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)
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
Altogether, working out the saddle point approximation to the l integral in (3.32) at
large l (large T2) gives
ZUv - exp + ecj (n = 2), (3.34)
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.
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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.