scholarly journals On polytopes and generalizations of the KLT relations

2020 ◽  
Vol 2020 (12) ◽  
Author(s):  
Nikhil Kalyanapuram
Keyword(s):  
Scattering Amplitudes ◽  
Large Class ◽  
Moduli Space ◽  
Differential Forms ◽  
Natural Generalization ◽  
Intersection Theory ◽  
Hyperplane Arrangements ◽  
Number Of Solutions ◽  
Scalar Theories ◽  
Double Copy

Abstract We combine the technology of the theory of polytopes and twisted intersection theory to derive a large class of double copy relations that generalize the classical relations due to Kawai, Lewellen and Tye (KLT). To do this, we first study a generalization of the scattering equations of Cachazo, He and Yuan. While the scattering equations were defined on ℳ0, n — the moduli space of marked Riemann spheres — the new scattering equations are defined on polytopes known as accordiohedra, realized as hyperplane arrangements. These polytopes encode as patterns of intersection the scattering amplitudes of generic scalar theories. The twisted period relations of such intersection numbers provide a vast generalization of the KLT relations. Differential forms dual to the bounded chambers of the hyperplane arrangements furnish a natural generalization of the Bern-Carrasco-Johansson (BCJ) basis, the number of which can be determined by counting the number of solutions of the generalized scattering equations. In this work the focus is on a generalization of the BCJ expansion to generic scalar theories, although we use the labels KLT and BCJ interchangeably.

scholarly journals Tevelev degrees and Hurwitz moduli spaces

2021 ◽  
pp. 1-32
Author(s):  
A. CELA ◽  
R. PANDHARIPANDE ◽  
J. SCHMITT
Keyword(s):  
Scattering Amplitudes ◽  
Exact Solutions ◽  
Projective Geometry ◽  
Moduli Space ◽  
Moduli Spaces ◽  
Intersection Theory ◽  
Dyck Paths ◽  
Simple Recursion ◽  
Almost All ◽  
Two Parameter

Abstract We interpret the degrees which arise in Tevelev’s study of scattering amplitudes in terms of moduli spaces of Hurwitz covers. Via excess intersection theory, the boundary geometry of the Hurwitz moduli space yields a simple recursion for the Tevelev degrees (together with their natural two parameter generalisation). We find exact solutions which specialise to Tevelev’s formula in his cases and connect to the projective geometry of lines and Castelnuovo’s classical count of $g^1_d$ ’s in other cases. For almost all values, the calculation of the two parameter generalisation of the Tevelev degree is new. A related count of refined Dyck paths is solved along the way.

scholarly journals Stringy canonical forms and binary geometries from associahedra, cyclohedra and generalized permutohedra

2020 ◽  
Vol 2020 (10) ◽  
Author(s):  
Song He ◽  
Zhenjie Li ◽  
Prashanth Raman ◽  
Chi Zhang
Keyword(s):  
Large Class ◽  
Configuration Space ◽  
Moduli Space ◽  
Cluster Algebras ◽  
Configuration Spaces ◽  
Minkowski Sum ◽  
Canonical Forms ◽  
Infinite Class ◽  
Special Cases ◽  
Generalized Associahedra

Abstract Stringy canonical forms are a class of integrals that provide α′-deformations of the canonical form of any polytopes. For generalized associahedra of finite-type cluster algebras, there exist completely rigid stringy integrals, whose configuration spaces are the so-called binary geometries, and for classical types are associated with (generalized) scattering of particles and strings. In this paper, we propose a large class of rigid stringy canonical forms for another class of polytopes, generalized permutohedra, which also include associahedra and cyclohedra as special cases (type An and Bn generalized associahedra). Remarkably, we find that the configuration spaces of such integrals are also binary geometries, which were suspected to exist for generalized associahedra only. For any generalized permutohedron that can be written as Minkowski sum of coordinate simplices, we show that its rigid stringy integral factorizes into products of lower integrals for massless poles at finite α′, and the configuration space is binary although the u equations take a more general form than those “perfect” ones for cluster cases. Moreover, we provide an infinite class of examples obtained by degenerations of type An and Bn integrals, which have perfect u equations as well. Our results provide yet another family of generalizations of the usual string integral and moduli space, whose physical interpretations remain to be explored.

scholarly journals A Family of Riesz Distributions for Differential Forms on Euclidian Space

2020 ◽  
Author(s):  
Matthias Fischmann ◽  
Bent Ørsted
Keyword(s):  
Fourier Transforms ◽  
Conformal Geometry ◽  
Differential Forms ◽  
Natural Generalization ◽  
Conformal Group ◽  
Meromorphic Continuation ◽  
New Family ◽  
Euclidian Space

