Perturbative and Geometric Analysis of the Quartic Kontsevich Model
Johannes Branahl, Alexander Hock, Raimar Wulkenhaar
备注：
33 pages, 13 figures. v2: distinction between our model and GKM, exact formula for genus zero closed graphs, example higher-order ramification: correct limit. Small corrections. v3: some material added to sec 5.1+5.3. v4: published version
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.
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 an…
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.
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.
In this paper I give an overview of mathematical structures appearing in perturbative algebraic quantum field theory (pAQFT) in the case of the massless scalar field on Minkowski spacetime. I also show how these relate to Kontsevich-Zagier periods. Next, I review the pAQFT version of the renormalization group flow and reformulate it in terms of Feynman graphs. This allows me to relate Kontsevich-Zagier periods to numbers appearing in computing the pAQFT $β$-function.
To lowest order of perturbation theory we show that an equivalence can be established between a $\cal PT$-symmetric generalized quartic anharmonic oscillator model and a Hermitian position-dependent mass Hamiltonian $h$. An important feature of $h$ is that it reveals a domain of couplings where the quartic potential could be attractive, vanishing or repulsive. We also determine the associated physical quantities.
In this paper we continue the perturbative analysis of the quartic Kontsevich model. We investigate meromorphic functions $Ω^{(0)}_m$ with $m=1,2$, that obey blobbed topological recursion. We calculate their expansions and check their equivalence to sums of ribbon graph weights, which are obtained with common methods of perturbation theory in QFT, up to fifth order in the coupling using Mathematica. Furthermore, we provide a catalog of permutation pairs $(α,σ)$, which encode all 5660 vacuum ribbon graphs that contribute to the free energy $\mathcal{F}^{(g)}$ with genus $g\geq 0$ up to fifth or…
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 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 matri…
We present an analytical calculation of periodic orbits in the homogeneous quartic oscillator potential. Exploiting the properties of the periodic Lam{é} functions that describe the orbits bifurcated from the fundamental linear orbit in the vicinity of the bifurcation points, we use perturbation theory to obtain their evolution away from the bifurcation points. As an application, we derive an analytical semiclassical trace formula for the density of states in the separable case, using a uniform approximation for the pitchfork bifurcations occurring there, which allows for full semiclassical qu…
This paper contains three results about generating functions for Lie-theoretic integration of Poisson brackets and their relation to quantization. In the first, we show how to construct a generating function associated to the germ of any local symplectic groupoid and we provide an explicit (smooth, non-formal) universal formula $S_π$ for integrating any Poisson structure $π$ on a coordinate space. The second result involves the relation to semiclassical quantization. We show that the formal Taylor expansion of $S_{tπ}$ around $t=0$ yields an extract of Kontsevich's star product formula based o…
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 …
Tensor field theory (TFT) focuses on quantum field theory aspects of random tensor models, a quantum-gravity-motivated generalisation of random matrix models. The TFT correlation functions have been shown to be classified by graphs that describe the geometry of the boundary states, the so-called boundary graphs. These graphs can be disconnected, although the correlation functions are themselves connected. In a recent work, the Schwinger-Dyson equations for an arbitrary albeit connected boundary were obtained. Here, we introduce the multivariable graph calculus in order to derive the missing eq…
