This paper shows that two important linear connections in mathematical physics, the Hitchin connection from geometric quantisation theory and the Knizhnik–Zamolodchikov connection from conformal field theory, coincide in the context of the moduli space of flat SU(N)-connections over the n-punctured sphere when the holonomies around the punctures have fixed conjugacy classes. This equivalence will allow geometric calculations and arguments to be carried over to topological quantum field theory, to allow explicit calculations.
This thesis presents a review of the background theory that goes into the calculation of instanton contribution via the Nekrasov Partition function. It discusses the mathematics required to formally understand gauge theory and introductory Yang–Mills theory. In particular, the classical solutions of Yang–Mills theory, known as instantons, are presented, and their geometry modulo gauge transformations is discussed by making use of the ADHM construction.
The importance of this space of classical solutions is underscored by demonstrating that N = 2 super Yang–Mills theory localises to the spaces of instantons. This path integral localisation is placed in the context along Duistermaat–Heckman localisation, as well as other related localisation theorems for finite-dimensional integrals, whose theory is elaborately discussed in the language of equivariant cohomology.
Motivated by the localisation to instanton spaces, the Seiberg–Witten theory of prepotentials on instanton spaces and the Nekrasov partition function are provided as tools to calculate integrals over instanton spaces. Seiberg–Witten theory reduces the problem to a geometric one, while the Nekrasov partition function uses Duistermaat–Heckman localisation in order to convert the problem to a combinatorial one. The link between these techniques, which is also connected to localisation, is briefly touched upon.
The celebrated van Benthem characterization theorem states that on Kripke structures modal logic is the bisimulation-invariant fragment of first-order logic. In this paper, we prove an analogue of the van Benthem characterization theorem for models based on descriptive general frames. This is an important class of general frames for which every modal logic is complete. These frames can be represented as Stone spaces equipped with a 'continuous' binary relation. The proof of our theorem generalizes Rosen's proof of the van Benthem theorem for finite frames and uses as an essential technique a new notion of descriptive unravelling. We also develop a basic model theory for descriptive general frames and show that in many ways it behaves like the model theory of finite structures. In particular, we prove the failure of the compactness theorem, of the Beth definability theorem, of the Craig interpolation theorem and of the upward Löwenheim–Skolem theorem.
This thesis investigates the modal and first-order model theory of the class of models over descriptive general frames. Descriptive general frames are Stone spaces with a suitable relation over which every modal logic is complete. The main theorem of this thesis is the van Benthem Characterisation Theorem for the class of descriptive general models. Moreover, a model-theoretic analysis is given to prove that many important results from classical model theory, including the Compactness Theorem for first-order logic and the upward Löwenheim–Skolem Theorem, fail on the class of descriptive general models. The main tool developed in this thesis is the descriptive unravelling, a version of the unravelling tree that is modified to be descriptive. A careful analysis of the operation is provided and three isomorphic constructions are given: a construction through duality theorems, a construction through a topological toolkit based on nets that is also developed, and an explicit construction in terms of finite and infinite paths.
This thesis investigates the universality of measurement-based quantum computation (QC_C), an alternative protocol for quantum computers using entanglement between prepared qubits and strategic measurements to execute quantum programs. After an introduction to the protocol, we explore the computational resources of entanglement in terms of several different measures of entanglement. In this context, we provide and prove convenient tricks for easily calculating complicated measures in specific situations. These are then applied to important clusters. Taking these conclusions, we discuss the connection of the calculations with estimations for classical simulation time for quantum circuit computation and QC_C. To do this, we use an intuitive translation between the two protocols. Subsequently, we introduce the Logic of Quantum Programs (LQP) to express both quantum circuit computation and QC_C in this language. Using this expression, we formalise the intuitive translation in LQP. Some syntactical proofs relating quantum computations are then given and finally the translation is discussed in terms of its potential to serve as an approach to syntactically prove the universality of QC_C.
Email: timhe[at]sdu[dot]dk
Email: tim[dot]henke[at]gmail[dot]com
Email: leastaction[at]timhenke[dot]nl
LinkedIn: My LinkedIn page
Address: Campusvej 55
Postal Code: 5230
City: Odense M
Country: Denmark
Office: Ø11-404b-1
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