30 research outputs found
Traintrack Calabi-Yaus from Twistor Geometry
We describe the geometry of the leading singularity locus of the traintrack
integral family directly in momentum twistor space. For the two-loop case,
known as the elliptic double box, the leading singularity locus is a genus one
curve, which we obtain as an intersection of two quadrics in .
At three loops, we obtain a K3 surface which arises as a branched surface over
two genus-one curves in . We present an
analysis of its properties. We also discuss the geometry at higher loops and
the supersymmetrization of the construction.Comment: 23 pages, 5 figure
On the Geometry of Null Polygons in Full N=4 Superspace
We discuss various formulations of null polygons in full, non-chiral N=4
superspace in terms of spacetime, spinor and twistor variables. We also note
that null polygons are necessarily fat along fermionic directions, a curious
fact which is compensated by suitable equivalence relations in physical
theories on this superspace.Comment: 25 pages, v2: comment on correlation functions adde
Smooth Wilson Loops in N=4 Non-Chiral Superspace
We consider a supersymmetric Wilson loop operator for 4d N=4 super Yang-Mills
theory which is the natural object dual to the AdS_5 x S^5 superstring in the
AdS/CFT correspondence. It generalizes the traditional bosonic 1/2 BPS
Maldacena-Wilson loop operator and completes recent constructions in the
literature to smooth (non-light-like) loops in the full N=4 non-chiral
superspace. This Wilson loop operator enjoys global superconformal and local
kappa-symmetry of which a detailed discussion is given. Moreover, the
finiteness of its vacuum expectation value is proven at leading order in
perturbation theory. We determine the leading vacuum expectation value for
general paths both at the component field level up to quartic order in
anti-commuting coordinates and in the full non-chiral superspace in suitable
gauges. Finally, we discuss loops built from quadric splines joined in such a
way that the path derivatives are continuous at the intersection.Comment: 44 pages. v2 Added some clarifying comments. Matches the published
versio
Null Polygonal Wilson Loops in Full N=4 Superspace
We compute the one-loop expectation value of light-like polygonal Wilson
loops in N=4 super-Yang-Mills theory in full superspace. When projecting to
chiral superspace we recover the known results for tree-level
next-to-maximally-helicity-violating (NMHV) scattering amplitude. The one-loop
MHV amplitude is also included in our result but there are additional terms
which do not immediately correspond to scattering amplitudes. We finally
discuss different regularizations and their Yangian anomalies.Comment: 55 pages, v2: reference adde
Steinmann Relations and the Wavefunction of the Universe
The physical principles of causality and unitarity put strong constraints on
the analytic structure of the flat-space S-matrix. In particular, these
principles give rise to the Steinmann relations, which require that the double
discontinuities of scattering amplitudes in partially-overlapping momentum
channels vanish. Conversely, at cosmological scales, the imprint of causality
and unitarity is in general less well understood---the wavefunction of the
universe lives on the future space-like boundary, and has all time evolution
integrated out. In the present work, we show how the flat-space Steinmann
relations emerge from the structure of the wavefunction of the universe, and
derive similar relations that apply to the wavefunction itself. This is done
within the context of scalar toy models whose perturbative wavefunction has a
first-principles definition in terms of cosmological polytopes. In particular,
we use the fact that the scattering amplitude is encoded in the scattering
facet of cosmological polytopes, and that cuts of the amplitude are encoded in
the codimension-one boundaries of this facet. As we show, the flat-space
Steinmann relations are thus implied by the non-existence of codimension-two
boundaries at the intersection of the boundaries associated with pairs of
partially-overlapping channels. Applying the same argument to the full
cosmological polytope, we also derive Steinmann-type constraints that apply to
the full wavefunction of the universe. These arguments show how the
combinatorial properties of cosmological polytopes lead to the emergence of
flat-space causality in the S-matrix, and provide new insights into the
analytic structure of the wavefunction of the universe
Rooting out letters:octagonal symbol alphabets and algebraic number theory
It is widely expected that NMHV amplitudes in planar, maximally
supersymmetric Yang-Mills theory require symbol letters that are not rationally
expressible in terms of momentum-twistor (or cluster) variables starting at two
loops for eight particles. Recent advances in loop integration technology have
made this an `experimentally testable' hypothesis: compute the amplitude at
some kinematic point, and see if algebraic symbol letters arise. We demonstrate
the feasibility of such a test by directly integrating the most difficult of
the two-loop topologies required. This integral, together with its rotated
image, suffices to determine the simplest NMHV component amplitude: the unique
component finite at this order. Although each of these integrals involve
algebraic symbol alphabets, the combination contributing to this amplitude
is---surprisingly---rational. We describe the steps involved in this analysis,
which requires several novel tricks of loop integration and also a considerable
degree of algebraic number theory. We find dramatic and unusual
simplifications, in which the two symbols initially expressed as almost ten
million terms in over two thousand letters combine in a form that can be
written in five thousand terms and twenty-five letters.Comment: 25 pages, 4 figures; detailed results available as ancillary file
Embedding Feynman integral (Calabi-Yau) geometries in weighted projective space
It has recently been demonstrated that Feynman integrals relevant to a wide
range of perturbative quantum field theories involve periods of Calabi-Yaus of
arbitrarily large dimension. While the number of Calabi-Yau manifolds of
dimension three or higher is considerable (if not infinite), those relevant to
most known examples come from a very simple class: degree- hypersurfaces in
-dimensional weighted projective space . In this
work, we describe some of the basic properties of these spaces and identify
additional examples of Feynman integrals that give rise to hypersurfaces of
this type. Details of these examples at three and four loops are included as
ancillary files to this work.Comment: 44 pages, 31 figures; detailed examples given in ancillary file.
Updated to reflect revisions for publicatio
All Two-Loop MHV Amplitudes in Multi-Regge Kinematics From Applied Symbology
Recent progress on scattering amplitudes has benefited from the mathematical
technology of symbols for efficiently handling the types of polylogarithm
functions which frequently appear in multi-loop computations. The symbol for
all two-loop MHV amplitudes in planar SYM theory is known, but explicit
analytic formulas for the amplitudes are hard to come by except in special
limits where things simplify, such as multi-Regge kinematics. By applying
symbology we obtain a formula for the leading behavior of the imaginary part
(the Mandelstam cut contribution) of this amplitude in multi-Regge kinematics
for any number of gluons. Our result predicts a simple recursive structure
which agrees with a direct BFKL computation carried out in a parallel
publication.Comment: 20 pages, 2 figures. v2: minor correction
Twistors, Harmonics and Holomorphic Chern-Simons
We show that the off-shell N=3 action of N=4 super Yang-Mills can be written
as a holomorphic Chern-Simons action whose Dolbeault operator is constructed
from a complex-real (CR) structure of harmonic space. We also show that the
local space-time operators can be written as a Penrose transform on the coset
SU(3)/(U(1) \times U(1)). We observe a strong similarity to ambitwistor space
constructions.Comment: 34 pages, 3 figures, v2: replaced with published version, v3: Added
referenc