Representations Of SO Research Articles

Extending global fits of 4D Composite Higgs Models with partially composite leptons

We perform the first convergent Bayesian global fits of 4D Composite Higgs Models with partially-composite third generation quarks and leptons based on the minimal SO(5) → SO(4) symmetry breaking pattern. We consider two models with the τ lepton and its associated neutrino in different representations of SO(5). Fitting each model with a wide array of experimental constraints allows us to analyse the Bayesian evidence and currently-observed fine-tuning of each model by calculating the Kullback-Leibler divergence between their respective priors and posteriors. Notably both models are found to be capable of satisfying all constraints simultaneously at the 3σ level at scales of < 5 TeV. From a Bayesian viewpoint of naturalness the model with leptons in the 14 and 10 representations is preferred over those in the 5 representation due to its lower fine-tuning. Finally, we consider the experimental signatures for the preferred parameters in these models, including lepton partner decay signatures and gluon-fusion produced Higgs signal strengths, and discuss their potential phenomenology at future high-luminosity LHC runs.

Supersymmetric domain walls in maximal 6D gauged supergravity III

We continue our study of gaugings the maximal N=(2,2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N=(2,2)$$\\end{document} supergravity in six dimensions with gauge groups obtained from decomposing the embedding tensor under R+×SO(4,4)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^+\ imes SO(4,4)$$\\end{document} subgroup of the global symmetry SO(5, 5). Supersymmetry requires the embedding tensor to transform in 144c\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ extbf{144}}_c$$\\end{document} representation of SO(5, 5). Under R+×SO(4,4)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^+\ imes SO(4,4)$$\\end{document} subgroup, this leads to the embedding tensor in (8±3,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$({\ extbf{8}}^{\\pm 3},$$\\end{document}8±1,56±1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ extbf{8}}^{\\pm 1},{\ extbf{56}}^{\\pm 1})$$\\end{document} representations. Gaugings in 8±3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ extbf{8}}^{\\pm 3}$$\\end{document} representations lead to a translational gauge group R8\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^8$$\\end{document} while gaugings in 8±1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ extbf{8}}^{\\pm 1}$$\\end{document} representations give rise to gauge groups related to the scaling symmetry R+.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^+.$$\\end{document} On the other hand, the embedding tensor in 56±1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ extbf{56}}^{\\pm 1}$$\\end{document} representations gives CSO(4-p,p,1)∼SO(4-p,p)⋉R4⊂SO(4,4)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$CSO(4-p,p,1)\\sim SO(4-p,p) < imes {\\mathbb {R}}^4\\subset SO(4,4)$$\\end{document} gauge groups with p=0,1,2.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p=0,1,2.$$\\end{document} More interesting gauge groups can be obtained by turning on more than one representation of the embedding tensor subject to the quadratic constraints. In particular, we consider gaugings in both 56-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ extbf{56}}^{-1}$$\\end{document} and 8+3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ extbf{8}}^{+3}$$\\end{document} representations giving rise to larger SO(5-p,p)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$SO(5-p,p)$$\\end{document} and SO(4-p,p+1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$SO(4-p,p+1)$$\\end{document} gauge groups for p=0,1,2.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p=0,1,2.$$\\end{document} In this case, we also give a number of half-supersymmetric domain wall solutions preserving different residual symmetries. The solutions for gaugings obtained only from 56-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ extbf{56}}^{-1}$$\\end{document} representation are also included in these results when the 8+3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ extbf{8}}^{+3}$$\\end{document} part is accordingly turned off.

Maximality Properties of Generalized Springer Representations of SO (N, ℂ)

Abstract Let $C$ be a unipotent class of $G=\textrm{SO}(N,\mathbb{C})$, $\mathcal{E}$ an irreducible $G$-equivariant local system on $C$. The generalized Springer representation $\rho (C,\mathcal{E})$ appears in the top cohomology of some variety. Let $\bar \rho (C,\mathcal{E})$ be the representation obtained by summing over all cohomology groups of this variety. It is well known that $\rho (C,\mathcal{E})$ appears in $\bar \rho (C,\mathcal{E})$ with multiplicity $1$ and that its Springer support $C$ is strictly minimal in the closure ordering among the Springer supports of the irreducbile subrepresentations of $\bar \rho (C,\mathcal{E})$. Suppose $C$ is parametrized by an orthogonal partition with only odd parts. We prove that $\bar \rho (C,\mathcal{E})$ (resp. $\textrm{sgn}\otimes \bar \rho (C,\mathcal{E})$) has a unique multiplicity 1 “maximal” subrepresentation $\rho ^{\textrm{max}}$ (resp. “minimal” subrepresentation $\textrm{sgn}\otimes \rho ^{\textrm{max}}$), where $\textrm{sgn}$ is the sign representation. These are analogues of results for $\textrm{Sp}(2n,\mathbb{C})$ by Waldspurger.

