Spin Foam Models with Finite Groups

Bähr, Benjamin; Dittrich, Bianca; Ryan, James P.

doi:10.1155/2013/549824

Cited by 41 publications

(81 citation statements)

References 257 publications

(539 reference statements)

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“…To finish, just as there are three jackets for each graph Γ, there are three distinct redefinitions of the form (15) at the level of the action, each one generating a different jacket.…”

Section: Jacketsmentioning

confidence: 99%

Tensor models and embedded Riemann surfaces

Ryan

2012

Phys. Rev. D

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Tensor models and, more generally, group field theories are candidates for higher-dimensional quantum gravity, just as matrix models are in the 2d setting. With the recent advent of a 1/Nexpansion for coloured tensor models, more focus has been given to the study of the topological aspects of their Feynman graphs. Crucial to the aforementioned analysis were certain subgraphs known as bubbles and jackets. We demonstrate in the 3d case that these graphs are generated by matrix models embedded inside the tensor theory. Moreover, we show that the jacket graphs represent (Heegaard) splitting surfaces for the triangulation dual to the Feynman graph. With this in hand, we are able to re-express the Boulatov model as a quantum field theory on these Riemann surfaces.Group field theories [1] and tensor models are higher dimensional analogues of matrix models [2]. Matrix integrals have been shown to provide a natural framework within which to frame a multitude of physical and mathematical questions ranging through the fields of statistical and condensed matter physics all the way to the more abstract enumeration of virtual knots and tangles. This highlights how apparently disparate physical and mathematical phenomena in fact share certain universal features.One particular facet that sparked considerable interest was the realization that matrix models could give a non-perturbative definition of 2d quantum gravity [3]. One considers a statistical ensemble of N ×N (oftentimes hermitian) matrices. The Feynman graphs arising in the perturbative expansion of the free energy describe discrete Riemann surfaces. Remarkably, the expansion can be ordered in powers of 1/N labelled by their topological invariant, the Euler characteristic. In the large-N limit, the 2-sphere dominates and moreover, one can tune the coupling constant so that in a double scaling limit one describes a continuum theory of 2d quantum gravity.Tensor models hope to reproduce the same successes that matrix models have enjoyed, with the ultimate aim of being viable candidates for quantum theories of gravity in higher dimensions. A stumbling block seemed to be that they generated a plethora of unwanted structures; not only simplicial manifolds, but simplicial pseudomanifolds [4]. Recently, after much work [5,11], a promising step has been made in that direction with the construction of a 1/N -topological expansion [6][7][8] for the so-called coloured tensor models [9,10]. In that context, it was shown that for arbitrary dimension d, only graphs corresponding to d-spheres arise at leading order in the 1/N -expansion. Central to this construction were the ribbon graphs associated to the Feynman graphs of the tensor model. These ribbon graphs are algebraic objects that capture the topological properties of the Feynman graphs. They contain two classes of subgraphs, known as bubbles [9] and jackets [11], which are of particular significance. While bubbles are easily identified as Riemann surfaces embedded in the dual triangulation, the topological properties of the jac...

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“…To finish, just as there are three jackets for each graph Γ, there are three distinct redefinitions of the form (15) at the level of the action, each one generating a different jacket.…”

Section: Jacketsmentioning

confidence: 99%

Tensor models and embedded Riemann surfaces

Ryan

2012

Phys. Rev. D

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“…This fact make it hard to control large scales in a spin-network basis. For this reason, the fusion basis is ideally suited for coarse-graining in lattice gauge theories and loop quantum gravity [12,13,15,[64][65][66][67][68][69][70][71][72][73].…”

Section: Discussionmentioning

confidence: 99%

“…Indeed, one can understand the q-deformation at root of unity characteristic of the TV representation, as in a certain sense turning SU(2) into a finite (quantum) group SU(2) q . Spin-foam models with finite groups are used to study the behaviour of spin-foams under coarse graining [13,[66][67][68][69], and we hope that the techniques developed here will be useful also in this context. We denote general elements of G by G, H, g, u, .…”

Section: Jhep02(2017)061mentioning

confidence: 99%

“…The fusion basis can also be useful in covariant (space-time) approaches to renormalization and coarse graining of lattice gauge theories and spin-foams [13,[66][67][68][69][70][71]. Here, the partition function associated to a space-time building block can be represented by a state on the boundary of this block [62,72,73].…”

Section: Jhep02(2017)061mentioning

confidence: 99%

“…One of our principal aims is studying the continuum limit and coarse-graining of spin-foam models [9,13,[66][67][68][69][70][71].…”

Section: Jhep02(2017)061mentioning

confidence: 99%

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Fusion basis for lattice gauge theory and loop quantum gravity

Delcamp

Dittrich

Riello

2017

J. High Energ. Phys.

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133

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We 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.

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