Five-dimensional fermionic Chern-Simons theory
HJE
Five-dimensional fermionic Chern-Simons theory
Dongsu Bak 0 1 3
Andreas Gustavsson 0 1 2
Gauge Theory, Topological Field Theories
0 Box 516 , SE-75120 Uppsala , Sweden
1 Seoul 02504 , South Korea
2 Department of Physics and Astronomy, Uppsala University
3 Physics Department, University of Seoul
We study 5d fermionic CS theory with a fermionic 2-form gauge potential. This theory can be obtained from 5d maximally supersymmetric YM theory by performing the maximal topological twist. We put the theory on a five-manifold and compute the partition function. We find that it is a topological quantity, which involves the Ray-Singer torsion of the five-manifold. For abelian gauge group we consider the uplift to the 6d theory and find a mismatch between the 5d partition function and the 6d index, due to the nontrivial dimensional reduction of a selfdual two-form gauge field on a circle. We also discuss an application of the 5d theory to generalized knots made of 2d sheets embedded in 5d.
BRST Quantization; Field Theories in Higher Dimensions; Supersymmetric
1 Introduction
2 Flat gauge fields on lens spaces
3 One-dimensional fermionic Chern-Simons
4 Five-dimensional fermionic Chern-Simons
4.1
Nonabelian gauge group
5 Higher-dimensional knot theory
6 Uplift to six dimensions
7 Dimensional reduction of selfdual two-form
A The Ray-Singer torsion
A.1 Explicit computations
B The Minakshisundaram-Pleijel theorem
C Partial gauge fixing by the Faddeev-Popov method
D Gauge fixing of zero modes
D.1 Bosonic zero mode gauge fixing
D.2 Fermionic zero mode gauge fixing
D.3 Fermionic zero mode gauge fixing, once again
E A review of 3d Chern-Simons perturbation theory
E.1
The dependence on the Chern-Simons level
E.2 The exact result
F Dimensional reduction of selfdual forms on a circle
1
Introduction
Chern-Simons theory in 3d whose classical action is given by
k Z
4π
2i
has a long history. The seminal paper [1] obtained the exact result for the partition function
for S3 by indirect methods. Later exact results have been obtained in [2–4] by various
– 1 –
methods (nonabelian localization [5], abelianization, supersymmetric localization [6]) on a
large class of three-manifolds. There have also been many works that have aimed to match
such exact results with corresponding perturbative results in the large k limit [7–12]. Thus
CS theory enables one to test path integral methods against known exact results.
We may generalize abelian CS theory to 2p+1 dimensions by taking the gauge potential
to be a p-form. When p is odd, the gauge field is bosonic. However, when p is even, a
bosonic gauge field leads to a CS term that is a total derivative since Ap∧dAp = 12 d(Ap∧Ap).
For even p we shall therefore take the p-form gauge field to be fermionic and then we have
a fermionic CS theory or FCS theory for short. In the first few dimensions these CS and
FCS actions, in Lorentzian signature and with canonical normalizations, are given by
HJEP02(18)37
The most general form of the gauge symmetry variations are
S1d =
S3d =
S5d =
S7d =
i Z
2
2
In addition to the usual exact forms, we shall also include the harmonic forms (χp)0 and
(λp)0 in order to have the most general closed forms by the Hodge decomposition [13]. The
fact that abelian CS theories in various dimensions form a sequence (1.2) suggests that
they could have some common features.
It is in general a quite difficult problem to generalize abelian higher rank gauge fields
to nonabelian gauge groups. However, in 5d we automatically solve this problem since 5d
FCS is obtained from 5d maximally supersymmetric YM theory (MSYM) by performing
the maximal twist. By this twist the SO(5) R-symmetry is identified with the SO(5)
Lorentz symmetry [
14, 15
]. The twist gives one scalar nilpotent supercharge, which we can
identify as the BRST charge associated with the two-form gauge symmetry, and the action
can be interpreted as a BRST gauge fixed action for nonabelian 5d FCS theory.
For 3d CS on lens space S3/Zp = L(p; 1), the exact partition function is known.
From this exact result we can extract the perturbative expansion in 1/k. For gauge group
– 2 –
G = SU(2), the resulting perturbative expansion for p odd, is1
result from the exact result presented in [2] by expanding it out in powers of 1/k but where
we suppress the next to leading orders in each sector labeled by ℓ = 0, 1, . . . , p −21 . This
result can be rewritten in the form
Z = e i4π dim(G)ηgrav Vol (HA(0) )
1
Here HA(ℓ) denotes the unbroken gauge group by the gauge field background and τℓ,SU(2)
denotes the Ray-Singer torsion of L(p; 1) associated with SU(2) gauge group and the
holonomy labeled by ℓ = 0, . . . , p − 1. For the lens space L(p; q1, · · · , qN−1) = S2N−1/Zp we have
where
τ0,SU(2) = (τ0)3
groups are HA(0) = SU(2) and HA(ℓ) = U(1), whose volumes are
We note that U(1) corresponds to the equator of SU(2) = S3. The radius shall be chosen as
in order to match with the exact result. We can see w (...truncated)