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Witten, in this paper, equation (3.42), says the integral over the complex plane of the imaginary part of a polynomial g(z)

$$\int dz d\bar z \exp( g(z)-\bar g (\bar z))$$

is convergent. How can I show it? Is there a good reference?

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1 Answer 1

What Witten write is:

Introduce a complex variable $z$ and a complex-valued polynomial $$g(z) = \sum_{j=0}^n a_n z^n.\qquad (2.7)$$ Now consider the integral $$Z_g=\int|d^2z|\exp\bigl(g(z)-\overline{g(z)}\bigr).\qquad (2.8)$$ This again is a convergent oscillatory integral and a closer analog of complex Chern-Simons theory, with $-ig(z)$ and $ig(z)$ corresponding to the terms $tW$ and $\widetilde{t}\overline{W}$ in the action (2.3).

We stress that no contour integral is intended in (2.8) --- a contour integral could scarcely be intended here as the integrand is not holomorphic! Rather, if $z=u+iv$, with real $u$, $v$, we integrate separately over $u$ and $v$, the integration measure being $|d^2z|=2\,du\,dv$.

(end of quote)

But this is plainly false: Take $g(z)=z$, then $$g(z)-\overline{g(z)}=u+iv-(u-iv)=2iv$$ and the integral $$\int e^{2iv} du dv$$ is not convergent.

Maybe he has forgotten to say that the integral is extended to a measurable set $M\subset \textbf{C}$ of finite Lebesgue measure.

I think a more complex polynomial will not solve the problem.

Perhaps he intended to integrate only with respect to $v$. But then the measure will not be as he said explicitly $du dv$ but only $dv$. The integral will not be absolutely convergent. But this may be salvaged if he consider a polynomial of degree $\ge2$ and take principal value of the integral

Then we may translate in the following way. If $g$ is a complex plynomial of degree $\ge2$ then $$\int_{-\infty}^{+\infty} e^{i\Im g(iv)} dv=\lim_{T\to+\infty}\int_{-T}^T e^{i\Im g(iv)} dv$$ exists. For example taking $g(z)=-\lambda(z^3+z)$, with $\lambda$ real,
we get $g(iv)=i\lambda( v^3/3 -v)$ so that we obtain as particular case Airy's integral $$\int_{-\infty}^{+\infty} e^{i\lambda(v^3/3-v)} dv\qquad (2.6)$$ As the proper Witten write in equation (2.6) of the paper.

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Thank you very much for your input! Could you suggest a reference where I can look for something related? I'm also OK if it converges as a distribution ... –  Yuji Tachikawa Jan 16 '13 at 5:23

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