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From simplicial Chern-Simons theory to the shadow invariant I

### Abstract

This is the first 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 will 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 Ref.
38
whose comments motivated me to look for an alternative approach for making sense of the RHS of Eq.
(2.42)
, which is less technical than the continuum approach in Refs.
36
,
38
, and
39
. This eventually led to the present paper and its sequel Ref.
40
. Moreover, I would like to thank Laurent Freidel for pointing out to me the widespread confusion about the “shift in k”-issue, cf. Remark 3.2 in Sec.
III A
above.

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 not only had a major impact on the presentation and overall structure of the present paper but also inspired me to reconsider the issue of discretizing the operator ∂
t
+ ad(B) appearing in
(5.13)
above. (This eventually led me to the operators
(5.5)
in Sec.
V B
above.)

Article outline:

I. INTRODUCTION
II. CHERN-SIMONS THEORY ON *M* = Σ × *S*
^{1} IN THE TORUS GAUGE
A. Chern-Simons theory
B. Torus gauge fixing
1. Motivation
2. Two heuristic concepts: “generalized” and “abstract” gauge fixing
3. Torus gauge fixing for non-compact *M* = Σ × *S*
^{1}
4. Torus gauge fixing for compact *M* = Σ × *S*
^{1}
C. Chern-Simons theory in the torus gauge
1. Application of Eq. (2.31) to Chern-Simons theory
2. Rewriting *S*_{CS}
(*A*
^{⊥} + *Bdt*)
3. The final heuristic formula
4. Some remarks
III. THE GENERAL SIMPLICIAL PROGRAM FOR CS THEORY AND *BF*
_{3}-THEORY
A. Overview
B. Potential applications
IV. A “SIMPLICIAL” DIFFERENTIAL GEOMETRIC FRAMEWORK
A. Chains and cochains
B. Simplicial curves, loops, and links
C. Simplicial ribbons and ribbon links
D. Some special (polyhedral) cell complexes
1. The cell complex ℤ_{
N
}
2. The cell complexes and
3. The cell complex
4. The cell complexes 𝒦 × ℤ_{
N
}, 𝒦′ × ℤ_{
N
}, and *q*𝒦 × ℤ_{
N
}
E. Discrete Hodge star operators
V. A SIMPLICIAL REALIZATION OF *WLO*(*L*)
A. Definition of the spaces , , , , and
B. Discrete analogue of the operator
1. The operators , , and
2. The operator *L*
^{(N)}(*B*)
C. Definition of
D. Definition of and
1. Simplicial loop case
2. Simplicial ribbon case
E. Definition of
F. Discrete version of 1_{
C
∞(Σ,𝔱
reg
)}(*B*)
G. Discrete version of the decomposition
H. Discrete versions of the two Gauss-type measures in Eq. (2.53)
I. Definition of and WLO_{
rig
}(*L*)
J. Two modifications
1. Modification (Mod1)
2. Modification (Mod2)
VI. THE MAIN RESULT
A. A special class of simplicial links
B. A special class of simplicial ribbon links
C. The main result
VII. OUTLOOK

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2016-10-25

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