Roaming moduli space using dynamical triangulations

Ambjørn, J.; Barkley, J.; Budd, Timothy

doi:10.1016/j.nuclphysb.2012.01.010

Cited by 14 publications

(19 citation statements)

References 47 publications

(62 reference statements)

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“…Thus in the discretized case we only have to look for such loops. Further, the harmonic forms which are important tools for analytic manifolds have very nice discretized analogies, and we can use these to construct a conformal mapping from the abstract triangulation to the complex plane [14,15]. We have shown an example of such a map in Fig.…”

Section: Is the Watabiki Formula Correct?mentioning

confidence: 99%

“…9. Example of a discrete analog of a harmonic map, used to map a triangulation of the torus consisting of equilateral triangles into the complex plane [14].…”

Section: Is the Watabiki Formula Correct?mentioning

confidence: 99%

“…The numerical expectation value L N of the length of the shortest noncontractible loop for triangulations of N triangles (the error bars are too small to display). The fit corresponds to L N = 0.45 N 1/3.56 [14].…”

Section: Is the Watabiki Formula Correct?mentioning

confidence: 99%

“…(14): it can be viewed as a conformal field theory with central charge c = d coupled to 2d quantum gravity. As we have seen, the theory degenerates into BP for c > 1.…”

Section: Is the Watabiki Formula Correct?mentioning

confidence: 99%

“…As we have seen, the theory degenerates into BP for c > 1. However, from (14) it is clear that we can formally perform an analytic continuation to c < 1. A special case is c = −2 because then the triangulations are weighted precisely by the determinant of the graph Laplacian, which can be represented as a sum over spanning trees of the given triangulations.…”

Section: Is the Watabiki Formula Correct?mentioning

confidence: 99%

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Two-dimensional Quantum Geometry

Ambjørn¹,

Budd²

2013

Acta Phys. Pol. B

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Section: Is the Watabiki Formula Correct?mentioning

confidence: 99%

“…9. Example of a discrete analog of a harmonic map, used to map a triangulation of the torus consisting of equilateral triangles into the complex plane [14].…”

Section: Is the Watabiki Formula Correct?mentioning

confidence: 99%

Section: Is the Watabiki Formula Correct?mentioning

confidence: 99%

“…(14): it can be viewed as a conformal field theory with central charge c = d coupled to 2d quantum gravity. As we have seen, the theory degenerates into BP for c > 1.…”

Section: Is the Watabiki Formula Correct?mentioning

confidence: 99%

Section: Is the Watabiki Formula Correct?mentioning

confidence: 99%

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Two-dimensional Quantum Geometry

Ambjørn¹,

Budd²

2013

Acta Phys. Pol. B

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Gauge invariant computable quantities in timelike Liouville theory

Maltz

2013

J. High Energ. Phys.

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Timelike Liouville theory admits the sphere S 2 as a real saddle point, about which quantum fluctuations can occur. An issue occurs when computing the expectation values of specific types of quantities, like the distance between points. The problem being that the gauge redundancy of the path integral over metrics is not completely fixed even after fixing to conformal gauge by imposing g µν = e 2 bφ g µν , where φ is the Liouville field and g µν is a reference metric. The physical metric g µν , and therefore the path integral over metrics still possesses a gauge redundancy due to invariance under SL 2 (C) coordinate transformations of the reference coordinates. This zero mode of the action must be dealt with before a perturbative analysis can be made. This paper shows that after fixing to conformal gauge, the remaining zero mode of the linearized Liouville action due to SL 2 (C) coordinate transformations can be dealt with by using standard Fadeev-Popov methods. Employing the gauge condition that the "dipole" of the reference coordinate system is a fixed vector, and then integrating over all values of this dipole vector. The "dipole" vector referring to how coordinate area is concentrated about the sphere; assuming the sphere is embedded in R 3 and centered at the origin, and the coordinate area is thought of as a charge density on the sphere. The vector points along the ray from the origin of R 3 to the direction of greatest coordinate area.A Green's function is obtained and used to compute the expectation value of the geodesic length between two points on the S 2 to second order in the Timelike Liouville coupling b. This quantity doesn't suffer from any power law or logarithmic divergences as a naïve power counting argument might suggest.

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Polyakov’s formulation of $2d$ bosonic string theory

2019

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Using probabilistic methods, we first define Liouville quantum field theory on Riemann surfaces of genus g 2 and show that it is a conformal field theory. We use the partition function of Liouville quantum field theory to give a mathematical sense to Polyakov's partition function of noncritical bosonic string theory [Po] (also called 2d bosonic string theory) and to Liouville quantum gravity. More specifically, we show the convergence of Polyakov's partition function over the moduli space of Riemann surfaces in genus g 2 in the case of D 1 boson. This is done by performing a careful analysis of the behavior of the partition function at the boundary of moduli space. An essential feature of our approach is that it is probabilistic and non perturbative. The interest of our result is twofold. First, to the best of our knowledge, this is the first mathematical result about convergence of string theories. Second, our construction describes conjecturally the scaling limit of higher genus random planar maps weighted by Conformal Field Theories: we make precise conjectures about this statement at the end of the paper.(1.13)for some constant C, where the determinants are defined using spectral zeta functions, P g is a first-order elliptic operator mapping 1-forms to trace-free symmetric 2-tensors, and J g is the Gram matrix of a fixed basis of ker P * g (see Section 5.1 further details). Then Belavin-Knizhnik [BeKn] and Wolpert [Wo2] proved that the integral (1.12) with D = 26 diverges at the boundary of (the compactification of) moduli space, a problematic fact in order to establish well-posedness of the partition function for critical D = 26 (bosonic) strings.Noncritical string theories are not formulated within the critical dimension D = 26, yet they are Weyl invariant. The idea, emerging once again from the paper [Po], is that for D = 26 the integral (1.9) possesses one further degree of freedom to be integrated over corresponding to the Weyl factor e ω in (1.11). For D 1, hence c M 1, Polyakov argued that integrating this factor requires using Liouville quantum field theory. In other words, applying once again the 1 The term "critical" refers in fact to the critical dimension D = 26 needed to get a Weyl invariant theory without quantizing the Weyl factor e ω in (1.11).

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Roaming moduli space using dynamical triangulations

Cited by 14 publications

References 47 publications

Two-dimensional Quantum Geometry

Two-dimensional Quantum Geometry

Gauge invariant computable quantities in timelike Liouville theory

Polyakov’s formulation of $2d$ bosonic string theory

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