Search

We define solvable quantum mechanical systems on a Hilbert space spanned by bipartite ribbon graphs with a fixed number of edges. The Hilbert space is also an associative algebra, where the product is derived from permutation group products. The existence and structure of this Hilbert space algebra has a number of consequences. The algebra product, which can be expressed in terms of integer ribbon graph reconnection coefficients, is used to define solvable Hamiltonians with eigenvalues expressed in terms of normalized characters of symmetric group elements and degeneracies given in terms of Kronecker coefficients, which are tensor product multiplicities of symmetric group representations. The square of the Kronecker coefficient for a triple of Young diagrams is shown to be equal to the dimension of a sub-lattice in the lattice of ribbon graphs. This leads to an answer to the long-standing question of a combinatoric interpretation of the Kronecker coefficients. As an avenue to explore quantum supremacy and its implications for computational complexity theory, we outline experiments to detect non-vanishing Kronecker coefficients for hypothetical quantum realizations/simulations of these quantum systems. The correspondence between ribbon graphs and Belyi maps leads to an interpretation of these quantum mechanical systems in terms of quantum membrane world-volumes interpolating between string geometries., Version 1 - 49 pages; 12 pages of Appendices; 1 figure; revision V2 - added Lemma 1 which simplifies the proof of the main theorems; revision V3 - published version

In this paper, we analyze the Bollob\'as and Riordan polynomial $\mathcal{R}$ for ribbon graphs with half-ribbons introduced in [Combinatorics, Probability and Computing 31, 507-549, 2022]. We prove the universality property of a multivariate version of $\mathcal{R}$ whereas $\mathcal{R}$ itself turns out to be universal for a subclass of ribbon graphs with half-ribbons. We also show that $\mathcal{R}$ can be defined on some equivalence classes of ribbon graphs involving half-ribbons moves and that the new polynomial is universal on these classes., Comment: 21 pages, 15 figures

Field theories with combinatorial non-local interactions such as tensor invariants are interesting candidates for describing a phase transition from discrete quantum-gravitational to continuum geometry. In the so-called cyclic-melonic potential approximation of a tensorial field theory on the $r$-dimensional torus it was recently shown using functional renormalization group techniques that no such phase transition to a condensate phase with a tentative continuum geometric interpretation is possible. Here, keeping the same approximation, we show how to overcome this limitation amending the theory by local degrees freedom on $\mathbb{R}^d$. We find that the effective $r-1$ dimensions of the torus part dynamically vanish along the renormalization group flow while the $d$ local dimensions persist up to small momentum scales. Consequently, for $d>2$ one can find a phase structure allowing also for phase transitions., 25+4 pages, 5 figures

Enhanced tensor field theories (eTFT) have dominant graphs that do not correspond to melonic diagrams of ordinary tensor field theories. They therefore describe pertinent candidates to escape the so-called branched polymer phase, the universal geometry found for tensor models. For generic rank $d$ of the tensor field, we compute the perturbative $\beta$-functions at one-loop of two just-renormalizable quartic eTFT coined by $+$ or $\times$, depending on their vertex weights. The models $+$ has two quartic coupling constants $(\lambda, \lambda_{+})$, and two 2-point couplings (mass, $Z_a$). Meanwhile, the model $\times$ has two quartic coupling constants $(\lambda, \lambda_{\times})$ and three 2-point couplings (mass, $Z_a$, $Z_{2a}$). At all orders, both models have a constant wave function renormalization: $Z=1$ and therefore no anomalous dimension. Despite such peculiar behavior, both models acquire nontrivial radiative corrections for the coupling constants. The RG flow of the model $+$ exhibits neither asymptotic freedom nor the ordinary Landau ghost of $\phi^4_4$ model: $\lambda_{+}$ is a fixed point and $\lambda$ has linear behavior in time scale $t = \log(k/k_0)$. In the UV, the mass behaves likewise namely is linear in $t$, whereas $Z_a$ decreases exponentially towards a constant value. For the model $\times$, both $\lambda$ and $\lambda_{\times}$ do not flow, all remaining 2-point coupling constants are linear functions of the time scale in the UV., Comment: 45 pages, 20 figures

