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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.
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Fractional differential (and difference) operators play a role in a number of diverse settings: integrable systems, mirror symmetry, Hurwitz numbers, the Bethe ansatz equations. We prove extensions of the three major results on algebras of commuting (ordinary) differentials operators to the setting of fractional differential operators: (1) the Burchnall-Chaundy theorem that a pair of commuting differential operators is algebraically dependent, (2) the classification of maximal commutative algebras of differential operators in terms of Satos theory and (3) the Krichever correspondence constructing those of rank 1 in an algebro-geometric way. Unlike the available proofs of the Burchnall-Chaundy theorem which use the action of one differential operator on the kernel of the other, our extension to the fractional case uses bounds on orders of fractional differential operators and growth of algebras, which also presents a new and much shorter proof of the original result. The second main theorem is achieved by developing a new tool of the spectral field of a point in Satos Grassmannian, which carries more information than the widely used notion of spectral curve of a KP solution. Our Krichever type correspondence for fractional differential operators is based on infinite jet bundles.
A formulation of $mathcal{N} = 2$ supersymmetric Yang-Mills theory with a spacetime-dependent gauge coupling allows to study the breaking of conformal symmetry at the quantum level. The theory has an energy-momentum tensor that is only conserved if an equation of motion for the coupling is imposed. It admits non-trivial solitons, among which the Wu-Yang monopole that can be regularized and turns out to be massless. On the other hand, the ordinary BPS monopole is only a solution in the large $N_c$ limit.
We consider the electroweak phase transition in the conformal extension of the standard model known as SU(2)cSM. Apart from the standard model particles, this model contains an additional scalar and gauge field that are both charged under the hidden SU(2)$_X$. This model generically exhibits a very strong phase transition that proceeds after a large amount of supercooling. We estimate the gravitational wave spectrum produced in this model and show that its amplitude and frequency fall within the observational window of LISA. We also discuss potential pitfalls and relevant points of improvement required to attain reliable estimates of the gravitational wave production in this - as well as in more general - class of models. In order to improve perturbativity during the early stages of transition that ends with bubble nucleation, we solve a thermal gap equation in the scalar sector inspired by the 2PI effective action formalism.
In this paper we study the existence of maximizers for two families of interpolation inequalities, namely a generalized Gagliardo-Nirenberg inequality and a new inequality involving the Riesz energy. Two basic tools in our argument are a generalization of Liebs Translation Lemma and a Riesz energy version of the Brezis--Lieb lemma.
We consider the inequalities of Gagliardo-Nirenberg and Sobolev in R^d, formulated in terms of the Laplacian Delta and of the fractional powers D^n := (-Delta)^(n/2) with real n >= 0; we review known facts and present novel results in this area. After illustrating the equivalence between these two inequalities and the relations between the corresponding sharp constants and maximizers, we focus the attention on the L^2 case where, for all sufficiently regular f : R^d -> C, the norm || D^j f||_{L^r} is bounded in terms of || f ||_{L^2} and || D^n f ||_{L^2} for 1/r = 1/2 - (theta n - j)/d, and suitable values of j,n,theta (with j,n possibly noninteger). In the special cases theta = 1 and theta = j/n + d/2 n (i.e., r = + infinity), related to previous results of Lieb and Ilyin, the sharp constants and the maximizers can be found explicitly; we point out that the maximizers can be expressed in terms of hypergeometric, Fox and Meijer functions. For the general L^2 case, we present two kinds of upper bounds on the sharp constants: the first kind is suggested by the literature, the second one is an alternative proposal of ours, often more precise than the first one. We also derive two kinds of lower bounds. Combining all the available upper and lower bounds, the Gagliardo-Nirenberg and Sobolev sharp constants are confined to quite narrow intervals. Several examples are given.
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