Aperiodic spin chains at the boundary of hyperbolic tilings

In view of making progress towards establishing a holographic duality for theories defined on a discrete tiling of the hyperbolic plane, we consider a recently proposed boundary spin chain Hamiltonian with aperiodic couplings that are chosen such as to reflect the inflation rule, i.e. the construction principle, of the bulk tiling. As a remnant of conformal symmetry, the spin degrees of freedom are arranged in multiplets of the dihedral group under which the bulk lattice is invariant. For the boundary Hamiltonian, we use strong-disorder RG techniques and evaluate correlation functions, the entanglement entropy and mutual information for the case that the ground state is in an aperiodic singlet phase. We find that two-point functions decay as a power-law with exponent equal to one. Furthermore, we consider the case that the spin variables transform in the fundamental representation of SO(N), leading to a gapless system, and find that the effective central charge obtained from the entanglement entropy scales as lnN, reflecting the number of local degrees of freedom. We also determine the dependence of this central charge on the parameters specifying the bulk tiling. Moreover, we obtain an analytical expression for the mutual information, according to which there is no phase transition at any finite value of the distance between the two intervals involved.

On the spectrum and structure constants of short operators in N=4 SYM at strong coupling

We study short operators in planar mathcal{N} = 4 SYM at strong coupling, for general spin and SO(6) symmetric traceless representations. At strong coupling their dimension grows like Delta sim 2sqrt{delta }{lambda}^{1/4} and their spectrum of degeneracies can be analysed by considering the massive spectrum of type II strings in flat space-time. We furthermore compute their structure constants with two arbitrary chiral primary operators. This is done by considering the four-point correlator of arbitrary chiral primary operators at strong coupling in planar mathcal{N} = 4 SYM, including the supergravity approximation plus the infinite tower of stringy corrections that contributes in the flat space limit. Our results are valid for generic rank n symmetric traceless representations of SO(6) and in particular for n ≫ 1, as long as n ≪ λ1/4.

Symmetries in Celestial CFTd

We use tools from conformal representation theory to classify the symmetries associated to conformally soft operators in celestial CFT (CCFT) in general dimensions d. The conformal multiplets in d > 2 take the form of celestial necklaces whose structure is much richer than the celestial diamonds in d = 2, it depends on whether d is even or odd and involves mixed-symmetric tensor representations of SO(d). The existence of primary descendants in CCFT multiplets corresponds to (higher derivative) conservation equations for conformally soft operators. We lay out a unified method for constructing the conserved charges associated to operators with primary descendants. In contrast to the infinite local symmetry enhancement in CCFT2, we find the soft symmetries in CCFTd>2 to be finite-dimensional. The conserved charges that follow directly from soft theorems are trivial in d > 2, while non trivial charges associated to (generalized) currents and stress tensor are obtained from the shadow transform of soft operators which we relate to (an analytic continuation of) a specific type of primary descendants. We aim at a pedagogical discussion synthesizing various results in the literature.

From celestial correlators to AdS, and back

We present a general relation between celestial correlation functions in d-dimensions and Witten diagrams in (d + 1)-dimensional Euclidean anti-de Sitter (EAdS) space, to all orders in perturbation theory. Contact diagram processes are proportional to contact Witten diagrams and particle exchanges can be recast as a continuum of particle exchanges in EAdS where the exchanged particles carrying unitary Principal Series representations of SO(d + 1, 1). One can then try to import familiar EAdS techniques to study the properties of celestial correlators. In this work we use this relation to infer the analytic structure of the spectral density in the conformal partial wave expansion of celestial correlators which, at least perturbatively, should be a meromorphic function of the spectral parameter. We also discuss non-perturbative constraints from unitarity in Euclidean Conformal Field Theory, which requires positivity of the spectral density. This extends similar relations recently uncovered between boundary correlation functions in de Sitter space and Witten diagrams in EAdS, suggesting that EAdS could play a central role in efforts towards holography for all lambdas.

