1,827 research outputs found
String Topology: Background and Present State
The data of a "2D field theory with a closed string compactification" is an
equivariant chain level action of a cell decomposition of the union of all
moduli spaces of punctured Riemann surfaces with each component compactified as
a pseudomanifold with boundary. The axioms on the data are contained in the
following assumptions. It is assumed the punctures are labeled and divided into
nonempty sets of inputs and outputs. The inputs are marked by a tangent
direction and the outputs are weighted by nonnegative real numbers adding to
unity. It is assumed the gluing of inputs to outputs lands on the
pseudomanifold boundary of the cell decomposition and the entire pseudomanifold
boundary is decomposed into pieces by all such factorings. It is further
assumed that the action is equivariant with respect to the toroidal action of
rotating the markings. A main result of compactified string topology is the
Theorem (closed strings): Each oriented smooth manifold has a 2D field theory
with a closed string compactification on the equivariant chains of its free
loop space mod constant loops. The sum over all surface types of the top
pseudomanifold chain yields a chain X satisfying the master equation dX + X*X =
0 where * is the sum over all gluings. This structure is well defined up to
homotopy.
The genus zero parts yields an infinity Lie bialgebra on the equivariant
chains of the free loop space mod constant loops. The higher genus terms
provide further elements of algebraic structure called a "quantum Lie
bialgebra" partially resolving the involutive identity.
There is also a compactified discussion and a Theorem 2 for open strings as
the first step to a more complete theory. We note a second step for knots.Comment: 39 pages, 17 figures, latex: compile twic
Homotopy theory of the master equation package applied to algebra and geometry: a sketch of two interlocking programs
We interpret mathematically the pair (master equation, solution of master
equation) up to equivalence, as the pair (a presentation of a free triangular
dga T over a combination operad O, dga map of T into C, a dga over O) up to
homotopy equivalence of dgOa maps, see Definition 1. We sketch two general
applications: I to the theory of the definition and homotopy theory of infinity
versions of general algebraic structures including noncompact frobenius
algebras and Lie bialgebras. Here the target C would be the total Hom complex
between various tensor products of another chain complex B, C = HomB, O
describes combinations of operations like composition and tensor product
sufficient to describe the algebraic structure and one says that B has the
algebraic structure in question. II to geometric systems of moduli spaces up to
deformation like the moduli of J holomorphic curves. Here C is some geometric
chain complex containing the fundamental classes of the moduli spaces of the
geometric problem. We also discuss analogues of homotopy groups and Postnikov
systems for maps and impediments to using them related to linear terms in the
master equation called anomalies.Comment: 7 page
Lattice Hydrodynamics
Using the combinatorics of two interpenetrating face centered cubic lattices
together with the part of calculus naturally encoded in combinatorial topology,
we construct from first principles a lattice model of 3D incompressible
hydrodynamics on triply periodic three space. Actually the construction applies
to every dimension, but has special duality features in dimension three.Comment: 5 page
Open and Closed String field theory interpreted in classical Algebraic Topology
There is an interpretation of open string field theory in algebraic topology.
An interpretation of closed string field theory can be deduced from this open
string theory to obtain as well the interpretation of open and closed string
field theory combined.Comment: 19 page
Topological conjugacy of circle diffeomorphisms
The classical criterion for a circle diffeomorphism to be topologically
conjugate to an irrational rigid rotation was given by A. Denjoy. In 1985, one
of us (Sullivan) gave a new criterion. There is an example satisfying Denjoy's
bounded variation condition rather than Sullivan's Zygmund condition and vice
versa. This paper will give the third criterion which is implied by either of
the above criteria
The Cumulant Bijection and Differential Forms
According to Jae Suk Park, physicists use "canonical coordinate systems" to
compute correlations in perturbative quantum field theories. One may interpret
these canonical coordinate systems as equivalences of generalized differential
Lie algebras. In this note we discuss these flattenings in one particular
setting and refer to them as "cumulant bijections". The main point we make is
that these cumulant bijections are functorial for deformation retracts. The
discussion is completely self contained and based on well known universal
properties
Teichm\"uller Theory and the Universal Period Mapping via Quantum Calculus and the Space on the Circle
The Universal Teichm\"uller Space, , is a universal parameter space for
all Riemann surfaces. In earlier work of the first author it was shown that one
can canonically associate infinite- dimensional period matrices to the
coadjoint orbit manifold -- which resides within
as the (Kirillov-Kostant) submanifold of ``smooth points'' of . We now
extend that period mapping to the entire Universal Teichm\"uller space
utilizing the theory of the Sobolev space . is an
equivariant injective holomorphic immersion of into Universal Siegel
Space, and we describe the Schottky locus utilizing Connes' ``quantum
calculus''. There are connections to string theory. Universal Teichm\"uller
Space contains also the separable complex submanifold -- the
Teichm\"uller space of the universal hyperbolic lamination. Genus-independent
constructions like the universal period mapping proceed naturally to live on
this completion of the classical Teichm\"uller spaces. We show that
carries a natural convergent Weil-Petersson pairing.Comment: 39 pages (TEX
A formula for topology/deformations and its significance
The formula is with
and , where ,
and in degrees , and 0 are the free generators of a completed
free graded Lie algebra . The coefficients are defined by
. The theorem is that (I)
this formula for on generators extends to a derivation of square
zero on , (II) the formula for is unique satisfying the
first property, once given the formulae for and ,
along with the condition that the "flow" generated by moves to in
unit time.
The immediate significance of this formula is that it computes the infinity
cocommutative coalgebra structure on the chains of the closed interval. It may
be derived and proved using the geometrical idea of flat connections and one
parameter groups or flows of gauge transformations. The deeper significance of
such general DGLAs which want to combine deformation theory and rational
homotopy theory is proposed as a research problem.Comment: 17 page
Structured vector bundles define differential K-theory
A equivalence relation, preserving the Chern-Weil form, is defined between
connections on a complex vector bundle. Bundles equipped with such an
equivalence class are called Structured Bundles, and their isomorphism classes
form an abelian semi-ring. By applying the Grothedieck construction one obtains
the ring K, elements of which, modulo a complex torus of dimension the sum of
the odd Betti numbers of the base, are uniquely determined by the corresponding
element of ordinary K and the Chern-Weil form. This construction provides a
simple model of differential K-theory, c.f.Hopkins-Singer (2005), as well as a
useful codification of vector bundles with connection.Comment: 23 page
Closed string operators in topology leading to Lie bialgebras and higher string algebra
This paper explains the conjectured algebraic duality between genus zero
Gromov-Witten theory and genus zero "Closed String topology". This duality in
another perspective is discussed on page 87 of the book "Frobenius manifold,
quantum cohomology, and moduli spaces" (by Yuri Manin). This paper also
discusses Fulton MacPherson strata
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