2D Quantum Gravity Research Articles

Matter coupled to 3d quantum gravity: one-loop unitarity

Abstract We expect quantum field theories for matter to acquire intricate corrections due to their coupling to quantum fluctuations of the gravitational field. This can be precisely worked out in 3d quantum gravity: after integrating out quantum gravity, matter fields are effectively described as noncommutative quantum field theories, with quantum-deformed Lorentz symmetries. An open question remains: Are such theories unitary or not? On the one hand, since these are effective field theories obtained after integrating out high energy degrees of freedom, we may expect the loss of unitarity. On the other hand, as rigorously defined field theories built with Lorentz symmetries and standing on their own, we naturally expect the conservation of unitarity. In an effort to settle this issue, we explicitly check unitarity for a scalar field at one-loop level in both Euclidean and Lorentzian space-time signatures. We find that unitarity requires adding an extra-term to the propagator of the noncommutative theory, corresponding to a massless mode and given by a representation with vanishing Plancherel measure, thus usually ignored in spinfoam path integrals for quantum gravity. This indicates that the inclusion of matter in spinfoam models, and more generally in quantum gravity, might be more subtle than previously thought.

Perturbative unorientable JT gravity and matrix models

We consider an orthogonal polynomial formulation of the double scaling limit of multicritical matrix models in the β = 1 Dyson-Wigner class. They capture the physics of 2D quantum gravity coupled to minimal matter on unorientable surfaces, otherwise called unoriented minimal strings. We derive a formula for the density of states valid to all orders in perturbation theory. We show how to define an interpolation between the multicritical models and that a certain interpolation among an infinite number of them provides an alternative definition of unoriented JT gravity. We discuss the strengths and weaknesses of our formulation.

Hartle-Hawking state and its factorization in 3d gravity

We study 3d quantum gravity with two asymptotically anti-de Sitter regions, in particular, using its relation with coupled Alekseev-Shatashvili theories and Liouville theory. Expressions for the Hartle-Hawking state, thermal 2n-point functions, torus wormhole correlators and Wheeler-DeWitt wavefunctions in different bases are obtained using the ZZ boundary states in Liouville theory. Exact results in 2d Jackiw-Teitelboim (JT) gravity are uplifted to 3d gravity, with two copies of Liouville theory in 3d gravity playing a similar role as Schwarzian theory in JT gravity. The connection between 3d gravity and the Liouville ZZ boundary states are manifested by viewing BTZ black holes as Maldacena-Maoz wormholes, with the two wormhole boundaries glued along the ZZ boundaries. In this work, we also study the factorization problem of the Hartle-Hawking state in 3d gravity. With the relevant defect operator that imposes the necessary topological constraint for contractibility, the trace formula in gravity is modified in computing the entanglement entropy. This trace matches with the one from von Neumann algebra considerations, further reproducing the Bekenstein-Hawking area formula from entanglement entropy. Lastly, we propose a calculation for off-shell geometrical quantities that are responsible for the ramp behavior in the late time two-point functions, which follows from the understanding of the Liouville FZZT boundary states in the context of 3d gravity, and the identification between Verlinde loop operators in Liouville theory and “baby universe” operators in 3d gravity.

Generalized Amit-Roginsky model from perturbations of 3D quantum gravity

Generalized Amit-Roginsky model from perturbations of 3D quantum gravity

Perturbing isoradial triangulations

We consider an infinite planar Delaunay graph \mathbf{G}_{\epsilon} which is obtained by locally deforming the coordinate embedding of a general isoradial graph \mathbf{G}_{\mathrm{cr}} , with respect to a real deformation parameter \epsilon . Using Kenyon’s exact and asymptotic results for the critical Green’s function on an isoradial graph, we calculate the leading asymptotics of the first- and second-order terms in the perturbative expansion of the log-determinant of the Laplace–Beltrami operator \Delta(\epsilon) , the David–Eynard Kähler operator \mathcal{D}(\epsilon) , and the conformal Laplacian {\underline{\boldsymbol{\Delta}}}(\epsilon) on the deformed Delaunay graph \mathbf{G}_{\epsilon} . We show that the scaling limits of the second-order bi-local term for both the Laplace–Beltrami and David–Eynard Kähler operators exist and coincide, with a shared value independent of the choice of the initial isoradial graph \mathbf{G}_{\mathrm{cr}} . Our results allow us to define a discrete analog of the stress-energy tensor for each of the three operators. Furthermore, we can identify a central charge ( c=-2 ) in the case of both the Laplace–Beltrami and David–Eynard Kähler operators. While the scaling limit is consistent with the stress-energy tensor and the value of the central charge for the Gaussian free field (GFF), the discrete central charge value of c=-2 for the David–Eynard Kähler operator is, however, at odds with the value of c=-26 expected by Polyakov’s theory of 2D quantum gravity; moreover, there are problems with convergence of the scaling limit of the discrete stress-energy tensor for the David–Eynard Kähler operator. The second-order bi-local term for the conformal Laplacian involves anomalous terms corresponding to the creation of discrete curvature dipoles in the deformed Delaunay graph \mathbf{G}_{\epsilon} ; we examine the difficulties in defining a convergent scaling limit in this case. Connections with some discrete statistical models at criticality are explored.