We investigate the universality classes of rank-4 colored bipartite U(1) tensor field models near the Gaussian fixed point with the functional renormalization group. In a truncation that contains all power counting relevant and marginal operators, we find a one-dimensional UV attractor that is connected with the Gaussian fixed point. Hence this is first evidence that the model could be asymptotically safe due to a mechanism similar to the one found in the Grosse-Wulkenhaar model, whose UV behavior near the Gaussian fixed point is also described by one-dimensional attractor that contains the Gaussian fixed point. However, the cancellation mechanism that is responsible for the simultaneous vanishing of the beta functions is new to tensor models, i.e. it does not occur in vector or matrix models., Latex, 4 pages + references

We introduce a tensorial group field theory endowed with weighted interaction terms of the form $p^{2a} ��^4$. The model can be seen as a field theory over $d=3,4$ copies of $U(1)$ where formal powers of Laplacian operators, namely $��^{a}$, $a>0$, act on tensorial $��^4$-interactions producing, after Fourier transform, $p^{2a}��^4$ 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, generally, the divergence degree is enhanced compared to the divergence degree of models without weighted vertices. The subleading terms in the partition function of the $��^4$ tensorial models become, in some cases, the dominant ones in the $p^{2a}��^4$ models. Finally, we explore sufficient conditions on the parameter $a$ yielding a list of potentially super-renormalizable $p^{2a}��^4$ models., 15 pages, 7 figures

We consider the parametric representation of the amplitudes of Abelian models in the so-called framework of rank $d$ Tensorial Group Field Theory. These models are called Abelian because their fields live on $U(1)^D$. We concentrate on the case when these models are endowed with particular kinetic terms involving a linear power in momenta. New dimensional regularization and renormalization schemes are introduced for particular models in this class: a rank 3 tensor model, an infinite tower of matrix models $��^{2n}$ over $U(1)$, and a matrix model over $U(1)^2$. For all divergent amplitudes, we identify a domain of meromorphicity in a strip determined by the real part of the group dimension $D$. From this point, the ordinary subtraction program is applied and leads to convergent and analytic renormalized integrals. Furthermore, we identify and study in depth the Symanzik polynomials provided by the parametric amplitudes of generic rank $d$ Abelian models. We find that these polynomials do not satisfy the ordinary Tutte's rules (contraction/deletion). By scrutinizing the "face"-structure of these polynomials, we find a generalized polynomial which turns out to be stable only under contraction., 69 pages, 35 figures

Classes of renormalizable models in the Tensorial Group Field Theory framework are investigated. The rank $d$ tensor fields are defined over $d$ copies of a group manifold $G_D=U(1)^D$ or $G_D= SU(2)^D$ with no symmetry and no gauge invariance assumed on the fields. In particular, we explore the space of renormalizable models endowed with a kinetic term corresponding to a sum of momenta of the form $p^{2a}$, $a\in ]0,1]$. This study is tailored for models equipped with Laplacian dynamics on $G_D$ (case $a=1$) but also for more exotic nonlocal models in quantum topology (case $0, Comment: 52 pages, 21 figures, 9 tables

Recently, a rank four tensor group field theory has been proved renormalizable. We provide here the key points on the renormalizability of this model and its UV asymptotic freedom., 7 pages, 3 Figures; Contribution to the XXIXth International Colloquium on Group-Theoretical Methods in Physics, Nankai, China, August 20-26, 2012

We identify the analogues of KMS state in topos theory. Topos KMS states can be viewed as classes of truth objects associated with a measure \mu^\rho (in one-to-one correspondence with an original KMS state \rho) which satisfies topos condition analogues to the ordinary KMS conditions. Topos KMS states can be defined both externally or internally according to whether the automorphism group of geometric morphisms acts externally on the topos structure, or the automorphism group is an object in the topos itself, in which case, it acts internally. We illustrate the formalism by a simple example on the Hilbert space $\Cl^3$.

