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Higherdimensional algebra and topological quantum field theory
 Jour. Math. Phys
, 1995
"... For a copy with the handdrawn figures please email ..."
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Cited by 138 (14 self)
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For a copy with the handdrawn figures please email
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
"... to appear in proceedings of the ..."
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 16 (2 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.
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 case G ..."
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Cited by 10 (5 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....
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 9 (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.
Orbifold string topology
"... Abstract. In this paper we study the string topology (á la ChasSullivan) of an orbifold. We define the string homology ring product at the level of the free loop space of the classifying space of an orbifold. We study its properties and do some explicit calculations. ..."
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Cited by 7 (3 self)
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Abstract. In this paper we study the string topology (á la ChasSullivan) of an orbifold. We define the string homology ring product at the level of the free loop space of the classifying space of an orbifold. We study its properties and do some explicit calculations.
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 5 (0 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
Morse field theory
, 2008
"... In this paper we define and study the moduli space of metricgraph flows in a manifold M. This is a space of smooth maps from a finite graph to M, which, when restricted to each edge, is a gradient flow line of a smooth (and generically Morse) function on M. Using the model of GromovWitten theory, ..."
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Cited by 3 (0 self)
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In this paper we define and study the moduli space of metricgraph flows in a manifold M. This is a space of smooth maps from a finite graph to M, which, when restricted to each edge, is a gradient flow line of a smooth (and generically Morse) function on M. Using the model of GromovWitten theory, with this moduli space replacing the space of stable holomorphic curves in a symplectic manifold, we obtain invariants, which are (co)homology operations in M. The invariants obtained in this setting are classical cohomology operations such as cup product, Steenrod squares, and StiefelWhitney classes. We show that these operations satisfy invariance and gluing properties that fit together to give the structure of a topological quantum field theory. By considering equivariant operations with respect to the action of the automorphism group of the graph, the field theory has more structure. It is analogous to a homological conformal field theory. In particular we show that classical relations such as the Adem relations and Cartan formulae are consequences of these field theoretic properties. These operations are defined and studied using two different methods. First, we use algebraic topological techniques to define appropriate virtual fundamental classes of these moduli spaces. This allows us to define the
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 3 (2 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.
STRING TOPOLOGY OF CLASSIFYING SPACES
, 2007
"... Let G be a finite group or a compact connected Lie group and let BG be its classifying space. Let LBG: = map(S 1, BG) be the free loop space of BG i.e. the space of continuous maps from the circle S 1 to BG. The purpose of this paper is to study the singular homology H∗(LBG) of this loop space. We ..."
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Cited by 2 (0 self)
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Let G be a finite group or a compact connected Lie group and let BG be its classifying space. Let LBG: = map(S 1, BG) be the free loop space of BG i.e. the space of continuous maps from the circle S 1 to BG. The purpose of this paper is to study the singular homology H∗(LBG) of this loop space. We prove that when taken with coefficients in a field the homology of LBG is a homological conformal field theory. As a byproduct of our main theorem, we get on the cohomology H ∗ (LBG) a BValgebra structure.