Commutative L-algebras and measure theory

Measure and integration theory for finitely additive measures, including vector-valued measures, is shown to be essentially covered by a class of commutative L-algebras, called measurable algebras. The domain and range of any measure is a commutative L-algebra. Each measurable algebra embeds into its structure group, an abelian group with a compatible lattice order, and each (general) measure extends uniquely to a monotone group homomorphism between the structure groups. On the other hand, any measurable algebra X is shown to be the range of an essentially unique measure on a measurable space, which plays the role of a universal covering. Accordingly, we exhibit a fundamental group of X, with stably closed subgroups corresponding to a special class of measures with X as target. All structure groups of measurable algebras arising in a classical context are archimedean. Therefore, they admit a natural embedding into a group of extended real-valued continuous functions on an extremally disconnected compact space, the Stone space of the measurable algebra. Extending Loomis’ integration theory for finitely additive measures, it is proved that, modulo null functions, each integrable function can be represented by a unique continuous function on the Stone space.

Galois coverings of one-sided bimodule problems

Applying geometric methods of 2-dimensional cell complex theory, we construct a Galois covering of a bimodule problem satisfying some structure, triangularity and finiteness conditions in order to describe the objects of finite representation type. Each admitted bimodule problem A is endowed with a quasi multiplicative basis. The main result shows that for a problem from the considered class having some finiteness restrictions and the schurian universal covering A', either A is schurian, or its basic bigraph contains a dotted loop, or it has a standard minimal non-schurian bimodule subproblem.

Notes on two-dimensional pure supersymmetric gauge theories

In this note we study IR limits of pure two-dimensional supersymmetric gauge theories with semisimple non-simply-connected gauge groups including SU(k)/ℤk, SO(2k)/ℤ2, Sp(2k)/ℤ2, E6/ℤ3, and E7/ℤ2 for various discrete theta angles, both directly in the gauge theory and also in nonabelian mirrors, extending a classification begun in previous work. We find in each case that there are supersymmetric vacua for precisely one value of the discrete theta angle, and no supersymmetric vacua for other values, hence supersymmetry is broken in the IR for most discrete theta angles. Furthermore, for the one distinguished value of the discrete theta angle for which supersymmetry is unbroken, the theory has as many twisted chiral multiplet degrees of freedom in the IR as the rank. We take this opportunity to further develop the technology of nonabelian mirrors to discuss how the mirror to a G gauge theory differs from the mirror to a G/K gauge theory for K a subgroup of the center of G. In particular, the discrete theta angles in these cases are considerably more intricate than those of the pure gauge theories studied in previous papers, so we discuss the realization of these more complex discrete theta angles in the mirror construction. We find that discrete theta angles, both in the original gauge theory and their mirrors, are intimately related to the description of centers of universal covering groups as quotients of weight lattices by root sublattices. We perform numerous consistency checks, comparing results against basic group-theoretic relations as well as with decomposition, which describes how two-dimensional theories with one-form symmetries (such as pure gauge theories with nontrivial centers) decompose into disjoint unions, in this case of pure gauge theories with quotiented gauge groups and discrete theta angles.

Complete integrability of the Benjamin–Ono equation on the multi-soliton manifolds

This paper is dedicated to proving the complete integrability of the Benjamin–Ono (BO) equation on the line when restricted to every N-soliton manifold, denoted by $$\mathcal {U}_N$$ U N . We construct generalized action–angle coordinates which establish a real analytic symplectomorphism from $$\mathcal {U}_N$$ U N onto some open convex subset of $${\mathbb {R}}^{2N}$$ R 2 N and allow to solve the equation by quadrature for any such initial datum. As a consequence, $$\mathcal {U}_N$$ U N is the universal covering of the manifold of N-gap potentials for the BO equation on the torus as described by Gérard–Kappeler (Commun Pure Appl Math, 2020. 10.1002/cpa.21896. arXiv:1905.01849). The global well-posedness of the BO equation on $$\mathcal {U}_N$$ U N is given by a polynomial characterization and a spectral characterization of the manifold $$\mathcal {U}_N$$ U N . Besides the spectral analysis of the Lax operator of the BO equation and the shift semigroup acting on some Hardy spaces, the construction of such coordinates also relies on the use of a generating functional, which encodes the entire BO hierarchy. The inverse spectral formula of an N-soliton provides a spectral connection between the Lax operator and the infinitesimal generator of the very shift semigroup.

Uniqueness of Complete Hypersurfaces in Weighted Riemannian Warped Products

In this paper, applying the weak maximum principle, we obtain the uniqueness results for the hypersurfaces under suitable geometric restrictions on the weighted mean curvature immersed in a weighted Riemannian warped product I × ρ M f n whose fiber M has f -parabolic universal covering. Furthermore, applications to the weighted hyperbolic space are given. In particular, we also study the special case when the ambient space is weighted product space and provide some results by Bochner’s formula. As a consequence of this parametric study, we also establish Bernstein-type properties of the entire graphs in weighted Riemannian warped products.

Positive energy representations of Sobolev diffeomorphism groups of the circle

We show that any positive energy projective unitary representation of $$\mathrm{Diff}_+(S^1)$$ Diff + ( S 1 ) extends to a strongly continuous projective unitary representation of the fractional Sobolev diffeomorphisms $$\mathcal {D}^s(S^1)$$ D s ( S 1 ) for any real $$s>3$$ s > 3 , and in particular to $$C^k$$ C k -diffeomorphisms $$\mathrm{Diff}_+^k(S^1)$$ Diff + k ( S 1 ) with $$k\ge 4$$ k ≥ 4 . A similar result holds for the universal covering groups provided that the representation is assumed to be a direct sum of irreducibles. As an application we show that a conformal net of von Neumann algebras on $$S^1$$ S 1 is covariant with respect to $$\mathcal {D}^s(S^1)$$ D s ( S 1 ) , $$s > 3$$ s > 3 . Moreover every direct sum of irreducible representations of a conformal net is also $$\mathcal {D}^s(S^1)$$ D s ( S 1 ) -covariant.

On macroscopic dimension of non-spin 4-manifolds

We prove that for 4-manifolds [Formula: see text] with residually finite fundamental group and non-spin universal covering [Formula: see text], the inequality [Formula: see text] implies the inequality [Formula: see text]. This allows us to complete the proof of Gromov’s Conjecture for 4-manifolds with abelian fundamental group.

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.

Fake Galois actions

We prove that for all non-abelian finite simple groups [Formula: see text], there exists a fake [Formula: see text]th Galois action on [Formula: see text] with respect to [Formula: see text], where [Formula: see text] is the universal covering group of [Formula: see text] and [Formula: see text] is any non-negative integer coprime to the order of [Formula: see text]. This is one of the two inductive conditions needed to prove an [Formula: see text]-modular analogue of the Glauberman–Isaacs correspondence.

Geometric Approach to Optimal Path Problem with Uncertain Arc Lengths

In this paper, the problem of finding the shortest paths, one of the most important problems in science and technology has been geometrically studied. Shortest path algorithm has been generalized to the shortest cycles in each homotopy class on a surface with arbitrary topology, using the universal covering space notion in the algebraic topology. Then, a general algorithm has been presented to compute the shortest cycles (geometrically rather than combinatorial) in each homotopy class. The algorithm can handle surface meshes with the desired topology, with or without boundary. It also provides a fundamental framework for other algorithms based on universal coverage space due to the capacity and flexibility of the framework.