New Dimensions for Wound Strings:The Modular Transform of Geometry to Topology Page: 5 of 29
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By modular invariance, the dominant contribution in the T2 -- 0 region must go like
ZUv ~ e-wT nmin/T2. (2.7)
So we have
ceff = -6a'miin. (2.8)
As a check, note that in the case of the type 0 theory, the lightest state is a tachyon
CV'mi = -2. This corresponds to celf = 12, the correct value once one subtracts the 2
dimensions cancelled out by the ghosts. Similarly in the bosonic string, the lightest state
is a tachyon with a'mti = -4, corresponding to celf = 24.
For models in which the mass squared is nonnegative, such as spacetime-supersymmetric
strings, there is no such exponential IR divergence in the partition function. Hence, by
modular invariance, there is also no surviving contribution of order eCOnst/T2. This requires
a dramatic cancellation going well beyond the necessary condition that the effective central
charges of the spacetime bosons and fermions agree, c = ce .
Theories which do have such IR divergences might naively be expected to universally
suffer from dramatic instabilities on equal footing with those in the bosonic string theory.
However, this is not the case; IR divergences can arise from modes, pseudotachyons, whose
condensation does not cause a large back reaction (as occurs in the standard case of infla-
tionary perturbation theory, as well as in more formal time-dependent string backgrounds
[10,11]). This is the case we will encounter in our examples. In the next section we will
study the UV and IR limits of the partition function, and the modular transformation
between them, in the case of spacetimes with compact negatively curved spatial slices.
3. Exponential Growth: Examples
Any compact manifold of negative sectional curvature has a fundamental group of
exponential growth , meaning that the number of independent elements grows exponen-
tially with minimal word length in any basis of generators of the group. It follows  that
the number of free homotopy classes also grows exponentially with geodesic length.
In this section, we reproduce this result in constant curvature examples using modular
invariance in string theory. This analysis will provide precise results for the IR and UV
limits of the partition function, and their modular equivalence, exhibiting these compact
negatively curved target spaces as supercritical backgrounds of string theory as suggested
in . We will then consider more general examples and applications.
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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/m1/5/: accessed January 21, 2019), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.