional Field Theory (EFT), collectively called ExFTs, have proven to be a remarkably powerful new framework for string and M-theory. Exceptional field theories were constructed on a case by case basis as often each EFT has its own idiosyncrasies. Intuitively though, an $E_{n-1(n-1)}$ EFT must be contained in an $E_{n(n)}$ ExFT. In this paper we propose a generalised Kaluza-Klein ansatz to relate different ExFTs. We then discuss in more detail the different aspects of the relationship between various ExFTs including the coordinates, section condition and (pseu…
We present scheme-independent calculations of the derivative of the beta function, denoted $β'_{IR}$, at a conformally invariant infrared (IR) fixed point, in several asymptotically free chiral gauge theories, namely SO($4k+2$) with $2 \le k \le 4$ with respective numbers $N_f$ of fermions in the spinor representation, and E$_6$ with fermions in the fundamental representation.
We propose new ADHM-like methods to compute the Coulomb branch instanton partition functions of 5d and 6d supersymmetric gauge theories, with certain exceptional gauge groups or exceptional matters. We study $G_2$ theories with $n_{\bf 7}\leq 3$ matters in ${\bf 7}$, and $SO(7)$ theories with $n_{\bf 8}\leq 4$ matters in the spinor representation ${\bf 8}$. We also study the elliptic genera of self-dual instanton strings of 6d SCFTs with exceptional gauge groups or matters, including all non-Higgsable atomic SCFTs with rank $2$ or $3$ tensor branches. Some of them are tested with topological v…
We consider the infrared and ultraviolet behaviour of the effective quantum field theory of a single $Z_2$ symmetric scalar field. In a previous paper we proved to all orders in perturbation theory the renormalizability of massive effective scalar field theory using Wilson's exact renormalization group equation. Here we show that away from exceptional momenta the massless theory is similarly renormalizable, and we prove detailed bounds on Green's functions as arbitrary combinations of exceptional Euclidean momenta are approached. As a corollary we also prove Weinberg's Theorem for the massive …
We generate by computer a basis of invariants for the fundamental representations of the exceptional Lie groups E(6) and E(7), up to degree 18. We discuss the relevance of this calculation for the study of supersymmetric gauge theories, and revisit the self-dual exceptional models. We study the chiral ring of G(2) to degree 13, as well as a few classical groups. The homological dimension of a ring is a natural estimator of its complexity and provides a guideline for identifying theories that have a good chance to be amenable to a solution.
We discuss theories with the exceptional centerless gauge group G(2), paying attention to confinement and the pattern of chiral symmetry breaking. Exploiting the Higgs mechanism to break the symmetry down to SU(3), we also present how the familiar features of confinement and chiral symmetry breaking of SU(3) gauge theories reemerge. G(2) gauge theories show up as an unusual theoretical framework to study SU(3) gauge theories without the ``luxury'' of a center.
We initiate the classification of nonrelativistic effective field theories (EFTs) for Nambu-Goldstone (NG) bosons, possessing a set of redundant, coordinate-dependent symmetries. Similarly to the relativistic case, such EFTs are natural candidates for "exceptional" theories, whose scattering amplitudes feature an enhanced soft limit, that is, scale with a higher power of momentum at long wavelengths than expected based on the mere presence of Adler's zero. The starting point of our framework is the assumption of invariance under spacetime translations and spatial rotations. The setup is nevert…
Curci and Ferrari found a unique BRS-invariant action for non-Abelian gauge theories which includes a mass term for the gauge bosons. I analyze this action. While the BRS operator is not nilpotent, the Zinn-Justin equation generalizes in a simple way so that the renormalization of the theory is consistent with the infrared regularization provided by the mass---infrared singularities and ultraviolet infinities are therefore clearly separated. Relations between renormalization constants are derived in dimensional regularization with minimal subtraction. Additional new symmetries allow a simple c…
The infrared behaviour of the $ϕ^3$-theory is discussed stressing analogies with the Witten-Seiberg story about $N=2$ $QCD$. Though the microscopic theory is apparently not integrable, the effective theory is shown to be integrable at classical level, and a general solution of it in terms of hypergeometric functions is obtained. An effective theory for the multiparticle soft scattering is sketched.
We study the structure of the flat space wavefunctional in scalar field theories with nonlinearly realized symmetries. These symmetries imply soft theorems that are satisfied by wavefunction coefficients in the limit where one of the external momenta is scaled to zero. After elucidating the structure of these soft theorems in the nonlinear sigma model, Dirac-Born-Infeld, and galileon scalar theories, we combine them with information about the singularity structure of the wavefunction to bootstrap the wavefunction coefficients of these theories. We further systematize this construction through …
The DBI and special galileon theories exhibit a conformal symmetry at unphysical values of the spacetime dimension. We find the Lagrangian form of this symmetry. The special conformal transformations are non-linearly realized on the fields, even though conformal symmetry is unbroken. Commuting the conformal transformations with the extended shift symmetries, we find new symmetries, which when taken together with the conformal and shift symmetries close into a larger algebra. For DBI this larger algebra is the conformal algebra of the higher dimensional bulk in the brane embedding view of DBI. …
We continue the construction of non-trivial vacua for gauge theories on the 3-torus, started in hep-th/9901154. Application of constructions based on twist in SU(N) with N > 2 produce more extra vacua in theories with exceptional groups. We calculate the relevant unbroken subgroups, and their contribution to the Witten index. We show that the extra vacua we find in the exceptional groups are sufficient to solve the Witten index problem for these groups.
