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Higherdimensional algebra and topological quantum field theory
 Jour. Math. Phys
, 1995
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Cited by 216 (13 self)
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For a copy with the handdrawn figures please email
Topological quantum field theories from compact Lie groups. arXiv:0905.0731
, 2009
"... Let G be a compact Lie group and BG a classifying space for G. Then a class in H n 1 BG; Z leads to an ndimensional topological quantum field theory (TQFT), at least for n�1,2,3. The theory for n�1is trivial, but we include it for completeness. The theory for n�2has some infinities if G is not a fi ..."
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Cited by 41 (2 self)
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Let G be a compact Lie group and BG a classifying space for G. Then a class in H n 1 BG; Z leads to an ndimensional topological quantum field theory (TQFT), at least for n�1,2,3. The theory for n�1is trivial, but we include it for completeness. The theory for n�2has some infinities if G is not a finite group; it is a topological limit of 2dimensional YangMills theory. The most direct analog for n�3 is an L 2 version of the topological quantum field theory based on the classical ChernSimons invariant, which is only partially defined. The TQFT constructed by Witten and ReshetikhinTuraev which goes by the name ‘ChernSimons theory ’ (sometimes ‘holomorphic ChernSimons theory ’ to distinguish it from the L 2 theory) is completely finite. The theories we construct here are extended, or multitiered, TQFTs which go all the way down to points. For the n�3 ChernSimons theory, which we term a ‘0123 theory ’ to emphasize the extension down to points, we only treat the cases where G is finite or G is a torus, the latter being one of the main novelties in this paper. In other words, for toral theories we provide an answer to the longstanding question: What does ChernSimons theory attach to a point? The answer is a bit subtle as ChernSimons is an anomalous field theory of oriented manifolds. 1 This framing anomaly was already flagged in Witten’s seminal paper [Wi]. Here we interpret the anomaly as an invertible
Knots and quantum gravity: progress and prospects, to appear in the proceedings of the Seventh Marcel Grossman Meeting on General Relativity, University of California at Riverside preprint available as grqc/9410018. 38
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Topological quantum field theory for CalabiYau threefolds and G_2manifolds
 ADV. THEOR. MATH. PHYS
, 2002
"... In the past two decades we witness many fruitful interactions between mathematics and physics. One example is the DonaldsonFloer theory for oriented four manifolds. Physical considerations leads to the discovery of the SeibergWitten theory which has profound impact to our understandings of four m ..."
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Cited by 22 (4 self)
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In the past two decades we witness many fruitful interactions between mathematics and physics. One example is the DonaldsonFloer theory for oriented four manifolds. Physical considerations leads to the discovery of the SeibergWitten theory which has profound impact to our understandings of four manifolds. Another example is the Mirror Symmetry for CalabiYau manifolds. This duality transformation in the string theory leads to many surprising predictions in the enumerative geometry.
Unoriented topological quantum field theory and link homology
"... We investigate link homology theories for stable equivalence classes of link diagrams on orientable surfaces. We apply.1C1/–dimensional unoriented topological quantum field theories to BarNatan’s geometric formalism to define new theories for stable equivalence classes. 57M25, 57R56; 81T40 1 ..."
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We investigate link homology theories for stable equivalence classes of link diagrams on orientable surfaces. We apply.1C1/–dimensional unoriented topological quantum field theories to BarNatan’s geometric formalism to define new theories for stable equivalence classes. 57M25, 57R56; 81T40 1
4Dimensional BF Theory as a Topological Quantum Field Theory
 Lett. Math. Phys
, 1996
"... Starting from a Lie group G whose Lie algebra is equipped with an invariant nondegenerate symmetric bilinear form, we show that 4dimensional BF theory with cosmological term gives rise to a TQFT satisfying a generalization of Atiyah's axioms to manifolds equipped with principal Gbundle. The c ..."
