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Raimar Wulkenhaar

Publications and source records attributed to Raimar Wulkenhaar.

At least 19 recordsLinked to original sources
arXiv2026

$q$-Deformed Topological Recursion: Quantum Curves and Non-perturbative Analysis

This work investigates the $q$-deformation of $(r,s)$-Airy structures and their realization via $q$-difference operators, providing a bridge between quantum spectral curves and integrable systems. We construct an all-order $q$-WKB solution for the matrix systems associated with the $q$-quantized curve $E_q(x,y)=0$. We demonstrate that the resulting non-perturbative connected $q$-amplitudes satisfy a set of shifted $q$-loop equations, which can be interpreted as the Ward identities of a $q$-deformed $\mathcal{W}(\mathfrak{gl}_r)$ algebra. Our main result provides a rigorous classification of admissible $(r,s,q)$ pairs and $q$-Casimir configurations that satisfy the $q$-topological type property. This ensures that the semi-classical expansion is uniquely governed by the $q$-topological recursion, offering new insights into the $q$-quantization of mirror curves and their underlying algebraic structures.

math-ph↗
arXiv2026

$q$-Deformed Topological Recursion, Weight Vectors and Algebraic Structures

We investigate a $q$-deformation of the shifted topological recursion, extending the construction of Belliard-Bouchard-Kramer-Nelson to the context of quantum algebras. Through the study of highest weight vectors in $q$-deformed of $\mathcal{W}$-algebra representations, we derive a $q$-analogue of the topological recursion and show it yields $q$-deformed quantum curves. This framework unifies various approaches to quantum integrability and provides new insights into the geometry of $q$-deformed moduli spaces.

math-ph↗
arXiv2025

Relationship between a $\Phi^4$ matrix model and harmonic oscillator systems

A Hermitian $\Phi^4$ matrix model with a Kontsevich-type kinetic term is studied. It was recently discovered that the partition function of this matrix model satisfies the Schr\"odinger equation of the $N$-body harmonic oscillator, and that eigenstates of the Virasoro operators can be derived from this partition function. We extend these results and obtain an explicit formula for such eigenstates in terms of the free energy. Furthermore, the Schr\"odinger equation for the $N$-body harmonic oscillator can also be reformulated in terms of connected correlation functions. The $U(1)^N$-symmetry allows us to derive loop equations.

hep-th↗
arXiv2025

Characterization of the $W_{1+\infty}$-n-algebra and applications

In this paper, we construct the $W_{1+\infty}$-n-algebras in the framework of the generalized quantum algebra. We characterize the $\mathcal{R}(p,q)$-multi-variable $W_{1+\infty}$-algebra and derive its $n$-algebra which is the generalized Lie algebra for $n$ even. Furthermore, we investigate the $\mathcal{R}(p,q)$-elliptic hermitian matrix model and determine a toy model for the generalized quantum $W_{\infty}$ constraints. Also, we deduce particular cases of our results.

math-ph↗
arXiv2025

Generalized super-$W_{1+\infty}$-$n$-algebra and Landau Problem

We investigate the $\mathcal{R}(p,q)$-super $n$-bracket and study their properties such that the generalized super Jacobi identity (GJSI). Furthermore, from the $\mathcal{R}(p,q)$-operators in a Supersymmetric Landau problem, we furnish the $\mathcal{R}(p,q)$-super $W_{1+\infty}$ $n$-algebra which obey the generalized super Jacobi identity (GSJI) for $n$ even. Also, we derive the $\mathcal{R}(p,q)$-super $W_{1+\infty}$ sub-$2n$-algebra and deduce particular cases induced by quantum algebras existing in the literature.

math-ph↗
arXiv2025

Stochastic quantization of $\lambda \phi_2^4$- theory in 2-d Moyal space

There is strong evidence for the conjecture that the $\lambda \phi^4$ QFT- model on 4-dimensional non-commutative Moyal space can be non-perturbatively constructed. As preparation, in this paper we construct the 2-dimensional case with the method of stochastic quantization. We show the local well-posedness and global well-posedness of the stochastic quantization equation, leading to a construction of the Moyal $\lambda \phi^4_2$ measure for any non-negative coupling constant $\lambda$.

math-ph↗
arXiv2024

A Note on BKP for the Kontsevich Matrix Model with Arbitrary Potential

We exhibit the Kontsevich matrix model with arbitrary potential as a BKP tau-function with respect to polynomial deformations of the potential. The result can be equivalently formulated in terms of Cartan-Plücker relations of certain averages of Schur $Q$-function. The extension of a Pfaffian integration identity of de Bruijn to singular kernels is instrumental in the derivation of the result.

