scholarly journals Two-dimensional gravity and intersection theory on moduli space

1990 ◽  
Vol 1 (1) ◽  
pp. 243-310 ◽  
Author(s):  
E. Witten
Keyword(s):  
Moduli Space ◽  
Intersection Theory ◽  
Two Dimensional ◽  
Dimensional Gravity

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 Developments in topological gravity

2018 ◽  
Vol 33 (30) ◽  
pp. 1830029 ◽  
Author(s):  
Robbert Dijkgraaf ◽  
Edward Witten
Keyword(s):  
Matrix Model ◽  
Moduli Space ◽  
Riemann Surfaces ◽  
Degrees Of Freedom ◽  
Intersection Theory ◽  
Two Dimensional ◽  
The Matrix ◽  
Dimensional Gravity ◽  
Topological Gravity ◽  
The Relationship

This note aims to provide an entrée to two developments in two-dimensional topological gravity — that is, intersection theory on the moduli space of Riemann surfaces — that have not yet become well known among physicists. A little over a decade ago, Mirzakhani discovered[Formula: see text] an elegant new proof of the formulas that result from the relationship between topological gravity and matrix models of two-dimensional gravity. Here we will give a very partial introduction to that work, which hopefully will also serve as a modest tribute to the memory of a brilliant mathematical pioneer. More recently, Pandharipande, Solomon, and Tessler3 (with further developments in Refs. 4–6) generalized intersection theory on moduli space to the case of Riemann surfaces with boundary, leading to generalizations of the familiar KdV and Virasoro formulas. Though the existence of such a generalization appears natural from the matrix model viewpoint — it corresponds to adding vector degrees of freedom to the matrix model — constructing this generalization is not straightforward. We will give some idea of the unexpected way that the difficulties were resolved.

scholarly journals Moduli space holography and the finiteness of flux vacua

2021 ◽  
Vol 2021 (10) ◽  
Author(s):  
Thomas W. Grimm
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Boundary Data ◽  
Hodge Theory ◽  
Two Dimensional ◽  
Hodge Conjecture ◽  
String Compactifications ◽  
Finiteness Result ◽  
Dimensional Gravity ◽  
Dimension One

Abstract A holographic perspective to study and characterize field spaces that arise in string compactifications is suggested. A concrete correspondence is developed by studying two-dimensional moduli spaces in supersymmetric string compactifications. It is proposed that there exist theories on the boundaries of each moduli space, whose crucial data are given by a Hilbert space, an Sl(2, ℂ)-algebra, and two special operators. This boundary data is motivated by asymptotic Hodge theory and the fact that the physical metric on the moduli space of Calabi-Yau manifolds asymptotes near any infinite distance boundary to a Poincaré metric with Sl(2, ℝ) isometry. The crucial part of the bulk theory on the moduli space is a sigma model for group-valued matter fields. It is discussed how this might be coupled to a two-dimensional gravity theory. The classical bulk-boundary matching is then given by the proof of the famous Sl(2) orbit theorem of Hodge theory, which is reformulated in a more physical language. Applying this correspondence to the flux landscape in Calabi-Yau fourfold compactifications it is shown that there are no infinite tails of self-dual flux vacua near any co-dimension one boundary. This finiteness result is a consequence of the constraints on the near boundary expansion of the bulk solutions that match to the boundary data. It is also pointed out that there is a striking connection of the finiteness result for supersymmetric flux vacua and the Hodge conjecture.

scholarly journals DISCRETE AND CONTINUUM APPROACHES TO THREE-DIMENSIONAL QUANTUM GRAVITY

1991 ◽  
Vol 06 (39) ◽  
pp. 3591-3600 ◽  
Author(s):  
HIROSI OOGURI ◽  
NAOKI SASAKURA
Keyword(s):  
Gauge Theory ◽  
Moduli Space ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Analogue Model ◽  
Gauge Invariant ◽  
Invariant Functions ◽  
Dimensional Lattice ◽  
Chern Simons ◽  
Dimensional Surface

It is shown that, in the three-dimensional lattice gravity defined by Ponzano and Regge, the space of physical states is isomorphic to the space of gauge-invariant functions on the moduli space of flat SU(2) connections over a two-dimensional surface, which gives physical states in the ISO(3) Chern–Simons gauge theory. To prove this, we employ the q-analogue of this model defined by Turaev and Viro as a regularization to sum over states. A recent work by Turaev suggests that the q-analogue model itself may be related to an Euclidean gravity with a cosmological constant proportional to 1/k2, where q=e2πi/(k+2).

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 New exact relations for steady irrotational two-dimensional gravity and capillary surface waves

2017 ◽  
Vol 376 (2111) ◽  
pp. 20170220 ◽  
Author(s):  
Didier Clamond
Keyword(s):  
Free Surface ◽  
Gravity Waves ◽  
Water Waves ◽  
Two Dimensional ◽  
Physical Plane ◽  
Theme Issue ◽  
Nonlinear Water Waves ◽  
Dimensional Gravity ◽  
Irrotational Motion ◽  
Exact Relations

Steady two-dimensional surface capillary–gravity waves in irrotational motion are considered on constant depth. By exploiting the holomorphic properties in the physical plane and introducing some transformations of the boundary conditions at the free surface, new exact relations and equations for the free surface only are derived. In particular, a physical plane counterpart of the Babenko equation is obtained. This article is part of the theme issue ‘Nonlinear water waves’.

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