Feynman diagrams, ribbon graphs, and topological recursion of Eynard-Orantin

Gopalakrishna, K.; Labelle, Patrick; Shramchenko, Vasilisa

doi:10.1007/jhep06(2018)162

Cited by 7 publications

(10 citation statements)

References 30 publications

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“…In particular, it would be interesting to study whether our work bears some relation to the generalized catalan numbers used in Ref. [17] in the context of the Eynard-Orantin topological recursion, which is another method for enumeration of Feynman diagrams. In this appendix, we write explicitly the terms of ( 46) and (72) that contribute to the asymptotic expansion until a = 6, for the case in which N = 1.…”

Section: Discussion and Perspectivesmentioning

confidence: 99%

“…[11] to get an exact formula and find an equivalence with the Arqués-Béraud formula for one-rooted maps (i.e., objects in algebraic topology) [15]. Particularly, equivalences between the counting of N -rooted maps and connected Feynman diagrams with 2N external legs have been established by means of a directed bijection between these two types of object [16] [17]. Exact formulas related to this algebraic curve topological theory have also been obtained, which, can also be used to count Feynman diagrams.…”

Section: Introductionmentioning

confidence: 99%

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A recursive enumeration of connected Feynman diagrams with an arbitrary number of external legs in the fermionic non-relativistic interacting gas

Castro¹,

Roditi²

2019

J. Phys. A: Math. Theor.

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In this work, we generalize a recursive enumerative formula for connected Feynman diagrams with two external legs. The Feynman diagrams are defined from a fermionic gas with a two-body interaction. The generalized recurrence is valid for connected Feynman diagrams with an arbitrary number of external legs and an arbitrary order. The recurrence formula terms are expressed in function of weak compositions of non-negative integers and partitions of positive integers in such a way that to each term of the recurrence correspond a partition and a weak composition. The foundation of this enumeration is the Wick theorem, permitting an easy generalization to any quantum field theory. The iterative enumeration is constructive and enables a fast computation of the number of connected Feynman diagrams for a large amount of cases. In particular, the recurrence is solved exactly for two and four external legs, leading to the asymptotic expansion of the number of different connected Feynman diagrams. FIG. 2:A rooted map embedded in the tori and the corresponding Feynman diagram. See all the correspondences for order m = 1, 2, 3 in ref. [15]

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Section: Discussion and Perspectivesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A recursive enumeration of connected Feynman diagrams with an arbitrary number of external legs in the fermionic non-relativistic interacting gas

Castro¹,

Roditi²

2019

J. Phys. A: Math. Theor.

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“…A novel graphical bijection between the Feynman and ribbon diagrams can be drawn. A comprehensive review on connections to other types of Feynman diagrams can be found in [38] and recent related works in [39,40]. The tree level diagram corresponds to the one root ribbon graph with zero edges which is a degenerate case.…”

Section: Relation To Rooted Ribbon Graphsmentioning

confidence: 99%

Double-logarithms in $$ \mathcal{N} $$ = 8 supergravity: impact parameter description & mapping to 1-rooted ribbon graphs

Vera

2019

J. High Energ. Phys.

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The set of double-logarithmic (DL) contributions (α t ln 2 s) n to the 4-graviton amplitude in N = 8 supergravity (SUGRA), with α being the gravitational coupling and (s, t) the Mandelstam invariants, is studied in impact parameter (ρ) representation. This sector of the amplitude shows interesting properties which shed light on the nature of quantum corrections in gravity. Besides having a convergent behaviour as s increases, which is not present in N < 4 SUGRA theories, there exists a critical line ρ c (s) above which the Born amplitude prevails. The short distance region ρ < ρ c (s) is dominated by the DL terms. As a consequence, when studied in terms of an eikonal approach in the forward limit, the scattering angle linked to the bending of the semiclassical trajectory of the graviton shows a transition from attractive gravity at large distances to a region at small ρ characterized by a repulsive DL contribution to the gravitational potential due to the gravitino content of the theory. In the complex angular momentum plane, this DL high energy asymptotics is driven by the rightmost pole singularity of a parabolic cylinder function. The resummation of DL quantum corrections in N = 8 SUGRA can be understood in terms of the counting of 1-rooted maps on orientable surfaces.

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“…, where the notations P and λ (m) are the same as in §2.3. This simply tells us that the abstract correlators F µ g are realized by the correlators of the Hermitian onematrix models if we assign Feynman rule (39). Therefore, the Hermitian one-matrix model is a refinement of the enumeration of fat graphs (see Example 3.1) where the degree of t encoded the number of faces in each graph.…”

Section: Hermitian One-matrix Model As a Realization Of The Abstract Qftmentioning

confidence: 99%

“…There is another point of view of the above realization. Instead of considering the realization of the abstract correlators by the Feynman rule (39), we may consider the realization of the abstract free energy and abstract partition function by the following Feynman rule:…”

Section: Hermitian One-matrix Model As a Realization Of The Abstract Qftmentioning

confidence: 99%

A formalism of abstract quantum field theory of summation of fat graphs

Wang,

Zhou

2021

Preprint

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In this work we present a formalism of abstract quantum field theory for fat graphs and its realizations. This is a generalization of an earlier work for stable graphs. We define the abstract correlators F µ g , abstract free energy Fg, abstract partition function Z, and abstract n-point functions Wg,n to be formal summations of fat graphs, and derive quadratic recursions using edge-contraction/vertex-splitting operators, including the abstract Virasoro constraints, an abstract cut-and-join type representation for Z, and a quadratic recursion for Wg,n which resembles the Eynard-Orantin topological recursion. When considering the realization by the Hermitian one-matrix models, we obtain the Virasoro constraints, a cut-and-join representation for the partition function Z Herm N which proves that Z Herm N is a tau-function of KP hierarchy, a recursion for n-point functions which is known to be equivalent to the E-O recursion, and a Schrödinger type-equation which is equivalent to the quantum spectral curve. We conjecture that in general cases the realization of the quadratic recursion for Wg,n is the E-O recursion, where the spectral curve and Bergmann kernel are constructed from realizations of W 0,1 and W 0,2 respectively using the framework of emergent geometry. Contents1. Introduction 1.1. Backgrounds 1.2. The formalism of abstract quantum field theories and their realizations 1.3. Description of main results 1.4. Plan of the paper 2. Abstract Quantum Field Theory for Fat Graphs 2.1. Graphs on oriented surfaces 2.2.

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Feynman diagrams, ribbon graphs, and topological recursion of Eynard-Orantin

Cited by 7 publications

References 30 publications

A recursive enumeration of connected Feynman diagrams with an arbitrary number of external legs in the fermionic non-relativistic interacting gas

A recursive enumeration of connected Feynman diagrams with an arbitrary number of external legs in the fermionic non-relativistic interacting gas

Double-logarithms in $$ \mathcal{N} $$ = 8 supergravity: impact parameter description & mapping to 1-rooted ribbon graphs

A formalism of abstract quantum field theory of summation of fat graphs

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