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string degrees of freedom (which go beyond general relativity) build up new effective di-
mensions, and these degrees of freedom become more important as the space shrinks.17
In more formal terms, it seems that a new type of string duality (perhaps best described
as a "D(imensional) duality") is emerging in the case of negatively curved compact target
We are grateful to A. Maloney for originally pointing out that the Selberg trace formula
provides a precise method for calculating the enhancement to ceff, and to O. Aharony for
many discussions on pseudotachyons. We are grateful to D. Green, A. Lawrence and
D. Morrison for their many insights in the related work . We thank A. Adams, M.
Freedman, G. Horowitz, S. Kachru, N. Lambert, L. McAllister, J. Polchinski, and A.
Tomasiello for useful discussions on this topic. E.S. and D.S. thank the KITP for hospitality
during the completion of this work, and gratefully acknowledge useful discussions in the
UCSB math/physics seminar on "Geometry, Topology, and Physics" run by D. Morrison.
This research was supported in part by the National Science Foundation under Grant No.
PHY99-07949. E.S. and D.S. are also supported in part by the DOE under contract DE-
AC03-76SF00515, by the NSF under contract 9870115, and by an FQXI grant. The work
of J.M. is supported in part by funds provided by the U.S. Department of Energy (D.O.E.)
under cooperative research agreement DE-FG0205ER41360.
17 It would be interesting to understand how this affects the question of summing over "three-
manifolds" (which has interesting subtleties to do with the distribution of volumes at fixed cur-
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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/m1/27/: accessed January 24, 2019), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.