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.The mathemati-cian might think that the use of expression 65 to de ne the invariant ratherthan the sum of gauge inequivalent solutions of equations 21 and 22 is a use-less complication, especially given that it is expressed in terms of non-rigorousfunctional integration.However, it must be understood that the quantum eldtheoretical formulation provides the physicist with appropriate tools and formalrules designed to the purpose of computing these functional integrals.Oftenit is precisely the fact that one can rephrase the computation as an integra-tion over all the unconstrained elds", rather than just the sum over the eldsconstrained" by the di erential equations, that makes it computable.It is theauthor's belief that a further study of the properties of the regularised Eulerclasses of Fredholm bundles will help to make some of these tools available tomathematicians as well.11.5 Quantum Field Theory and Floer homologyThe three dimensional invariant and the Floer homology were rst introducedwithin the quantum eld theoretic formalism 10.It is well known from Atiyah'sformulation of quantum eld theory 2 , 3 that we can think of a quantum eldtheory as a functor which associates to a closed three manifold a vector spaceand to a four-manifold with boundary an element in the vector space attached tothe boundary.A pairing of the vector space with its dual corresponds to gluingtwo four-manifolds along their boundaries.The numbers that results via thispairing are invariants of the di erentiable structure of a closed 4-manifold.In156 our case the Floer homology is the vector space associated to a three-manifold.However, our case does not entirely t into Atiyah's de nition.In fact we haveseen that there is a subtle problem of metric dependence in the Seiberg WittenFloer homology: a phenomenon that did not appear in Donaldson theory.Instanton homology also didn't quite t into Atiyah's formulation, but fora di erent reason.In that case the main di culty was to extend the de nitionof the Floer homology from the case of homology spheres to all closed three-manifolds.To a large extent this problem was overcome following two di erentstrategies: the Fukaya-Floer 14 homology or the equivariant Floer homology5.The latter can be used also in Seiberg Witten theory to deal with a similarproblem; and the equivariant formulation turns out to be e ective also in dealingwith the metric dependence problem.A Fukaya-Floer complex for SeibergWitten theory has been considered in 11 in connection to the formulation ofrelative invariants.The problem of the metric dependence also has a physical formulation.De nition 11.12 A quantum eld theory is determined by a manifold X anda Fredholm bundle E; s de ned by means of geometric data on the manifoldX metrics, connections and sections of some vector bundles, etc.The ex-pectation values" of a quantum eld theory are the Euler numbers obtained bycapping the regularised Euler class s E with cohomology classes of total degreeInd Ds , which also encode some geometric data of the manifold X.A quantumeld theory is called topological" if the expectation values are independent ofthe metric on X.We distinguish two kinds of topological quantum eld theories see theoverview 23 , 24.De nition 11.13 A topological quantum eld theory is said to be of Schwarztype if it satis es the condition that the variation of the section s of the Fredholmbundle E with respect to a one parameter family of metrics on X is zero and thatthe cohomology classes that are capped with the regularised Euler class s E arealso chosen in a way that is independent of the metric on X.A typical example of topological quantum eld theory of Schwarz type isChern-Simons theory.De nition 11.14 A topological quantum eld theory is said to be of Wittentype if it satis es the condition that the cohomology classes capped with s Eare independent of the metric on X and the variation of the Mathai Quillenform 65 with respect to a one parameter family of metrics on X is an exactform.An example of topological quantum eld theory of Witten type is the twistedYang-Mills theory that reproduces Donaldson polynomials as expectation val-ues 35.Seiberg Witten gauge theory also ts into this second type.In the157 three-dimensional case when b1 Y 0 the invariant is metric independent evenif the section 67 and the functional 32 depend on the metric.This result canbe rephrased exactly in terms of the property that de nes topological quantumeld theories of Witten type
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Tematy
IndexPrzykladowe sprawozdanie finansowe on line
00157 f3b6d2f1cd90311243c9d1a5e
L Frank Baum Oz 29 The Wishinghorse of Oz
Coulter Catherine Gwiazda 03 Gwiazda z nefrytu
Kimball Kristin Brudna robota
Kaim Karolina Cykl Sfora 01 Czarownica
Marek Grzybowski, Janusz Tomaszewski red.nauk. Logistyka, komunikacja, bezpieczeństwo. Wybrane problemy
Rudazow Aleksander Arcymag, t.1
Moje miejsce na ziemi Adler Elizabeth
Lennox Judith Duchy przeszłoÂści