A recent rank 4 tensor field model generating 4D simplicial manifolds has been proved to be renormalizable at all orders of perturbation theory [arXiv:1111.4997 [hep-th]]. The model is built out of $��^6$ ($��^6_{(1/2)}$), $��^4$ ($��^4_{(1)}$) interactions and an anomalous term ($��^4_{(2)}$). The $��$-functions of this model are evaluated at two and four loops. We find that the model is asymptotically free in the UV for both the main $��^6_{(1/2)}$ interactions whereas it is safe in the $��^4_{(1)}$ sector. The remaining anomalous term turns out to possess a Landau ghost., 31 pages, 31 figures; improved version

We prove that the rank 3 analogue of the tensor model defined in [arXiv:1111.4997 [hep-th]] is renormalizable at all orders of perturbation. The proof is given in the momentum space. The one-loop $��$- and $��$-functions of the model are also determined. We find that the model with a unique coupling constant for all interactions and a unique wave function renormalization is asymptotically free in the UV., 42 pages, 14 figures; improved version, some statements corrected, more comments

Enhanced quantization is an improved program for overcoming difficulties which may arise during an ordinary canonical quantization procedure. We review here how this program applies for a particle on circle., Comment: 6 pages; Contribution to the XXIXth International Colloquium on Group-Theoretical Methods in Physics, Nankai, China, August 20-26, 2012

This paper deals with an extension of our previous work [J. Phys. A: Math. Theor. {\bf 40} F817] by considering an alternative construction of canonical and deformed vector coherent states (VCSs) of the Gazeau-Klauder type associated with generalized spin-orbit Hamiltonians. We define an annihilation operator which takes into account the finite dimensional space of states induced by the $k$-photon transition processes of the two-level atom interacting with the single-mode radiation field. The class of nonlinear VCSs (NVCSs) corresponding to the action of the annihilation operator is deduced and expressed in terms of generalized displacement operators. Various NVCSs including their "dual" counterparts are also discussed. Still by using the Hilbert space structure, a new family of NVCSs parameterized by unit vectors of the $S^3$ sphere has been identified without making use of the annihilation operator., Typo corrected, references added

The $��$-functions of O(N) and U(N) invariant Grosse-Wulkenhaar models are computed at one loop using the matrix basis. In particular, for ``parallel interactions", the model is proved asymptotically free in the UV limit for $N >1$, and has a triviality problem or Landau ghost for $N, 14 pages, 6 Figures

We consider, in a superspace, new operator dependent noncommutative (NC) geometries of the nonlinear quantum Hall limit related to classes of f-deformed Landau operators in the spherical harmonic well. Different NC coordinate algebras are determined using unitary representation spaces of Fock-Heisenberg tensored algebras and of the Schwinger-Fock realisation of the su(1,1) Lie algebra. A reduced model allowing an underlying N=2 superalgebra is also discussed., Comment: Contribution to the Proceedings of the Fifth International Workshop on Contemporary Problems in Mathematical Physics, October 27 - November 2, 2007, Cotonou (Republic of Benin), 6 pages

Recently, a new type of renormalizable $��^{\star 4}_{4}$ scalar model on the Moyal space was proved to be perturbatively renormalizable. It is translation-invariant and introduces in the action a $a/(��^2p^2)$ term. We calculate here the $��$ and $��$ functions at one-loop level for this model. The coupling constant $��_��$ function is proved to have the same behaviour as the one of the $��^4$ model on the commutative $\mathbb{R}^4$. The $��_a$ function of the new parameter $a$ is also calculated. Some interpretation of these results are done., 13 pages, 3 figures