Showing papers by "Harald Grosse published in 2019"

Solution of all quartic matrix models

Abstract: We consider the quartic analogue of the Kontsevich model, which is defined by a measure $\exp(-N\,\mathrm{Tr}(E\Phi^2+(\lambda/4)\Phi^4)) d\Phi$ on Hermitean $N \times N$-matrices, where $E$ is any positive matrix and $\lambda$ a scalar. We prove that the two-point function is a rational function evaluated at roots of another rational function $J$ constructed from the spectrum of $E$. This rationality is strong support for the conjecture that the quartic analogue of the Kontsevich model is integrable. We also solve the large-$N$ limit to unbounded operators $E$. The renormalised two-point function is given by an integral formula involving a regularisation of $J$.

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

Abstract: Previously the exact solution of the planar sector of the self-dual $\Phi^4$-model on 4-dimensional Moyal space was established up to the solution of a Fredholm integral equation. This paper solves, for any coupling constant $\lambda>-\frac{1}{\pi}$, 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 $4-2\frac{\arcsin(\lambda\pi)}{\pi}$ for $|\lambda| 0$ avoids the triviality problem of the matricial $\Phi^4_4$-model. We also establish the power series approximation of the Fredholm solution to all orders in $\lambda$. 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.

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

Abstract: Let $F_g(t)$ be the generating function of intersection numbers on the moduli spaces $\overline{\mathcal{M}}_{g,n}$ of complex curves of genus $g$ As by-product of a complete solution of all non-planar correlation functions of the renormalised $\Phi^3$-matrical QFT model, we explicitly construct a Laplacian $\Delta_t$ on the space of formal parameters $t_i$ satisfying $\exp(\sum_{g\geq 2} N^{2-2g}F_g(t))=\exp((-\Delta_t+F_2(t))/N^2)1$ 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 $\Phi^3$-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

How Prof. Zeidler Supported Our Research on Exact Solution of Quantum Field Theory Toy Models

Abstract: Over many years, we developed the construction of the ϕ4-model on four-dimensional Moyal space The solution of the related matrix model $\mathcal {Z}[E,J]=\int d{\Phi } \exp (\text {tr}(J{\Phi }- E{\Phi }^{2} -\frac {\lambda }{4} {\Phi }^{4}))$ is given in terms of the solution of a non-linear equation for the 2-point function and the eigenvalues of E The resulting Schwinger functions in position space are symmetric and invariant under the full Euclidean group Locality is fulfilled The Schwinger 2-point function is reflection positive in special cases