Breaking Operators Research Articles

Searches for violation of Lorentz invariance in top quark pair production using dilepton events in 13 TeV proton-proton collisions

A search for violation of Lorentz invariance in the production of top quark pairs (tt‾) is presented. The measured normalized differential tt‾ production cross section, as a function of the sidereal time, is examined for potential modulations induced by Lorentz-invariance breaking operators in an effective field theory extension of the standard model (SM). The cross section is measured from collision events collected by the CMS detector at a center-of-mass-energy of 13 TeV, corresponding to an integrated luminosity of 77.8 fb−1, and containing one electron and one muon. The results are found to be compatible with zero, in agreement with the SM, and are used to place upper limits at 68% confidence level on the magnitude of the Lorentz-violating couplings ranging from 1–8×10−3. This is the first precision test of the isotropy in special relativity with top quarks at the LHC, restricting further the bounds on such couplings by up to two orders of magnitude with respect to previous searches conducted at the Tevatron.

Symmetry breaking operators for the reductive dual pair (U[formula omitted],U[formula omitted])

We consider the dual pair (G,G′)=(Ul,Ul′) in the symplectic group Sp2ll′(R). Fix a Weil representation of the metaplectic group Sp˜2ll′(R). Let G˜ and G˜′ be the preimages of G and G′ in Sp˜2ll′(R), and let Π⊗Π′ be a genuine irreducible representation of G˜×G˜′. We study the Weyl symbol fΠ⊗Π′ of the (unique up to a possibly zero constant) symmetry breaking operator (SBO) intertwining the Weil representation with Π⊗Π′. This SBO coincides with the orthogonal projection of the space of the Weil representation onto its Π-isotypic component and also with the orthogonal projection onto its Π′-isotypic component. Hence fΠ⊗Π′ can be computed in two different ways, one using Π and the other using Π′. By matching the results, we recover Weyl’s theorem stating that Π⊗Π′ occurs in the Weil representation with multiplicity at most one and we also recover the complete list of the representations Π⊗Π′ occurring in Howe’s correspondence.

Branching problem for tensoring two Verma modules and its application to differential symmetry breaking operators

Kobayashi–Pevzner discovered in [Differential symmetry breaking operators: II. Rankin–Cohen operators for symmetric pairs, Selecta Math. (N.S.) 22(2) (2016) 847–911, MR 3477337] that the failure of the multiplicity-one property in the fusion rule of Verma modules of [Formula: see text] occurs exactly when the Rankin–Cohen brackets vanish, and classified all the corresponding parameters. In this paper, we provide yet another characterization for these parameters, and give a precise description of indecomposable components of the tensor product. Furthermore, we discuss when the tensor products of two Verma modules are isomorphic to each other for semisimple Lie algebras [Formula: see text].

Electrodynamics with violations of Lorentz and U(1) gauge symmetries and their Hamiltonian structures* *Supported by the National Key Research and Development Program of China (2020YFC2201503, the Zhejiang Provincial Natural Science Foundation of China (LR21A050001, LY20A050002) and the National Natural Science Foundation of China (12275238, 11675143). B.-F. Li is supported by the National Natural Science Foundation of China (NNSFC) (12005186)

This study aims to investigate Lorentz/U(1) gauge symmetry-breaking electrodynamics in the framework of the standard-model extension and analyze the Hamiltonian structure for the theory with a specific dimension of Lorentz breaking operators. For this purpose, we consider a general quadratic action of the modified electrodynamics with Lorentz/gauge-breaking operators and calculate the number of independent components of the operators at different dimensions in gauge invariance and breaking. With this general action, we then analyze how Lorentz/gauge symmetry-breaking can change the Hamiltonian structure of the theories by considering Lorentz/gauge-breaking operators with dimension as examples. We show that the Lorentz-breaking operators with gauge invariance do not change the classes of the theory constrains and the number of physical degrees of freedom of the standard Maxwell electrodynamics. When U(1) gauge symmetry-breaking operators are present, the theories generally lack a first-class constraint and have one additional physical degree of freedom compared to the standard Maxwell electrodynamics.

Generating operators of symmetry breaking—From discrete to continuous

Based on the “generating operator” of the Rankin–Cohen brackets introduced in Kobayashi–Pevzner [arXiv:2306.16800], we present a method to construct various fundamental operators with continuous parameters such as invariant trilinear forms on infinite-dimensional representations, the Fourier and the Poisson transforms on the anti-de Sitter space, and integral symmetry breaking operators for the fusion rules, among others, out of a countable set of differential symmetry breaking operators.

