1,623 research outputs found
Loop Equations and the Topological Phase of Multi-Cut Matrix Models
We study the double scaling limit of mKdV type, realized in the two-cut
Hermitian matrix model. Building on the work of Periwal and Shevitz and of
Nappi, we find an exact solution including all odd scaling operators, in terms
of a hierarchy of flows of matrices. We derive from it loop
equations which can be expressed as Virasoro constraints on the partition
function. We discover a ``pure topological" phase of the theory in which all
correlation functions are determined by recursion relations. We also examine
macroscopic loop amplitudes, which suggest a relation to 2D gravity coupled to
dense polymers.Comment: 24p
Combinatorial Gradient Fields for 2D Images with Empirically Convergent Separatrices
This paper proposes an efficient probabilistic method that computes
combinatorial gradient fields for two dimensional image data. In contrast to
existing algorithms, this approach yields a geometric Morse-Smale complex that
converges almost surely to its continuous counterpart when the image resolution
is increased. This approach is motivated using basic ideas from probability
theory and builds upon an algorithm from discrete Morse theory with a strong
mathematical foundation. While a formal proof is only hinted at, we do provide
a thorough numerical evaluation of our method and compare it to established
algorithms.Comment: 17 pages, 7 figure
Two-dimensional state sum models and spin structures
The state sum models in two dimensions introduced by Fukuma, Hosono and Kawai
are generalised by allowing algebraic data from a non-symmetric Frobenius
algebra. Without any further data, this leads to a state sum model on the
sphere. When the data is augmented with a crossing map, the partition function
is defined for any oriented surface with a spin structure. An algebraic
condition that is necessary for the state sum model to be sensitive to spin
structure is determined. Some examples of state sum models that distinguish
topologically-inequivalent spin structures are calculated.Comment: 43 pages. Mathematica script in ancillary file. v2: nomenclature of
models and their properties changed, some proofs simplified, more detailed
explanations. v3: extended introduction, presentational improvements; final
versio
Recursion between Mumford volumes of moduli spaces
We propose a new proof, as well as a generalization of Mirzakhani's recursion
for volumes of moduli spaces. We interpret those recursion relations in terms
of expectation values in Kontsevich's integral, i.e. we relate them to a Ribbon
graph decomposition of Riemann surfaces. We find a generalization of
Mirzakhani's recursions to measures containing all higher Mumford's kappa
classes, and not only kappa1 as in the Weil-Petersson case.Comment: Latex, 18 page
A paradigm of open/closed duality: Liouville D-branes and the Kontsevich model
We argue that topological matrix models (matrix models of the Kontsevich
type) are examples of exact open/closed duality. The duality works at finite N
and for generic `t Hooft couplings. We consider in detail the paradigm of the
Kontsevich model for two-dimensional topological gravity. We demonstrate that
the Kontsevich model arises by topological localization of cubic open string
field theory on N stable branes. Our analysis is based on standard worldsheet
methods in the context of non-critical bosonic string theory. The stable branes
have Neumann (FZZT) boundary conditions in the Liouville direction. Several
generalizations are possible.Comment: v2: References added; a new section with generalization to non-zero
bulk cosmological constant; expanded discussion on topological localization;
added some comment
Quantum Tetrahedra
We discuss in details the role of Wigner 6j symbol as the basic building
block unifying such different fields as state sum models for quantum geometry,
topological quantum field theory, statistical lattice models and quantum
computing. The apparent twofold nature of the 6j symbol displayed in quantum
field theory and quantum computing -a quantum tetrahedron and a computational
gate- is shown to merge together in a unified quantum-computational SU(2)-state
sum framework
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