A bound on chaos from stability

We explore the quantum chaos of the coadjoint orbit action of diffeomorphism group of S1. We study quantum fluctuation around a saddle point to evaluate the soft mode contribution to the out-of-time-ordered correlator. We show that the stability condition of the semi-classical analysis of the coadjoint orbit found in [1] leads to the upper bound on the Lyapunov exponent which is identical to the bound on chaos proven in [2]. The bound is saturated by the coadjoint orbit Diff(S1)/SL(2) while the other stable orbit Diff(S1)/U(1) where the SL(2, ℝ) is broken to U(1) has non-maximal Lyapunov exponent.

Complexity measures in QFT and constrained geometric actions

We study the conditions under which, given a generic quantum system, complexity metrics provide actual lower bounds to the circuit complexity associated to a set of quantum gates. Inhomogeneous cost functions — many examples of which have been recently proposed in the literature — are ruled out by our analysis. Such measures are shown to be unrelated to circuit complexity in general and to produce severe violations of Lloyd’s bound in simple situations. Among the metrics which do provide lower bounds, the idea is to select those which produce the tightest possible ones. This establishes a hierarchy of cost functions and considerably reduces the list of candidate complexity measures. In particular, the criterion suggests a canonical way of dealing with penalties, consisting in assigning infinite costs to directions not belonging to the gate set. We discuss how this can be implemented through the use of Lagrange multipliers. We argue that one of the surviving cost functions defines a particularly canonical notion in the sense that: i) it straightforwardly follows from the standard Hermitian metric in Hilbert space; ii) its associated complexity functional is closely related to Kirillov’s coadjoint orbit action, providing an explicit realization of the “complexity equals action” idea; iii) it arises from a Hamilton-Jacobi analysis of the “quantum action” describing quantum dynamics in the phase space canonically associated to every Hilbert space. Finally, we explain how these structures provide a natural framework for characterizing chaos in classical and quantum systems on an equal footing, find the minimal geodesic connecting two nearby trajectories, and describe how complexity measures are sensitive to Lyapunov exponents.

Gradedness of the set of rook placements in A n−1

A rook placement is a subset of a root system consisting of positive roots with pairwise non-positive inner products. To each rook placement in a root system one can assign the coadjoint orbit of the Borel subgroup of a reductive algebraic group with this root system. Degenerations of such orbits induce a natural partial order on the set of rook placements. We study combinatorial structure of the set of rook placements in An− 1 with respect to a slightly different order and prove that this poset is graded.

Comparison and continuity of Wick-type star products on certain coadjoint orbits

In this paper, we discuss continuity properties of the Wick-type star product on the 2-sphere, interpreted as a coadjoint orbit. Star products on coadjoint orbits in general have been constructed by different techniques. We compare the constructions of Alekseev–Lachowska and Karabegov, and we prove that they agree in general. In the case of the 2-sphere, we establish the continuity of the star product, thereby allowing for a completion to a Fréchet algebra.

Minimal-dimensional representations of reduced enveloping algebras for

Let $\mathfrak{g}=\mathfrak{g}\mathfrak{l}_{N}(\Bbbk )$ , where $\Bbbk$ is an algebraically closed field of characteristic $p>0$ , and $N\in \mathbb{Z}_{{\geqslant}1}$ . Let $\unicode[STIX]{x1D712}\in \mathfrak{g}^{\ast }$ and denote by $U_{\unicode[STIX]{x1D712}}(\mathfrak{g})$ the corresponding reduced enveloping algebra. The Kac–Weisfeiler conjecture, which was proved by Premet, asserts that any finite-dimensional $U_{\unicode[STIX]{x1D712}}(\mathfrak{g})$ -module has dimension divisible by $p^{d_{\unicode[STIX]{x1D712}}}$ , where $d_{\unicode[STIX]{x1D712}}$ is half the dimension of the coadjoint orbit of $\unicode[STIX]{x1D712}$ . Our main theorem gives a classification of $U_{\unicode[STIX]{x1D712}}(\mathfrak{g})$ -modules of dimension $p^{d_{\unicode[STIX]{x1D712}}}$ . As a consequence, we deduce that they are all parabolically induced from a one-dimensional module for $U_{0}(\mathfrak{h})$ for a certain Levi subalgebra $\mathfrak{h}$ of $\mathfrak{g}$ ; we view this as a modular analogue of Mœglin’s theorem on completely primitive ideals in $U(\mathfrak{g}\mathfrak{l}_{N}(\mathbb{C}))$ . To obtain these results, we reduce to the case where $\unicode[STIX]{x1D712}$ is nilpotent, and then classify the one-dimensional modules for the corresponding restricted $W$ -algebra.