The special galileon and Dirac-Born-Infeld (DBI) theories are effective field theories of a single scalar field that have many interesting properties in flat space. These theories can be extended to all maximally symmetric spaces, where their algebras of shift symmetries are simple. We study aspects of the curved space versions of these theories: for the special galileon, we find a new compact expression for its Lagrangian in de Sitter space and a field redefinition that relates it to the previous, more complicated formulation. This field redefinition reduces to the well-studied galileon duali…
We find low energy equivalences between $N=2$ supersymmetric gauge theories with different simple gauge groups with and without matter. We give a construction of equivalences based on subgroups and find all examples with maximal simple subgroups. This is used to solve some theories with exceptional gauge groups $G_2$ and $F_4$. We are also able to solve an $E_6$ theory on a codimension one submanifold of its moduli space.
Recently proposed exceptional field theories (EFTs) making manifest the duality $E_{n(n)}$ symmetry, first observed as nonlinearly realized symmetries of the maximal $d=3,4,...,9$ supergravity ($n=11-d$) and containing 11D and type IIB supergravity as sectors, were formulated in enlarged spacetimes. In the case of $E_{7(7)}$ EFT such an enlarged spacetime can be identified with the bosonic body of the $d=4$ central charge superspace $Σ^{(60|32)}$, the ${\cal N}=8$ $d=4$ superspace completed by 56 additional bosonic coordinates associated to central charges of the maximal $d=4$ supersymmetry al…
We study non-perturbatively, via the Schwinger-Dyson equations, the leading infrared behavior of the pressure in the ladder approximation. This problem is discussed firstly in the context of a thermal scalar field theory, and the analysis is then extended to the Yang-Mills theory at high temperatures. Using the Feynman gauge, we find a system of two coupled integral equations for the gluon and ghost self-energies, which is solved analytically. The solutions of these equations show that the contributions to the pressure, when calculated in the ladder approximation, are finite in the infrared do…
We present two new one-parameter families of scheme transformations and apply these to study the scheme dependence of the infrared zero in the beta function of an asymptotically free non-Abelian gauge theory up to four-loop order. Our results provide a further quantitative measure of this scheme dependence, showing that for moderate values of the gauge coupling and the parameter specifying the scheme transformation, this dependence is relatively mild. We also remark on a generalized multi-parameter family of rational scheme transformations.
We classify four-dimensional effective field theories (EFTs) with enhanced soft limits, which arise due to non-linearly realised symmetries on the Goldstone modes of such theories. We present an algorithm for deriving all possible algebras that can be non-linearly realised on a set of Goldstone modes with canonical propagators, linearly realised Poincaré symmetries and interactions at weak coupling. We then perform a full classification of the cases with multiple scalars or multiple spin-$1/2$ fermions as the Goldstone modes. In each case there are only a small number of algebras consistent wi…
A method of exact all-order summation of leading infrared logarithms in two dimensional massless $Φ^4$-type non-renormalizable effective field theories (EFTs) is developed. The method is applied to the ${\rm O}(N)$-symmetric EFT, which is a two-dimensional sibling of the four dimensional ${\rm O}(N+1)/{\rm O}(N)$ sigma-model. For the first time the exact all-order summation of the $\left(E^{2} \ln(1/E)\right)^n$ contributions (chiral logarithms) for the $2 \to 2$ scattering amplitudes is performed in closed analytical form. The cases when the resulting amplitudes turn to be meromorphic functio…
We consider interesting Seiberg dualities for $Usp$ gauge theories with an antisymmetric, $8$ fundamentals and no superpotential. We reduce to three dimensions and systematically analyze deformations triggered by real and complex masses, reaching a plethora of $\mathcal{N}\!=\!2$ dualities for $U(N)$ and $Usp(2N)$ gauge theories, possibly with monopole superpotentials and Chern-Simons interactions. Special cases of these "exceptional dualities" are: supersymmetry enhancement dualities, "duality appetizers" and many known dualities relating rank-$1$ gauge groups. The $4d$ $\mathcal{N}\!=\!1$ $U…
Recently a prescription to compute the superconformal index for all theories of class S was proposed. In this paper we discuss some of the physical information which can be extracted from this index. We derive a simple criterion for the given theory of class S to have a decoupled free component and for it to have enhanced flavor symmetry. Furthermore, we establish a criterion for the "good", the "bad", and the "ugly" trichotomy of the theories. After interpreting the prescription to compute the index with non-maximal flavor symmetry as a residue calculus we address the computation of the index…
The coset spaces E$_8$/SO(10)$\times$H$_F$ allow complex structures which can account for three quark-lepton generations including right-handed neutrinos. We show that in the context of supersymmetric SO(10) gauge theories in 6 dimensions they also provide the Higgs fields which are needed to break the electroweak and ${B-L}$ gauge symmetries, and to generate small neutrino masses via the seesaw mechanism.
