Sharp Fourier type and cotype with respect to compact semisimple Lie groups

We perform a comprehensive analysis of the symmetry-resolved (SR) entanglement entropy (EE) for one single interval in the ground state of a 1 + 1D conformal field theory (CFT), that is invariant under an arbitrary finite or compact Lie group, G. We utilize the boundary CFT approach to study the total EE, which enables us to find the universal leading order behavior of the SREE and its first correction, which explicitly depends on the irreducible representation under consideration and breaks the equipartition of entanglement. We present two distinct schemes to carry out these computations. The first relies on the evaluation of the charged moments of the reduced density matrix. This involves studying the action of the defect-line, that generates the symmetry, on the boundary states of the theory. This perspective also paves the way for discussing the infeasibility of studying symmetry resolution when an anomalous symmetry is present. The second scheme draws a parallel between the SREE and the partition function of an orbifold CFT. This approach allows for the direct computation of the SREE without the need to use charged moments. From this standpoint, the infeasibility of defining the symmetry-resolved EE for an anomalous symmetry arises from the obstruction to gauging. Finally, we derive the symmetry-resolved entanglement spectra for a CFT invariant under a finite symmetry group. We revisit a similar problem for CFT with compact Lie group, explicitly deriving an improved formula for U(1) resolved entanglement spectra. Using the Tauberian formalism, we can estimate the aforementioned EE spectra rigorously by proving an optimal lower and upper bound on the same. In the abelian case, we perform numerical checks on the bound and find perfect agreement.

We study anti-self-dual Yang–Mills instantons on {mathbb {R}}^{3}times S^{1}, also known as calorons, and their behaviour under collapse of the circle factor. In this limit, we make explicit the decomposition of calorons in terms of constituent pieces which are essentially charge 1 monopoles. We give a gluing construction of calorons in terms of the constituents and use it to compute the dimension of the moduli space. The construction works uniformly for structure group an arbitrary compact semi-simple Lie group.

In this paper we present symbolic criteria for invariant operators on compact topological groups $G$ characterising the Schatten-von Neumann classes $S_{r}(L^{2}(G))$ for all $0<r\leq\infty$. Since it is known that for pseudo-differential operators criteria in terms of kernels may be less effective (Carleman's example), our criteria are given in terms of the operators' symbols defined on the noncommutative analogue of the phase space $G\times\hat{G}$, where $G$ is a compact topological (or Lie) group and $\hat{G}$ is its unitary dual. We also show results concerning general non-invariant operators as well as Schatten properties on Sobolev spaces. A trace formula is derived for operators in the Schatten class $S_{1}(L^{2}(G))$. Examples are given for Bessel potentials associated to sub-Laplacians (sums of squares) on compact Lie groups, as well as for powers of the sub-Laplacian and for other non-elliptic operators on SU(2)$\simeq\mathbb S^3$ and on SO(3).

In this paper we show that sets of polynomials in the components of (2j + 1)-dimensional vectors, solutions of certain invariant partial differential equations, form bases for all the irreducible representations of the unitary group U2j+1. These polynomials will play, for the group U2j+1, the same role that the solid spherical harmonics (themselves polynomials in the components of a three-dimensional vector) play for the rotation group R3. With the help of these polynomials we define and determine the reduced Wigner coefficients for the unitary groups, which we then use to derive the Wigner coefficients of U2j+1 by a factorization procedure. An ambiguity remains in the explicit expression for the Wigner coefficients as the Kronecker product of two irreducible representations of U2j+1 is not, in general, multiplicity-free. We show how to eliminate this ambiguity with the help of operators that serve to characterize completely the rows of representations of unitary groups for a particular chain of subgroups. The procedure developed to determine the polynomial bases of U2j+1 seems, in principle, generalizable to arbitrary semisimple compact Lie groups.

The geometric quantization on a homogeneous manifold is studied. For any quantizable function f , the stochastical expression for the unitary group exp(i tQ λ(f)) generated by the quantized operator Q λ( f ) is established. As an application, a Feynman-Kac formula for the compact semisimple Lie group is rederived.

Let [Formula: see text] be a compact Lie group acting on a smooth manifold [Formula: see text]. In this paper, we consider Meinrenken’s [Formula: see text]-equivariant bundle gerbe connections on [Formula: see text] as objects in a 2-groupoid. We prove this 2-category is equivalent to the 2-groupoid of gerbe connections on the differential quotient stack associated to [Formula: see text], and isomorphism classes of [Formula: see text]-equivariant gerbe connections are classified by degree 3 differential equivariant cohomology. Finally, we consider the existence and uniqueness of conjugation-equivariant gerbe connections on compact semisimple Lie groups.

Stability results for approximation by positive definite functions on SO(3)

Functions generated by blocks were introduced by M. Taibleson and G. Weiss in the setting of the one-dimensional torus T T [TW1]. They showed that these functions formed a space "close" to the class of integrable functions for which we have almost everywhere convergence of Fourier series. Together with S. Lu [LTW] they extended the theory to the n n -dimensional torus where this convergence result (for Bochner-Riesz means at the critical index) is valid provided we also restrict ourselves to L log ⁡ L L\log L . In this paper we show that this restriction is not needed if the underlying domain is a compact semisimple Lie group (or certain more general spaces of a homogeneous type). Other considerations (for example, these spaces form an interesting family of quasi-Banach spaces; they are connected with the notion of entropy) guide one in their study. We show how this point of view can be exploited in the setting of more general underlying domains.

