Path integral over conformally self-dual geometries

The path integral for higher-derivative quantum gravity with torsion is considered. Applying the methods of two-dimensional quantum gravity, this path integral is analyzed in the limit of conformally self-dual metrics. A scaling law for fixed-volume geometry is obtained.

It is shown that one can formulate Euclidean two-dimensional quantum gravity as the scaling limit of an ordinary statistical system. Scaling relations can be derived and the critical exponents have simple geometric interpretations. In addition it is possible to calculate reparametrization-invariant correlation functions as functions of the geodesic distance. The definitions for pure gravity generalize to matter systems coupled to two-dimensional quantum gravity. As long as the central charge c of the matter field is less than 1, it is still possible to solve the coupled system explicitly. For mean-field theory is reliable, and again we can solve the mean-field equations using the discretized approach. Finally, the discretized approach makes sense even in higher-dimensional spacetime. Although analytic solutions are still missing in the higher-dimensional case, numerical studies reveal an interesting structure and allow the identification of a fixed point where we can hope to define a genuine non-perturbative theory of four-dimensional quantum gravity.

Two-dimensional quantum gravity is identified as a second-class system which we convert into a first-class system via the Batalin-Fradkin (BF) procedure. Using the extended phase space method, we then formulate the theory in most general class of gauges. The conformal gauge action suggested by David, Distler and Kawai is derived from a first principle. We find a local, light-cone gauge action whose Becchi-Rouet-Stora-Tyutin invariance implies Polyakov's curvature equation $\partial_{-}R=\partial_{-}^{3}g_{++}=0$, revealing the origin of the $SL(2,R)$ Kac-Moody symmetry. The BF degree of freedom turns out be dynamically active as the Liouville mode in the conformal gauge, while in the light-cone gauge the conformal degree of freedom plays that r{\^o}le. The inclusion of the cosmological constant term in both gauges and the harmonic gauge-fixing are also considered.

We study the trace-anomaly-induced dynamics of the conformal factor of four-dimensional (4D) quantum gravity. The resulting effective scalar theory is ultraviolet renormalizable, and possesses a nontrivial, infrared stable fixed point. The exact anomalous scaling dimension of the conformal factor at the critical point is derived. We argue that this theory describes 4D gravity at large distances and provides a framework for a dynamical solution of the cosmological-constant problem.

Gravity coupled to a scalar field in two dimensions is analyzed in the conformal gauge. We obtain the solution of a scalar field. The residual symmetry associated with the scalar field is found from the invariance of the action and the covariance of the equation of motion of a scalar field. This symmetry is naturally related to the fact that the energy-momentum tensor satisfies the transformation rule of the usual second-rank tensor under $\mathrm{SL}(2,R)\ensuremath{\bigotimes}\mathrm{SL}(2,R)$ group transformations for the coordinates ${x}^{+}$ and ${x}^{\ensuremath{-}}$.

In many quantum gravity approaches, the cosmological constant is introduced by deforming the gauge group into a quantum group. In three dimensions, the quantization of the Chern-Simons formulation of gravity provided the first example of such a deformation. The Turaev-Viro model, which is an example of a spin-foam model, is also defined in terms of a quantum group. By extension, it is believed that in four dimensions, a quantum group structure could encode the presence of $\mathrm{\ensuremath{\Lambda}}\ensuremath{\ne}0$. In this article, we introduce by hand the quantum group ${\mathcal{U}}_{q}(\mathfrak{s}u(2))$ into the loop quantum gravity (LQG) framework; that is, we deal with ${\mathcal{U}}_{q}(\mathfrak{s}u(2))$-spin networks. We explore some of the consequences, focusing in particular on the structure of the observables. Our fundamental tools are tensor operators for ${\mathcal{U}}_{q}(\mathfrak{s}u(2))$. We review their properties and give an explicit realization of the spinorial and vectorial ones. We construct the generalization of the $\mathrm{U}(N)$ formalism in this deformed case, which is given by the quantum group ${\mathcal{U}}_{q}(\mathfrak{u}(N))$. We are then able to build geometrical observables, such as the length, area or angle operators, etc. We show that these operators characterize a quantum discrete hyperbolic geometry in the three-dimensional LQG case. Our results confirm that a quantum group structure in LQG can be a tool to introduce a nonzero cosmological constant into the theory. Our construction is both relevant for three-dimensional Euclidian quantum gravity with a negative cosmological constant and four-dimensional Lorentzian quantum gravity with a positive cosmological constant.

We show how to formulate a lattice gauge theory whose naive continuum limit corresponds to two-dimensional (Euclidean) quantum gravity including a positive cosmological constant. More precisely the resultant continuum theory corresponds to gravity in a first-order formalism in which the local frame and spin connection are treated as independent fields. Recasting this lattice theory as a tensor network allows us to study the theory at strong coupling without encountering a sign problem. In two dimensions this tensor network is exactly soluble and we show that the system has a series of critical points that occur for pure imaginary coupling and are associated with first order phase transitions. We then augment the action with a Yang-Mills term which allows us to control the lattice spacing and show how to apply the tensor renormalization group to compute the free energy and look for critical behavior. Finally we perform an analytic continuation in the gravity coupling in this extended model and show that its critical behavior in a certain scaling limit depends only on the topology of the underlying lattice. We also show how the lattice gauge theory can be naturally generalized to generate the Polyakov or Liouville action for two dimensional quantum gravity.

Operator product expansion in two-dimensional quantum gravity

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Shaken, but not stirred—Potts model coupled to quantum gravity

The Feynman path integral has revolutionized modern approaches to quantum physics. Although the path integral formalism has proven very successful and spawned several approximation schemes, the direct evaluation of real-time path integrals is still extremely expensive and numerically delicate due to its high-dimensional and oscillatory nature. We propose an efficient method for the numerical evaluation of the real-time world-line path integral for theories where the potential is dominated by a quadratic at infinity. This is done by rewriting the high-dimensional oscillatory integral in terms of a series of low-dimensional oscillatory integrals, that we efficiently evaluate with Picard-Lefschetz theory or approximate with the eikonal approximation. Subsequently, these integrals are stitched together with a series of fast Fourier transformations to recover the lattice regularized Feynman path integral. Our method directly applies to problems in quantum mechanics, the word-line quantization of quantum field theory, and quantum gravity. Published by the American Physical Society 2025

The ϵ-expansion of two-dimensional quantum gravity

Further evidence of a smooth phase in 4D simplicial quantum gravity

String vacuum backgrounds with covariantly constant null Killing vector and two-dimensional quantum gravity

We present new path integral studies on the Polyakov noncritical and Nambu–Goto critical string theories and their applications to [Formula: see text] interquark potential. We also evaluate the long distance asymptotic behavior of the interquark potential on the Nambu–Goto string theory with an extrinsic term in Polyakov’s string at [Formula: see text]. We also propose an alternative and a new view to covariant Polyakov’s string path integral with a fourth-order two-dimensional quantum gravity, is an effective stringy description for [Formula: see text] at the deep infrared region.