19,490 research outputs found
Global residues for sparse polynomial systems
We consider families of sparse Laurent polynomials f_1,...,f_n with a finite
set of common zeroes Z_f in the complex algebraic n-torus. The global residue
assigns to every Laurent polynomial g the sum of its Grothendieck residues over
the set Z_f. We present a new symbolic algorithm for computing the global
residue as a rational function of the coefficients of the f_i when the Newton
polytopes of the f_i are full-dimensional. Our results have consequences in
sparse polynomial interpolation and lattice point enumeration in Minkowski sums
of polytopes.Comment: Typos corrected, reference added, 13 pages, 5 figures. To appear in
JPA
Unification of Residues and Grassmannian Dualities
The conjectured duality relating all-loop leading singularities of n-particle
N^(k-2)MHV scattering amplitudes in N=4 SYM to a simple contour integral over
the Grassmannian G(k,n) makes all the symmetries of the theory manifest. Every
residue is individually Yangian invariant, but does not have a local space-time
interpretation--only a special sum over residues gives physical amplitudes. In
this paper we show that the sum over residues giving tree amplitudes can be
unified into a single algebraic variety, which we explicitly construct for all
NMHV and N^2MHV amplitudes. Remarkably, this allows the contour integral to
have a "particle interpretation" in the Grassmannian, where higher-point
amplitudes can be constructed from lower-point ones by adding one particle at a
time, with soft limits manifest. We move on to show that the connected
prescription for tree amplitudes in Witten's twistor string theory also admits
a Grassmannian particle interpretation, where the integral over the
Grassmannian localizes over the Veronese map from G(2,n) to G(k,n). These
apparently very different theories are related by a natural deformation with a
parameter t that smoothly interpolates between them. For NMHV amplitudes, we
use a simple residue theorem to prove t-independence of the result, thus
establishing a novel kind of duality between these theories.Comment: 56 pages, 11 figures; v2: typos corrected, minor improvement
T-Branes and Monodromy
We introduce T-branes, or "triangular branes," which are novel non-abelian
bound states of branes characterized by the condition that on some loci, their
matrix of normal deformations, or Higgs field, is upper triangular. These
configurations refine the notion of monodromic branes which have recently
played a key role in F-theory phenomenology. We show how localized matter
living on complex codimension one subspaces emerge, and explain how to compute
their Yukawa couplings, which are localized in complex codimension two. Not
only do T-branes clarify what is meant by brane monodromy, they also open up a
vast array of new possibilities both for phenomenological constructions and for
purely theoretical applications. We show that for a general T-brane, the
eigenvalues of the Higgs field can fail to capture the spectrum of localized
modes. In particular, this provides a method for evading some constraints on
F-theory GUTs which have assumed that the spectral equation for the Higgs field
completely determines a local model.Comment: 110 pages, 5 figure
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