TY - JOUR

T1 - Developments in topological gravity

AU - Dijkgraaf, Robbert

AU - Witten, Edward

N1 - Funding Information:
We thank D. Freed, R. Penner, and J. Solomon for comments on the manuscript. Research of EW is supported in part by NSF Grant PHY-1606531.
Publisher Copyright:
© 2018 World Scientific Publishing Company.

PY - 2018/10/30

Y1 - 2018/10/30

N2 - 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 discovered1,2 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.

AB - 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 discovered1,2 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.

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U2 - 10.1142/S0217751X18300296

DO - 10.1142/S0217751X18300296

M3 - Article

AN - SCOPUS:85056616626

VL - 33

JO - International Journal of Modern Physics A

JF - International Journal of Modern Physics A

SN - 0217-751X

IS - 30

M1 - 1830029

ER -