From scalar fields on quantum spaces to blobbed topological recursion

Branahl, Johannes; Grosse, Harald; Hock, Alexander; Wulkenhaar, Raimar

doi:10.48550/arxiv.2110.11789

Cited by 5 publications

(7 citation statements)

References 37 publications

(50 reference statements)

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“…We refer to [BGHW21] for a more detailed introduction to the model, the progress achieved in recent years and the relation of the QKM to other models like the Kontsevich model [Kon92] and the Hermitian 1-and 2-matrix model [Eyn16,CEO06]. As already mentioned, this paper extends some of the results obtained in [BHW21] and hence will refer to and cite from the latter on multiple occasions.…”

Section: Introductionmentioning

confidence: 61%

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Two Approaches For a Perturbative Expansion in Blobbed Topological Recursion

Lindner¹

2022

Preprint

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In this paper we continue the perturbative analysis of the quartic Kontsevich model. We investigate meromorphic functions Ω (0) m with m = 1, 2, that obey blobbed topological recursion, by calculating their ribbon graph expansions with a Mathematica program using two complementary methods. In doing so, we determine all vacuum ribbon graphs that contribute to the free energy with genus g = 0 up to fourth order. This is a first step towards complete automation of the calculation of graph expansions in the planar sector.

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Section: Introductionmentioning

confidence: 61%

“…a result, which is mentioned in [BGHW21] and based on investigations by Bernardi and Fusy [BF17]. Here l h := 1 2 i n i is half the sum of all boundary lengths n i and #n e := 3l h + 2b + 2n − 4 is the total number of edges.…”

Section: Counting Graphsmentioning

confidence: 94%

Two Approaches For a Perturbative Expansion in Blobbed Topological Recursion

Lindner¹

2022

Preprint

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“…The original aim of the quartic Kontsevich model was to construct an exactly solvable quantum field theory on non-commutative spaces, where the property of exact solvability might be explained by the appearance of topological recursion. We will not go into the details of the background (and refer the interested reader to a new overview of the accomplishments in the last two decades, to find in [BGHW21]) and only take concepts into account that will be necessary to understand this paper. Consider an integral over self-adjoint N × N -matrices living in…”

Section: The Quartic Kontsevich Modelmentioning

confidence: 99%

Genus one free energy contribution to the quartic Kontsevich model

Branahl¹,

Hock²

2021

Preprint

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We prove a formula for the genus one free energy F (1) of the quartic Kontsevich model for arbitrary ramification by working out a boundary creation operator for blobbed topological recursion. We thus investigate the differences in F (1) compared with its generic representation for ordinary topological recursion. In particular, we clarify the role of the Bergman τ -function in blobbed topological recursion. As a by-product, we show that considering the holomorphic additions contributing to ω g,1 or not gives a distinction between the enumeration of bipartite and non-bipartite quadrangulations of a genus-g surface.

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“…Initially, the tensor track is an attempt to quantize gravity in dimension D > 2 by combining random tensor models to discrete geometry and the renormalisation group [8]. The tensor track lies at the crossroad of several closely related approaches to quantize gravity, most notably causal dynamical triangulations [9], quantum field theory on non-noncommutative spaces [10][11][12], and group field theory [13].…”

Section: Introductionmentioning

confidence: 99%

The Tensor Track VII: From Quantum Gravity to Artificial Intelligence

Ouerfelli¹,

Rivasseau²,

Tamaazousti³

2022

Preprint

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Assuming some familiarity with quantum field theory and with the tensor track approach that one of us presented in the previous series "Tensor Track I-VI", we provide, as usual, the developments in quantum gravity of the last two years. Next we present in some detail two algorithms inspired by Random Tensor Theory which has been developed in the quantum gravity context. One is devoted to the detection and recovery of of a signal in a random tensor (that can be associated to the noise) with new theoretical guarantees for more general cases such as tensors with different dimensions. The other, SMPI, is more ambitious but maybe less rigorous; it is devoted to significantly and fundamentally improve the performance of algorithms for Tensor principal component analysis but without complete theoretical guarantees yet. Then we sketch all sorts of application relevant to information theory and artificial intelligence and provide their corresponding bibliography.

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From scalar fields on quantum spaces to blobbed topological recursion

Abstract: We review the construction of the λφ 4 -model on noncommutative geometries via exact solutions of Dyson-Schwinger equations and explain how this construction relates via (blobbed) topological recursion to problems in algebraic and enumerative geometry.

Cited by 5 publications

References 37 publications

Two Approaches For a Perturbative Expansion in Blobbed Topological Recursion

Two Approaches For a Perturbative Expansion in Blobbed Topological Recursion

Genus one free energy contribution to the quartic Kontsevich model

The Tensor Track VII: From Quantum Gravity to Artificial Intelligence

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