It has been recently noted that the diffeomorphism covariance of a Chiral Conformal QFT in the vacuum sector automatically ensures Möbius covariance in all charged sectors. In this article it is shown that the diffeomorphism covariance and the positivity of the energy in the vacuum sector even ensure the positivity of the energy in the charged sectors.
The main observation of this paper is that the positivity of the energy -- at least in case of a Chiral Conformal QFT -- is a local concept: it is related to the fact that the energy density, when smeared with some local nonnegative test functions, remains bounded from below (with the bound depending on the test function).
The presented proof relies in an essential way on recently developed methods concerning the smearing of the stress-energy tensor on nonsmooth functions.
Consider a correlated Gaussian random energy model built by successively adding one particle (spin) into the system and imposing the positivity of the associated covariance matrix. We show that the validity of a recently isolated condition ensuring the existence of the thermodynamic limit forces the covariance matrix to exhibit the Parisi replica symmetry breaking scheme with a convexity condition on the matrix elements.
We show that for a conformal local net of observables on the circle, the split property is automatic. Both full conformal covariance (i.e. diffeomorphism covariance) and the circle-setting play essential roles in this fact, while by previously constructed examples it was already known that even on the circle, Möbius covariance does not imply the split property. On the other hand, here we also provide an example of a local conformal net living on the two-dimensional Minkowski space, which - although being diffeomorphism covariant - does not have the split property.
A new method for computing exact conformal partial wave expansions is developed and applied to approach the problem of Hilbert space (Wightman) positivity in a non-perturbative four-dimensional quantum field theory model. The model is based on the assumption of global conformal invariance on compactified Minkowski space. Bilocal fields arising in the harmonic decomposition of the operator product expansion prove to be a powerful instrument in exploring the field content. In particular, in the theory of a field of dimension 4 which has the properties of a (gauge invariant) Lagrangian, the scala…
This PhD thesis focuses on local conformal nets of von Neumann algebras on the circle. For a more detailed description of its content and of the results published within, see its preface.
Every locally normal representation of a local chiral conformal quantum theory is covariant with respect to global conformal transformations, if this theory is diffeomorphism covariant in its vacuum representation. The unitary, strongly continuous representation implementing conformal symmetry is constructed; it consists of operators which are inner in a global sense for the representation of the quantum theory. The construction is independent of positivity of energy and applies to all locally normal representations irrespective of their statistical dimensions (index).
In the present paper we review in a fibre bundle context the covariant and massless canonical representations of the Poincare' group as well as certain unitary representations of the conformal group (in 4 dimensions). We give a simplified proof of the well-known fact that massless canonical representations with discrete helicity extend to unitary and irreducible representations of the conformal group mentioned before. Further we give a simple new proof that massless free nets for any helicity value are covariant under the conformal group. Free nets are the result of a direct (i.e. independent …
The concept of reflection positivity has its origins in the work of Osterwalder--Schrader on constructive quantum field theory and duality between unitary representations of the euclidean motion group and the Poincare group. On the mathematical side this duality can be made precise as follows. If $\g$ is a Lie algebra with an involutive automorphism $τ$. Decompose $\g = \fh \oplus \fq = \ker(τ- \1) \oplus \ker(τ+ \1)$ into $τ$-eigenspaces and let $\g^c := \fh \oplus i \fq$. At the core of the notion of reflection positivity is the idea that this duality can sometimes be implemented on the leve…
Region bounds, error margins and certified positivity in the numerical conformal bootstrap; Or the first mathematically rigorous bounds on the 3d Ising CFT critical exponents
For $γ\in (0,2)$, $U\subset \mathbb C$, and an instance $h$ of the Gaussian free field (GFF) on $U$, the $γ$-Liouville quantum gravity (LQG) surface associated with $(U,h)$ is formally described by the Riemannian metric tensor $e^{γh} (dx^2 + dy^2)$ on $U$. Previous work by the authors showed that one can define a canonical metric (distance function) $D_h$ on $U$ associated with a $γ$-LQG surface. We show that this metric is conformally covariant in the sense that it respects the coordinate change formula for $γ$-LQG surfaces. That is, if $U,\widetilde{U}$ are domains, $ϕ\colon U \to \widetild…
We consider 5-point functions in conformal field theories in d > 2 dimensions. Using weight-shifting operators, we derive recursion relations which allow for the computation of arbitrary conformal blocks appearing in 5-point functions of scalar operators, reducing them to a linear combination of blocks with scalars exchanged. We additionally derive recursion relations for the conformal blocks which appear when one of the external operators in the 5-point function has spin 1 or 2. Our results allow us to formulate positivity constraints using 5-point functions which describe the expectation val…
Extending previous work on 2 -- and 3 -- point functions, we study the 4 -- point function and its conformal block structure in conformal quantum mechanics CFT$_1$, which realizes the SO(2,1) symmetry group. Conformal covariance is preserved even though the operators with which we work need not be primary and the states are not conformally invariant. We find that only one conformal block contributes to the four-point function. We describe some further properties of the states that we use and we construct dynamical evolution generated by the compact generator of SO(2.1).
We formulate conformal field theory in the setting of algebraic quantum field theory as Haag-Kastler nets of local observable algebras with diffeomorphism covariance on the two-dimensional Minkowski space. We then obtain a decomposition of a two-dimensional theory into two chiral theories. We give the first classification result of such chiral theories with representation theoretic invariants. That is, we use the central charge as the first invariant, and if it is less than 1, we obtain a complete classification. Our classification list contains a new net which does not seem to arise from the …
We provide a detailed construction of the quantum theory of the massless scalar field on 2-dimensional, globally-hyperbolic (in particular, Lorentzian) manifolds using the framework of perturbative algebraic quantum field theory. From this we subalgebras of observables isomorphic to the Heisenberg and Virasoro algebras on the Einstein cylinder. We also show how the conformal version of general covariance, as first introduced by Pinamonti as an extension of the construction due to Brunetti, Fredenhagen and Verch, may be applied to the concept of natural Lagrangians in order to obtain a simple c…
We show that any solitonic representation of a conformal (diffeomorphism covariant) net on S^1 has positive energy and construct an uncountable family of mutually inequivalent solitonic representations of any conformal net, using nonsmooth diffeomorphisms. On the loop group nets, we show that these representations induce representations of the subgroup of loops compactly supported in S^1 \ {-1} which do not extend to the whole loop group. In the case of the U(1)-current net, we extend the diffeomorphism covariance to the Sobolev diffeomorphisms D^s(S^1), s > 2, and show that the positive-energ…
Rindler positivity is a property that holds in any relativistic Quantum Field Theory and implies an infinite set of inequalities involving the exponential of the Rényi mutual information $I_n(A_i,\bar{A}_j)$ between $A_i$ and $\bar{A}_j$, where $A_i$ is a spacelike region in the right Rindler wedge and $\bar{A}_j$ is the wedge reflection of $A_j$. We explore these inequalities in order to get local inequalities for $I_n(A,\bar{A})$ as a function of the distance between $A$ and its mirror region $\bar{A}$. We show that the assumption, based on the cluster property of the vacuum, that $I_n$ goes…
Conformal invariance of the six-vertex model's height function
Université de Genève（日内瓦大学）；Institut des Hautes Études Scientifiques（高等科学研究所）；CNRS, Sorbonne Université, LPSM（法国国家科学研究中心，索邦大学，LPSM）；Université de Fribourg（弗里堡大学）
