Bundle Gerbes for Chern-Simons and Wess-Zumino-Witten Theories
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Citations
Geometric cycles, index theory and twisted K-homology
Multiplicative bundle gerbes with connection
A higher stacky perspective on Chern-Simons theory
Thom isomorphism and Push-forward map in twisted K-theory
Categorified Symplectic Geometry and the String Lie 2-Algebra
References
The Yang-Mills equations over Riemann surfaces
Topological Gauge Theories and Group Cohomology
Characteristic forms and geometric invariants
Loop Spaces, Characteristic Classes and Geometric Quantization
Related Papers (5)
Frequently Asked Questions (14)
Q2. What is the choice of a finite dimensional principal G-bundle?
Given a principal G-bundle P with a connection A over M and an integer n > max{5, dimM}, there is a choice of nconnected finite dimensional principal G-bundle En with a connection
Q3. What is the corresponding left-invariant closed 3-form on BG?
The corresponding left-invariant closed 3-form ωφ on G is an integer multiple of the standard 3-form < θ, [θ, θ] > where θ is the left-invariant MaurerCartan form on G and < , > is the symmetric bilinear form on the Lie algebra of G defined by Φ ∈ I2(G).
Q4. What is the holonomy of the bundle gerbe G over 0?
The authors can construct a flat G-bundle over Σ0,3 ×Σ with boundary orientation given in such a way that the usual holonomies for flat G-bundle are σ1, σ2 and σ1 · σ2 respectively.
Q5. What is the proof of the definition of a Deligne characteristic class?
Note that the characteristic classes only depend on the underlying topological principal G-bundle, in order to define a Deligne cohomology valued characteristic class, the authors will restrict ourselves to differentiable principal G-bundles.
Q6. What is the space of 2-morphisms between two stable isomorphisms?
Note that the space of 2-morphisms between two stable isomorphisms is one-to-one corresponding to the space of line bundles over M .Consider the face operators πi : X [n] → X [n−1] on the simplicial manifold X• ={Xn = X [n+1]}.
Q7. What is the holonomy of the so-called B field?
It implies that for the transgressive Wess-Zumino-Witten models, the so-called B field satisfies a certain integrality condition.
Q8. What is the definition of a three dimensional Chern-Simons gauge theory?
The authors make the following definitions:(1) A three dimensional Chern-Simons gauge theory with gauge group G is defined to be a Deligne characteristic class of degree 3 for a principal Gbundle with connection.
Q9. what is the proof of a class of principal G-bundles?
The proof is: given a characteristic class c, the authors have, of course, c(EG) ∈ H∗(BG,Z) and conversely if ξ ∈ H∗(BG,Z) is given, then defining cξ(P ) = f∗(ξ) for any classifying map f : M → BG gives rise to a characteristic class for the isomorphism class of principal G-bundles defined by the classifying map f .
Q10. What is the definition of a bundle gerbe on a simplicial manifold?
The authors begin with the definition of a simplicial bundle gerbe as in [44] on a simplicial manifold X• = {Xn}n≥0 with face operators di : Xn+1 → Xn (i = 0, 1, · · · , n+ 1).
Q11. What is the equivalence class of a bundle 2-gerbe with connection and?
Recall that for a connection on a line bundle over M , a gauge transformation is given by a smooth function M → U(1), and an extended gauge transformation for a bundle gerbe with connection and curving is given by a line bundle with connection over M .
Q12. What is the relevance of the Chern-Simons gauge theory?
The relevance of Chern-Simons gauge theory has been noted by many authors, starting with Ramadas-Singer-Weitsman [43] and recently Dupont-Johansen [20], who used gauge covariance of the Chern-Simons functional to give a geometric2000 Mathematics Subject Classification.
Q13. What is the commutative diagram of a homotopy class?
This commutative diagram gives rise to a homotopy class of maps ψ̂ : G→ K(Z, 3) which determines a classcψ ∈ H 3(G,Z) ∼= [G,K(Z, 3)].
Q14. What is the role of Deligne cohomology in topological field theories?
The role of Deligne cohomology as an ingredient in topological field theories goes back to [27] and the authors add a new feature in section 2 by using Deligne cohomologyvalued characteristic classes for principal G-bundles with connection.