New Dimensions for Wound Strings:The Modular Transform of Geometry to Topology Page: 3 of 29
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infrared spectrum and the winding strings provides a straightforward means for consis-
tently analyzing the one-loop partition function in a time-dependent background.1 One
interesting outcome is that this check of modular invariance works timeslice by timeslice,
i.e., it works before performing the path integration over the time variable.
It is interesting also to ask what the effective dimensionality becomes at small radius.
In the case of genus h Riemann surfaces, numerous diagnostics - including a variant of
Buscher's worldsheet path integral argument for T-duality, D-brane systems, and algebro-
geometric genericity arguments - all suggest that the system can naturally cross over to a
supercritical theory on the 2h-dimensional Jacobian torus, with a rolling tachyon producing
a transition from this to the large radius expanding space .2
Altogether, we are being led to a generalization of T-duality and the Calabi-
Yau/Landau-Ginzburg correspondence applicable to the generic case of curved compact
target spaces. This provides a new mechanism by which spatial dimensions emerge, arising
in a novel way from nontrivial topology.
The paper is organized as follows. We start by reviewing the modular invariance rela-
tion between the UV and IR limits of the partition function. Next we analyze these limits
for solutions of the equations of motion with compact spatial slices of constant negative
curvature, using their realization as freely acting orbifolds of hyperbolic n-dimensional
space. We find precise agreement with the modular invariance prediction for the relation
between the deep UV and deep IR contributions to the partition function. We consider a
more subtle case of exponential growth, obtained from compact quotients of the Sol geom-
etry, and a subtle case of power law growth, obtained from compact quotients of the Nil
geometry. In all these homogeneous cases, modular invariance and the growth of effective
dimensions checks out explicitly. We then discuss more general cases, analogies to other
mechanisms for growing dimensions in string theory, and applications to mathematical and
physical questions in the last two sections.
1 See, e.g., [6,7] for previous works computing and interpreting a full partition function in a
Lorentzian-signature target space. There remain significant subleties with these models partly
to do with their construction as a global orbifold of Minkowski space which introduces extra
challenges in the very early time physics. In the present work we will evade these subtleties by
working in a regime where IR quantum field theory and semiclassical cosmic strings are easily
controlled, and by avoiding a global orbifold construction.
2 More generally, for higher dimensional examples the Albanese variety arises at small radius.
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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/m1/3/: accessed March 21, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.