In this paper,
https://projecteuclid.org/euclid.cmp/1104248307
on path-integral quantization of Chern-Simons theory, on page 434 (equation 4.17), the authors used fermions to interpret wedge product and contractions of differential forms.
Let M be a manifold, with local coordinate xi. For any differential form a∈Ω(M), one has the operations
ψi:a→dxi∧a,
and
χj:a→a(∂j).
One has the Clifford algebra
{ψi,χj}=δij,{ψi,ψj}={χi,χj}=0
Define the Witten-index (−1)F as
(−1)F:ω→(−1)qw,for∀ω∈Ωq(M).
Then one has the relation (equation 4.17)
∗ψi∗=(−1)Fχi,∗χi∗=ψi(−1)F
where ∗ must be a Hodge star operator (I will assume that there is a Riemannian metric on M so that ∗2=1.)
Can anybody explain to me how to derive the relations (4.17)
I also posted my question here: https://physics.stackexchange.com/q/439992/185558
New Edition
I calculated this by myself but I cannot obtain the correct (−1)F factor.
Let ω∈Ωq(M) be a differential form on M. In local coordinates, one has
ω=1q!ωi1⋯iqdxi1∧⋯∧dxiq
Hodge star operator is defined as
∗:Ωq(M)→Ωn−q(M)
such that ∗2=1.
One has
(∗ω)j1⋯jn−q=1q!ϵi1⋯iqj1⋯jn−qωi1⋯iq
where the ϵ symbol is raised by the metric tensor. Therefore, one has
∗ω=1(n−q)!(1q!ϵi1⋯iqj1⋯jn−qωi1⋯iq)dxj1∧⋯∧dxjn−q
Then, one has
ψi∗ω=dxi∧∗ω
=1(n−q+1)!((n−q+1)!(n−q)!q!ϵi1⋯iqj1⋯jn−qωi1⋯iq)dxi∧dxj1∧⋯∧dxjn−q
Applying the Hodge star operator again, one has
(∗ψi∗ω)k1⋯kq−1=1(n−q+1)!ϵij1⋯jn−qk1⋯kq−1(ψi∗ω)ij1⋯jn−q
Thus, one has
(∗ψi∗ω)k1⋯kq−1=1(n−q)!q!ϵij1⋯jn−qk1⋯kq−1ϵi1⋯iqj1⋯jn−qωi1⋯iq
Rearranging indices of ϵ tensors, one has
ϵij1⋯jn−qk1⋯kq−1ϵi1⋯iqj1⋯jn−q=(−1)(q−1)(n−q)ϵik1⋯kq−1j1⋯jn−qϵi1⋯iqj1⋯jn−q
Using contraction rules of ϵ tensor, one has
∗ψi∗=(−1)(q−1)(n−q)χi
I expect to have (−1)q. Where did I make mistakes?