Gravitational action for a massive Majorana fermion in 2d quantum gravity

We compute the gravitational action of a free massive Majorana fermion coupled to two-dimensional gravity on compact Riemann surfaces of arbitrary genus. The structure is similar to the case of the massive scalar. The small-mass expansion of the gravitational yields the Liouville action at zeroth order, and we can identify the Mabuchi action at first order. While the massive Majorana action is a conformal deformation of the massless Majorana CFT, we find an action different from the one given by the David-Distler-Kawai (DDK) ansatz.

3D Quantum Gravity Partition Function at Three Loops

3D Quantum Gravity Partition Function at Three Loops

Solving 3d gravity with Virasoro TQFT

We propose a precise reformulation of 3d quantum gravity with negative cosmological constant in terms of a topological quantum field theory based on the quantization of the Teichmüller space of Riemann surfaces that we refer to as “Virasoro TQFT”. This TQFT is similar, but importantly not equivalent, to SL(2, \mathbb{R}ℝ) Chern-Simons theory. This sharpens the folklore that 3d gravity is related to SL(2, \mathbb{R}ℝ) Chern-Simons theory into a precise correspondence, and resolves some well-known issues with this lore at the quantum level. Our proposal is computationally very useful and provides a powerful tool for the further study of 3d gravity. In particular, we explain how together with standard TQFT surgery techniques this leads to a fully algorithmic procedure for the computation of the gravity partition function on a fixed topology exactly in the central charge. Mathematically, the relation leads to many nontrivial conjectures for hyperbolic 3-manifolds, Virasoro conformal blocks and crossing kernels.

Holography for bulk states in 3D quantum gravity

In this work we discuss the holographic description of states in the Hilbert space of (2+1)-dimensional quantum gravity, living on a time slice in the bulk. We focus on pure gravity coupled to pointlike sources for heavy spinning particles. We develop a formulation where the equations for the backreacted metric reduce to two decoupled Liouville equations with delta-function sources under pseudosphere boundary conditions. We show that both the semiclassical wavefunction and the gravity solution are determined by a universal object, namely a classical Virasoro vacuum block on the sphere. In doing so we derive a version of Polyakov’s conjecture, as well as an existence criterion, for classical Liouville theory on the pseudosphere. We also discuss how some of these results are modified when considering closed universes with compact spatial slices.

An inner product for 4D quantum gravity and the Chern–Simons–Kodama state

We demonstrate that reality conditions for the Ashtekar connection imply a non-trivial measure for the inner product of gravitational states in the polarization where the Ashtekar connection is diagonal, and we express the measure as the determinant of a certain first-order differential operator. This result opens the possibility to perform a non-perturbative analysis of the quantum gravity scalar product. In this polarization, the Chern–Simons–Kodama state, which solves the constraints of quantum gravity for a certain factor ordering, and which has de Sitter space as a semiclassical limit, is perturbatively non-normalizable with respect to the naïve inner product. Our work reopens the question of whether this state might be normalizable when the correct non-perturbative inner product and choice of integration contour are taken into account. As a first step, we perform a semi-classical treatment of the measure by evaluating it on the round three-sphere, viewed as a closed spatial slice of de Sitter. The result is a simple, albeit divergent, infinite product that might serve as a regulator for a more complete treatment of the problem. Additionally, our results suggest deep connections between the problem of computing the norm of the CSK state in quantum gravity and computing the Chern–Simons partition function for a complex group.

