COMBINATORICS OF THE MODULAR GROUP II THE KONTSEVICH INTEGRALS

We study algebraic aspects of Kontsevich integrals as generating functions for intersection theory over moduli space and review the derivation of Virasoro and KdV constraints.

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On polytopes and generalizations of the KLT relations

Journal of High Energy Physics ◽

10.1007/jhep12(2020)057 ◽

2020 ◽

Vol 2020 (12) ◽

Cited By ~ 1

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.

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Subgroups of infinite index in the modular group II

Glasgow Mathematical Journal ◽

10.1017/s0017089500004523 ◽

1981 ◽

Vol 22 (1) ◽

pp. 101-118 ◽

Cited By ~ 5

Author(s):

W. W. Stothers

Keyword(s):

Modular Group ◽

Negative Integer ◽

Infinite Index ◽

Group Ii

Let H be a subgroup of Γ, the modular group. Let h be the number of orbits of under the action of H. In each orbit, the stabilizers are H-conjugate. Let U be the mapping z↦z + 1. Each stabilizer is Γ-conjugate to 〈Uc〉 for some non-negative integer c. The integer c is the cusp-width of the orbit. Let h0 be the number of orbits with non-trivial stabilizer, i.e. with c>0. The sequence (c(1), …, c(h0)) of non-zero cuspwidths is the cusp-split of H. Clearly, h0<h, and h∞ = h−h0 is the number of orbits with trivial stabilizer.

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The moduli space of bilevel-6 abelian surfaces

Nagoya Mathematical Journal ◽

10.1017/s0027763000008394 ◽

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.

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

Modern Physics Letters A ◽

10.1142/s0217732390002420 ◽

1990 ◽

Vol 05 (26) ◽

pp. 2127-2134 ◽

Cited By ~ 4

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.

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