Abstract In this paper, we introduce a new family of operator-valued distributions on Euclidian space acting by convolution on differential forms. It provides a natural generalization of the important Riesz distributions acting on functions, where the corresponding operators are $(-\Delta )^{-\alpha /2}$, and we develop basic analogous properties with respect to meromorphic continuation, residues, Fourier transforms, and relations to conformal geometry and representations of the conformal group.

scholarly journals The moduli space of bilevel-6 abelian surfaces

2002 ◽  
Vol 168 ◽  
pp. 113-125
Author(s):  
G. K. Sankaran ◽  
J. G. Spandaw
Keyword(s):  
Moduli Space ◽  
Modular Group ◽  
Symplectic Group ◽  
Differential Forms ◽  
Kodaira Dimension ◽  
Cusp Forms ◽  
Abelian Surfaces

AbstractWe show that the moduli space of abelian surfaces with polarisation of type (1,6) and a bilevel structure has positive Kodaira dimension and indeed pg ≥ 3. To do this we show that three of the Siegel cusp forms with character for the paramodular symplectic group constructed by Gritsenko and Nikulin are cusp forms without character for the modular group associated to this moduli problem. We then calculate the divisors of the corresponding differential forms, using information about the fixed loci of elements of the paramodular group previously obtained by Brasch.

scholarly journals Extension theorems for differential forms and Bogomolov–Sommese vanishing on log canonical varieties

2009 ◽  
Vol 146 (1) ◽  
pp. 193-219 ◽  
Author(s):  
Daniel Greb ◽  
Stefan Kebekus ◽  
Sándor J. Kovács
Keyword(s):  
Differential Forms ◽  
Natural Generalization ◽  
Resolution Of Singularities ◽  
Vanishing Theorem ◽  
Exceptional Set ◽  
Extension Theorems ◽  
Normal Variety ◽  
Log Canonical ◽  
Smooth Locus

AbstractGiven a normal variety Z, a p-form σ defined on the smooth locus of Z and a resolution of singularities $\pi : \widetilde {Z} \to Z$, we study the problem of extending the pull-back π*(σ) over the π-exceptional set $E \subset \widetilde {Z}$. For log canonical pairs and for certain values of p, we show that an extension always exists, possibly with logarithmic poles along E. As a corollary, it is shown that sheaves of reflexive differentials enjoy good pull-back properties. A natural generalization of the well-known Bogomolov–Sommese vanishing theorem to log canonical threefold pairs follows.

INTERSECTION THEORY AND TWO-DIMENSIONAL GRAVITY AT GENUS 3 AND 4

1990 ◽  
Vol 05 (26) ◽  
pp. 2127-2134 ◽  
Author(s):  
JAMES H. HORNE
Keyword(s):  
Moduli Space ◽  
Intersection Theory ◽  
Gravity Theory ◽  
Two Dimensional ◽  
Intersection Numbers ◽  
Dimensional Gravity

We show that the k = 1 two-dimensional gravity amplitudes at genus 3 agree precisely with the results from intersection theory on moduli space. Predictions for the genus 4 intersection numbers follow easily from the two-dimensional gravity theory.

scholarly journals Lie polynomials and a twistorial correspondence for amplitudes

2021 ◽  
Vol 111 (6) ◽  
Author(s):  
Hadleigh Frost ◽  
Lionel Mason
Keyword(s):  
Moduli Space ◽  
Twistor Space ◽  
Riemann Sphere ◽  
Momentum Conservation ◽  
Mathematical Framework ◽  
Penrose Transform ◽  
Invariant Description ◽  
Logarithmic Singularities ◽  
Gravity Theories ◽  
Double Copy

AbstractWe review Lie polynomials as a mathematical framework that underpins the structure of the so-called double copy relationship between gauge and gravity theories (and a network of other theories besides). We explain how Lie polynomials naturally arise in the geometry and cohomology of $$\mathcal {M}_{0,n}$$ M 0 , n , the moduli space of n points on the Riemann sphere up to Mobiüs transformation. We introduce a twistorial correspondence between the cotangent bundle $$T^*_D\mathcal {M}_{0,n}$$ T D ∗ M 0 , n , the bundle of forms with logarithmic singularities on the divisor D as the twistor space, and $$\mathcal {K}_n$$ K n the space of momentum invariants of n massless particles subject to momentum conservation as the analogue of space–time. This gives a natural framework for Cachazo He and Yuan (CHY) and ambitwistor-string formulae for scattering amplitudes of gauge and gravity theories as being the corresponding Penrose transform. In particular, we show that it gives a natural correspondence between CHY half-integrands and scattering forms, certain $$n-3$$ n - 3 -forms on $$\mathcal {K}_n$$ K n , introduced by Arkani-Hamed, Bai, He and Yan (ABHY). We also give a generalization and more invariant description of the associahedral $$n-3$$ n - 3 -planes in $$\mathcal {K}_n$$ K n introduced by ABHY.

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