A proposal for 3d quantum gravity and its bulk factorization

Recent progress in AdS/CFT has provided a good understanding of how the bulk spacetime is encoded in the entanglement structure of the boundary CFT. However, little is known about how spacetime emerges directly from the bulk quantum theory. We address this question in an effective 3d quantum theory of pure gravity, which describes the high temperature regime of a holographic CFT. This theory can be viewed as a q-deformation and dimensional uplift of JT gravity. Using this model, we show that the Bekenstein-Hawking entropy of a two-sided black hole equals the bulk entanglement entropy of gravitational edge modes. In the conventional Chern-Simons description, these black holes correspond to Wilson lines in representations of PSL(2, ℝ) ⨂ PSL(2, ℝ). We show that the correct calculation of gravitational entropy suggests we should interpret the bulk theory as an extended topological quantum field theory associated to the quantum semi-group {textrm{SL}}_q^{+}left(2,mathbb{R}right)bigotimes {textrm{SL}}_q^{+}left(2,mathbb{R}right) . Our calculation suggests an effective description of bulk microstates in terms of collective, anyonic degrees of freedom whose entanglement leads to the emergence of the bulk spacetime.

Transition Amplitudes in 3D Quantum Gravity: Boundaries and Holography in the Coloured Boulatov Model

We consider transition amplitudes in the coloured simplicial Boulatov model for three-dimensional Riemannian quantum gravity. First, we discuss aspects of the topology of coloured graphs with non-empty boundaries. Using a modification of the standard rooting procedure of coloured tensor models, we then write transition amplitudes systematically as topological expansions. We analyse the transition amplitudes for the simplest boundary topology, the 2-sphere, and prove that they factorize into a sum entirely given by the combinatorics of the boundary spin network state and that the leading order is given by graphs representing the closed 3-ball in the large N limit. This is the first step towards a more detailed study of the holographic nature of coloured Boulatov-type GFT models for topological field theories and quantum gravity.

A flow in the forest

Using the matrix-forest theorem and the Parisi-Sourlas trick we formulate and solve a one-matrix model with non-polynomial potential which provides perturbation theory for massive spinless fermions on dynamical planar graphs. This is a lattice version of 2d quantum gravity coupled to massive spinless fermions. Our model equivalently describes the ensemble of spanning forests on the same graphs. The solution is formulated in terms of an elliptic curve. We then focus on a near-critical scaling limit when both the graphs and the trees in the forests are macroscopically large. In this limit we obtain one-point scaling functions (condensates), parameterized in terms of the Lambert function. Our results provide a rare example where one can explore the flow between two gravity models — in this case, the theories of conformal matter coupled to 2d gravity with c = −2 (large trees regime) and c = 0 (small trees regime). We also compute the disc partition functions with Dirichlet and Neumann boundary conditions in the same critical limit.

3D quantum gravity from holomorphic blocks

Three-dimensional gravity is a topological field theory, which can be quantized as the Ponzano-Regge state-sum model built from the ${3nj}$ symbols of the recoupling of the SU(2) representations, in which spins are interpreted as quantized edge lengths in Planck units. It describes the flat spacetime as the gluing of three-dimensional cells with a fixed boundary metric encoding length scale. In this paper, we revisit the Ponzano-Regge model formulated in terms of spinors and rewrite the quantum geometry of 3D cells with holomorphic recoupling symbols. These symbols, known as Schwinger's generating function for the ${6j}$ symbols, are simply the squared inverse of the partition function of the 2D Ising model living on the boundary of the 3D cells. They can furthermore be interpreted, in their critical regime, as scale-invariant basic elements of geometry. We show how to glue them together into a discrete topological quantum field theory. This reformulation of the path integral for 3D quantum gravity, with a rich pole structure of the elementary building blocks, opens a new door toward the study of phase transitions and continuum limits in 3D quantum gravity and offers a new twist on the construction of a duality between 3D quantum gravity and a 2D conformal theory.

2D quantum gravity partition function on the fluctuating sphere

Motivated by recent works on the connection between 2D quantum gravity and timelike Liouville theory, we revisit the latter and clarify some aspects of the computation of its partition function: we present a detailed computation of the Liouville partition function on the fluctuating sphere at finite values of the central charge. The results for both the spacelike theory and the timelike theory are given, and their properties analyzed. We discuss the derivation of the partition function from the DOZZ formula, its derivation using the Coulomb gas approach, a semiclassical computation of it using the fixed area saddle point, and, finally, we arrive to an exact expression for the timelike partition function whose expansion can be compared with the 3-loop perturbative calculations reported in the literature. We also discuss the connection to the 2D black hole and other related topics.