math-ph↗
arXiv2023

Real symmetric $Φ^4$-matrix model as Calogero-Moser model

We study a real symmetric $Φ^4$-matrix model whose kinetic term is given by $\mathrm{Tr}( E Φ^2)$, where $E$ is a positive diagonal matrix without degenerate eigenvalues. We show that the partition function of this matrix model corresponds to a zero-energy solution of a Schödinger type equation with Calogero-Moser Hamiltonian. A family of differential equations satisfied by the partition function is also obtained from the Virasoro algebra.

hep-th↗
arXiv2023

Generalized Heisenberg-Virasoro algebra and matrix models from quantum algebra

In this paper, we construct the Heisenberg-Virasoro algebra in the framework of the $\mathcal{R}(p,q)$-deformed quantum algebras. Moreover, the $\mathcal{R}(p,q)$-Heisenberg-Witt $n$-algebras is also investigated. Furthermore, we generalize the notion of the elliptic hermitian matrix models. We use the constraints to evaluate the $\mathcal{R}(p,q)$-differential operators of the Virasoro algebra and generalize it to higher order differential operators. Particular cases corresponding to quantum algebras existing in literature are deduced.

math.QA↗
arXiv2023

Blobbed topological recursion from extended loop equations

We consider the $N\times N$ Hermitian matrix model with measure $d\mu_{E,\lambda}(M)=\frac{1}{Z} \exp(-\frac{\lambda N}{4} \mathrm{tr}(M^4)) d\mu_{E,0}(M)$, where $d\mu_{E,0}$ is the Gaussian measure with covariance $\langle M_{kl}M_{mn}\rangle=\frac{\delta_{kn}\delta_{lm}}{N(E_k+E_l)}$ for given $E_1,...,E_N>0$. It was previously understood that this setting gives rise to two ramified coverings $x,y$ of the Riemann sphere strongly tied by $y(z)=-x(-z)$ and a family $\omega^{(g)}_{n}$ of meromorphic differentials conjectured to obey blobbed topological recursion due to Borot and Shadrin. We develop a new approach to this problem via a system of six meromorphic functions which satisfy extended loop equations. Two of these functions are symmetric in the preimages of $x$ and can be determined from their consistency relations. An expansion at $\infty$ gives global linear and quadratic loop equations for the $\omega^{(g)}_{n}$. These global equations provide the $\omega^{(g)}_{n}$ not only in the vicinity of the ramification points of $x$ but also in the vicinity of all other poles located at opposite diagonals $z_i+z_j=0$ and at $z_i=0$. We deduce a recursion kernel representation valid at least for $g\leq 1$.

math-ph↗
arXiv2022

Intersection theory of the complex quartic Kontsevich model

We expand correlation functions of the Langmann-Szabo-Zarembo (LSZ) model in terms of intersection numbers on the moduli space of complex curves. This provides an explicit, physically motivated example for the expansion of correlation functions generated by Chekhov-Eynard-Orantin topological recursion. To this end, we unify notation as well as different conventions present in the literature and use a set of moduli of the spectral curve adapted to the physically motivated model. The presentation focuses on an illustrative, step-by-step comprehension of the work.

math-ph↗
arXiv2022

A Laplacian to compute intersection numbers on $\bar{\mathcal{M}}_{g,n}$ and correlation functions in NCQFT

Let $F_g(t)$ be the generating function of intersection numbers on the moduli spaces $\bar{\mathcal{M}}_{g,n}$ of complex curves of genus $g$. As by-product of a complete solution of all non-planar correlation functions of the renormalised $Φ^3$-matrical QFT model, we explicitly construct a Laplacian $Δ_t$ on the space of formal parameters $t_i$ satisfying $\exp(\sum_{g\geq 2} N^{2-2g}F_g(t))=\exp((-Δ_t+F_2(t))/N^2)1$ for any $N>0$. The result is achieved via Dyson-Schwinger equations from noncommutative quantum field theory combined with residue techniques from topological recursion. The genus-$g$ correlation functions of the $Φ^3$-matricial QFT model are obtained by repeated application of another differential operator to $F_g(t)$ and taking for $t_i$ the renormalised moments of a measure constructed from the covariance of the model.

math-ph↗
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.

math-ph↗
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.

math-ph↗
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.

math-ph↗