AQFTUK 9

Maria Stella Adamo (FAU Erlangen-Nürnberg): Osterwalder-Schrader axioms for unitary full VOAs 

Full Vertex Operator Algebras (VOAs) can be seen as extensions of commuting chiral and anti-chiral VOAs, introduced to describe compact 2D conformal field theories, which may include also irrational theories, e.g., extensions of Heisenberg VOAs. We introduce unitarity for a reasonable class of full VOAs and, under the assumption of polynomial energy bound and polynomial spectral density, we show a conformal version of the OS axioms, including the linear growth condition, for n-point correlation functions constructed from full vertex operators in the full VOAs. 

The celebrated Osterwalder-Schrader (OS) reconstruction results provide conditions to be verified by Euclidean n-point correlation functions to produce distributions that verify the Wightman axioms, which ultimately give rise to a quantum field theory in the Wightman formalism. Thus, lastly we will sketch the main ideas behind the reconstruction of Wightman fields provided by the OS reconstruction result on one side, and directly constructed through fields in the full VOA on the other side.

Ben Graham (York): Towards a Description of Measurement for Interacting Quantum Fields

The framework for measurement of quantum fields due to Fewster and Verch was developed to describe arbitrary local measurements in quantum field theory (QFT) on potentially curved spacetimes, using methods of algebraic QFT.  In this approach, a QFT of interest is measured indirectly using a second QFT, the probe, modelled by a dynamical coupling of the theories in a compact spacetime region. A natural question arises when studying specific examples of Lagrangian field theories, namely: How should one handle the computations involved a system-probe interaction that isn't explicitly solvable, for example, scalar fields exhibiting self interaction, rather than being free fields? This talk explores how the C*-algebraic framework of Buchholz and Fredenhagen, in addition to the perturbative algebraic approach, can provide insights into addressing such questions.

Filippo Nava (York): Interplay between boundary conditions and the Lorentzian Wetterich equation 

In the framework of the functional renormalization group and of the perturbative, algebraic approach to quantum field theory, D'Angelo et. al. have derived a Lorentzian version of a flow equation à la Wetterich, which can be used to study non-linear, quantum scalar field theories on a globally hyperbolic spacetime. In this talk, I'll show that the realm of validity of this result can be extended to study interacting scalar field theories on globally hyperbolic manifolds with a timelike boundary. By considering the specific examples of half Minkowski spacetime and of the Poincaré patch of Anti-de Sitter, I'll show that the form of the Lorentzian Wetterich equation is strongly dependent on the boundary conditions assigned to the underlying field theory. In addition, using a numerical approach, we have been able to provide strong evidence that there is a qualitative and not only a quantitative difference in the associated flow and we highlight this feature by considering Dirichlet and Neumann boundary conditions on half Minkowski spacetime. Based on the joint work with C. Dappiaggi and L. Sinibaldi 2401.07130. 

Sebastiano Carpi (Rome Tor Vergata): Vertex operator algebras, Teichmüller modular forms and the monster

Classical modular functions and modular forms are  meromorphic functions on the upper half plane H satisfying certain functional equations related to the action of the modular group SL(2,Z) on H. They play an important role in number theory and they are deeply related to the geometry of the moduli space of genus one compact complex curves. Vertex operator algebras (VOAs) can be seen as an axiomatization of chiral conformal (quantum) field theories in two space-time dimensions (chiral CFT). An important example is given by the Frenkel-Lepowsky-Meurman moonshine VOA. It is a holomorphic VOA i.e. a VOA with trivial representation theory.  Its genus one partition function can be directly related to the Klein modular function j and its automorphism group is the Fisher-Griess monster group. This gives an explanation of McKay's observation relating the Klein modular function j and the representation theory of the monster group and led Borcherds to prove the Conway-Norton moonshine conjecture. More generally the genus one partition function of a holomorphic VOA with central charge c gives rise to a modular form of weight c/2. This modular form is an important invariant of the VOA but in general it is not sufficient to determine it, as shown by examples coming from isospectral self-dual positive definite even lattices. Almost fifty years ago Friedan and Shenker conjectured, on physical grounds, that the collection of all genus g partition functions of a two-dimensional CFT completely determines the theory. Correspondingly, they argued that CFTs can be completely described in terms of the geometry of the moduli spaces of genus g compact complex curves. Teichmüller modular forms are higher genus generalizations of classical modular forms. In this talk I will review some recent results of an ongoing joint work with Giulio Codogni. If V is a holomorphic VOA of central charge c we show that the genus g partition function of V gives rise in a natural way to a Teichmüller modular form of weight c/2. This gives strong constraints on the partition functions of holomorphic VOAs. Moreover, we clarify the relation between unitary VOAs having the same genus g partition function for all g and give various examples in which the collection of all the genus g partition functions determines the VOA. Finally, we relate the important open problem of the uniqueness of the moonshine VOA with a weak form of the Harrison-Morrison slope conjecture about the geometry of the module spaces of compact Riemann surfaces.