Light ray fluctuations in simplicial quantum gravity

A non-perturbative study on the quantum fluctuations of light ray propagation through a quantum region of spacetime is long overdue. Within the theory of Lorentzian simplicial quantum gravity, we compute the probabilities for a test light ray to land at different locations after travelling through a symmetry-reduced box region in 2, 3 and 4 spacetime dimensions. It is found that for fixed boundary conditions, light ray fluctuations are generically large when all coupling constants are relatively small in absolute value. For fixed coupling constants, as the boundary size is decreased light ray fluctuations first increase and then decrease in a 2D theory with the cosmological constant, Einstein–Hilbert and R-squared terms. While in 3D and 4D theories with the cosmological constant and Einstein–Hilbert terms, as the boundary size is decreased light ray fluctuations just increase. Incidentally, when studying 2D quantum gravity we show that the global time-space duality with the cosmological constant and Einstein–Hilbert terms noted previously also holds when arbitrary even powers of the Ricci scalar are added. We close by discussing how light ray fluctuations can be used in obtaining the continuum limit of non-perturbative Lorentzian quantum gravity.

Self-duality of the $6j$-symbol and Fisher zeros for the tetrahedron

The relation between the 2d Ising partition function and spin network evaluations, reflecting a bulk-boundary duality between the 2d Ising model and 3d quantum gravity, promises an exchange of results and methods between statistical physics and quantum geometry. We apply this relation to the case of the tetrahedral graph. First, we find that the high/low temperature duality of the 2d Ising model translates into a new self-duality formula for Wigner’s 6j -symbol from the theory of spin recoupling. Second, we focus on the duality between the large spin asymptotics of the 6j -symbol and Fisher zeros. Using the Ponzano–Regge formula for the asymptotics for the 6j -symbol at large spins in terms of the tetrahedron geometry, we obtain a geometric formula for the zeros of the (inhomogeneous) Ising partition function in terms of triangle angles and dihedral angles in the tetrahedron. While it is well known that the 2d intrinsic geometry can be used to parametrize the critical point of the Ising model, e.g., on isoradial graphs, it is the first time to our knowledge that the extrinsic geometry is found to also be relevant. This outlines a method towards a more general geometric parametrization of the Fisher zeros for the 2d Ising model on arbitrary graphs.

Time-space duality in 2D quantum gravity

An important task faced by all approaches of quantum gravity is to incorporate superpositions and quantify quantum uncertainties of spacetime causal relations. We address this task in 2D. By identifying a global Z 2 symmetry of 1 + 1D quantum gravity, we show that gravitational path integral configurations come in equal amplitude pairs with timelike and spacelike relations exchanged. As a consequence, any two points are equally probable to be timelike and spacelike separated in a Universe without boundary conditions. In the context of simplicial quantum gravity we identify a local symmetry of the action which shows that even with boundary conditions causal uncertainties are generically present. Depending on the boundary conditions, causal uncertainties can still be large and even maximal.

Disc partition function of 2d R2 gravity from DWG matrix model

We compute the sum over flat surfaces of disc topology with arbitrary number of conical singularities. To that end, we explore and generalize a specific case of the matrix model of dually weighted graphs (DWG) proposed and solved by one of the authors, M. Staudacher and Th. Wynter. Namely, we compute the sum over quadrangulations of the disc with certain boundary conditions, with parameters controlling the number of squares (area), the length of the boundary and the coordination numbers of vertices. The vertices introduce conical defects with angle deficit given by a multiple of π, corresponding to positive, zero or negative curvature. Our results interpolate between the well-known 2d quantum gravity solution for the disc with fluctuating 2d metric and the regime of “almost flat” surfaces with all the negative curvature concentrated on the boundary. We also speculate on possible ways to study the fluctuating 2d geometry with AdS2 background instead of the flat one.

W_{1+∞} Algebra and the Celestial Sphere: Infinite Towers of Soft Graviton, Photon, and Gluon Symmetries.

It is shown that the infinite tower of tree-level plus-helicity soft graviton symmetries in asymptotically flat 4D quantum gravity can be organized into a single chiral 2D Kac-Moody symmetry based on the wedge algebra of w_{1+∞}, which naturally acts on the celestial sphere at null infinity. The infinite towers of soft photon or gluon symmetries also transform irreducibly under w_{1+∞}.