Fourfolds of Weil type and the spinor map

Recent papers by Markman and O’Grady give, besides their main results on the Hodge conjecture and on hyperkähler varieties, surprising and explicit descriptions of families of abelian fourfolds of Weil type with trivial discriminant. They also provide a new perspective on the well-known fact that these abelian varieties are Kuga Satake varieties for certain weight two Hodge structures of rank six.In this paper we give a pedestrian introduction to these results. The spinor map, which is defined using a half-spin representation of SO(8), is used intensively. For simplicity, we use basic representation theory and we avoid the use of triality.

Axions, WIMPs, proton decay and observable r in SO(10)

We explore some experimentally testable predictions of an SO(10) axion model which includes two 10-plets of fermions in order to resolve the axion domain wall problem. The axion symmetry can be safely broken after inflation, so that the isocurvature perturbations associated with the axion field are negligibly small. An unbroken gauge Z_2 symmetry in SO(10) ensures the presence of a stable WIMP-like dark matter, a linear combination of the electroweak doublets in the fermion 10-plets and an SO(10) singlet fermion with mass sim 62.5 ; textrm{GeV}; (1 ; textrm{TeV}) when it is mostly the singlet (doublet) fermion, that co-exists with axion dark matter. We also discuss gauge coupling unification, proton decay, inflation with non-minimal coupling to gravity and leptogenesis. With the identification of the SM singlet Higgs field in the 126 representation of SO(10) as inflaton, the magnetic monopoles are inflated away, and we find 0.963 lesssim n_s lesssim 0.965 and 0.003 lesssim r lesssim 0.036, where n_s and r denote the scalar spectral index and tensor-to-scalar ratio, respectively. These predictions can be tested in future experiments such as CMB-S4.

Ternary algebras associated with irreducible tensor representations of SO(3) and the quark model

We show that each irreducible tensor representation of weight 2 of the rotation group in the space of rank 3 tensors over three-dimensional space gives rise to an associative algebra with unity. We find the algebraic conditions to be satisfied by the generators of these algebras. Part of these relations is of binary, and another part is of ternary type. The structure of the latter one is based on the use of the cyclic group [Formula: see text] generated by the primitive cubic root of unity given by [Formula: see text]. The subspace of each algebra spanned by triple products of generators is five-dimensional (5D) and is identical with the space of irreducible tensor representation of weight 2 of the rotation group [Formula: see text]. We define a Hermitian scalar product in this 5D subspace and construct its orthonormal basis in terms of triple products of generators. Then we find an explicit formula for the Lie algebra homomorphism [Formula: see text]. We suggest that the algebras constructed in this way, with binary and ternary constitutive relations, may find applications in the quark model and the Grand Unified Theories.

Particles of a de Sitter Universe

The de Sitter spacetime is a maximally symmetric spacetime. It is one of the vacuum solutions to Einstein equations with a cosmological constant. It is the solution with a positive cosmological constant and describes a universe undergoing accelerated expansion. Among the possible signs for a cosmological constant, this solution is relevant for primordial and late-time cosmology. In the case of a zero cosmological constant, studies on the representations of its isometry group have led to a broader understanding of particle physics. The isometry group of d+1-dimensional de Sitter is the group SO(d+1,1), whose representations are well known. Given this insight, what can we learn about the elementary degrees of freedom in a four dimensional de Sitter universe by exploring how the unitary irreducible representations of SO(4,1) present themselves in cosmological setups? This article aims to summarize recent advances along this line that benefit towards a broader understanding of quantum field theory and holography at different signs of the cosmological constant. Particular focus is given to the manifestation of SO(4,1) representations at the late-time boundary of de Sitter. The discussion is concluded by pointing towards future questions at the late-time boundary and the static patch with a focus on the representations.

