# New Dimensions for Wound Strings:The Modular Transform of Geometry to Topology Page: 13 of 29

Let us now turn to the test of modular invariance. Plugging our result for mmin
(3.28) into the modular relation (2.7) gives the small-T2 behavior demanded by modular
invariance:
(cv'(n - 1)2 rccGrit
ZUV ~ exp y nt2 2 + 6T . (3.29)
where ceft is the value of the effective central charge in the critical theory on Minkowski
space (reviewed in 2 above). Comparing to (2.6) yields
r 3c'(n - 1)2
ceff - cef 2t2 (3.30)
This means that modular invariance can be satisfied in the late-time limit only if the
compact negatively curved manifold M provides a new contribution to the exponential
density of states. In the superstring this further requires the new contribution to survive
the sum over bosonic and fermionic contributions, an issue we will discuss below in 3.4.
The projection of II, by freely acting isometries I' has several effects, which modify
both the IR and UV limits of the spectrum. First, it compactifies the space, so that
new modes become normalizable; this introduces a new state into the IR spectrum which
causes a divergence in the one-loop partition function as we saw in 3.2. Second, it restricts
unwound modes to invariant states. Third, it introduces new sectors of winding strings;
we will see that these winding strings contribute a Hagedorn density of states to the UV
limit of the spectrum in just the right way to provide the requisite effective central charge
(3.30).7
The partition function includes a sum over winding sectors and a sum over states
arising from transverse motion of the winding strings, as well as a sum over eigenstates of
Lo, Lo in each winding sector (which we will refer to as "oscillator modes"). The mass of
each winding mode is
mw = R2 e (3.31)
when the string winds a cycle of length l = 27R.
7 In our case the winding strings which contribute to the exponential density are in one-to-one
correspondence with free homotopy classes (conjugacy classes of the fundamental group). In more
general examples, one might find additional contributions from metastable closed geodesics that
are not supported by topology. This does not happen in cases where the sectional curvature is
bounded above by a negative value [16], in particular in our cases of constant curvature.

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