Conformally covariant differential symmetry breaking operators for a vector bundle of rank 3 on S3

We construct and give a complete classification of all the differential symmetry breaking operators [Formula: see text], between the spaces of smooth sections of a vector bundle of rank [Formula: see text] over the [Formula: see text]-sphere [Formula: see text], and a line bundle over the [Formula: see text]-sphere [Formula: see text]. In particular, we give necessary and sufficient conditions on the tuple of parameters [Formula: see text] for which these operators exist.

Gravitational reheating at strong coupling

We use the gauge/gravity correspondence to study the gravitational reheating of strongly coupled gauge theories in the rapid exit from the long inflationary (de Sitter) phase. We estimate the maximal reheating temperature of holographic models in d-spatial dimensions, whose scale invariance is explicitly broken by a source term for an operator \U0001d4aa∆ of dimension frac{d}{2}+frac{1}{2}<Delta <d+1 , as well as the top-down \U0001d4a9 = 2* and the cascading gauge theories. The reheating is most efficient when the inflationary phase Hubble constant is much larger than the mass scale of the conformal symmetry breaking of the theory, and for theories with the conformal symmetry breaking operators that are close to marginal.

A geometrical point of view for branching problems for holomorphic discrete series of conformal Lie groups

This paper is devoted to branching problems for holomorphic discrete series representations of a conformal group [Formula: see text] of a tube domain [Formula: see text] over a symmetric cone [Formula: see text] restricted to the conformal group [Formula: see text] of a tube domain [Formula: see text] holomorphically embedded in [Formula: see text]. The goal of this work is the explicit construction of the symmetry breaking and holographic operators in this geometrical setting. We answer this program with the introduction of another functional model for holomorphic discrete series representations. This model leads to a geometrical interpretation and a close relation to the theory of orthogonal polynomials for such branching problems. This program is illustrated by three cases. First, we consider the [Formula: see text]-fold tensor product of holomorphic discrete series of the universal covering of [Formula: see text]. Then, it is tested on the restrictions of a member of the scalar-valued holomorphic discrete series of the conformal group [Formula: see text] to the subgroup [Formula: see text], and finally to the subgroup [Formula: see text].

Computation of Weighted Bergman Inner Products on Bounded Symmetric Domains and Restriction to Subgroups

Let $(G,G_1)$ be a symmetric pair of holomorphic type, and we consider a pair of Hermitian symmetric spaces $D_1=G_1/K_1\subset D=G/K$, realized as bounded symmetric domains in complex vector spaces $\mathfrak{p}^+_1\subset\mathfrak{p}^+$ respectively. Then the universal covering group $\widetilde{G}$ of $G$ acts unitarily on the weighted Bergman space $\mathcal{H}_\lambda(D)\subset\mathcal{O}(D)$ on $D$. Its restriction to the subgroup $\widetilde{G}_1$ decomposes discretely and multiplicity-freely, and its branching law is given explicitly by Hua-Kostant-Schmid-Kobayashi's formula in terms of the $K_1$-decomposition of the space $\mathcal{P}(\mathfrak{p}^+_2)$ of polynomials on the orthogonal complement $\mathfrak{p}^+_2$ of $\mathfrak{p}^+_1$ in $\mathfrak{p}^+$. The object of this article is to compute explicitly the inner product $\big\langle f(x_2),{\rm e}^{(x|\overline{z})_{\mathfrak{p}^+}}\big\rangle_\lambda$ for $f(x_2)\in\mathcal{P}(\mathfrak{p}^+_2)$, $x=(x_1,x_2)$, $z\in\mathfrak{p}^+=\mathfrak{p}^+_1\oplus\mathfrak{p}^+_2$. For example, when $\mathfrak{p}^+$, $\mathfrak{p}^+_2$ are of tube type and $f(x_2)=\det(x_2)^k$, we compute this inner product explicitly by introducing a multivariate generalization of Gauss' hypergeometric polynomials ${}_2F_1$. Also, as an application, we construct explicitly $\widetilde{G}_1$-intertwining operators (symmetry breaking operators) $\mathcal{H}_\lambda(D)|_{\widetilde{G}_1}\to\mathcal{H}_\mu(D_1)$ from holomorphic discrete series representations of $\widetilde{G}$ to those of $\widetilde{G}_1$, which are unique up to constant multiple for sufficiently large $\lambda$.

