Aldo Riello

CRITICAL BEHAVIOR OF COLORED TENSOR MODELS IN THE LARGE N LIMIT

Colored tensor models have been recently shown to admit a large N expansion, whose leading order encodes a sum over a class of colored triangulations of the D-sphere. The present paper investigates in details this leading order. We show that the relevant triangula- tions proliferate like a species of colored trees. The leading order is therefore summable and exhibits a critical behavior, independent of the dimension. A continuum limit is reached by tuning the coupling constant to its critical value while inserting an infinite number of pairs of D-simplices glued together in a specific way. We argue that the dominant triangulations are branched polymers. Tensor models (1-3) and group field theories (4-7) are the natural generalization of matrix models (8, 9) implementing in a consistent way the sum over random triangulations in dimensions higher than two. They are notoriously hard to control analytically and one usually resorts to numerical simulations (10-12). Progress has recently been made in the analytic control of tensor models with the advent of the 1/N expansion (13-15) of colored (16-18) tensor models. This expansion synthetizes several alternative evaluations of graph amplitudes in tensor models (19-27) and provides a straightforward generalization of the familiar genus expansion of matrix models (28, 29) in arbitrary dimension. The coloring of the fields allows one to address previously inaccessible questions in tensor models like the implementation of the diffeomorphism symmetry (27, 30) in the Boulatov model or the identification of embedded matrix models (31). The symmetries of tensor models have recently been studied using n-ary algebras (32, 33). This paper is the first in a long series of studies of the implications of the 1/N expansion in colored tensor models. We present here a complete analysis of the leading order in the large N limit in arbitrary dimensions, indexed by graphs of spherical topology (14). To perform the study of this leading order one needs to address the following two questions • What is the combinatorics of the Feynman graphs contributing to the leading order, i.e. the higher dimensional extension of the notion of planar graphs? Unlike in matrix models, where planarity and spherical topology are trivially related, this question is non trivial in tensor models. In particular not all triangulations of the sphere contribute to the leading order. • Is the series of the leading order summable with a non zero radius of convergence? If this is the case, then, in the large N limit, the model exhibits a critical behavior whose critical exponents one needs to compute.

Self-energy of the Lorentzian Engle-Pereira-Rovelli-Livine and Freidel-Krasnov model of quantum gravity

We calculate the most divergent contribution to non-degenerate sectorof the self-energy (or “melonic”) graph in the context of the Lorentzian EPRL-FK Spin Foam model of Quantum Gravity. We find that such a contribution is logarithmically divergent in the cut-off over the SU (2)-representation spins when one chooses the face amplitude guaranteeing the facesplitting invariance of the foam. We also find that the dependence on the boundary data is different from that of the bare propagator. This fact has its origin in the non-commutativity of the EPRL-FK Yγ-map with the projector onto SL(2,C)-invariant states. In the course of the paper, we discuss in detail the approximations used during the calculations, its geometrical interpretation as well as the physical consequences of our result. ar X iv :1 30 2. 17 81 v2 [ gr -q c] 1 3 M ar 2 01 3

SL(2,C) Chern–Simons theory, a non-planar graph operator, and 4D quantum gravity with a cosmological constant: Semiclassical geometry

Abstract We study the expectation value of a nonplanar Wilson graph operator in SL ( 2 , C ) Chern–Simons theory on S 3 . In particular we analyze its asymptotic behavior in the double-scaling limit in which both the representation labels and the Chern–Simons coupling are taken to be large, but with fixed ratio. When the Wilson graph operator has a specific form, motivated by loop quantum gravity, the critical point equations obtained in this double-scaling limit describe a very specific class of flat connection on the graph complement manifold. We find that flat connections in this class are in correspondence with the geometries of constant curvature 4-simplices. The result is fully non-perturbative from the perspective of the reconstructed geometry. We also show that the asymptotic behavior of the amplitude contains, at the leading order, an oscillatory part proportional to the Regge action for the single 4-simplex in the presence of a cosmological constant. In particular, the cosmological term contains the full-fledged curved volume of the 4-simplex. Interestingly, the volume term stems from the asymptotics of the Chern–Simons action. This can be understood as arising from the relation between Chern–Simons theory on the boundary of a region, and a theory defined by an F 2 action in the bulk. Another peculiarity of our approach is that the sign of the curvature of the reconstructed geometry, and hence of the cosmological constant in the Regge action, is not fixed a priori, but rather emerges semiclassically and dynamically from the solution of the equations of motion. In other words, this work suggests a relation between 4-dimensional loop quantum gravity with a cosmological constant and SL ( 2 , C ) Chern–Simons theory in 3 dimensions with knotted graph defects.

Spacetime thermodynamics without hidden degrees of freedom

A celebrated result by Jacobson is the derivation of Einsteins equations from Unruhs temperature, the Bekenstein-Hawking entropy and the Clausius relation. This has been repeatedly taken as evidence for an interpretation of Einsteins equations as equations of state for unknown degrees of freedom underlying the metric. We show that a different interpretation of Jacobson result is possible, which does not imply the existence of additional degrees of freedom, and follows only from the quantum properties of gravity. We introduce the notion of quantum gravitational Hadamard states, which give rise to the full local thermodynamics of gravity.

