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.

Coupled dynamics of populations supported by discrete sites and their continuum limit

2010 ◽  
Vol 466 (2121) ◽  
pp. 2561-2586 ◽  
Author(s):  
Timothy R. Field ◽  
Robert J. A. Tough
Keyword(s):  
Evolution Equation ◽  
Closed Form ◽  
Generating Function ◽  
Rate Equation ◽  
Continuum Limit ◽  
Infinite Chain ◽  
Migration Process ◽  
The Mean ◽  
The Continuum ◽  
Birth Death

The illumination of single population behaviour subject to the processes of birth, death and immigration has provided a basis for the discussion of the non-Gaussian statistical and temporal correlation properties of scattered radiation. As a first step towards the modelling of its spatial correlations, we consider the populations supported by an infinite chain of discrete sites, each subject to birth, death and immigration and coupled by migration between adjacent sites. To provide some motivation, and illustrate the techniques we will use, the migration process for a single particle on an infinite chain of sites is introduced and its diffusion dynamics derived. A certain continuum limit is identified and its properties studied via asymptotic analysis. This forms the basis of the multi-particle model of a coupled population subject to single site birth, death and immigration processes, in addition to inter-site migration. A discrete rate equation is formulated and its generating function dynamics derived. This facilitates derivation of the equations of motion for the first- and second-order cumulants, thus generalizing the earlier results of Bailey through the incorporation of immigration at each site. We present a novel matrix formalism operating in the time domain that enables solution of these equations yielding the mean occupancy and inter-site variances in the closed form. The results for the first two moments at a single time are used to derive expressions for the asymptotic time-delayed correlation functions, which relates to Glauber’s analysis of an Ising model. The paper concludes with an analysis of the continuum limit of the birth–death–immigration–migration process in terms of a path integral formalism. The continuum rate equation and evolution equation for the generating function are developed, from which the evolution equation of the mean occupancy is derived, in this limit. Its solution is provided in closed form.

scholarly journals ON THE CONTINUUM LIMIT OF THE CONFORMAL MATRIX MODELS

1993 ◽  
Vol 08 (18) ◽  
pp. 3107-3137 ◽  
Author(s):  
A. MIRONOV ◽  
S. PAKULIAK
Keyword(s):  
Matrix Models ◽  
Continuum Limit ◽  
Scaling Limit ◽  
Universality Class ◽  
Partition Functions ◽  
Discrete Level ◽  
New Class ◽  
Double Scaling Limit ◽  
The Continuum

The double scaling limit of a new class of the multi-matrix models proposed in Ref. 1, which possess the W-symmetry at the discrete level, is investigated in detail. These models are demonstrated to fall into the same universality class as the standard multi-matrix models. In particular, the transformation of the W-algebra at the discrete level into the continuum one of the papers2 is proposed and the corresponding partition functions compared. All calculations are demonstrated in full in the first nontrivial case of W(3)-constraints.

scholarly journals Relative Quasimaps and Mirror Formulae

2020 ◽  
Author(s):  
Luca Battistella ◽  
Navid Nabijou
Keyword(s):  
Generating Function ◽  
Moduli Space ◽  
Toric Variety ◽  
Recursion Formula ◽  
Genus Zero ◽  
Relative Invariant ◽  
Lefschetz Theorem ◽  
Virtual Class

Abstract We construct and study the theory of relative quasimaps in genus zero, in the spirit of Gathmann. When $X$ is a smooth toric variety and $Y$ is a smooth very ample hypersurface in $X$, we produce a virtual class on the moduli space of relative quasimaps to $(X,Y)$, which we use to define relative quasimap invariants. We obtain a recursion formula which expresses each relative invariant in terms of invariants of lower tangency, and apply this formula to derive a quantum Lefschetz theorem for quasimaps, expressing the restricted quasimap invariants of $Y$ in terms of those of $X$. Finally, we show that the relative $I$-function of Fan–Tseng–You coincides with a natural generating function for relative quasimap invariants, providing mirror-symmetric motivation for the theory.

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.

N = ½ SUPERSTRING IN ZERO DIMENSION AND THE SPONTANEOUS BREAKDOWN OF THE SUPERSYMMETRY

1992 ◽  
Vol 07 (32) ◽  
pp. 2979-2989 ◽  
Author(s):  
SHIN’ICHI NOJIRI
Keyword(s):  
Matrix Models ◽  
Random Matrix ◽  
Continuum Limit ◽  
Scaling Limit ◽  
Partition Functions ◽  
Spontaneous Breakdown ◽  
Random Matrix Models ◽  
Double Scaling Limit ◽  
The Continuum ◽  
String Theories

We propose random matrix models which have N=1/2 supersymmetry in zero dimension. The supersymmetry breaks down spontaneously. It is shown that the double scaling limit can be defined in these models and the breakdown of the supersymmetry remains in the continuum limit. The exact non-trivial partition functions of the string theories corresponding to these matrix models are also obtained.

