Asymptotic estimate of a perturbative Chern-Simons integral

Itaru Mitoma Dept.of Math. Saga University

Chern-Simons integral is a kind of the Feynman integral on G-valued 1-forms on a 3dimensional compact oriented manifold.

∫A F (A)e^{L(A)}D(A),

L(A) = −\frac{ik}{4π} T r ∫_{M} A ⋀ dA + \frac{2}{3}A ⋀ A ⋀ A.

Fix the Lorentz Gäge and use the method of super fields. Then we get the path integral representation where L(A) has an isolated critical point A_{0}.

∫_{C′} ∫_{C′} ∫_{Φ} ∫_{{A∈A:(D^{0})^{∗}A=0}} F (A_{0} + A)e^{L(A_{0})}

e^{ik((A,ϕ),Q_{0}(A,ϕ))_{+}− \frac{ik}{4π} ∫_{M} T r \frac{2}{3} A ⋀ A ⋀ A−∫_{M} T rc′∗D^{0}∗D_{A}c′}

D(A)D(ϕ)D(c′)D(ĉ′).

Here， ϕ, c′, ĉ′ are external bosonic 3-form, fermionic 0-form and 3-form respectively and Q_{0} is a Dirac operator.

Since formally

∫_{C′} ∫_{C′} e^{− ∫_{M} T rc′∗D^{0}∗D_{A}c′}D(ĉ′)D(c′)

= det ∗D^{0} ∗ D_{A}.

= det Δ det_{R,E} ∗D^{0} ∗ D_{A},

setting x = (A, ϕ), x_{A} = A and defining

F̃(A_{0} + x_{A}) = F (A_{0} + x_{A}) det_{R,E} ∗D^{0} ∗ D_{x_{A}},

we arrive at the normalized Chern-Simons integral

= \frac{1}{Z} ∫_{(D^{0})^{∗}x_{A}=0} F̃(A_{0} + x_{A})e^{ik(x,Q_{0}x)_{+}− \frac{ik}{4π} ∫_{M} T r \frac{2}{3} x_{A} ⋀ x_{A} ⋀ x_{A}}D(x)

where

Z = ∫_{(D^{0})^{∗}x_{A}=0} e^{ik(x,Q_{0}x)_{+}}D(x).

For any natural number N, we have

\frac{1}{Z} ∫_{(D^{0})^{∗}x_{A}=0} F̃(A_{0} + x_{A})e^{ik(x,Q_{0}x)_{+}− \frac{ik}{4π} ∫_{M} T r \frac{2}{3} x_{A} ⋀ x_{A} ⋀ x_{A}}D(x)

= \frac{1}{Z} ∑_{r=1}^{N} ∫_{(D^{0})^{∗}x_{A}=0} F̃(A_{0} + x_{A})(−\frac{ik}{4π} ∫_{M} T r\frac{2}{3}x_{A} ⋀ x_{A} ⋀ x_{A})^{r}/r!e^{ik(x,Q_{0}x)_{+}}D(x)

+\frac{1}{Z} ∫_{(D^{0})^{∗}x_{A}=0} F̃(A_{0} + x_{A})\{ ∑_{r=N+1}^{∞}(−\frac{ik}{4π} ∫_{M} T r\frac{2}{3}x_{A} ⋀ x_{A} ⋀ x_{A})^{r}/r!\}e^{ik(x,Q_{0}x)_{+}}D(x)

In this talk, by estimating the above each term through the Wiener space realization, we show the asymptotic expansion of the normalized Chern-Simons integral.