arXiv subjects

Raimar Wulkenhaar

Publications and source records attributed to Raimar Wulkenhaar.

At least 37 records · Page 2Linked to original sources

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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arXiv2021

An algebraic approach to a quartic analogue of the Kontsevich model

We consider an analogue of Kontsevich's matrix Airy function where the cubic potential $\mathrm{Tr}(Φ^3)$ is replaced by a quartic term $\mathrm{Tr}(Φ^4)$. Cumulants of the resulting measure are known to decompose into cycle types for which a recursive system of equations can be established. We develop a new, purely algebraic geometrical solution strategy for the two initial equations of the recursion, based on properties of Cauchy matrices. These structures led in subsequent work to the discovery that the quartic analogue of the Kontsevich model obeys blobbed topological recursion.

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arXiv2019

The Leutwyler-Smilga relation on the lattice

According to the Leutwyler-Smilga relation, in Quantum Chromodynamics (QCD) the topological susceptibility vanishes linearly with the quark masses. Calculations of the topological susceptibility in the context of lattice QCD, extrapolated to zero quark masses, show a remnant non-zero value as a lattice artefact. Employing the Atiyah-Singer theorem in the framework of Symanzik's effective action and chiral perturbation theory, we show the validity of the Leutwyler-Smilga relation in lattice QCD with lattice artefacts of order $a^2$ in the lattice spacing $a$.

hep-lat↗

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

Lambert-W solves the noncommutative $Φ^4$-model

The closed Dyson-Schwinger equation for the 2-point function of the noncommutative $λϕ^4_2$-model is rearranged into the boundary value problem for a sectionally holomorphic function in two variables. We prove an exact formula for a solution in terms of Lambert's $W$-function. This solution is holomorphic in $λ$ inside a domain which contains $(-1/\log 4,\infty)$. Our methods include the Hilbert transform, perturbation series and Lagrange-Bürmann resummation.

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arXiv2018

Nonperturbative evaluation of the partition function for the real scalar quartic QFT on the Moyal plane at weak coupling

The remarkable properties of the real scalar quartic quantum field theory on the Moyal plane in combination with its similarity to the Kontsevich model make the model's partition function an interesting object to study. However, direct evalua- tions is obstructed by the intertwining of the field's various modes. A factorization procedure to circumvent this problem is proposed and discussed here in the context of the real scalar quartic qft on the Moyal plane. This factorization consists of integrating against the asymptotic volume of the diagonal subpolytope of symmet- ric stochastic matrices. This volume has been determined to this end. Using this method the partition function for regime of weak coupling is computed. Using the same method it is as well possible to determine the partition function and free energy density for other regimes.

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arXiv2018

Correlation functions of $\mathrm{U}(N)$-tensor models and their Schwinger-Dyson equations

We analyse the correlation functions of $\mathrm{U}(N)$-tensor models (or complex tensor models), which turn out to be classified by boundary graphs, and use the Ward-Takahashi identity and the graph calculus developed in [Commun. Math. Phys. (2018) 358: 589] in order to derive the complete tower of exact, analytic Schwinger-Dyson equations for correlation functions with connected boundary graphs. We write them explicitly for ranks $D=3$ and $D=4$. Throughout, we follow a non-perturbative approach to Tensor (Group) Field Theories. We propose the extension of this program to the Gurau-Witten model, a holographic tensor model based on the Sachdev-Ye-Kitaev model (SYK model).

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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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arXiv2017

The $Φ^3_4$ and $Φ^3_6$ matricial QFT models have reflection positive two-point function

We extend our previous work (on $D=2$) to give an exact solution of the $Φ^3_D$ large-$\mathcal{N}$ matrix model (or renormalised Kontsevich model) in $D=4$ and $D=6$ dimensions. Induction proofs and the difficult combinatorics are unchanged compared with $D=2$, but the renormalisation - performed according to Zimmermann - is much more involved. As main result we prove that the Schwinger 2-point function resulting from the $Φ^3_D$-QFT model on Moyal space satisfies, for real coupling constant, reflection positivity in $D=4$ and $D=6$ dimensions. The Källén-Lehmann mass spectrum of the associated Wightman 2-point function describes a scattering part $|p|^2 \geq 2μ^2$ and an isolated fuzzy mass shell around $|p|^2=μ^2$.

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arXiv2017

Exact solution of matricial $Φ^3_2$ quantum field theory

We apply a recently developed method to exactly solve the $Φ^3$ matrix model with covariance of a two-dimensional theory, also known as regularised Kontsevich model. Its correlation functions collectively describe graphs on a multi-punctured 2-sphere. We show how Ward-Takahashi identities and Schwinger-Dyson equations lead in a special large-$\mathcal{N}$ limit to integral equations that we solve exactly for all correlation functions. Remarkably, these functions are analytic in the $Φ^3$ coupling constant, although bounds on individual graphs justify only Borel summability. The solved model arises from noncommutative field theory in a special limit of strong deformation parameter. The limit defines ordinary 2D Schwinger functions which, however, do not satisfy reflection positivity.

