Real or complex tensor model observables, the backbone of the tensor theory space, are classical (unitary, orthogonal, symplectic) Lie group invariants. These observables represent as colored graphs, and that representation gives an handle to study their combinatorial, topological and algebraic properties. We give here an overview of the symmetric group-theoretic formulation of the enumeration of unitary and orthogonal invariant observables which turns out to bear a rich structure. From their counting formulae, one finds a correspondence with topological field theory on 2-cellular complexes that brings other interpretations of the same countings. Furthermore, tensor model observables span an algebra that turns out to be semi-simple. Dealing with complex tensors, we discuss the representation theoretic base of the algebra making explicit its Wedderburn-Artin decomposition. The real case is more subtle as a base of its Wedderburn-Artin decomposition is yet unknown.
In this paper, we use the exact renormalisation in the context of tensor models and tensorial group field theories. As a byproduct, we rederive Gaussian universality for random tensors and provide a general power counting for Abelian tensorial field theories with a closure constraint, leading us to a only five renormalizable theories.
We introduce a tensorial group field theory endowed with weighted interaction terms of the form $p^{2a} ϕ^4$. The model can be seen as a field theory over $d=3,4$ copies of $U(1)$ where formal powers of Laplacian operators, namely $Δ^{a}$, $a>0$, act on tensorial $ϕ^4$-interactions producing, after Fourier transform, $p^{2a}ϕ^4$ interactions. Using multi-scale analysis, we provide a power counting theorem for this type of models. A new quantity depending on the incidence matrix between vertices and faces of Feynman graphs is invoked in the degree of divergence of amplitudes. As a result, gener…
$O(N)$ invariants are the observables of real tensor models. We use regular colored graphs to represent these invariants, the valence of the vertices of the graphs relates to the tensor rank. We enumerate $O(N)$ invariants as $d$-regular graphs, using permutation group techniques. We also list their generating functions and give (software) algorithms computing their number at an arbitrary rank and an arbitrary number of vertices. As an interesting property, we reveal that the algebraic structure which organizes these invariants differs from that of the unitary invariants. The underlying topolo…
Representation theory provides a suitable framework to count and classify invariants in tensor models. We show that there are two natural ways of counting invariants, one for arbitrary rank of the gauge group and a second, which is only valid for large N. We construct bases of invariant operators based on the counting, and compute correlators of their elements. The basis associated with finite N diagonalizes the two-point function of the theory and it is analogous to the restricted Schur basis used in matrix models. We comment on future lines of investigation.
The tensor rank decomposition is a useful tool for the geometric interpretation of the tensors in the canonical tensor model (CTM) of quantum gravity. In order to understand the stability of this interpretation, it is important to be able to estimate how many tensor rank decompositions can approximate a given tensor. More precisely, finding an approximate symmetric tensor rank decomposition of a symmetric tensor $Q$ with an error allowance $Δ$ is to find vectors $ϕ^i$ satisfying $\|Q-\sum_{i=1}^R ϕ^i\otimes ϕ^i\cdots \otimes ϕ^i\|^2 \leq Δ$. The volume of all possible such $ϕ^i$ is an interest…
We introduce simple group-theoretic techniques for classifying conformally-invariant tensor-structures. With them, we classify tensor structures of general n-point functions of non-conserved operators, and $n\geq 4$-point functions of general conserved currents, with or without permutation symmetries, and in any spacetime dimension d. (The case n=3 for conserved operators will appear in subsequent work.) Our techniques are useful for bootstrap applications. The rules we derive simultaneously count tensor structures for flat-space scattering amplitudes in d+1 dimensions.
The melonic sector has been proven to be dominant in tensor models at large N. This is true as long as the observables we consider, composites of 2n tensors, are small. That is, if n is much smaller than N. In this paper, I argue that, in order to recover geometries (and then gravity) in the continuum limit, n must grow like N. In that case, I provide examples where non-melonic contributions overcome the total sum in the computation of the expectation value of certain observables.
Tensor models are generalizations of matrix models and as such, it is a natural question to ask whether they satisfy some form of the topological recursion. The world of unitary-invariant observables is however much richer in tensor models than in matrix models. It is therefore a priori unclear which set of observables could satisfy the topological recursion. Such a set of observables was identified a few years ago in the context of the quartic melonic model by the first author and Dartois. It was shown to satisfy an extension of the topological recursion introduced by Borot and called the blo…
We investigate the gauge-invariant observables constructed by smearing the graviton and inflaton fields by compactly supported tensors at linear order in general single-field inflation. These observables correspond to gauge-invariant quantities that can be measured locally. In particular, we show that these observables are equivalent to (smeared) local gauge-invariant observables such as the linearised Weyl tensor, which have better infrared properties than the graviton and inflaton fields. Special cases include the equivalence between the compactly supported gauge-invariant graviton observabl…
In the recently proposed non-local theory of quantum gravity one can avoid massive tensor ghosts at the tree level by a special choice of the non-local form factor between the two Ricci tensors. We show that at the quantum level this theory has an infinite amount of massive unphysical states, mostly corresponding to complex poles.
