2,687 research outputs found
Boundaries of Amplituhedra and NMHV Symbol Alphabets at Two Loops
In this sequel to arXiv:1711.11507 we classify the boundaries of amplituhedra
relevant for determining the branch points of general two-loop amplitudes in
planar super-Yang-Mills theory. We explain the connection to
on-shell diagrams, which serves as a useful cross-check. We determine the
branch points of all two-loop NMHV amplitudes by solving the Landau equations
for the relevant configurations and are led thereby to a conjecture for the
symbol alphabets of all such amplitudes.Comment: 42 pages, 6 figures, 8 tables; v2: minor corrections and improvement
Simplicial and Cellular Trees
Much information about a graph can be obtained by studying its spanning
trees. On the other hand, a graph can be regarded as a 1-dimensional cell
complex, raising the question of developing a theory of trees in higher
dimension. As observed first by Bolker, Kalai and Adin, and more recently by
numerous authors, the fundamental topological properties of a tree --- namely
acyclicity and connectedness --- can be generalized to arbitrary dimension as
the vanishing of certain cellular homology groups. This point of view is
consistent with the matroid-theoretic approach to graphs, and yields
higher-dimensional analogues of classical enumerative results including
Cayley's formula and the matrix-tree theorem. A subtlety of the
higher-dimensional case is that enumeration must account for the possibility of
torsion homology in trees, which is always trivial for graphs. Cellular trees
are the starting point for further high-dimensional extensions of concepts from
algebraic graph theory including the critical group, cut and flow spaces, and
discrete dynamical systems such as the abelian sandpile model.Comment: 39 pages (including 5-page bibliography); 5 figures. Chapter for
forthcoming IMA volume "Recent Trends in Combinatorics
Crystals, instantons and quantum toric geometry
We describe the statistical mechanics of a melting crystal in three
dimensions and its relation to a diverse range of models arising in
combinatorics, algebraic geometry, integrable systems, low-dimensional gauge
theories, topological string theory and quantum gravity. Its partition function
can be computed by enumerating the contributions from noncommutative instantons
to a six-dimensional cohomological gauge theory, which yields a dynamical
realization of the crystal as a discretization of spacetime at the Planck
scale. We describe analogous relations between a melting crystal model in two
dimensions and N=4 supersymmetric Yang-Mills theory in four dimensions. We
elaborate on some mathematical details of the construction of the quantum
geometry which combines methods from toric geometry, isospectral deformation
theory and noncommutative geometry in braided monoidal categories. In
particular, we relate the construction of noncommutative instantons to deformed
ADHM data, torsion-free modules and a noncommutative twistor correspondence.Comment: 33 pages, 5 figures; Contribution to the proceedings of "Geometry and
Physics in Cracow", Jagiellonian University, Cracow, Poland, September 21-25,
2010. To be published in Acta Physica Polonica Proceedings Supplemen
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