Pome

Proposition 17.16.29A3E893F6A601

Proposition 17.16.29.A3E893F6A601 The pullback $\pi ^*v_ t$ is the full Hermitian Perron segment and is stationary under all smooth fixed-endpoint variations. Its distance and action are

\[ \begin{aligned} d_ B(0,\pi ^*v_1)^2& =\frac{2[(\sum _ ja_ j)^2+\sum _ ja_ j^2]}{n+1},\\ \mathscr S(\pi ^*v)& =\mathcal A_ B(\pi ^*v) =\frac{(\sum _ ja_ j)^2+\sum _ ja_ j^2}{n+1}. \end{aligned} \]

Proof. On the strip, the holomorphic map $F([z],s)=[\mathrm e^{a_0s/2}z_0:\cdots :\mathrm e^{a_{n-1}s/2}z_{n-1}]$ satisfies $\operatorname {pr}_ Y^*\kappa +\mathrm{dd}^{\mathrm{c}}V=F^*\kappa $. The imaginary-time phases are unitary, so $V$ depends only on $\operatorname{Re}s$. This form has rank $d$, strictly positive spatial slices, and zero $(d+1)$st power. The classical bounded Kähler Dirichlet uniqueness identifies it with the base Perron segment; proposition 17.16.16, proposition 17.16.22 give the full segment and its stationarity.

For $x_ j=|z_ j|^2/\sum |z_ j|^2$, the initial velocity is $\sum _ ja_ jx_ j$. A standard complex Gaussian vector has independent exponential squared coordinates, and its projective direction has distribution $\kappa ^ d$. Dividing those coordinates by their sum gives the uniform simplex distribution, with moments

\[ \int _ Yx_ i^2\kappa ^ d=\frac2{n(n+1)}, \qquad \int _ Yx_ ix_ j\kappa ^ d=\frac1{n(n+1)}\quad (i\ne j). \]

The smooth distance formula and (17.16.20) give the claimed squared distance; the action is half of it. $\square$

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