Encoding Curved Tetrahedra in Face Holonomies: Phase Space of Shapes from Group-Valued Moment Maps

We present a generalization of Minkowski’s classic theorem on the reconstruction of tetrahedra from algebraic data to homogeneously curved spaces. Euclidean notions such as the normal vector to a face are replaced by Levi–Civita holonomies around each of the tetrahedron’s faces. This allows the reconstruction of both spherical and hyperbolic tetrahedra within a unified framework. A new type of hyperbolic simplex is introduced in order for all the sectors encoded in the algebraic data to be covered. Generalizing the phase space of shapes associated to flat tetrahedra leads to group-valued moment maps and quasi-Poisson spaces. These discrete geometries provide a natural arena for considering the quantization of gravity including a cosmological constant. This becomes manifest in light of their relation with the spin-network states of loop quantum gravity. This work therefore provides a bottom-up justification for the emergence of deformed gauge symmetries and quantum groups in covariant loop quantum gravity in the presence of a cosmological constant.

Fusion basis for lattice gauge theory and loop quantum gravity

A bstractWe introduce a new basis for the gauge-invariant Hilbert space of lattice gauge theory and loop quantum gravity in (2 + 1) dimensions, the fusion basis. In doing so, we shift the focus from the original lattice (or spin-network) structure directly to that of the magnetic (curvature) and electric (torsion) excitations themselves. These excitations are classified by the irreducible representations of the Drinfel’d double of the gauge group, and can be readily “fused” together by studying the tensor product of such representations. We will also describe in detail the ribbon operators that create and measure these excitations and make the quasi-local structure of the observable algebra explicit. Since the fusion basis allows for both magnetic and electric excitations from the onset, it turns out to be a precious tool for studying the large scale structure and coarse-graining flow of lattice gauge theories and loop quantum gravity. This is in neat contrast with the widely used spin-network basis, in which it is much more complicated to account for electric excitations, i.e. for Gauß constraint violations, emerging at larger scales. Moreover, since the fusion basis comes equipped with a hierarchical structure, it readily provides the language to design states with sophisticated multi-scale structures. Another way to employ this hierarchical structure is to encode a notion of subsystems for lattice gauge theories and (2 + 1) gravity coupled to point particles. In a follow-up work, we have exploited this notion to provide a new definition of entanglement entropy for these theories.

On entanglement entropy in non-Abelian lattice gauge theory and 3D quantum gravity

A bstractEntanglement entropy is a valuable tool for characterizing the correlation structure of quantum field theories. When applied to gauge theories, subtleties arise which prevent the factorization of the Hilbert space underlying the notion of entanglement entropy. Borrowing techniques from extended topological field theories, we introduce a new definition of entanglement entropy for both Abelian and non-Abelian gauge theories. Being based on the notion of excitations, it provides a completely relational way of defining regions. Therefore, it naturally applies to background independent theories, e.g. gravity, by circumventing the difficulty of specifying the position of the entangling surface. We relate our construction to earlier proposals and argue that it brings these closer to each other. In particular, it yields the non-Abelian analogue of the ‘magnetic centre choice’, as obtained through an extended-Hilbert-space method, but applied to the recently introduced fusion basis for 3D lattice gauge theories. We point out that the different definitions of entanglement entropy can be related to a choice of (squeezed) vacuum state.

How to detect an anti-spacetime

Is it possible, in principle, to measure the sign of the Lapse? We show that fermion dynamics distinguishes spacetimes having the same metric but different tetrads, for instance a Lapse with opposite sign. This sign might be a physical quantity not captured by the metric. We discuss its possible role in quantum gravity.

The observer’s ghost: notes on a field space connection

A bstractWe introduce a functional covariant differential as a tool for studying field space geometry in a manifestly covariant way. We then touch upon its role in gauge theories and general relativity over bounded regions, and in BRST symmetry. Due to the Gribov problem, we argue that our formalism — allowing for a non-vanishing functional curvature — is necessary for a global treatment of gauge-invariance in field space. We conclude by suggesting that the structures we introduce satisfactorily implement the notion of a (non-asymptotic) observer in gauge theories and general relativity.

Quasi-local holographic dualities in non-perturbative 3d quantum gravity II – From coherent quantum boundaries to BMS 3 characters

Abstract We analyze the partition function of three-dimensional quantum gravity on the twisted solid torus and the ensuing dual field theory. The setting is that of a non-perturbative model of three-dimensional quantum gravity, the Ponzano–Regge model, which is here reviewed in a self-contained manner and then used to compute quasi-local amplitudes for its boundary states. In this second paper of the series, we choose a particular class of boundary spin network states which impose Gibbons–Hawking–York boundary conditions to the partition function. The peculiarity of these states is to encode a two-dimensional quantum geometry peaked around a classical quadrangulation of the finite toroidal boundary. Thanks to the topological properties of three-dimensional gravity, the theory easily projects onto the boundary while crucially still keeping track of the topological properties of the bulk. This produces, at the non-perturbative level, a specific non-linear sigma-model on the boundary, akin to a Wess–Zumino–Novikov–Witten model, whose classical equations of motion can be used to reconstruct different bulk geometries: the expected classical one is accompanied by other “quantum” solutions. The classical regime of the sigma-model becomes reliable in the limit of large boundary spins, which coincides with the semi-classical limit of the boundary geometry. In a 1-loop approximation around the solutions to the classical equations of motion, we recover (with corrections due to the non-classical bulk geometries) results obtained in the past via perturbative quantum General Relativity and through the study of characters of the BMS3 group. The exposition is meant to be completely self-contained.