New Dimensions for Wound Strings:The Modular Transform of Geometry to Topology Page: 4 of 29
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2. Structure of the One-loop Amplitude
Let us use the conventions of Polchinski's book [9]. The one-loop vacuum amplitude
in perturbative closed string theory takes the form
2 Z1(Fi) = Tr d2( (-1)F LO (2.1)
12 2
where T is the modular parameter of the worldsheet torus, F is the spacetime fermion
number, q = e2?TZ, and Lo+Lo are the worldsheet Hamiltonian and momentum generators.
Here Zi(T) is defined so as to be modular invariant: Zi(T) = Zi(T + 1) = Zi(-1/T).
The T2 -- oc region of the integral is dominated by the deep IR part of the spectrum;
Z1 can be expanded as
Zl/T2 - Lo ,minLOmin (2.2)
Let us first review the IR and UV behavior of ZI in the 10-dimensional classical
backgrounds of the superstring (including the tachyonic cases) and the 26-dimensional
unstable background of the bosonic string. In terms of the spacetime masses and momenta,
for level-matched contributions we have
Lo = (k2 + m2)a'/4 = Lo, (2.3)
so the IR limit of the partition function has an integrand going like3
ZIR -1 Tr-TOG ~ e-T2 mmin (2.4)
The T2 -- 0 region is dominated by the deep UV. We can compute directly the UV
asymptotics of the torus amplitude using the density of states for large masses,
p(m) ~ em' a 2ceg/3 (2.5)
where ceff is the effective central charge. Summing over m with this measure by saddle
point approximation gives
I 2 0f
ZUV - Z1 T2-O ~ t p(m)e--T2C' m _ e 672 . (2.6)
m
3 Here, as will be our custom, we ignore the subleading polynomial dependence of the asymp-
totic form. Equivalently, our statements that two functions are asymptotic should be interpreted
as statements about the corresponding logarithms.3
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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]. (https://digital.library.unt.edu/ark:/67531/metadc877426/m1/4/: accessed April 18, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.