Topological quantum field theories

By A. Schwarz. 2000

540 citations recorded by the source corpus.

Following my plenary lecture on ICMP2000 I review my results concerning two closely related topics: topological quantum field theories and the problem of quantization of gauge theories. I start with old results (first examples of topological quantum field theories were constructed in my papers in late seventies) and I come to some new results, that were not published yet.

Subfields: Mathematical Physics, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/a7f833dfd9b16d892d01f4874c42765181ec2034

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Citations in the corpus

S-duality and 2d topological QFT (2009, background): “In this section we determine explicitly the structure constants and the metric of the TQFT and write them in terms of elliptic Beta integrals.”
Reshetikhin-Turaev invariants of Seifert 3-manifolds for classical simple Lie algebras, and their asymptotic expansions (2002, background): “AMS Classification numbers Primary: 57M27 Secondary: 17B37, 18D10, 41A60 Keywords: Quantum invariants, Seifert manifolds, quantum groups, modular tensor categories, asymptotic expansions”
Traces in symmetric monoidal categories (2011, background): “An n-dimensional topological field theory [2] with values in a symmetric monoidal category C (such as Vectk) is a strong symmetric monoidal functor Z : nCob → C.”
A functor-valued invariant of tangles (2001, background): “In the original Atiyah’s definition [4] the target category for a TQFT is the category of modules over a (commutative) ring Λ.”
On the Classification of Topological Field Theories (2009, methodology): “The collection of all such functors is naturally organized into another (∞, 1)category, which we will denote by Fun([1],C).”
Quantum Quandaries: a Category-Theoretic Perspective (2004, unknown): “ar X iv :q ua nt -p h/ 04 04 04 0v 2 1 4 A”
Local subsystems in gauge theory and gravity (2016, background): “In the axiomatic approach to TQFT, every closed codimension-1 surface Σ is assigned a Hilbert space Z(Σ), and each codimension-0 surface M with boundary ∂M is associated with a state in the Hilbert space Z(∂M) [52].”
Quantum picturalism (2009, background): “Following Atiyah in [75], topological quantum field theories can be succinctly defined as monoidal functors from nCob into FHilb, where a functor is a map both on objects and on morphisms which preserves composition and tensor [22,23,76].”
Frobenius Algebras and 2-D Topological Quantum Field Theories (2004, background): “Atiyah [5]), an n-dimensional topological quantum field theory is a rule A which to each closed oriented manifold Σ (of dimension n − 1) associates a vector space ΣA , and to each oriented n-manifold whose boundary is Σ associates a vector in ΣA .”
Topological quantum field theory structure on symplectic cohomology (2010, background): “A TQFT [4] is a tensor functor from the category of 2-dimensional cobordisms between 1-manifolds to the category of vector spaces.”
On the superconformal index of Argyres–Douglas theories (2015, background): “Moreover, since two-dimensional YM theory is invariant under area-preserving diffeomorphisms, it becomes a TQFT (in the sense of [53]) in the limit we take the area of Cg,m to zero.”
Notes for a brief history of quantum gravity (2000, background): “Formalized by Atiyah [66], the idea of TQFT will have beautiful developments, and will strongly affect later development in quantum gravity.”
On the structure of open-closed topological field theory in two-dimensions (2000, background): “I consider the abstract framework of such systems along the lines of [19, 28, 29] (see [22] for a review).”
Topological sigma-models with H-flux and twisted generalized complex manifolds (2004, background): “Witten in 1988 that one can construct interesting examples of topological field theories by “twisting” supersymmetric field theories [16].”