201 research outputs found
Large N 2D Yang-Mills Theory and Topological String Theory
We describe a topological string theory which reproduces many aspects of the
1/N expansion of SU(N) Yang-Mills theory in two spacetime dimensions in the
zero coupling (A=0) limit. The string theory is a modified version of
topological gravity coupled to a topological sigma model with spacetime as
target. The derivation of the string theory relies on a new interpretation of
Gross and Taylor's ``\Omega^{-1} points.'' We describe how inclusion of the
area, coupling of chiral sectors, and Wilson loop expectation values can be
incorporated in the topological string approach.Comment: 95 pages, 15 Postscript figures, uses harvmac (Please use the "large"
print option.) Extensive revisions of the sections on topological field
theory. Added a compact synopsis of topological field theory. Minor typos
corrected. References adde
Evolution equation for the structure function g_2(x,Q^2)
We perform an extensive study of the scale dependence of flavor-singlet
contributions to the structure function g_2(x,Q^2) in polarized deep-inelastic
scattering. We find that the mixing between quark-antiquark-gluon and
three-gluon twist-3 operators only involves the three-gluon operator with the
lowest anomalous dimension and is weak in other cases. This means, effectively,
that only those three-gluon operators with the lowest anomalous dimension for
each moment are important, and allows to formulate a simple two-component
parton-like description of g_2(x,Q^2) in analogy with the conventional
description of twist-2 parton distributions. The similar simplification was
observed earlier for the nonsinglet distributions, although the reason is in
our case different.Comment: 53 pages, 10 figures, LaTeX styl
Positivity of the universal pairing in 3 dimensions
Associated to a closed, oriented surface S is the complex vector space with
basis the set of all compact, oriented 3-manifolds which it bounds. Gluing
along S defines a Hermitian pairing on this space with values in the complex
vector space with basis all closed, oriented 3-manifolds. The main result in
this paper is that this pairing is positive, i.e. that the result of pairing a
nonzero vector with itself is nonzero. This has bearing on the question of what
kinds of topological information can be extracted in principle from unitary 2+1
dimensional TQFTs.
The proof involves the construction of a suitable complexity function c on
all closed 3-manifolds, satisfying a gluing axiom which we call the topological
Cauchy-Schwarz inequality, namely that c(AB) <= max(c(AA),c(BB)) for all A,B
which bound S, with equality if and only if A=B. The complexity function c
involves input from many aspects of 3-manifold topology, and in the process of
establishing its key properties we obtain a number of results of independent
interest. For example, we show that when two finite volume hyperbolic
3-manifolds are glued along an incompressible acylindrical surface, the
resulting hyperbolic 3-manifold has minimal volume only when the gluing can be
done along a totally geodesic surface; this generalizes a similar theorem for
closed hyperbolic 3-manifolds due to Agol-Storm-Thurston.Comment: 83 pages, 21 figures; version 3: incorporates referee's comments and
correction
The mathematical foundations of quantum field theory
Imperial Users onl
Q(sqrt(-3))-Integral Points on a Mordell Curve
We use an extension of quadratic Chabauty to number fields,recently developed by the author with Balakrishnan, Besser and M ̈uller,combined with a sieving technique, to determine the integral points overQ(√−3) on the Mordell curve y2 = x3 − 4
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