Bubble Divergences from Cellular Cohomology

Bonzom, Valentin; Smerlak, Matteo

doi:10.1007/s11005-010-0414-4

Cited by 71 publications

(128 citation statements)

References 28 publications

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“…Hence, reflects the orbit volume of this translational gauge symmetry. It is this symmetry that leads to divergences in the BF-theory partition functions for the "continuous" Lie groups [105,106]. 16 …”

Section: Topological Modelsmentioning

confidence: 99%

“…As a result, the amplitude within square brackets is the BF-theory amplitude associated to the topological manifold represented by G. It may be evaluated to obtain some G-dependent factor of (actually directly dependent on the second Betti number of G [233][234][235]), which allows the graphs to be organised into a 1/ -expansion. A more in-depth analysis has yet to be completed, although these preliminary results are very promising.…”

Section: Non-iid Modelsmentioning

confidence: 99%

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Spin Foam Models with Finite Groups

2013

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Spin foam models, loop quantum gravity, and group field theory are discussed as quantum gravity candidate theories and usually involve a continuous Lie group. We advocate here to consider quantum gravity-inspired models with finite groups, firstly as a test bed for the full theory and secondly as a class of new lattice theories possibly featuring an analogue diffeomorphism symmetry. To make these notes accessible to readers outside the quantum gravity community, we provide an introduction to some essential concepts in the loop quantum gravity, spin foam, and group field theory approach and point out the many connections to the lattice field theory and the condensed-matter systems.

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Section: Topological Modelsmentioning

confidence: 99%

Section: Non-iid Modelsmentioning

confidence: 99%

Spin Foam Models with Finite Groups

2013

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“…Its relation to gravity comes about when one changes to a field φ : SU(2) ×3 → C. The resulting SU(2) BF amplitude is known to arise as a quantization of a first order form of 3d gravity on the graph Γ. Furthermore, moving to a non-abelian (Lie) group, introduces many subtleties in placing the amplitude in a form similar to (22), but this has been successfully accomplished in a series of works [17].…”

Section: Corollary the Following Relation Holdsmentioning

confidence: 99%

“…Unsurprisingly, this is not the case here, since the amplitude captures the topology of the ambient 3d triangulation. In some sense the best way to solve the constraint is to reconstruct the 3d manifold and perform the analysis of [17]. Having said that, the power behind the reformulation is that the Boulatov model can now be expressed at the level of the action as a quantum field theory with support on the jackets.…”

Section: Jacket Field Theorymentioning

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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“…In fact, an impressive amount of new results has been obtained recently in this respect, in particular, for so-called ''colored'' group field theories [29]. These go from exact power counting results [30][31][32][33][34] to perturbative scaling bounds [35,36] and first steps in computing radiative corrections [37], from properties of the combinatorial structures generated [38][39][40] to the important proof that manifolds of spherical topology dominate in the limit of large cutoff (the analogue of the large N limit of matrix models) in any dimension, at least for topological models (not yet for 4d gravity models) [41][42][43]. Finally, the critical behavior of such models has been analyzed [44], including a study of Schwinger-Dyson equations in this regime.…”

Section: Introductionmentioning

confidence: 99%

Bounding bubbles: The vertex representation of 3d group field theory and the suppression of pseudomanifolds

Carrozza

Oriti

2012

Phys. Rev. D

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Based on recent work on simplicial diffeomorphisms in colored group field theories, we develop a representation of the colored Boulatov model, in which the group field theory (GFT) fields depend on variables associated to vertices of the associated simplicial complex, as opposed to edges. On top of simplifying the action of diffeomorphisms, the main advantage of this representation is that the GFT Feynman graphs have a different stranded structure, which allows a direct identification of subgraphs associated to bubbles, and their evaluation is simplified drastically. As a first important application of this formulation, we derive new scaling bounds for the regularized amplitudes, organized in terms of the genera of the bubbles, and show how the pseudomanifold configurations appearing in the perturbative expansion are suppressed as compared to manifolds. Moreover, these bounds are proved to be optimal.

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Bubble Divergences from Cellular Cohomology

Cited by 71 publications

References 28 publications

Spin Foam Models with Finite Groups

Spin Foam Models with Finite Groups

Tensor models and embedded Riemann surfaces

Bounding bubbles: The vertex representation of 3d group field theory and the suppression of pseudomanifolds

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