9/15. Eigenstates of open strings on nonflat complex manifolds
A spatial gauge transformation of the 3+1 large spacetime dimensions
rotates the spacetime frame to \(\color{gold}{t}\)-bit up.
The 26 dimensions
\(X^\kappa\)
in which the bosonic strings propagate comprise
\[
26
=
\underset{\textrm{spacetime}}{1{+}1}
+
\underset{\textrm{fermionic D10-brane}}{10}
+
\underset{\textrm{bosonic 14-torus}}{14}
\ .
\]
Gauge transform the string metric to be conformally flat.
Wick rotate the 2 string coordinates
\(\sigma^\alpha \equiv \{ \tau , \sigma \}\)
to the conformally Euclidean complex plane
(\(L\) is a fundamental scale, the string scale),
\[
z \equiv e^{( \tau + i \sigma ) / L}
\ , \quad
\tilde{z} \equiv e^{( \tau - i \sigma ) / L}
\ .
\]
In curved spacetime, the Nambu-Goto string action implies the
string wave equation
\[
\partial_z \partial_{\tilde{z}} X^\kappa
+
\Gamma^\kappa_{\mu\nu}
\partial_z X^\mu
\partial_{\tilde{z}} X^\nu
=
0
\ .
\]
In Minkowski spacetime all coordinate connections
\(\Gamma^\kappa_{\mu\nu}\) vanish,
and the general solution of the wave equation is a sum
\(X^\kappa(z) + \tilde{X}{}^\kappa(\tilde{z})\)
of right- and left-moving modes.
But
in a curved complex manifold the wave equation also has
open-string solutions in which all modes are right-moving,
functions only of \(z\).
Expand right-moving modes as
\(X^\kappa(z)\) as
\[
X^\kappa(z)
=
x^\kappa
+
i L
\biggl(
( k^\kappa - l^\kappa ) \ln z
+
\sum_{m \neq 0}
\frac{\alpha_m^\kappa}{m} z^m
\biggr)
\ ,
\]
where
\(k^\kappa\) and \(l^\kappa\) are dual momentum and winding modes
on the 14-torus,
and
\(\alpha_m^\kappa\) are creation (\(m < 0\)) and destruction (\(m > 0\))
operators for a ladder of Kaluza-Klein modes for each coordinate \(\kappa\).
The \(z\) can be treated as a Fourier mode \(z = e^{i \theta}\),
with conjugate \(1/z = e^{-i \theta}\), and
quantization of open-string eigenmodes carries through
in a curved complex manifold the same as in Minkowski space.
10/15. Gauge and gravitational anomalies vanish
