Quantum product, topological recursion relations, and the Virasoro conjecture

2019 ◽  
Author(s):  
Xiaobo Liu
Keyword(s):  
Recursion Relations ◽  
Topological Recursion ◽  
Virasoro Conjecture

scholarly journals Super Quantum Airy Structures

2020 ◽  
Vol 380 (1) ◽  
pp. 449-522
Author(s):  
Vincent Bouchard ◽  
Paweł Ciosmak ◽  
Leszek Hadasz ◽  
Kento Osuga ◽  
Błażej Ruba ◽  
...  
Keyword(s):  
Classification Scheme ◽  
Graphical Representation ◽  
Free Energies ◽  
Recursion Relations ◽  
Gauge Transformations ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Frobenius Algebras ◽  
Topological Recursion

Abstract We introduce super quantum Airy structures, which provide a supersymmetric generalization of quantum Airy structures. We prove that to a given super quantum Airy structure one can assign a unique set of free energies, which satisfy a supersymmetric generalization of the topological recursion. We reveal and discuss various properties of these supersymmetric structures, in particular their gauge transformations, classical limit, peculiar role of fermionic variables, and graphical representation of recursion relations. Furthermore, we present various examples of super quantum Airy structures, both finite-dimensional—which include well known superalgebras and super Frobenius algebras, and whose classification scheme we also discuss—as well as infinite-dimensional, that arise in the realm of vertex operator super algebras.

Topological recursion relations on $$\overline M _{3,2} $$

2015 ◽  
Vol 58 (9) ◽  
pp. 1909-1922 ◽  
Author(s):  
Takashi Kimura ◽  
Xiaobo Liu
Keyword(s):  
Recursion Relations ◽  
Topological Recursion

scholarly journals New topological recursion relations

2010 ◽  
Vol 20 (3) ◽  
pp. 479-494 ◽  
Author(s):  
Xiaobo Liu ◽  
Rahul Pandharipande
Keyword(s):  
Recursion Relations ◽  
Topological Recursion

scholarly journals Open topological recursion relations in genus 1 and integrable systems

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Oscar Brauer Gomez ◽  
Alexandr Buryak
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Explicit Formula ◽  
Integrable Systems ◽  
Spatial Variable ◽  
Partial Differential ◽  
Recursion Relations ◽  
Topological Recursion

Abstract The paper is devoted to the open topological recursion relations in genus 1, which are partial differential equations that conjecturally control open Gromov-Witten invariants in genus 1. We find an explicit formula for any solution analogous to the Dijkgraaf-Witten formula for a descendent Gromov-Witten potential in genus 1. We then prove that at the approximation up to genus 1 the exponent of an open descendent potential satisfies a system of explicitly constructed linear evolutionary PDEs with one spatial variable.

Runtime Aggregation of Recursion Relations

1989 ◽  
Author(s):  
Harry Berryman ◽  
Joel Salz
Keyword(s):  
Recursion Relations

scholarly journals Shell-crossing in a ΛCDM Universe

2020 ◽  
Vol 501 (1) ◽  
pp. L71-L75
Author(s):  
Cornelius Rampf ◽  
Oliver Hahn
Keyword(s):  
Dark Matter ◽  
Perturbation Theory ◽  
Large Scale ◽  
Cold Dark Matter ◽  
Three Dimensional ◽  
Random Initial Data ◽  
Recursion Relations ◽  
Indispensable Tool ◽  
First Time ◽  
Very High

ABSTRACT Perturbation theory is an indispensable tool for studying the cosmic large-scale structure, and establishing its limits is therefore of utmost importance. One crucial limitation of perturbation theory is shell-crossing, which is the instance when cold-dark-matter trajectories intersect for the first time. We investigate Lagrangian perturbation theory (LPT) at very high orders in the vicinity of the first shell-crossing for random initial data in a realistic three-dimensional Universe. For this, we have numerically implemented the all-order recursion relations for the matter trajectories, from which the convergence of the LPT series at shell-crossing is established. Convergence studies performed at large orders reveal the nature of the convergence-limiting singularities. These singularities are not the well-known density singularities at shell-crossing but occur at later times when LPT already ceased to provide physically meaningful results.

scholarly journals New ideas for handling of loop and angular integrals in D-dimensions in QCD

2021 ◽  
Vol 2021 (6) ◽  
Author(s):  
Valery E. Lyubovitskij ◽  
Fabian Wunder ◽  
Alexey S. Zhevlakov
Keyword(s):  
Computer Algebra ◽  
Cross Sections ◽  
Orthogonal Basis ◽  
Computer Algebra Systems ◽  
Covariant Formalism ◽  
Linear Combinations ◽  
Recursion Relations ◽  
New Ideas ◽  
Rotational Invariant ◽  
Geometric Picture

Abstract We discuss new ideas for consideration of loop diagrams and angular integrals in D-dimensions in QCD. In case of loop diagrams, we propose the covariant formalism of expansion of tensorial loop integrals into the orthogonal basis of linear combinations of external momenta. It gives a very simple representation for the final results and is more convenient for calculations on computer algebra systems. In case of angular integrals we demonstrate how to simplify the integration of differential cross sections over polar angles. Also we derive the recursion relations, which allow to reduce all occurring angular integrals to a short set of basic scalar integrals. All order ε-expansion is given for all angular integrals with up to two denominators based on the expansion of the basic integrals and using recursion relations. A geometric picture for partial fractioning is developed which provides a new rotational invariant algorithm to reduce the number of denominators.

scholarly journals Open associahedra and scattering forms

2020 ◽  
Vol 2020 (12) ◽  
Author(s):  
Aidan Herderschee ◽  
Fei Teng
Keyword(s):  
Scattering Amplitudes ◽  
Canonical Form ◽  
Recursion Relations ◽  
Fiber Product ◽  
New Phenomena

Abstract We continue the study of open associahedra associated with bi-color scattering amplitudes initiated in ref. [1]. We focus on the facet geometries of the open associahedra, uncovering many new phenomena such as fiber-product geometries. We then provide novel recursion procedures for calculating the canonical form of open associahedra, generalizing recursion relations for bounded polytopes to unbounded polytopes.

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