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Alexander Hock

Publications and source records attributed to Alexander Hock.

28 records · Page 2Linked to original sources
arXiv2022

Blobbed topological recursion of the quartic Kontsevich model I: Loop equations and conjectures

We provide strong evidence for the conjecture that the analogue of Kontsevich's matrix Airy function, with the cubic potential $\mathrm{Tr}(Φ^3)$ replaced by a quartic term $\mathrm{Tr}(Φ^4)$, obeys the blobbed topological recursion of Borot and Shadrin. We identify in the quartic Kontsevich model three families of correlation functions for which we establish interwoven loop equations. One family consists of symmetric meromorphic differential forms $ω_{g,n}$ labelled by genus and number of marked points of a complex curve. We reduce the solution of all loop equations to a straightforward but lengthy evaluation of residues. In all evaluated cases, the $ω_{g,n}$ consist of a part with poles at ramification points which satisfies the universal formula of topological recursion, and of a part holomorphic at ramification points for which we provide an explicit residue formula.

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arXiv2022

On the $x$-$y$ Symmetry of Correlators in Topological Recursion via Loop Insertion Operator

Topological Recursion generates a family of symmetric differential forms (correlators) from some initial data $(Σ,x,y,B)$. We give a functional relation between the correlators of genus $g=0$ generated by the initial data $(Σ,x,y,B)$ and by the initial data $(Σ,y,x,B)$, where $x$ and $y$ are interchanged. The functional relation is derived with the loop insertion operator by computing a functional relation for some intermediate correlators. Additionally, we show that our result is equivalent to the recent result of \cite{Borot:2021thu} in case of $g=0$. Consequently, we are providing a simplified functional relation between generating series of higher order free cumulants and moments in higher order free probability.

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arXiv2021

Genus one free energy contribution to the quartic Kontsevich model

We prove a formula for the genus one free energy $\mathcal{F}^{(1)}$ 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 $\mathcal{F}^{(1)}$ 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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arXiv2021

Nested Catalan tables and a recurrence relation in noncommutative quantum field theory

Correlation functions in a dynamic quartic matrix model are obtained from the two-point function through a recurrence relation. This paper gives the explicit solution of the recurrence by mapping it bijectively to a two-fold nested combinatorial structure each counted by Catalan numbers. These `nested Catalan tables' have a description as diagrams of non-crossing chords and threads.

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arXiv2021

Perturbative and Geometric Analysis of the Quartic Kontsevich Model

The analogue of Kontsevich's matrix Airy function, with the cubic potential $\operatorname{Tr}\big(Φ^3\big)$ replaced by a quartic term $\operatorname{Tr}\big(Φ^4\big)$ with the same covariance, provides a toy model for quantum field theory in which all correlation functions can be computed exactly and explicitly. In this paper we show that distinguished polynomials of correlation functions, themselves given by quickly growing series of Feynman ribbon graphs, sum up to much simpler and highly structured expressions. These expressions are deeply connected with meromorphic forms conjectured to obey blobbed topological recursion. Moreover, we show how the exact solutions permit to explore critical phenomena in the quartic Kontsevich model.

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arXiv2021

Blobbed topological recursion of the quartic Kontsevich model II: Genus=0

We prove that the genus-0 sector of the quartic analogue of the Kontsevich model is completely governed by an involution identity which expresses the meromorphic differential $ω_{0,n}$ at a reflected point $ιz$ in terms of all $ω_{0,m}$ with $m\leq n$ at the original point $z$. We prove that the solution of the involution identity obeys blobbed topological recursion, which confirms a previous conjecture about the quartic Kontsevich model.

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arXiv2020

Matrix Field Theory

This thesis studies matrix field theories, which are a special type of matrix models. First, the different types of applications are pointed out, from (noncommutative) quantum field theory over 2-dimensional quantum gravity up to algebraic geometry with explicit computation of intersection numbers on the moduli space of complex curves. The Kontsevich model, which has proved the Witten conjecture, is the simplest example of a matrix field theory. Generalisations of this model will be studied, where different potentials and the spectral dimension (defined by the asymptotics of the external matrix) are introduced. Because they are naturally embedded into a Riemann surface, the correlation functions are graded by the genus and the number of boundary components. The renormalisation procedure of quantum field theory leads to finite UV-limit. We provide a method to determine closed Schwinger-Dyson equations with the usage of Ward-Takahashi identities in the continuum limit. The cubic (Kontsevich model) and the quartic (Grosse-Wulkenhaar model) potentials are studied separately. For the cubic potential, we show that the renormalisation procedure is compatible with topological recursion (TR). This means that the exact results computed by TR coincide perturbatively with the graph expansion renormalised by Zimmermann's forest formula. For the quartic model, the first correlation function (2-point function) is computed exactly. We give hints that the quartic model has structurally the same properties as the hermitian 2-matrix model with genus zero spectral curve.

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arXiv2019

Solution of the self-dual $Φ^4$ QFT-model on four-dimensional Moyal space

Previously the exact solution of the planar sector of the self-dual $Φ^4$-model on 4-dimensional Moyal space was established up to the solution of a Fredholm integral equation. This paper solves, for any coupling constant $λ>-\frac{1}π$, the Fredholm equation in terms of a hypergeometric function and thus completes the construction of the planar sector of the model. We prove that the interacting model has spectral dimension $4-2\frac{\arcsin(λπ)}π$ for $|λ|<\frac{1}π$. It is this dimension drop which for $λ>0$ avoids the triviality problem of the matricial $Φ^4_4$-model. We also establish the power series approximation of the Fredholm solution to all orders in $λ$. The appearing functions are hyperlogarithms defined by iterated integrals, here of alternating letters $0$ and $-1$. We identify the renormalisation parameter which gives the same normalisation as the ribbon graph expansion.

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arXiv2018

Noncommutative 3-colour scalar quantum field theory model in 2D

We introduce the 3-colour noncommutative quantum field theory model in two dimensions. For this model we prove a generalised Ward-Takahashi identity, which is special to coloured noncommutative QFT models and has no underlying continuous symmetry. It reduces to the usual Ward-Takahashi identity in a particular case. The Ward-Takahashi identity is used to simplify the Schwinger-Dyson equations for the 2-point function and the N-point function. The absence of any renormalisation conditions in the large $(\mathcal{N},V)$-limit in 2D leads to a recursive integral equation for the 2-point function, which we solve perturbatively to sixth order in the coupling constant.

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