Hamiltonian Expression of Curvature Tensors in the York Canonical Basis: II) The Weyl Tensor, Weyl Scalars, the Weyl Eigenvalues and the Problem of the Observables of the Gravitational Field
We study invariant operators in general tensor models. We show that representation theory provides an efficient framework to count and classify invariants in tensor models. In continuation and completion of our earlier work, we present two natural ways of counting invariants, one for arbitrary rank of the group and another valid for large rank. We construct basis of invariant operators based on the counting, and compute correlators of their elements. The basis associated with finite rank diagonalizes two-point function. It is analogous to the restricted Schur basis used in matrix models. We sh…
We show that the counting of observables and correlators for a 3-index tensor model are organized by the structure of a family of permutation centralizer algebras. These algebras are shown to be semi-simple and their Wedderburn-Artin decompositions into matrix blocks are given in terms of Clebsch-Gordan coefficients of symmetric groups. The matrix basis for the algebras also gives an orthogonal basis for the tensor observables which diagonalizes the Gaussian two-point functions. The centres of the algebras are associated with correlators which are expressible in terms of Kronecker coefficients…
The simple $m^2ϕ^2$ potential as an inflationary model is coming under increasing tension with limits on the tensor-to-scalar ratio $r$ and measurements of the scalar spectral index $n_s$. Cubic Galileon interactions in the context of the Horndeski action can potentially reconcile the observables. However, we show that this cannot be achieved with only a constant Galileon mass scale because the interactions turn off too slowly, leading also to gradient instabilities after inflation ends. Allowing for a more rapid transition can reconcile the observables but moderately breaks the slow-roll appr…
In the context of scalar-tensor theories of gravity we compute the third-order corrected spectral indices in the slow-roll approximation. The calculation is carried out by employing the Green's function method for scalar and tensor perturbations in both the Einstein and Jordan frames. Then, using the interrelations between the Hubble slow-roll parameters in the two frames we find that the frames are equivalent up to third order. Since the Hubble slow-roll parameters are related to the potential slow-roll parameters, we express the observables in terms of the latter which are manifestly invaria…
Permutation group algebras, and their generalizations called permutation centralizer algebras (PCAs), play a central role as hidden symmetries in the combinatorics of large $N$ gauge theories and matrix models with manifest continuous gauge symmetries. Polynomial functions invariant under the manifest symmetries are the observables of interest and have applications in AdS/CFT. We compute such correlators in the presence of matrix or tensor witnesses, which by definition, can include a matrix or tensor field appearing as a coupling in the action (i.e a spurion) or as a classical (un-integrated)…
Observables and Unconstrained Spin Tensor Dynamics in General Relativity from Scattering Amplitudes
Pennsylvania State University（宾夕法尼亚州立大学）；University of California at Los Angeles（加州大学洛杉矶分校）；Université de Genève（日内瓦大学）；University of Copenhagen（哥本哈根大学）；Max Planck Institute for Gravitational Physics (Albert Einstein Institute)（马克斯·普朗克引力物理研究所（爱因斯坦研究所））；Fudan University（复旦大学）
We construct the general permutation invariant Gaussian 2-matrix model for matrices of arbitrary size $D$. The parameters of the model are given in terms of variables defined using the representation theory of the symmetric group $S_D$. A correspondence is established between the permutation invariant polynomial functions of the matrix variables (the observables of the model) and directed colored graphs, which sheds light on stability properties in the large $D$ counting of these invariants. The refined counting of the graphs is given in terms of double cosets involving permutation groups defi…
The counting of the dimension of the space of $U(N) \times U(N) \times U(N)$ polynomial invariants of a complex $3$-index tensor as a function of degree $n$ is known in terms of a sum of squares of Kronecker coefficients. For $n \le N$, the formula can be expressed in terms of a sum of symmetry factors of partitions of $n$ denoted $Z_3(n)$. We derive the large $n$ all-orders asymptotic formula for $ Z_3(n)$ making contact with high order results previously obtained numerically. The derivation relies on the dominance in the sum, of partitions with many parts of length $1$. The dominance of othe…
The leading order dynamics of the type IIB Large Volume Scenario is characterised by the interplay between $α'$ and non-perturbative effects which fix the overall volume and all local blow-up modes leaving (in general) several flat directions. In this paper we show that, in an arbitrary Calabi-Yau with at least one blow-up mode resolving a point-like singularity, any remaining flat directions can be lifted at subleading order by the inclusions of higher derivative $α'$ corrections. We then focus on simple fibred cases with one remaining flat direction which can behave as an inflaton if its pot…
Using $δN$ formalism, in the context of a generic multi-field inflation driven on a non-flat field space background, we revisit the analytic expressions of the various cosmological observables such as scalar/tensor power spectra, scalar/tensor spectral tilts, non-Gaussianity parameters, tensor-to-scalar ratio, and the various runnings of these observables. In our backward formalism approach, the subsequent expressions of observables automatically include terms beyond the leading order slow-roll expansion correcting many of the expression at subleading order. To connect our analysis properly wi…
