New Dimensions for Wound Strings:The Modular Transform of Geometry to Topology Page: 22 of 29
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Following , we can consider the late-time Ricci flow in a very supercritical limit of
our system, as in the above discussion of the Sol geometry in @3.5, which introduced the
relevant framework and notation. In the Nil case, the string-frame metric is at large t
1/3 1 1 2 t1/3
ds2 = -dt2 + Q1/3 (dx + -ydz - -zdy + dy2 + dz2) + dx_, (4.1)
leading to a 4d Einstein frame metric
Q1/2 / 1__ 1 t/
dstdE = e2Qt ydt2 + 7> (dx + 2ydz - zdy 2 1/3 (dy dz2)) (4.2)
and a scalar Laplacian
/'1 \ ti/3
e2QtV2n = - a/t - + 2Q) t + 3 Q/3 (4.
7< / (asi + a, (-Ya. zay)axq ! (y2 z2)a2,h).
We are interested in the possibility of an enhanced IR divergence in the compact
quotient of the Nil geometry. Such new pseudotachyons would require normalizable modes
for which Hubble friction dominates at late times. Since the x direction shrinks, modes
with momentum in the kx direction blueshift, with gradient energy dominating. Let us
therefore consider kx = 0 modes. For these, (4.3) boils down to
- - + 2Q Om + t Q 3 (a + ae). (4.4)
As in the Sol case, we can read off a new contribution to the Hubble friction arising from
the Nil directions (the first term in square brackets). In contrast to the Sol case, here the
new contribution to the Hubble friction squared is subdominant to the gradient energy
in the y, z directions. This means these modes should not gain a new pseudotachyonic
contribution in addition to the usual one from the supercritical linear dilaton, which is
consistent with the absence of exponential growth of the fundamental group. Also as in
Sol, one obtains a similar result for the field canonically normalized with respect to the
proper time eQtI/Q.
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McGreevy, John; Silverstein, Eva & Starr, David. New Dimensions for Wound Strings:The Modular Transform of Geometry to Topology, article, December 18, 2006; [Menlo Park, California]. (digital.library.unt.edu/ark:/67531/metadc877426/m1/22/: accessed January 23, 2019), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.