New Dimensions for Wound Strings:The Modular Transform of Geometry to Topology Page: 17 of 29
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will loosely refer to as the oscillator levels). In the unwound sector, the SUSY breaking
is detected only by paths which propagate around a nontrivial cycle introduced by the
projection to a compact space, since the parent theory was simply flat spacetime. These
contributions are suppressed by a factor of e-R(t)m for a cycle of length R(t), so summing
over these high oscillator levels weighted by the Hagedorn density em"r 2ci/3 does not
yield a divergence in itself. The same might be expected for fixed winding number at high
oscillator level in R(t)/l1, since the winding is a subleading contribution to the sum over
random walks traced out by the string at high oscillator level (though to check this would
require a more detailed accounting of the way in which the asymptotic SUSY cancellation
of oscillator levels works). The sum over winding sectors at fixed oscillator level (say the
zero mode) probes the supersymmetry breaking directly, and there is no reason to expect
a similar exponential suppression.
It would be interesting to work out the fermion contributions in more detail. An
important issue in that regard is the contribution of the worldsheet fermion determinants
to the semiclassical path integral. Since the IR propagator for a given bosonic field is
the same in the superstring as it is in the bosonic theory, this determinant must not
change the leading l dependence in the expression (3.37). This seems plausible to us
since the dependence on the winding string length 1 arises most directly in the boundary
conditions on the bosonic embedding coordinates X , and its role in the fermionic sector
is more indirect. A full calculation of the fermionic contributions might make use of known
variants of the Selberg trace formula involving superparticles. Such formulas would need
to be used with care, however, to properly take into account the embedding of the spatial
slices into the full spacetime.
The application of our correspondence to the superstring has the feature that the
critical contribution to celf vanishes, so that the winding mode contribution is the only
source for celf. As we just discussed, we expect this contribution to survive the supertrace
over bosons and fermions. Given this, the "ten dimensional" superstring theory on a
(large) finite-volume compact hyperbolic target space is in fact a (slightly) supercritical
background of string theory.
Milnor's theorem  applies to arbitrary manifolds of negative sectional curvature.
The proof uses a result of Gunther that for any manifold M of strictly negative sectional
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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/m1/17/: accessed November 18, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.