A CRITICAL MATRIX MODEL AT c=1

1991 ◽  
Vol 06 (03) ◽  
pp. 259-270 ◽  
Author(s):  
JACQUES DISTLER ◽  
CUMRUN VAFA
Keyword(s):  
Free Energy ◽  
Euler Characteristic ◽  
Matrix Model ◽  
Moduli Space ◽  
Riemann Surfaces ◽  
Continuum Theory ◽  
Critical Limit ◽  
Logarithmic Corrections

By taking the critical limit of Penner’s matrix model we obtain a continuum theory whose free energy at genus-g is the Euler characteristic of moduli space of Riemann surfaces of genus-g. The exponents, and the appearance of logarithmic corrections suggest that we are dealing with a theory at c=1.

THE (ORBIFOLD) EULER CHARACTERISTIC OF THE MODULI SPACE OF CURVES AND THE CONTINUUM LIMIT OF PENNER'S CONNECTED GENERATING FUNCTION

1991 ◽  
Vol 03 (03) ◽  
pp. 285-300 ◽  
Author(s):  
NOUREDDINE CHAIR
Keyword(s):  
Free Energy ◽  
Matrix Models ◽  
Euler Characteristic ◽  
Generating Function ◽  
Moduli Space ◽  
Continuum Limit ◽  
Recursion Relation ◽  
Moduli Space Of Curves ◽  
Genus Zero ◽  
The Continuum

The generating function that gives rise to the orbifold Euler characteristic of the moduli space of punctured compact Rieman surfaces [Formula: see text], g ≥ 0 is derived explicitly. In the derivation, we show that we do not need to use the three-term recursion relation for the orthogonal polynomials. Also the continuum limit of Penner's connected generating function is considered and is shown to be formally equivalent to the free energy obtained recently by Distler and Vafa which exhibits the logarithmic divergences found for genus zero and one in D = 1 matrix models. Finally, it is shown that the free energy and its s-derivatives are nothing but the continuum limit of a certain generating function introduced by Harer and Zagier in obtaining the true Euler characteristic with any number of punctures,[Formula: see text], s ≥ 0.

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.

A Primer on Mapping Class Groups (PMS-49)

2017 ◽  
Author(s):  
Benson Farb ◽  
Dan Margalit
Keyword(s):  
Graduate Students ◽  
Moduli Space ◽  
Riemann Surfaces ◽  
Class Group ◽  
Mapping Class ◽  
Torelli Group ◽  
Surface Bundles ◽  
Surface Homeomorphisms ◽  
Low Dimensional

The study of the mapping class group Mod(S) is a classical topic that is experiencing a renaissance. It lies at the juncture of geometry, topology, and group theory. This book explains as many important theorems, examples, and techniques as possible, quickly and directly, while at the same time giving full details and keeping the text nearly self-contained. The book is suitable for graduate students. It begins by explaining the main group-theoretical properties of Mod(S), from finite generation by Dehn twists and low-dimensional homology to the Dehn–Nielsen–Baer–theorem. Along the way, central objects and tools are introduced, such as the Birman exact sequence, the complex of curves, the braid group, the symplectic representation, and the Torelli group. The book then introduces Teichmüller space and its geometry, and uses the action of Mod(S) on it to prove the Nielsen-Thurston classification of surface homeomorphisms. Topics include the topology of the moduli space of Riemann surfaces, the connection with surface bundles, pseudo-Anosov theory, and Thurston's approach to the classification.

scholarly journals The Volume of the Quiver Vortex Moduli Space

2021 ◽  
Author(s):  
Kazutoshi Ohta ◽  
Norisuke Sakai
Keyword(s):  
Gauge Theory ◽  
Moduli Space ◽  
Riemann Surfaces ◽  
Gauge Theories ◽  
Target Space ◽  
Contour Integral ◽  
Quiver Gauge Theory ◽  
Volume Formula ◽  
Volume Operator ◽  
Space Volume

Abstract We study the moduli space volume of BPS vortices in quiver gauge theories on compact Riemann surfaces. The existence of BPS vortices imposes constraints on the quiver gauge theories. We show that the moduli space volume is given by a vev of a suitable cohomological operator (volume operator) in a supersymmetric quiver gauge theory, where BPS equations of the vortices are embedded. In the supersymmetric gauge theory, the moduli space volume is exactly evaluated as a contour integral by using the localization. Graph theory is useful to construct the supersymmetric quiver gauge theory and to derive the volume formula. The contour integral formula of the volume (generalization of the Jeffrey-Kirwan residue formula) leads to the Bradlow bounds (upper bounds on the vorticity by the area of the Riemann surface divided by the intrinsic size of the vortex). We give some examples of various quiver gauge theories and discuss properties of the moduli space volume in these theories. Our formula are applied to the volume of the vortex moduli space in the gauged non-linear sigma model with CPN target space, which is obtained by a strong coupling limit of a parent quiver gauge theory. We also discuss a non-Abelian generalization of the quiver gauge theory and “Abelianization” of the volume formula.

scholarly journals Vortices over Riemann surfaces and dominated splittings

2021 ◽  
pp. 1-26
Author(s):  
THOMAS METTLER ◽  
GABRIEL P. PATERNAIN
Keyword(s):  
Euler Characteristic ◽  
Tangent Bundle ◽  
Riemann Surfaces ◽  
Dominated Splitting ◽  
Vortex Equations ◽  
Special Cases ◽  
Unit Tangent Bundle ◽  
Anosov Flows

Abstract We associate a flow $\phi $ with a solution of the vortex equations on a closed oriented Riemannian 2-manifold $(M,g)$ of negative Euler characteristic and investigate its properties. We show that $\phi $ always admits a dominated splitting and identify special cases in which $\phi $ is Anosov. In particular, starting from holomorphic differentials of fractional degree, we produce novel examples of Anosov flows on suitable roots of the unit tangent bundle of $(M,g)$ .

Free Energy of Granular Materials in Static Equilibrium

1979 ◽  
Vol 46 (4) ◽  
pp. 944-945 ◽  
Author(s):  
M. Shahinpoor ◽  
G. Ahmadi
Keyword(s):  
Free Energy ◽  
Granular Materials ◽  
Continuum Theory ◽  
Gravitational Effect ◽  
Experimental Results ◽  
Static Equilibrium ◽  
The Continuum

We employ the continuum theory of granular materials due to Goodman and Cowin and some experimental results due to P. G. Nutting to arrive at a functional from for the free energy of granular materials in static equilibrium. The results obtained indicate the dominance of gravitational effect, modify and enlarge the results previously obtained by J. T. Jenkins.

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