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From simplicial Chern-Simons theory to the shadow invariant II

### Abstract

This is the second of a series of papers in which we introduce and study a rigorous “simplicial” realization of the non-Abelian Chern-Simons path integral for manifolds M of the form M = Σ × S
^{1} and arbitrary simply connected compact structure groups G. More precisely, we introduce, for general links L in M, a rigorous simplicial version WLO
rig
(L) of the corresponding Wilson loop observable WLO(L) in the so-called “torus gauge” by Blau and Thompson [Nucl. Phys. B 408(2), 345–390 (1993)]. For a simple class of links L, we then evaluate WLO
rig
(L) explicitly in a non-perturbative way, finding agreement with Turaev’s shadow invariant
.

© 2015 AIP Publishing LLC

Received 26 September 2014
Accepted 28 January 2015
Published online 06 March 2015

Acknowledgments:
I want to thank the anonymous referee of my paper
^{11}
whose comments motivated me to look for an alternative approach for making sense of the RHS of (the original version of) Eq.
(2.7)
, which is less technical than the continuum approach in Refs.
9–12
. This eventually led to Ref.
13
and the present paper.

I am also grateful to Jean-Claude Zambrini for several comments which led to improvements in the presentation of the present paper.

Finally, it is a great pleasure for me to thank Benjamin Himpel for many useful and important comments and suggestions which had a major impact on the presentation and overall structure of the present paper.

Article outline:

I. INTRODUCTION
II. THE BASIC HEURISTIC FORMULA
A. Basic spaces
B. The heuristic Wilson loop observables
C. The basic heuristic formula
III. SIMPLICIAL REALIZATION OF WLO(*L*)
A. Review of the simplicial setup
B. The basic spaces
1. The decomposition
C. Discrete analogue of the operator
D. Definition of
E. Definition of
F. Definition of
G. Discrete version of 1_{
C
∞(Σ,𝔱
reg
)}(*B*)
H. Discrete versions of the two Gauss-type measures in Eq. (2.7)
I. Definition of and WLO_{
rig
}(*L*)
J. Two modifications
1. Modification (Mod1)
2. Modification (Mod2)
K. The main result
IV. OSCILLATORY GAUSS-TYPE MEASURES ON EUCLIDEAN SPACES
A. Basic definitions
B. Three propositions
V. PROOF OF THEOREM 3.5
A. Some preparations
1. Computation of det(*L*
^{(N)}(*B*))
2. Some consequences of conditions (NCP)’ and (NH)’
B. Step 1: Performing the integration in Eq. (3.21)
C. Step 2: Performing the -integration in (5.18)
D. Step 3: Some simplifications
E. Step 4: Performing the remaining limit procedures ∫^{∼}⋯*db*, **∑**
_{
y∈I
}, and *s* → 0 in Eq. (5.44)
F. Step 5: Rewriting Det^{
disc
}(*B*) in Eq. (5.53)
G. Step 6: Comparison of WLO_{
rig
}(*L*) with the shadow invariant
VI. A BRIEF COMMENT REGARDING GENERAL SIMPLICIAL RIBBON LINKS
VII. TRANSITION TO THE “*BF*
_{3}-THEORY POINT OF VIEW”
A. Motivation
B. The “*BF*
_{3}-theory point of view”
Step 1: “Group doubling”
Step 2: Linear change of variable
C. Simplification of some of the notation in Sec. VII B
D. Discretization of Eq. (7.7)
VIII. DISCUSSION AND OUTLOOK

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