Path integral for quantum Mabuchi K-energy

@article{Lacoin2018PathIF,
  title={Path integral for quantum Mabuchi K-energy},
  author={Hubert Lacoin and R{\'e}mi Rhodes and Vincent Vargas},
  journal={Duke Mathematical Journal},
  year={2018}
}
We construct a path integral based on the coupling of the Liouville action and the Mabuchi K-energy on a one-dimensional complex manifold. To the best of our knowledge this is the first rigorous construction of such an object and this is done by means of probabilistic tools. Both functionals play an important role respectively in Riemannian geometry (in the case of surfaces) and K\"ahler geometry. As an output, we obtain a path integral whose Weyl anomaly displays the standard Liouville anomaly… 

The Fyodorov–Bouchaud formula and Liouville conformal field theory

  • G. Remy
  • Mathematics
    Duke Mathematical Journal
  • 2020
In a remarkable paper in 2008, Fyodorov and Bouchaud conjectured an exact formula for the density of the total mass of (sub-critical) Gaussian multiplicative chaos (GMC) associated to the Gaussian

Integrability of Boundary Liouville Conformal Field Theory

  • G. RemyT. Zhu
  • Mathematics
    Communications in Mathematical Physics
  • 2022
Liouville conformal field theory (LCFT) is considered on a simply connected domain with boundary, specializing to the case where the Liouville potential is integrated only over the boundary of the

The distribution of Gaussian multiplicative chaos on the unit interval

We consider the sub-critical Gaussian multiplicative chaos (GMC) measure defined on the unit interval [0,1] and prove an exact formula for the fractional moments of the total mass of this measure.

Volume of metric balls in Liouville quantum gravity

We study the volume of metric balls in Liouville quantum gravity (LQG). For $\gamma \in (0,2)$, it has been known since the early work of Kahane (1985) and Molchan (1996) that the LQG volume of

A family of probability distributions consistent with the DOZZ formula: towards a conjecture for the law of 2D GMC

  • D. Ostrovsky
  • Mathematics
    Probability and Mathematical Physics
  • 2021
A three parameter family of probability distributions is constructed such that its Mellin transform is defined over the same domain as the 2D GMC on the Riemann sphere with three insertion points

Another probabilistic construction of ${\Phi}^{2n}$ in dimension 2

The main input of this note is to provide an alternative probabilistic approach to the ${\Phi}^{2n}$ theory in dimension 2, based on concentration phenomenon of martingales associated to polynomials

Electron. Commun. Probab. 26 (2021), article no. 19, https://doi.org/10.1214/21-ECP389

This note provides an alternative probabilistic approach to the Φ theory in dimension 2. The key idea is to study the concentration phenomenon of martingales associated to polynomials of Gaussian

References

SHOWING 1-10 OF 51 REFERENCES

Local Conformal Structure of Liouville Quantum Gravity

In 1983 Belavin, Polyakov, and Zamolodchikov (BPZ) formulated the concept of local conformal symmetry in two dimensional quantum field theories. Their ideas had a tremendous impact in physics and

Liouville quantum gravity and KPZ

AbstractConsider a bounded planar domain D, an instance h of the Gaussian free field on D, with Dirichlet energy (2π)−1∫D∇h(z)⋅∇h(z)dz, and a constant 0≤γ<2. The Liouville quantum gravity measure on

Quantum Liouville Theory in the Background Field Formalism I. Compact Riemann Surfaces

AbstractUsing Polyakov’s functional integral approach and the Liouville action functional defined in [ZT87c] and [TT03a], we formulate quantum Liouville theory on a compact Riemann surface X of genus

Conformal Ward and BPZ Identities for Liouville quantum field theory

In this work, we continue the constructive probabilistic approach to the Liouville Quantum Field theory (LQFT) started in [8]. We give a rigorous construction of the stress energy tensor in LQFT and

Liouville quantum gravity and the Brownian map II: Geodesics and continuity of the embedding

We endow the √ 8/3-Liouville quantum gravity sphere with a metric space structure and show that the resulting metric measure space agrees in law with the Brownian map. Recall that a Liouville quantum

Liouville Quantum Gravity on the Riemann Sphere

In this paper, we rigorously construct Liouville Quantum Field Theory on the Riemann sphere introduced in the 1981 seminal work by Polyakov. We establish some of its fundamental properties like

Liouville quantum gravity and the Brownian map III: the conformal structure is determined

Previous works in this series have shown that an instance of a $$\sqrt{8/3}$$ 8 / 3 -Liouville quantum gravity (LQG) sphere has a well-defined distance function, and that the resulting metric measure

The Fyodorov–Bouchaud formula and Liouville conformal field theory

  • G. Remy
  • Mathematics
    Duke Mathematical Journal
  • 2020
In a remarkable paper in 2008, Fyodorov and Bouchaud conjectured an exact formula for the density of the total mass of (sub-critical) Gaussian multiplicative chaos (GMC) associated to the Gaussian
...