The 𝒫(𝜑)₂ Model on de Sitter Space

In 1975 Figari, Høegh-Krohn and Nappi constructed theP(φ)2{\mathscr P}(\varphi )_2model on the de Sitter space. Here we complement their work with new results, which connect this model to various areas of mathematics. In particular, i.) we discuss the causal structure of de Sitter space and the induces representations of the Lorentz group. We show that the UIRs ofSO0(1,2)SO_0(1,2)for both the principal and the complementary series can be formulated on Hilbert spaces whose functions are supported on a Cauchy surface. We describe the free classical dynamical system in both its covariant and canonical form, and present the associated quantum one-particle KMS structures in the sense of Kay (1985). Furthermore, we discuss the localisation properties of one-particle wave functions and how these properties are inherited by the algebras of local observables.ii.) we describe the relations between the modular objects (in the sense of Tomita-Takesaki theory) associated to wedge algebras and the representations of the Lorentz group. We connect the representations of SO(1,2) to unitary representations ofSO(3)SO(3)on the Euclidean sphere, and discuss how theP(φ)2{\mathscr P}(\varphi )_2interaction can be represented by a rotation invariant vector in the Euclidean Fock space. We present a novel Osterwalder-Schrader reconstruction theorem, which shows that physicalinfrared problemsare absent on de Sitter space. As shown in Figari, Høegh-Krohn, and Nappi (1975), the ultraviolet problems are resolved just like on flat Minkowski space. We state the Haag–Kastler axioms for theP(φ)2{\mathscr P}(\varphi )_2model and we explain how thegeneratorsof the boosts and the rotations for the interacting quantum field theory arise from thestress-energy tensor. Finally, we show that the interacting quantum fields satisfy theequations of motionin their covariant form. In summary, we argue that the de SitterP(φ)2{\mathscr P}(\varphi )_2model is the simplest and most explicit relativistic quantum field theory, which satisfies basic expectations, like covariance, particle creation, stability and finite speed of propagation.

A Numerical Calculation Algorithm of the Inclination Function Based on the Representation of SO(3) Group

A Numerical Calculation Algorithm of the Inclination Function Based on the Representation of SO(3) Group

Modification as a linguistic ‘relationship’: Ajust soproblem in Functional Discourse Grammar

AbstractThis study traces the relationship between two erstwhile separate linguistic elements,justandso, within the framework of Functional Discourse Grammar (FDG). In keeping with FDG’s form-oriented function-to-form approach, the study proceeds semasiologically by, first, examining the uses of relatively independent forms (i.e.the focus particlejustmodifyingsoas a degree word and a manner proform), then turning to more tightly-knit structures (i.e. just soas a subordinator of purpose and condition), and finally, looking at the fixed expressionjust soused as a part of a pragmatic marker. Using data from theCorpus of Contemporary American Englishand theCorpus of Historical American English, we argue that the different meanings ofjust soraise a number of issues related to the analysis of modification in FDG, namely the status and function of the modifierjustin the constructions under discussion and the concomitant representation ofso. Furthermore, the analysis shows that FDG can model very precisely the interplay of semantic and pragmatic information in the stages whenjust sois still compositional, withjustproviding interpersonal (i.e.pragmatic) andsorepresentational (i.e.semantic) information, as well as its development into the non-compositional and purely pragmatic discourse markerjust so you know.

’t Hooft expansion of SO(N) and Sp(N) mathcal{N} = 4 SYM revisited

We study the ’t Hooft expansion of d = 4 mathcal{N} = 4 supersymmetric Yang-Mills (SYM) theory with the gauge group SO(N) or Sp(N). We consider the 1/N5 expansion with fixed gsN5, where gs denotes the string coupling of bulk type IIB string theory on AdS5× mathbbm{RP} 5 and N5 refers to the RR 5-form flux through mathbbm{RP} 5. N5 differs from N due to a shift coming from the RR charge of O3-plane. As an example, we consider the 1/N5 expansion of the free energy of mathcal{N} = 4 SYM on S4 and the 1/2 BPS circular Wilson loops in the fundamental representation of SO(N) or Sp(N). We find that the 1/N5 expansion is more “closed string like” than the ordinary 1/N expansion.

Anatomy of scalar mediated proton decays in SO(10) models

Realistic models based on the renormalizable grand unified theories have varieties of scalars, many of which are capable of mediating baryon (B) and lepton (L) number non-conserving processes. We identify all such scalar fields residing in 10, overline{mathbf{126}} and 120 dimensional irreducible representations of SO(10) which can induce baryon and lepton number violating interactions through the leading order d = 6 and d = 7 operators. Explicitly computing their couplings with the standard model fermions, we derive the effective operators including the possibility of mixing between the scalars stemming from a given representation. We find that such interactions at d = 6 are mediated by only three sets of scalars: T(3, 1, −1/3), mathcal{T} (3, 1, −4/3) and mathbbm{T} (3, 3, −1/3) and their conjugates. In the models with 10 and overline{mathbf{126}} , only the first has appropriate couplings to mediate the proton decay. While mathcal{T} and mathbbm{T} can induce baryon number violating interactions when 120 is present, mathcal{T} does not contribute to the proton decay at tree level because of its flavour antisymmetric coupling. Three additional colour triplets and their conjugates can mediate nucleon decay via d = 7 operators which violate also the B − L. We give general expressions for partial widths of proton in terms of the fundamental Yukawa couplings and use these results to explicitly compute the proton lifetime and branching ratios for the minimal non-supersymmetric SO(10) model based on 10 and overline{mathbf{126}} Higgs. We find that the proton preferably decays into overline{nu} K+ or μ+K0 and list several distinct features of scalar mediated proton decay. If the latter dominates over the gauge mediated contributions, the proton decay spectrum provides a direct probe to the flavour structure of the underlying grand unified theory.