We study four dimensional N=1 gauge theories that arise on the worldvolume of D3-branes probing complex cones over del Pezzo surfaces. Global symmetries of the gauge theories are made explicit by using a correspondence between bifundamental fields in the quivers and divisors in the underlying geometry. These global symmetries are hidden, being unbroken when all inverse gauge couplings of the quiver theory vanish. In the broken phase, for finite gauge couplings, only the Cartan subalgebra is manifest as a global symmetry. Superpotentials for these models are constructed using global symmetry in…
Infrared divergences in the perturbative expansion of gauge theory amplitudes and cross sections have been a focus of theoretical investigations for almost a century. New insights still continue to emerge, as higher perturbative orders are explored, and high-precision phenomenological applications demand an ever more refined understanding. This review aims to provide a pedagogical overview of the subject. We briefly cover some of the early historical results, we provide some simple examples of low-order applications in the context of perturbative QCD, and discuss the necessary tools to extend …
Infrared singularities in massless gauge theories are known since the foundation of quantum field theories. The root of this problem can be tracked back to the very definition of these long-range interacting theories such as QED. It can be shown that singularities are caused by the massless degrees of freedom (i.e. the photons in the case of QED). In the Bloch-Nordsieck model the absence of the infrared catastrophe can be shown exactly by the complete summation of the radiative corrections. In this paper we will give the idea of the derivation of the Bloch-Nordsieck propagator, that describes …
We discuss the ultraviolet (UV) to infrared (IR) evolution of asymptotically free chiral gauge theories. Various types of IR behavior are considered, including confinement without spontaneous chiral symmetry breaking, and formation of bilinear fermion condensates that preserve or dynamically break the chiral gauge invariance. We compare different schemes for the study of this evolution and the resultant IR structure of the theories, including a conjectured inequality based on the counting of degrees of freedom. We note some new patterns of UV to IR evolution.
We analyze the ultraviolet to infrared evolution and nonperturbative properties of asymptotically free ${\rm SU}(N) \otimes {\rm SU}(N-4) \otimes {\rm U}(1)$ chiral gauge theories with $N_f$ copies of chiral fermions transforming according to $([2]_N,1)_{N-4} + ([\bar 1]_N,[\bar 1]_{N-4})_{-(N-2)} + (1,(2)_{N-4})_N$, where $[k]_N$ and $(k)_N$ denote the antisymmetric and symmetric rank-$k$ tensor representations of SU($N$) and the rightmost subscript is the U(1) charge. We give a detailed discussion for the lowest nondegenerate case, $N=6$. These theories can exhibit both self-breaking of a st…
We study theories with the exceptional gauge group G(2). The 14 adjoint "gluons" of a G(2) gauge theory transform as {3}, {3bar} and {8} under the subgroup SU(3), and hence have the color quantum numbers of ordinary quarks, anti-quarks and gluons in QCD. Since G(2) has a trivial center, a "quark" in the {7} representation of G(2) can be screened by "gluons". As a result, in G(2) Yang-Mills theory the string between a pair of static "quarks" can break. In G(2) QCD there is a hybrid consisting of one "quark" and three "gluons". In supersymmetric G(2) Yang-Mills theory with a {14} Majorana "gluin…
We address the construction of manifest U-duality invariant generalized diffeomorphisms. The closure of the algebra requires an extension of the tangent space to include a tensor hierarchy indicating the existence of an underlying unifying structure, compatible with E_{11} and Borcherds algebras constructions. We begin with four-dimensional gauged maximal supergravity, and build a generalized Lie derivative that encodes all the gauge transformations of the theory. A generalized frame is introduced, which accommodates for all the degrees of freedom, including the tensor hierarchy. The generaliz…
We calculate the ultraviolet to infrared evolution and analyze possible types of infrared behavior for several asymptotically free chiral gauge theories with gauge group SU($N$) and massless chiral fermions transforming according to a symmetric rank-2 tensor representation $S$ and $N+4$ copies (flavors) of a conjugate fundamental representation $\bar F$, together with a vectorlike subsector with chiral fermions in higher-dimensional representation(s). We construct and study three such chiral gauge theories. These have respective vectorlike subsectors comprised of (a) $p$ copies of fermions in …