An efficient construction of lattice points FM of any density and any admissible symmetry is described in a finite region F⊃FM of a real n‐dimensional Euclidean space. The shape of F and the lattice symmetry of FM is determined by a compact semisimple Lie group of rank n underlying the construction. The density of FM is fixed by our choice of a positive integer M, where 1⩽M<∞. The Lie group allows one to introduce systems of special functions discretely orthogonal on FM together with the possibility of expanding digital data on FM into finite series of such special functions.

Flat connections on punctured surfaces and geodesic polygons in a Lie group

The Bott–Borel–Weil theorem (BBW) is a statement about a complex homogeneous space X=G/R where G is a compact semisimple Lie group and R is the centralizer of a torus in G. One knows that BBW is equivalent to the determination of how R operates on the cohomology of a certain nilpotent Lie algebra of antiholomorphic tangent vectors operating on an arbitrary irreducible G-module. Upon replacing the complex structure with a space S of spinors BBW is equivalent to a statement about the eigenvalues of the Casimir operator of R in the module S ⊗ Vλ where Vλ is the irreducible G-module with highest weight λ. But then the complex structure is eliminated and the problem makes sense for any compact subgroup R ⊂ G. We solve the problem in the case where R and G have the same rank. The very special case where R is the centralizer of a torus then yields BBW. Involved in the general case (i.e. arbitrary R) is a new (at least to mathematicians) Dirac operator with a cubic term whose square is expressed in terms of the Casimir operator. Actually the spectral resolution of the Dirac operator is determined in the general case. The kernel of the Dirac operator is only given in the equal rank case. A key feature in that case is that, associated to λ, there is a multiplet of representations of R, having rather striking properties, and cardinality equal to the Euler number of G/R.

Pluriclosed metrics on compact semisimple Lie groups

Let G be a compact semisimple Lie group, g its Lie algebra, (,) an AdG-invariant inner product on g, and M an adjoint orbit in 9. In this article, if (M,(,)|M) is Kähler with respect to its canonical complex structure, then we give, for a closed minimal Lagrangian submanifold L⊂M, upper bounds on the first positive eigenvalue λ1(L) of the Laplacian ΔL, which acts on C∞(L), and lower bounds on the volume of L. In particular, when (M,(,)|M) is Kähler-Einstein, (p=cω, where p and ω are Ricci form and Kähler form of (M,(,)|M) with respect to the canonical complex structure respectively, and c is a positive constant,) we prove λ1(L)≤c. Combining with a result of Oh [5], we can see that L is Hamiltonian stable if and only if λ1(L)=c.

A flag manifold is a homogeneous space M = G/K, where G is a compact semisimple Lie group, and K the centralizer of a torus in G. Equivalently, M can be identified with the adjoint orbit Ad (G)w of an element w in the Lie algebra of G. We present several aspects of flag manifolds, such as their classification in terms of painted Dynkin diagrams, T-roots and G-invariant metrics, and Kähler metrics. We give a Lie-theoretic expression of the Ricci tensor in M, hence reducing the Einstein equation on flag manifolds into an algebraic system of equations, which can be solved in several cases. A flag manifold is also a complex manifold, and this dual representation as a real and a complex manifold is related to a similar property of an infinite-dimensional manifold, the loop space, which in fact can be viewed as a "universal" flag manifold.

A generalized metric on a manifold M M , i.e., a pair ( g , H ) (g,H) , where g g is a Riemannian metric and H H a closed 3 3 -form, is a fixed point of the generalized Ricci flow if and only if ( g , H ) (g,H) is Bismut Ricci flat: H H is g g -harmonic and R c ( g ) = 1 4 H g 2 Rc(g)=\tfrac {1}{4}H_g^2 . On any homogeneous space M = G / K M=G/K , where G = G 1 × G 2 G=G_1\times G_2 is a compact semisimple Lie group with two simple factors, under some mild assumptions, we exhibit a Bismut Ricci flat G G -invariant generalized metric, which is proved to be unique among a 4 4 -parameter space of metrics in many cases, including when K K is neither abelian nor semisimple. On the other hand, if K K is simple and the standard metric is Einstein on both G 1 / π 1 ( K ) G_1/\pi _1(K) and G 2 / π 2 ( K ) G_2/\pi _2(K) , we give a one-parameter family of Bismut Ricci flat G G -invariant generalized metrics on G / K G/K and show that it is most likely pairwise non-homothetic by computing the ratio of Ricci eigenvalues. This is proved to be the case for every space of the form M = G × G / Δ K M=G\times G/\Delta K and for M 35 = S O ( 8 ) × S O ( 7 ) / G 2 M^{35}=SO(8)\times SO(7)/G_2 .