Inversion of Rankin–Cohen operators via Holographic Transform

L’analyse des problèmes de branchement des restrictions des représentations fait émerger le concept de transformation de brisure de symétrie et celui de transformation holographique. Les opérateurs de brisure de symétrie diminuent le nombre de variables dans les modèles géométriques tandis que les opérateurs holographiques l’augmentent. Plusieurs développements en série ou intégrale de l’analyse classique sont des cas particuliers de telles transformations.

Branching laws for discrete series of some affine symmetric spaces

In this paper we study branching laws for certain unitary representations. This is done on the smooth vectors via a version of the {\it period integrals}, studied in number theory, and also closely connected to the {\it symmetry-breaking operators}, introduced by T.~Kobayashi. We exhibit non-vanishing symmetry breaking operators for the restriction of a representation $\Pi$ in the discrete spectrum for real hyperboloids to representations of smaller orthogonal groups. In the last part we discuss some conjectures for the restriction of representations in Arthur packets containing the representation $\Pi$ and the corresponding Arthur-Vogan packets to smaller orthogonal groups; these are inspired by the Gross-Prasad conjectures.

Conformal symmetry breaking differential operators on differential forms

We study conformal symmetry breaking differential operators which map differential forms on $\mathbb{R}^n$ to differential forms on a codimension one subspace $\mathbb{R}^{n-1}$. These operators are equivariant with respect to the conformal Lie algebra of the subspace $\mathbb{R}^{n-1}$. They correspond to homomorphisms of generalized Verma modules for ${\mathfrak so}(n,1)$ into generalized Verma modules for ${\mathfrak so}(n+1,1)$ both being induced from fundamental form representations of a parabolic subalgebra. We apply the F-method to derive explicit formulas for such homomorphisms. In particular, we find explicit formulas for the generators of the intertwining operators of the related branching problems restricting generalized Verma modules for ${\mathfrak so}(n+1,1)$ to ${\mathfrak so}(n,1)$. As consequences, we find closed formulas for all conformal symmetry breaking differential operators in terms of the first-order operators $d$, $\delta$, $\bar{d}$ and $\bar{\delta}$ and certain hypergeometric polynomials. A dominant role in these studies will be played by two infinite sequences of symmetry breaking differential operators which depend on a complex parameter $\lambda$. These will be termed the conformal first and second type symmetry breaking operators. Their values at special values of $\lambda$ appear as factors in two systems of factorization identities which involve the Branson-Gover operators of the Euclidean metrics on $\mathbb{R}^n$ and $\mathbb{R}^{n-1}$ and the operators $d$, $\delta$, $\bar{d}$ and $\bar{\delta}$ as factors, respectively. Moreover, they are shown to naturally recover the gauge companion and $Q$-curvature operators of the Euclidean metric on the subspace $\mathbb{R}^{n-1}$, respectively.

Towards a Swampland Global Symmetry Conjecture using weak gravity

It is widely believed and in part established that exact global symmetries are inconsistent with quantum gravity. One then expects that approximate global symmetries can be quantitatively constrained by quantum gravity or swampland arguments. We provide such a bound for an important class of global symmetries: Those arising from a gauged U(1) with the vector made massive via Higgsing with an axion. The latter necessarily couples to instantons, and their action can be constrained, using both the electric and magnetic version of the axionic weak gravity conjecture, in terms of the cutoff of the theory. As a result, instanton-induced symmetry breaking operators with a suppression factor not smaller than exp⁡(−MP2/Λ2) are present, where Λ is a cutoff of the 4d effective theory. We provide a general argument and clarify the meaning of Λ. Simple 4d and 5d models are presented to illustrate this, and we recall that this is the standard way in which things work out in string compactifications with brane instantons. The relation of our constraint to bounds that can be derived from wormholes or gravitational instantons and to those motivated by black-hole effects at finite temperature are discussed, and we present a generalization of the Giddings-Strominger wormhole solution to the case of a gauge-derived U(1) global symmetry. Finally, we discuss potential loopholes to our arguments.

