A quantum algorithm for approximating efficiently three–manifold topological invariants in the framework of $SU(2)$ Chern–Simons–Witten (CSW) topological quantum field theory at finite values of the coupling constant $k$ is provided. The model of computation adopted is the $q$-deformed spin network model viewed as a quantum recognizer in the sense of [1], where each basic unitary transition function can be efficiently processed by a standard quantum circuit.

This achievement is an extension of the algorithm for approximating polynomial invariants of colored oriented links found in "Spin networks, quantum automata and link invariants" and "An efficient quantum algorithm for colored Jones polynomials." Thus all the significant quantities — partition functions and observables — of quantum CSW theory can be processed efficiently on a quantum computer, reflecting the intrinsic, field-theoretic solvability of such theory at finite $k$.

The paper is supplemented by a critical overview of the basic conceptual tools underlying the construction of quantum invariants of links and three–manifolds and connections with algorithmic questions that arise in geometry and quantum gravity models are discussed.

A variational principle is introduced which minimizes an action formulated for configurations of vacuum Dirac seas. The action is analyzed in position and momentum space. We relate the corresponding Euler– Lagrange equations to the notion of state stability. Examples of numerical minimizers are constructed and discussed.

In Sections 1–5 of this paper an axiomatic formulation of general theory of rativity (GR) is given and studied. Here use is made of the concept of pre-radar charts. These charts have “infinitesimally” the same properties as the true radar charts used in space–time theory. Their existence in GR has far-reaching consequences which are discussed throughout the paper. For the sake of simplicity and convenience I consider only such material systems the state of which is defined by a velocity field, a mass density and a temperature field. But the main results hold also for more complex systems. It follows from the axiomatics that the pre-radar charts define an atlas for the space–time manifold and that, in addition, they generate the metric, the velocity field and the displacement of the matter. Therefore, they are called generating functions. They act like “potentials”. In Section 6 it is shown that the existence of pre-radar charts allows to simplify the original axiomatics drasticly. But the two versions of GR are equivalent. In Sections 7 and 8 the so-called inverse problem is treated. This means the question whether it is possible to define preradar charts, i.e., generating functions in arbitrary space–times. This problem is subtle. A local general constructive solution of it is presented. Sufficient conditions for the existence of global solutions are given. The aim of the last sections is formulating GR as a scalar field theory. The basic structural elements of it are a generating function, a generalized density and a generalized temperature. One of the axioms of this theory is a generalized Einstein equation that determines the generating function directly. It is shown that basic concepts like orientation, time orientation and isometry are expressible in terms of generating functions. At the end of the paper six problems are formulated which are still unsolved and can act as a stimulant for further research.

The finite symplectic group $Sp(2g)$ over the field of two elements has a natural representation on the vector space of Siegel modular forms of given weight for the principal congruence subgroup of level two. In this paper we decompose this representation, for various (small) values of the genus and the level, into irreducible representations. As a consequence we obtain uniqueness results for certain modular forms related to the superstring measure, a better understanding of certain modular forms in genus three studied by D’Hoker and Phong as well as a new construction of Miyawaki’s cusp form of weight twelve in genus three.

Tractor calculus is a powerful tool for analyzing Weyl invariance; although fundamentally linked to the Cartan connection, it may also be arrived at geometrically by viewing a conformal manifold as the space of null rays in a Lorentzian ambient space. For dimension d conformally flat manifolds we show that the $(d + 2)$-dimensional Fefferman–Graham ambient space corresponds to the momentum space of a massless scalar field. Hence on the one hand the null cone parameterizes physical momentum excitations, while on the other hand, null rays correspond to points in the underlying conformal manifold. This allows us to identify a fundamental set of tractor operators with the generators of conformal symmetries of a scalar field theory in a momentum representation. Moreover, these constitute the minimal representation of the non-compact conformal Lie symmetry algebra of the scalar field with Kostant–Kirillov dimension $d + 1$. Relaxing the conformally flat requirement, we find that while the conformal Lie algebra of tractor operators is deformed by curvature corrections, higher relations in the enveloping algebra corresponding to the minimal representation persist. We also discuss potential applications of these results to physics and conformal geometry.

We provide a quantum path integral definition of an ’t Hooft loop operator, which inserts a point-like monopole in a four-dimensional gauge theory. We explicitly compute the expectation value of the circular ’t Hooft operators in $N = 4$ super Yang–Mills with arbitrary gauge group $G$ up to next to leading order in perturbation theory. We also compute in the strong coupling expansion the expectation value of the circular Wilson loop operators. The result of the computation of an ’t Hooft loop operator in the weak coupling expansion exactly reproduces the strong coupling result of the conjectured dual Wilson loop operator under the action of $S$-duality. This paper demonstrates — for the first time — that correlation functions in $N = 4$ super Yang–Mills admit the action of $S$-duality.