GENERALIZED TODA AND VOLTERRA MODELS

1992 ◽  
Vol 07 (20) ◽  
pp. 4855-4869 ◽  
Author(s):  
J. AVAN
Keyword(s):  
Matrix Models ◽  
Differential System ◽  
Continuum Limit ◽  
Automorphism Groups ◽  
Volterra Models ◽  
The Mean ◽  
The Continuum

The mean procedure of Faddeev-Reshetikhin with non-Abelian automorphism groups is applied to construct generalizations of the open Toda sl(n, C) chain. These models admit a consistent reduction to integrable generalized Volterra models. An example of such models is analyzed: it leads in the continuum limit to the [Formula: see text] Hirota differential system, associated with two-matrix models of discrete gravity. The continuum limit of the general Volterra models and their relation with discretized versions of Wn-algebra are analyzed.

scholarly journals THE WEINGARTEN MODEL À LA POLYAKOV

1992 ◽  
Vol 07 (18) ◽  
pp. 1651-1660 ◽  
Author(s):  
SIMON DALLEY
Keyword(s):  
Quantum Gravity ◽  
Matrix Models ◽  
Critical Behavior ◽  
Continuum Limit ◽  
Hermitian Matrix ◽  
Gauge Model ◽  
Vertex Models ◽  
Lattice Gauge ◽  
Genus Expansion ◽  
The Continuum

The Weingarten lattice gauge model of Nambu-Goto strings is generalized to allow for fluctuations of an intrinsic worldsheet metric through a dynamical quadrilation. The continuum limit is taken for c≤1 matter, reproducing the results of Hermitian matrix models to all orders in the genus expansion. For the compact c=1 case the vortices are Wilson lines, whose exclusion leads to the theory of non-interacting fermions. As a by-product of the analysis one finds the critical behavior of SOS and vertex models coupled to 2D quantum gravity.

scholarly journals Stability in the homology of Deligne–Mumford compactifications

2021 ◽  
Vol 157 (12) ◽  
pp. 2635-2656
Author(s):  
Philip Tosteson
Keyword(s):  
Asymptotic Behavior ◽  
Generating Function ◽  
Moduli Space ◽  
Vector Spaces ◽  
Moduli Space Of Curves ◽  
Contravariant Functor ◽  
Finite Sets ◽  
Projective Lines

Abstract Using the theory of ${\mathbf {FS}} {^\mathrm {op}}$ modules, we study the asymptotic behavior of the homology of ${\overline {\mathcal {M}}_{g,n}}$ , the Deligne–Mumford compactification of the moduli space of curves, for $n\gg 0$ . An ${\mathbf {FS}} {^\mathrm {op}}$ module is a contravariant functor from the category of finite sets and surjections to vector spaces. Via copies that glue on marked projective lines, we give the homology of ${\overline {\mathcal {M}}_{g,n}}$ the structure of an ${\mathbf {FS}} {^\mathrm {op}}$ module and bound its degree of generation. As a consequence, we prove that the generating function $\sum _{n} \dim (H_i({\overline {\mathcal {M}}_{g,n}})) t^n$ is rational, and its denominator has roots in the set $\{1, 1/2, \ldots, 1/p(g,i)\},$ where $p(g,i)$ is a polynomial of order $O(g^2 i^2)$ . We also obtain restrictions on the decomposition of the homology of ${\overline {\mathcal {M}}_{g,n}}$ into irreducible $\mathbf {S}_n$ representations.

ON THE DIMERIZATION OF LINEAR POLYMERS

1989 ◽  
Vol 03 (02) ◽  
pp. 125-133 ◽  
Author(s):  
C. ARAGÃO DE CARVALHO
Keyword(s):  
Free Energy ◽  
Effective Potential ◽  
Ground State ◽  
Finite Temperature ◽  
Continuum Limit ◽  
Linear Polymers ◽  
Renormalization Condition ◽  
Loop Approximation ◽  
The One ◽  
The Continuum

We use the continuum limit of the Su-Schrieffer-Heeger model for linear polymers to construct its effective potential (Gibbs free energy) both at zero and finite temperature. We study both trans and cis-polymers. Our results show that, depending on a renormalization condition to be extracted from experiment, there are several possibilities for the minima of the dimerized ground state of cis-polymers. All calculations are done in the one-loop approximation.

scholarly journals DECONSTRUCTION AND OTHER APPROACHES TO SUPERSYMMETRIC LATTICE FIELD THEORIES

2006 ◽  
Vol 21 (15) ◽  
pp. 3039-3093 ◽  
Author(s):  
JOEL GIEDT
Keyword(s):  
Moduli Space ◽  
Continuum Limit ◽  
The Self ◽  
Field Theories ◽  
Phase Problem ◽  
Complex Phase ◽  
Lattice Field ◽  
The Continuum ◽  
Fermion Action

This paper contains both a review of recent approaches to supersymmetric lattice field theories and some new results on the deconstruction approach. The essential reason for the complex phase problem of the fermion determinant is shown to be derivative interactions that are not present in the continuum. These irrelevant operators violate the self-conjugacy of the fermion action that is present in the continuum. It is explained why this complex phase problem does not disappear in the continuum limit. The fermion determinant suppression of various branches of the classical moduli space is explored, and found to be supportive of previous claims regarding the continuum limit.

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