Topological quantum field theory
In a topological field theory, correlation functions do not depend on the metric of spacetime.Topological field theories are not very interesting on flat Minkowski spacetime used in particle physics.A topological sigma model targets infinite-dimensional projective space, and if such a thing could be defined it would have countably infinitely many degrees of freedom.)For instance, in the BF model, the spacetime is a two-dimensional manifold M, the observables are constructed from a two-form F, an auxiliary scalar B, and their derivatives.The independence of the stress-energy tensor Tαβ of the system from the metric depends on whether the BRST-operator is closed.Witten-type TQFTs arise if the following conditions are satisfied: As an example (Linker 2015): Given a 2-form fieldAtiyah originally proposed the axioms of a topological quantum field theory (TQFT) in dimension d defined over a ground ring Λ as following: These data are subject to the following axioms (4 and 5 were added by Atiyah): Remark.The fifth axiom shows that where on the right we compute the norm in the hermitian (possibly indefinite) metric.Physically (2) + (4) are related to relativistic invariance while (3) + (5) are indicative of the quantum nature of the theory.The main feature of topological QFTs is that H = 0, which implies that there is no real dynamics or propagation along the cylinder Σ × I.The reason why a theory with a zero Hamiltonian can be sensibly formulated resides in the Feynman path integral approach to QFT.Let us extend Sn to a compact Lie group G and consider "integrable" orbits for which the symplectic structure comes from a line bundle, then quantization leads to the irreducible representations V of G. This is the physical interpretation of the Borel–Weil theorem or the Borel–Weil–Bott theorem.We should consider periodic boundary conditions given by closed loops in a compact symplectic manifold X.Floer has given a rigorous treatment, i.e. Floer homology, based on Witten's Morse theory ideas; for the case when the boundary conditions are over the interval instead of being periodic, the path initial and end-points lie on two fixed Lagrangian submanifolds.This might not have been considered strictly topological quantum field theory at the time because Hilbert spaces are infinite dimensional.Quantizing these produces the Hilbert spaces of the theory of irreducible (projective) representations of LG.As a result, the partition function in such theories depends on complex structure, thus it is not purely topological.The integer multiple k, called the level, is a parameter of the theory and k → ∞ gives the classical limit.The details have been described by Witten who shows that the partition function for a (framed) link in the 3-sphere is just the value of the Jones polynomial for a suitable root of unity.The theory can be defined over the relevant cyclotomic field, see Atiyah (1988) harvtxt error: no target: CITEREFAtiyah1988 (help).Donaldson has defined the integer invariant of smooth 4-manifolds by using moduli spaces of SU(2)-instantons.Witten (1988a) has produced a super-symmetric Lagrangian which formally reproduces the Donaldson theory.The Hamiltonian version of the theory has been developed by Floer in terms of the space of connections on a 3-manifold.As above, regard two morphisms in Bordn as equivalent if they are homotopic, and form the quotient category hBordn.A TQFT on n-dimensional manifolds is then a functor from hBordn to the category of vector spaces, which maps disjoint unions of bordisms to their tensor product.For example, for (1 + 1)-dimensional bordisms (2-dimensional bordisms between 1-dimensional manifolds), the map associated with a pair of pants gives a product or coproduct, depending on how the boundary components are grouped – which is commutative or cocommutative, while the map associated with a disk gives a counit (trace) or unit (scalars), depending on the grouping of boundary components, and thus (1+1)-dimension TQFTs correspond to Frobenius algebras.Furthermore, we can consider simultaneously 4-dimensional, 3-dimensional and 2-dimensional manifolds related by the above bordisms, and from them we can obtain ample and important examples.Stochastic (partial) differential equations (SDEs) are the foundation for models of everything in nature above the scale of quantum degeneracy and coherence and are essentially Witten-type TQFTs.This supersymmetry preserves the continuity of phase space by continuous flows, and the phenomenon of supersymmetric spontaneous breakdown by a global non-supersymmetric ground state encompasses such well-established physical concepts as chaos, turbulence, 1/f and crackling noises, self-organized criticality etc.
gauge theorymathematical physicsquantum field theorytopological invariantsknot theoryfour-manifoldsalgebraic topologymoduli spacesalgebraic geometryDonaldsonWittenKontsevichFields Medalscondensed matter physicstopologically orderedfractional quantum Hallstring-netstrongly correlated quantum liquidcorrelation functionsmetricspacetimeMinkowski spacetimecontracted to a pointRiemann surfacesdefined on spacetimesbackground-independentsigma modelcountably infinitelypartition functionsBF modelA. SchwarzChern–Simons theoryknot invariantstopological Yang–Mills theoryBRST-operatorstring theoryLie derivativeobservablesfunctional derivativeHaar measureAtiyahconformal field theoryfunctorcategorycobordismsvector spacescommutative ringhomotopydiffeomorphismshermitianadjointHilbert spaceHamiltonianvacuum expectation valuestatistical mechanicspartition functionFeynman path integralLagrangianCasson invariantDonaldson invariantGromov's theoryFloer homologyJones–Witten theorysymmetric groupsymplectic manifoldphase spaceline bundleBorel–Weil theoremBorel–Weil–Bott theoremholonomyrepresentation theoryLie groupspseudo holomorphic mapsholomorphic mapsKähler manifoldMorse theoryLagrangian submanifoldsGromov–Witten invariantHolomorphicloop groupcomplex structureChern–Simons functionJones polynomialcyclotomic fieldRiemann surfaceGauss–Bonnet theoremSeiberg–Witten gauge theorysubmanifoldsconnectedhomotopicsymmetric monoidal functorequivalence of categoriesFrobenius algebraspair of pantsmonoidal categorytopological string theoryquantum knot invariantsSupersymmetric theory of stochastic dynamicsStochastic (partial) differential equationsBRST supersymmetryexterior derivativeturbulencecracklingself-organized criticalityQuantum topologyTopological defectTopological entropy in physicsTopological orderTopological quantum numberTopological quantum computerArithmetic topologyCobordism hypothesisAtiyah, MichaelPublications Mathématiques de l'IHÉSSchwarz, AlbertSegal, GraemeBibcodeWitten, EdwardQuantum field theoriesAlgebraic QFTAxiomatic QFTLattice field theoryNoncommutative QFTQFT in curved spacetimeSupergravityThermal QFT