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Cited by 16 (6 self)
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Starting from a Lie group G whose Lie algebra is equipped with an invariant nondegenerate symmetric bilinear form, we show that 4dimensional BF theory with cosmological term gives rise to a TQFT satisfying a generalization of Atiyah's axioms to manifolds equipped with principal Gbundle. The case G = GL(4; R) is especially interesting because every 4manifold is then naturally equipped with a principal Gbundle, namely its frame bundle. In this case, the partition function of a compact oriented 4manifold is the exponential of its signature, and the resulting TQFT is isomorphic to that constructed by Crane and Yetter using a state sum model, or by Broda using a surgery presentation of 4manifolds. 1 Introduction In comparison to the situation in 3 dimensions, topological quantum field theories (TQFTs) in 4 dimensions are poorly understood. This is ironic, because the subject was initiated by an attempt to understand Donaldson theory in terms of a quantum field theory in 4 dimensions....
Superconnections and Parallel Transport
, 2006
"... Abstract. This note addresses the construction of a notion of parallel transport along superpaths arising from the concept of a superconnection on a vector bundle over a manifold M. A superpath in M is, loosely speaking, a path in M together with an odd vector field in M along the path. We also deve ..."
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Cited by 15 (3 self)
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Abstract. This note addresses the construction of a notion of parallel transport along superpaths arising from the concept of a superconnection on a vector bundle over a manifold M. A superpath in M is, loosely speaking, a path in M together with an odd vector field in M along the path. We also develop a notion of parallel transport associated with a connection (a.k.a. covariant derivative) on a vector bundle over a supermanifold which is a direct generalization of the classical notion of parallel transport for connections over manifolds. 1.
Holomorphic Quantization of Linear Field Theory in the General Boundary Formulation
"... Abstract. We present a rigorous quantization scheme that yields a quantum field theory in general boundary form starting from a linear field theory. Following a geometric quantization approach in the Kähler case, state spaces arise as spaces of holomorphic functions on linear spaces of classical sol ..."
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Cited by 14 (7 self)
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Abstract. We present a rigorous quantization scheme that yields a quantum field theory in general boundary form starting from a linear field theory. Following a geometric quantization approach in the Kähler case, state spaces arise as spaces of holomorphic functions on linear spaces of classical solutions in neighborhoods of hypersurfaces. Amplitudes arise as integrals of such functions over spaces of classical solutions in regions of spacetime. We prove the validity of the TQFTtype axioms of the general boundary formulation under reasonable assumptions. We also develop the notions of vacuum and coherent states in this framework. As a first application we quantize evanescent waves in Klein–Gordon theory. Key words: geometric quantization; topological quantum field theory; coherent states; foundations of quantum theory; quantum field theory 2010 Mathematics Subject Classification: 57R56; 81S10; 81T05; 81T20 1
Orbifolding Frobenius algebras
, 2000
"... Abstract. We study the general theory of Frobenius algebras with group actions. These structures arise when one is studying the algebraic structures associated to a geometry stemming from a physical theory with a global finite gauge group, i.e. orbifold theories. In this context, we introduce and ax ..."
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Cited by 13 (1 self)
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Abstract. We study the general theory of Frobenius algebras with group actions. These structures arise when one is studying the algebraic structures associated to a geometry stemming from a physical theory with a global finite gauge group, i.e. orbifold theories. In this context, we introduce and axiomatize these algebras. Furthermore, we define geometric cobordism categories whose functors to the category of vector spaces are parameterized by these algebras. The theory is also extended to the graded and super–graded cases. As an application, we consider Frobenius algebras having some additional properties making them more tractable. These properties are present in Frobenius algebras arising as quotients of Jacobian ideal, such as those having their origin in quasi–homogeneous singularities and their symmetries.
Twodimensional quantum YangMills theory with corners
, 2006
"... The solution of quantum YangMills theory on arbitrary compact twomanifolds is well known. We bring this solution into a TQFTlike form and extend it to include corners. Our formulation is based on an axiomatic system that we hope is flexible enough to capture actual quantum field theories also in ..."
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Cited by 11 (7 self)
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The solution of quantum YangMills theory on arbitrary compact twomanifolds is well known. We bring this solution into a TQFTlike form and extend it to include corners. Our formulation is based on an axiomatic system that we hope is flexible enough to capture actual quantum field theories also in higher dimensions. We motivate this axiomatic system from a formal SchrödingerFeynman quantization procedure. We also discuss the physical meaning of unitarity, the concept of vacuum, (partial) Wilson loops and nonorientable surfaces.