scholarly journals Two-dimensional topological gravity and intersection theory on the moduli space of holomorphic bundles

1991 ◽  
Vol 260 (3-4) ◽  
pp. 303-310 ◽  
Author(s):  
T.P. Killingback
Keyword(s):  
Moduli Space ◽  
Intersection Theory ◽  
Two Dimensional ◽  
Topological Gravity ◽  
Holomorphic Bundles

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.

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 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.

ON AN ANALYTIC PROOF OF A RESULT BY DONALDSON

1996 ◽  
Vol 07 (01) ◽  
pp. 1-17
Author(s):  
GUANG-YUAN GUO
Keyword(s):  
Moduli Space ◽  
Analytic Proof ◽  
One To One ◽  
Holomorphic Bundles

We give an analytic proof of a result by Donaldson which asserts that there is a one to one correspondence between the moduli space of framed instantons on S4 and the moduli space of holomorphic bundles over CP2 trivialized along a line.

scholarly journals On the geometry of bifurcation currents for quadratic rational maps

2014 ◽  
Vol 35 (5) ◽  
pp. 1369-1379 ◽  
Author(s):  
FRANÇOIS BERTELOOT ◽  
THOMAS GAUTHIER
Keyword(s):  
Projective Space ◽  
Moduli Space ◽  
Complex Projective Space ◽  
The Self ◽  
Rational Maps ◽  
Two Dimensional ◽  
Lelong Numbers ◽  
Complex Projective

We describe the behaviour at infinity of the bifurcation current in the moduli space of quadratic rational maps. To this purpose, we extend it to some closed, positive $(1,1)$-current on a two-dimensional complex projective space and then compute the Lelong numbers and the self-intersection of the extended current.

Moduli Space of Flat Connections as a Poisson Manifold

1997 ◽  
Vol 11 (26n27) ◽  
pp. 3195-3206 ◽  
Author(s):  
V. V. Fock ◽  
A. A. Rosly
Keyword(s):  
Riemann Surface ◽  
Lie Groups ◽  
Poisson Structure ◽  
Moduli Space ◽  
Poisson Manifold ◽  
Gauge Fields ◽  
Two Dimensional ◽  
Flat Connections ◽  
Lattice Gauge ◽  
Poisson Lie Groups

In this talk we describe the Poisson structure of the moduli space of flat connections on a two dimensional Riemann surface in terms of lattice gauge fields and Poisson–Lie groups.

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