Holographic transform for tensor product of holomorphic discrete series

We study holographic operators associated with Rankin–Cohen brackets which are symmetry breaking operators for the restriction of tensor products of holomorphic discrete series of the universal covering of [Formula: see text]. Furthermore, we investigate a geometrical interpretation of these operators and their relations to classical Jacobi polynomials.

Symmetry breaking operators for real reductive groups of rank one

For a pair of real reductive groups G′⊂G we consider the space HomG′(π|G′,τ) of intertwining operators between spherical principal series representations π of G and τ of G′, also called symmetry breaking operators. Restricting to those pairs (G,G′) where dim⁡HomG′(π|G′,τ)<∞ and G and G′ are of real rank one, we classify all symmetry breaking operators explicitly in terms of their distribution kernels. This generalizes previous work by Kobayashi–Speh for (G,G′)=(O(1,n+1),O(1,n)) to the reductive pairs(G,G′)=(U(1,n+1;F),U(1,m+1;F)×F) with F=C,H,O and F<U(n−m;F). In most cases, all symmetry breaking operators can be constructed using one meromorphic family of distributions whose poles and residues we describe in detail. In addition to this family, there may occur some sporadic symmetry breaking operators which we determine explicitly.

The compact picture of symmetry-breaking operators for rank-one orthogonal and unitary groups

We present a method to calculate intertwining operators between the underlying Harish-Chandra modules of degenerate principal series representations of a semisimple Lie group $G$ and a semisimple subgroup $G'$, and between their composition factors. Our method describes the restriction of these operators to the $K'$-isotypic components, $K'\subseteq G'$ a maximal compact subgroup, and reduces the representation theoretic problem to an infinite system of scalar equations of a combinatorial nature. For rank one orthogonal and unitary groups and spherical principal series representations we calculate these relations explicitly and use them to classify intertwining operators. We further show that in these cases automatic continuity holds, i.e. every intertwiner between the Harish-Chandra modules extends to an intertwiner between the Casselman--Wallach completions, verifying a conjecture by Kobayashi. Altogether, this establishes the compact picture of the recently studied symmetry breaking operators for orthogonal groups by Kobayashi--Speh, gives new proofs of their main results and extends them to unitary groups.

Knapp-Stein Type Intertwining Operators for Symmetric Pairs II. - The Translation Principle and Intertwining Operators for Spinors

For a symmetric pair $(G,H)$ of reductive groups we extend to a large class of generalized principal series representations our previous construction of meromorphic families of symmetry breaking operators. These operators intertwine between a possibly vector-valued principal series of $G$ and one for $H$ and are given explicitly in terms of their integral kernels. As an application we give a complete classification of symmetry breaking operators from spinors on a Euclidean space to spinors on a hyperplane, intertwining for a double cover of the conformal group of the hyperplane.

Shallow and deep neural network training by water wave optimization

It is well known that the performance of artificial neural networks (ANNs) is significantly affected by their structure design and parameter selection, for which traditional training methods have drawbacks such as long training times, over-fitting, and premature convergence. Evolutionary algorithms (EAs) have provided an effective tool for ANN parameter optimization. However, simultaneously optimizing ANN structures and parameters remains a difficult problem. In this study, we adapt water wave optimization (WWO), a relatively new EA, for optimizing both the parameters and structures of ANNs, including classical shallow ANNs and deep neural networks (DNNs). We use a variable-dimensional solution encoding to represent both the structure and parameters of an ANN, and adapt WWO propagation, refraction, and breaking operators to efficiently evolve variable-dimensional solutions to solve the complex network optimization problems. Computational experiments on a variety of benchmark datasets show that the WWO algorithm achieves a very competitive performance compared to other popular gradient-based algorithms and EAs.

Bernstein-Sato identities and conformal symmetry breaking operators

We present Bernstein-Sato identities for scalar-, spinor- and differential form-valued distribution kernels on Euclidean space associated to conformal symmetry breaking operators. The associated Bernstein-Sato operators lead to partially new formulae for conformal symmetry breaking differential operators on functions, spinors and differential forms.

Double Gegenbauer expansion of |s–t|α

ABSTRACTMotivated by the study of symmetry breaking operators for indefinite orthogonal groups, we give a Gegenbauer expansion of the two variable function in terms of the ultraspherical polynomials and . Generalization, specialization, and limits of the expansion are also discussed.