Abstract: Among the scalar field matrix models with coupling to an external field more results are known for the cubic (Kontesevich) model than for the quartic (Grosse-Wulkenhaar) model (e.g. integrability). We construct a Hybric cubic-quartic matrix model with four and three point interactions, the later is coupled to a positive matrix M and makes the model solvable. First, we derive Feynman rules and
calculate the perturbative expansions of some multipoint functions. Second, we calculate the path integral of the partition function and obtain exact solutions for the one point function with one boundary, the two point functions with one as well as two boundaries and the n-point functions with n boundaries. They include contributions from non-planar as well as higher genus Feynman diagrams.
Research in Teams Project:
The quartic matrix model arises by restricting the noncommutative λΦ^4-model to finite matrices. In previous work we achieved an understanding of many structures and details. All correlation functions can be algebraically obtained from a family ω(g,n) of meromorphic differentials, labelled by genus g and number n of marked points of a Riemann surface.
It is essentially proved that the ω(g,n) satisfy linear and quadratic loop equations and thus obey blobbed topological recursion. The loop equations are completely known for g=0 and g=1; for higher g only up to terms holomorphic in certain ramification points. From the loop equations one can solve the ω(g,n), and then all correlation functions, exactly.
The RIT project studies the widely open question whether the exact solvability of the quartic matrix model is related to integrability.
Integrability means that the ω(g,0), with corrections for g=0 and g=1, provide a τ - function for a Hirota equation. For an important subclass of blobbed topological recursion, this is automatic. The quartic matrix model does not belong to the subclass. The RIT will investigate whether tools and results of the subclass extend to the model under consideration.
Research Team:
Harald Grosse (U of Vienna), Naoyuki Kanomata (Tokyo U of Science), Akifumi Sako (Tokyo U of Science), Raimar Wulkenhaar (U Münster)