New Dimensions for Wound Strings:The Modular Transform of Geometry to Topology Page: 12 of 29
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In any perturbative string theory, the minimal mass squared obtained in the daughter
theory is that in the parent theory mmi , offset by the gap -(n - 1)2/(412).
For example, we can formally apply our methods to the bosonic string theory, whose
lowest lying mode in the flat spacetime (3.2) is a tachyon of mass squared mT = -4/a'.
Once we project to the daughter space Mn = IHaI/F, the minimal mass squared in the
bosonic theory drops to
()2 4 (n _ 1)2
mmin (t) a 4t2 (3.26)
3.3. Modular Transform and UV Limit
In the previous subsection, we found a gap in the spectrum of spatial modes at
gap _ (n - 1)2 (3.27)
t2 4t2 (
This led to a minimal mass squared
2 (n - 1)2 (0) 2 (328)
mmin - 4t2 + mmin
where min2 is the minimal mass squared in Minkowski space (which is zero in the super-
string, and equal to mT = -4/a' in the bosonic string). We obtained this by evaluating
the partition function in the deep IR limit, where low-energy effective field theory applies.
Now we would like to evaluate the partition function in the deep UV limit, as in (2.6), to
check modular invariance.
Before implementing this, we must address one subtlety. In general, the string par-
tition function in a consistent background is only guaranteed to be conformally invariant
(and hence modular invariant) after performing the path integral over all the embedding
coordinates, including t; in the Hamiltonian formalism, equivalently, modular invariance
only holds in general after taking the trace in (2.1). Using our complete basis of functions
(3.15)(3.18), we will find that our test of modular invariance works before performing the
t integral. A priori, this is not necessary, but it is a sufficient condition for passing this
test of modular invariance.6
6 The fact that it works for every t may follow from a more extensive analysis of one-loop
non-vacuum S-matrix amplitudes involving a set of different scattering processes each of which
preserve modular invariance but which localize interactions at different times. It is somewhat
reminiscent of "fiber-wise" string dualities.
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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/m1/12/: accessed January 22, 2019), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.