Pseudo-Riemannian geometry encodes information geometry in optimal transport

Optimal transport and information geometry both study geometric structures on spaces of probability distributions. Optimal transport characterizes the cost-minimizing movement from one distribution to another, while information geometry originates from coordinate invariant properties of statistical inference. Their relations and applications in statistics and machine learning have started to gain more attention. In this paper we give a new differential-geometric relation between the two fields. Namely, the pseudo-Riemannian framework of Kim and McCann, which provides a geometric perspective on the fundamental Ma–Trudinger–Wang (MTW) condition in the regularity theory of optimal transport maps, encodes the dualistic structure of statistical manifold. This general relation is described using the framework of c-divergence under which divergences are defined by optimal transport maps. As a by-product, we obtain a new information-geometric interpretation of the MTW tensor on the graph of the transport map. This relation sheds light on old and new aspects of information geometry. The dually flat geometry of Bregman divergence corresponds to the quadratic cost and the pseudo-Euclidean space, and the logarithmic $$L^{(\alpha )}$$ L ( α ) -divergence introduced by Pal and the first author has constant sectional curvature in a sense to be made precise. In these cases we give a geometric interpretation of the information-geometric curvature in terms of the divergence between a primal-dual pair of geodesics.

Three dimensional mirror symmetry beyond ADE quivers and Argyres-Douglas theories

Mirror symmetry, a three dimensional $$ \mathcal{N} $$ N = 4 IR duality, has been studied in detail for quiver gauge theories of the ADE-type (as well as their affine versions) with unitary gauge groups. The A-type quivers (also known as linear quivers) and the associated mirror dualities have a particularly simple realization in terms of a Type IIB system of D3-D5-NS5-branes. In this paper, we present a systematic field theory prescription for constructing 3d mirror pairs beyond the ADE quiver gauge theories, starting from a dual pair of A-type quivers with unitary gauge groups. The construction involves a certain generalization of the S and the T operations, which arise in the context of the SL(2, ℤ) action on a 3d CFT with a U(1) 0-form global symmetry. We implement this construction in terms of two supersymmetric observables — the round sphere partition function and the superconformal index on S2 × S1. We discuss explicit examples of various (non-ADE) infinite families of mirror pairs that can be obtained in this fashion. In addition, we use the above construction to conjecture explicit 3d $$ \mathcal{N} $$ N = 4 Lagrangians for 3d SCFTs, which arise in the deep IR limit of certain Argyres-Douglas theories compactified on a circle.

Heisenberg Doubles for Snyder-Type Models

A Snyder model generated by the noncommutative coordinates and Lorentz generators closes a Lie algebra. The application of the Heisenberg double construction is investigated for the Snyder coordinates and momenta generators. This leads to the phase space of the Snyder model. Further, the extended Snyder algebra is constructed by using the Lorentz algebra, in one dimension higher. The dual pair of extended Snyder algebra and extended Snyder group is then formulated. Two Heisenberg doubles are considered, one with the conjugate tensorial momenta and another with the Lorentz matrices. Explicit formulae for all Heisenberg doubles are given.

On a Dual Pair of Multiobjective Interval-Valued Variational Control Problems

In this paper, by using the new concept of (ϱ,ψ,ω)-quasiinvexity associated with interval-valued path-independent curvilinear integral functionals, we establish some duality results for a new class of multiobjective variational control problems with interval-valued components. More concretely, we formulate and prove weak, strong, and converse duality theorems under (ϱ,ψ,ω)-quasiinvexity hypotheses for the considered class of optimization problems.

On Discrete Time Wilson Systems

In this paper, we define the discrete time Wilson frame (DTW frame) for l 2 ℤ and discuss some properties of discrete time Wilson frames. Also, we give an interplay between DTW frames and discrete time Gabor frames. Furthermore, a necessary and a sufficient condition for the DTW frame in terms of Zak transform are given. Moreover, the frame operator for the DTW frame is obtained. Finally, we discuss dual pair of frames for discrete time Wilson systems and give a sufficient condition for their existence.

Contact Dual Pairs

We introduce and study the notion of contact dual pair adopting a line bundle approach to contact and Jacobi geometry. A contact dual pair is a pair of Jacobi morphisms defined on the same contact manifold and satisfying a certain orthogonality condition. Contact groupoids and contact reduction are the main sources of examples. Among other properties, we prove the characteristic leaf correspondence theorem for contact dual pairs that parallels the analogous result of Weinstein for symplectic dual pairs.

Freely adjoining monoidal duals

Given a monoidal category $\mathcal C$ with an object J, we construct a monoidal category $\mathcal C[{J^ \vee }]$ by freely adjoining a right dual ${J^ \vee }$ to J. We show that the canonical strong monoidal functor $\Omega :\mathcal C \to \mathcal C[{J^ \vee }]$ provides the unit for a biadjunction with the forgetful 2-functor from the 2-category of monoidal categories with a distinguished dual pair to the 2-category of monoidal categories with a distinguished object. We show that $\Omega :\mathcal C \to \mathcal C[{J^ \vee }]$ is fully faithful and provide coend formulas for homs of the form $\mathcal C[{J^ \vee }](U,\,\Omega A)$ and $\mathcal C[{J^ \vee }](\Omega A,U)$ for $A \in \mathcal C$ and $U \in \mathcal C[{J^ \vee }]$ . If ${\rm{N}}$ denotes the free strict monoidal category on a single generating object 1, then ${\rm{N[}}{{\rm{1}}^ \vee }{\rm{]}}$ is the free monoidal category Dpr containing a dual pair – ˧ + of objects. As we have the monoidal pseudopushout $\mathcal C[{J^ \vee }] \simeq {\rm{Dpr}}{{\rm{ + }}_{\rm{N}}}\mathcal C$ , it is of interest to have an explicit model of Dpr: we provide both geometric and combinatorial models. We show that the (algebraist’s) simplicial category Δ is a monoidal full subcategory of Dpr and explain the relationship with the free 2-category Adj containing an adjunction. We describe a generalization of Dpr which includes, for example, a combinatorial model Dseq for the free monoidal category containing a duality sequence X0 ˧ X1 ˧ X2 ˧ … of objects. Actually, Dpr is a monoidal full subcategory of Dseq.

Dual pair correspondence in physics: oscillator realizations and representations

We study general aspects of the reductive dual pair correspondence, also known as Howe duality. We make an explicit and systematic treatment, where we first derive the oscillator realizations of all irreducible dual pairs: (GL(M, ℝ), GL(N, ℝ)), (GL(M, ℂ), GL(N, ℂ)), (U∗(2M), U∗(2N)), (U (M+, M−), U (N+, N−)), (O(N+, N−), Sp (2M, ℝ)), (O(N, ℂ), Sp(2M, ℂ)) and (O∗(2N ), Sp(M+, M−)). Then, we decompose the Fock space into irreducible representations of each group in the dual pairs for the cases where one member of the pair is compact as well as the first non-trivial cases of where it is non-compact. We discuss the relevance of these representations in several physical applications throughout this analysis. In particular, we discuss peculiarities of their branching properties. Finally, closed-form expressions relating all Casimir operators of two groups in a pair are established.