AdS (super)projectors in three dimensions and partial masslessness

We derive the transverse projection operators for fields with arbitrary integer and half-integer spin on three-dimensional anti-de Sitter space, AdS3. The projectors are constructed in terms of the quadratic Casimir operators of the isometry group SO(2, 2) of AdS3. Their poles are demonstrated to correspond to (partially) massless fields. As an application, we make use of the projectors to recast the conformal and topologically massive higher-spin actions in AdS3 into a manifestly gauge-invariant and factorised form. We also propose operators which isolate the component of a field that is transverse and carries a definite helicity. Such fields correspond to irreducible representations of SO(2, 2). Our results are then extended to the case of mathcal{N} = 1 AdS3 supersymmetry.

Exploring the landscape for soft theorems of nonlinear sigma models

We generalize soft theorems of the nonlinear sigma model beyond the mathcal{O} (p2) amplitudes and the coset of SU(N) × SU(N)/SU(N). We first discuss the universal flavor ordering of the amplitudes for the Nambu-Goldstone bosons, so that we can reinterpret the known mathcal{O} (p2) single soft theorem for SU(N) × SU(N)/SU(N) in the context of a general symmetry group representation. We then investigate the special case of the fundamental representation of SO(N), where a special flavor ordering of the “pair basis” is available. We provide novel amplitude relations and a Cachazo-He-Yuan formula for such a basis, and derive the corresponding single soft theorem. Next, we extend the single soft theorem for a general group representation to mathcal{O} (p4), where for at least two specific choices of the mathcal{O} (p4) operators, the leading non-vanishing pieces can be interpreted as new extended theory amplitudes involving bi-adjoint scalars, and the corresponding soft factors are the same as at mathcal{O} (p2). Finally, we compute the general formula for the double soft theorem, valid to all derivative orders, where the leading part in the soft momenta is fixed by the mathcal{O} (p2) Lagrangian, while any possible corrections to the subleading part are determined by the mathcal{O} (p4) Lagrangian alone. Higher order terms in the derivative expansion do not contribute any new corrections to the double soft theorem.

Conformal wave expansions for flat space amplitudes

The extended BMS algebra contains a conformal subgroup that acts on the celestial sphere as SO(1, 3). It is of interest to perform mode expansions of free fields in Minkowski spacetime that realize this symmetry in a simple way. In the present work we perform such a mode expansion for massive scalar fields using the unitary principal series representations of SO(1, 3) with a view to developing a holographic approach to gravity in asymptotically flat spacetime. These mode expansions are also of use in studying holography in three-dimensional de Sitter spacetime.

Universal Approximations of Invariant Maps by Neural Networks

We describe generalizations of the universal approximation theorem for neural networks to maps invariant or equivariant with respect to linear representations of groups. Our goal is to establish network-like computational models that are both invariant/equivariant and provably complete in the sense of their ability to approximate any continuous invariant/equivariant map. Our contribution is three-fold. First, in the general case of compact groups we propose a construction of a complete invariant/equivariant network using an intermediate polynomial layer. We invoke classical theorems of Hilbert and Weyl to justify and simplify this construction; in particular, we describe an explicit complete ansatz for approximation of permutation-invariant maps. Second, we consider groups of translations and prove several versions of the universal approximation theorem for convolutional networks in the limit of continuous signals on euclidean spaces. Finally, we consider 2D signal transformations equivariant with respect to the group SE(2) of rigid euclidean motions. In this case we introduce the “charge–conserving convnet”—a convnet-like computational model based on the decomposition of the feature space into isotypic representations of SO(2). We prove this model to be a universal approximator for continuous SE(2)—equivariant signal transformations.