8,775 research outputs found
Open string theory and planar algebras
In this note we show that abstract planar algebras are algebras over the
topological operad of moduli spaces of stable maps with Lagrangian boundary
conditions, which in the case of the projective line are described in terms of
real rational functions. These moduli spaces appear naturally in the
formulation of open string theory on the projective line. We also show two
geometric ways to obtain planar algebras from real algebraic geometry, one
based on string topology and one on Gromov-Witten theory. In particular,
through the well known relation between planar algebras and subfactors, these
results establish a connection between open string theory, real algebraic
geometry, and subfactors of von Neumann algebras.Comment: 13 pages, LaTeX, 7 eps figure
Complex Curve of the Two Matrix Model and its Tau-function
We study the hermitean and normal two matrix models in planar approximation
for an arbitrary number of eigenvalue supports. Its planar graph interpretation
is given. The study reveals a general structure of the underlying analytic
complex curve, different from the hyperelliptic curve of the one matrix model.
The matrix model quantities are expressed through the periods of meromorphic
generating differential on this curve and the partition function of the
multiple support solution, as a function of filling numbers and coefficients of
the matrix potential, is shown to be the quasiclassical tau-function. The
relation to softly broken N=1 supersymmetric Yang-Mills theories is discussed.
A general class of solvable multimatrix models with tree-like interactions is
considered.Comment: 36 pages, 10 figures, TeX; final version appeared in special issue of
J.Phys. A on Random Matrix Theor
Correspondences between projective planes
We characterize integral homology classes of the product of two projective
planes which are representable by a subvariety.Comment: Improved readability, 14 page
Elliptic Feynman integrals and pure functions
We propose a variant of elliptic multiple polylogarithms that have at most
logarithmic singularities in all variables and satisfy a differential equation
without homogeneous term. We investigate several non-trivial elliptic two-loop
Feynman integrals with up to three external legs and express them in terms of
our functions. We observe that in all cases they evaluate to pure combinations
of elliptic multiple polylogarithms of uniform weight. This is the first time
that a notion of uniform weight is observed in the context of Feynman integrals
that evaluate to elliptic polylogarithms.Comment: 47 page
Moduli of Tropical Plane Curves
We study the moduli space of metric graphs that arise from tropical plane
curves. There are far fewer such graphs than tropicalizations of classical
plane curves. For fixed genus , our moduli space is a stacky fan whose cones
are indexed by regular unimodular triangulations of Newton polygons with
interior lattice points. It has dimension unless or .
We compute these spaces explicitly for .Comment: 31 pages, 25 figure
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