555 research outputs found
LOCALIC COMPLETION OF GENERALIZED METRIC SPACES I
Following Lawvere, a generalized metric space (gms) is a set X equipped with a metric map from X2 to the interval of upper reals (approximated from above but not from below) from 0 to ∞ inclusive, and satisfying the zero self-distance law and the triangle inequality.
We describe a completion of gms’s by Cauchy filters of formal balls. In terms of Lawvere’s approach using categories enriched over [0,∞], the Cauchy filters are equivalent to flat left modules.
The completion generalizes the usual one for metric spaces. For quasimetrics it is equivalent to the Yoneda completion in its netwise form due to K¨unzi and Schellekens and thereby gives a new and explicit characterization of the points of the Yoneda completion.
Non-expansive functions between gms’s lift to continuous maps between the completions.
Various examples and constructions are given, including finite products.
The completion is easily adapted to produce a locale, and that part of the work is constructively valid. The exposition illustrates the use of geometric logic to enable
point-based reasoning for locales
On uniform canonical bases in lattices and other metric structures
We discuss the notion of \emph{uniform canonical bases}, both in an abstract
manner and specifically for the theory of atomless lattices. We also
discuss the connection between the definability of the set of uniform canonical
bases and the existence of the theory of beautiful pairs (i.e., with the finite
cover property), and prove in particular that the set of uniform canonical
bases is definable in algebraically closed metric valued fields
D3-instantons, Mock Theta Series and Twistors
The D-instanton corrected hypermultiplet moduli space of type II string
theory compactified on a Calabi-Yau threefold is known in the type IIA picture
to be determined in terms of the generalized Donaldson-Thomas invariants,
through a twistorial construction. At the same time, in the mirror type IIB
picture, and in the limit where only D3-D1-D(-1)-instanton corrections are
retained, it should carry an isometric action of the S-duality group SL(2,Z).
We prove that this is the case in the one-instanton approximation, by
constructing a holomorphic action of SL(2,Z) on the linearized twistor space.
Using the modular invariance of the D4-D2-D0 black hole partition function, we
show that the standard Darboux coordinates in twistor space have modular
anomalies controlled by period integrals of a Siegel-Narain theta series, which
can be canceled by a contact transformation generated by a holomorphic mock
theta series.Comment: 42 pages; discussion of isometries is amended; misprints correcte
Automorphic Instanton Partition Functions on Calabi-Yau Threefolds
We survey recent results on quantum corrections to the hypermultiplet moduli
space M in type IIA/B string theory on a compact Calabi-Yau threefold X, or,
equivalently, the vector multiplet moduli space in type IIB/A on X x S^1. Our
main focus lies on the problem of resumming the infinite series of D-brane and
NS5-brane instantons, using the mathematical machinery of automorphic forms. We
review the proposal that whenever the low-energy theory in D=3 exhibits an
arithmetic "U-duality" symmetry G(Z) the total instanton partition function
arises from a certain unitary automorphic representation of G, whose Fourier
coefficients reproduce the BPS-degeneracies. For D=4, N=2 theories on R^3 x S^1
we argue that the relevant automorphic representation falls in the quaternionic
discrete series of G, and that the partition function can be realized as a
holomorphic section on the twistor space Z over M. We also offer some comments
on the close relation with N=2 wall crossing formulae.Comment: 25 pages, contribution to the proceedings of the workshop "Algebra,
Geometry and Mathematical Physics", Tjarno, Sweden, 25-30 October, 201
- …