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arXiv2016

The Moyal Sphere

We construct a family of constant curvature metrics on the Moyal plane and compute the Gauss-Bonnet term for each of them. They arise from the conformal rescaling of the metric in the orthonormal frame approach. We find a particular solution, which corresponds to the Fubini-Study metric and which equips the Moyal algebra with the geometry of a noncommutative sphere.

math.QA↗

arXiv2015

Solvable 4D noncommutative QFT: phase transitions and quest for reflection positivity

We provide further analytical and first numerical results on the solvable $λϕ^4_4$-NCQFT model. We prove that for $λ<0$ the singular integral equation has a unique solution, whereas for $λ>0$ there is considerable freedom. Furthermore we provide integral formulae for partial derivatives of the matrix 2-point function, which are the key to investigate reflection positivity. The numerical implementation of these equations gives evidence for phase transitions. The derivative of the finite wavefunction renormalisation with respect to $λ$ is discontinuous at $λ_c \approx -0.39$. This leads to singularities in higher correlation functions for $λ<λ_c$. The phase $λ>0$ is not yet under control because of the freedom in the singular integral equation. Reflection positivity requires that the two-point function is Stieltjes. Implementing Widder's criteria for Stieltjes functions we exclude reflection positivity outside the phase $[λ_c,0]$. For the phase $λ_c<λ\leq 0$ we show that refining the discrete approximation we satisfy Widder to higher and higher order. This is clear evidence, albeit no proof, of reflection positivity in that phase.

hep-th↗

arXiv2015

Correlation functions of just renormalizable tensorial group field theory: The melonic approximation

The $D$-colored version of tensor models has been shown to admit a large $N$-limit expansion. The leading contributions result from so-called melonic graphs which are dual to the $D$-sphere. This is a note about the Schwinger-Dyson equations of the tensorial $φ^{4}_{5}$-model (with propagator $1/{\bf p}^{2}$) and their melonic approximation. We derive the master equations for two- and four-point correlation functions and discuss their solution.

hep-th↗

arXiv2015

On the fixed point equation of a solvable 4D QFT model

The regularisation of the $λϕ^4_4$-model on noncommutative Moyal space gives rise to a solvable QFT model in which all correlation functions are expressed in terms of the solution of a fixed point problem. We prove that the non-linear operator for the logarithm of the original problem satisfies the assumptions of the Schauder fixed point theorem, thereby completing the solution of the QFT model.

math-ph↗

arXiv2014

Construction of the Φ^4_4-quantum field theory on noncommutative Moyal space

We review our recent construction of the $ϕ^4$-model on four-dimensional Moyal space. A milestone is the exact solution of the quartic matrix model $Z[E,J]=\int dΦ\exp(tr(JΦ- EΦ^2 -(λ/4) Φ^4))$ in terms of the solution of a non-linear equation for the 2-point function and the eigenvalues of $E$. The $β$-function vanishes identically. For the Moyal model, the theory of Carleman type singular integral equations reduces the construction to a fixed point problem. Its numerical solution reveals a second-order phase transition at $λ_c\approx-0.396$ and a phase transition of infinite order at $λ=0$. The resulting Schwinger functions in position space are symmetric and invariant under the full Euclidean group. They are only sensitive to diagonal matrix correlation functions, and clustering is violated. The Schwinger 2-point function is reflection positive iff the diagonal matrix 2-point function is a Stieltjes function. Numerically this seems to be the case for coupling constants $λ\in [λ_c,0]$.

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arXiv2013

Solvable limits of a 4D noncommutative QFT

In previous work we have shown that the (θ->\infty)-limit of ϕ^4_4-quantum field theory on noncommutative Moyal space is an exactly solvable matrix model. In this paper we translate these results to position space. We show that the Schwinger functions are symmetric and invariant under the full Euclidean group. The Schwinger functions only depend on matrix correlation functions at coinciding indices per topological sector, and clustering is violated. We prove that Osterwalder-Schrader reflection positivity of the Schwinger two-point function is equivalent to the question whether the diagonal matrix two-point function is a Stieltjes function. Numerical investigations suggest that this can at best be expected for the wrong sign of the coupling constant. The corresponding Wightman functions would describe particles which interact without momentum transfer. The theory differs from a free theory by the presence of non-trivial topological sectors.

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arXiv2013

Self-dual noncommutative ϕ^4-theory in four dimensions is a non-perturbatively solvable and non-trivial quantum field theory

We study quartic matrix models with partition function Z[E,J]=\int dM \exp(trace(JM-EM^2-(λ/4)M^4)). The integral is over the space of Hermitean NxN-matrices, the external matrix E encodes the dynamics, λ>0 is a scalar coupling constant and the matrix J is used to generate correlation functions. For E not a multiple of the identity matrix, we prove a universal algebraic recursion formula which gives all higher correlation functions in terms of the 2-point function and the distinct eigenvalues of E. The 2-point function itself satisfies a closed non-linear equation which must be solved case by case for given E. These results imply that if the 2-point function of a quartic matrix model is renormalisable by mass and wavefunction renormalisation, then the entire model is renormalisable and has vanishing β-function. As main application we prove that Euclidean ϕ^4-quantum field theory on four-dimensional Moyal space with harmonic propagation, taken at its self-duality point and in the infinite volume limit, is exactly solvable and non-trivial. This model is a quartic matrix model, where E has for N->\infty the same spectrum as the Laplace operator in 4 dimensions. Using the theory of singular integral equations of Carleman type we compute (for N->\infty and after renormalisation of E,λ) the free energy density (1/volume)\log(Z[E,J]/Z[E,0]) exactly in terms of the solution of a non-linear integral equation. Existence of a solution is proved via the Schauder fixed point theorem. The derivation of the non-linear integral equation relies on an assumption which we verified numerically for coupling constants 0<λ\leq (1/π).

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