• Corpus ID: 119682962


  author={Daniel S. Freed},
  • D. Freed
  • Published 4 June 1992
  • Mathematics
There is a large mathematical literature on classical mechanics and field theory, especially on the relationship to symplectic geometry. One might think that the classical Chern-Simons theory, which is topological and so has vanishing hamiltonian, is completely trivial. However, this theory exhibits interesting geometry that is usually absent from ordinary field theories. (The same is true on the quantum level; topological quantum field theories exhibit geometric properties not usually seen in… 
Classical Chern-simons Theory, Part 2
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Topological Quantum Field Theories from Compact Lie Groups
It is a long-standing question to extend the definition of 3-dimensional Chern-Simons theory to one which associates values to 1-manifolds with boundary and to 0-manifolds. We provide a solution in
Lectures on Topological Quantum Field Theory
What follows are lecture notes about Topological Quantum Field Theory. While the lectures were aimed at physicists, the content is highly mathematical in its style and motivation. The subject of
A Higher Stacky Perspective on Chern–Simons Theory
The first part of this text is a gentle exposition of some basic constructions and results in the extended prequantum theory of Chern–Simons-type gauge field theories. We explain in some detail how
Classical Chern-Simons on manifolds with spin structure
We construct a 2+1 dimensional classical gauge theory on manifolds with spin structure whose action is a refinement of the Atiyah-Patodi- Singer eta-invariant for twisted Dirac operators. We
Twisted Equivariant Matter
We show how general principles of symmetry in quantum mechanics lead to twisted notions of a group representation. This framework generalizes both the classical threefold way of real/complex/
v 2 8 J un 1 99 3 REVISED VERSION Higher Algebraic Structures and Quantization
Very Abstract. We derive (quasi-)quantum groups in 2+1 dimensional topological field theory directly from the classical action and the path integral. Detailed computations are carried out for the
A higher Chern-Weil derivation of AKSZ sigma-models
Chern–Weil theory provides for each invariant polynomial on a Lie algebra 𝔤 a map from 𝔤-connections to differential cocycles whose volume holonomy is the corresponding Chern–Simons theory action
A Chern-Simons action for noncommutative spaces in general with the example SU_q(2)
Witten constructed a topological quantum field theory with the Chern-Simons action as Lagrangian. We define a Chern-Simons action for 3-dimensional spectral triples. We prove gauge invariance of the
Dirac Charge Quantization and Generalized Differential Cohomology
The main new result here is the cancellation of global anomalies in the Type I superstring, with and without D-branes. Our argument here depends on a precise interpretation of the 2-form abelian


Loop Spaces, Characteristic Classes and Geometric Quantization
This book deals with the differential geometry of manifolds, loop spaces, line bundles and groupoids, and the relations of this geometry to mathematical physics. Recent developments in mathematical
Higher algebraic structures and quantization
We derive (quasi-)quantum groups in 2+1 dimensional topological field theory directly from the classical action and the path integral. Detailed computations are carried out for the Chern-Simons
Topological gauge theories and group cohomology
We show that three dimensional Chern-Simons gauge theories with a compact gauge groupG (not necessarily connected or simply connected) can be classified by the integer cohomology groupH4(BG,Z). In a
Some comments on Chern-Simons gauge theory
Following M. F. Atiyah and R. Bott [AB] and E. Witten [W], we consider the space of flat connections on the trivialSU(2) bundle over a surfaceM, modulo the space of gauge transformations. We describe
Quantum field theory and the Jones polynomial
It is shown that 2+1 dimensional quantum Yang-Mills theory, with an action consisting purely of the Chern-Simons term, is exactly soluble and gives a natural framework for understanding the Jones
On the rigidity theorems of Witten
In this paper we prove the rigidity theorems predicted by Witten in 1986, about the index of certain elliptic operators on manifolds with an S1 action [W]. Witten's insight was the culmination of an
Stable and unitary vector bundles on a compact Riemann surface
Let X be a compact Riemann surface of genus g _ 2. A holomorphic vector bundle on X is said to be unitary if it arises from a unitary representation of the fundamental group of X. We prove in this
The Yang-Mills equations over Riemann surfaces
  • M. Atiyah, R. Bott
  • Mathematics
    Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences
  • 1983
The Yang-Mills functional over a Riemann surface is studied from the point of view of Morse theory. The main result is that this is a ‘perfect' functional provided due account is taken of its gauge
Moduli of vector bundles on curves with parabolic structures
Let H be the upper half plane and F a discrete subgroup of AutH. When H mo d F is compact, one knows that the moduli space of unitary representations of F has an algebraic interpretation (cf. [7] and
Taming the Conformal Zoo