# New Dimensions for Wound Strings:The Modular Transform of Geometry to Topology Page: 14 of 29

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We are studying the one-loop partition function in the limit where it degenerates into

a long thin tube. In the previous section, we studied this in the channel where worldsheet

time ran around the long direction of the tube, which is dominated by IR modes. In order

to study the UV density of states, including the contribution of long winding strings, we

need to analyze the diagram in the opposite channel.

Consider a rectangular Euclidean worldsheet torus, with the worldsheet space direction

61 ranging from 0 to 27T2, and the worldsheet time direction c-2 ranging from 0 to 27.

T2 is related to the modular parameter T2 of the previous section by a modular tranform

T2 = 1/T2. That is, to study the UV behavior T2 -- 0, we are interested in the asymptotic

behavior as T2 -- oc of this diagram.8

Let us start with the case n = 2, where the simplest version of the Selberg trace formula

computes for each value of t the path integral we are interested in, in the degeneration

limit T2 -- oo, as a sum over periodic orbits. Up to an overall constant, the sum over

windings and their transverse motions contribute to ZUv as

dlp(l) 1 _2/(4?';) (n = 2), (3.32)

J smh(l/2t)

where p(l) is the density of periodic geodesics. The latter is given [3,17,18] by

p(l) = el/t (n = 2) (3.33)

(up to powers of l). Here we are using the fact that t is the curvature radius of M at

time t.

Altogether, working out the saddle point approximation to the l integral in (3.32) at

large l (large T2) gives

/(C 7crit

ZUv - exp + ecj (n = 2), (3.34)

4t2T2 6T2

where ceJf arises from the usual sum over oscillator contributions. This result (3.34)

reproduces directly in the winding channel the enhancement to the effective central charge

needed to satisfy the modular invariance requirement (3.29).9

8 We can see the basic effect we need without keeping track of worldsheet angles determined

by Ti, which we have set to zero for simplicity. As discussed further below, at least in familiar

examples the exponential growth of the level-matched and non-level-matched spectra are the same.

9 Here we are computing the density of states for physical winding strings, whereas the trace

appearing in the partition function (2.1) includes all states in the CFT, whether or not they are

level matched. In the familiar case of flat space the density of states at high level scales the same

for level-matched and non-level-matched states (though the subleading polynomial dependence

differs). Our check will work for the level-matched states, which leads us to suspect that they

scale the same as the non-level-matched states also in our system.13

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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/m1/14/: accessed January 17, 2019), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.