Research PapersNo Access

A cut-off tubular geometry of loop space

Partha Mukhopadhyay

The Institute of Mathematical Sciences, C.I.T. Campus, Taramani, Chennai 600113, India

https://doi.org/10.1142/S0129055X17500064Cited by:0 (Source: Crossref)

Abstract

Motivated by the computation of loop space quantum mechanics as indicated in [14], here we seek a better understanding of the tubular geometry of loop space $ℒ ℳ$ corresponding to a Riemannian manifold $ℳ$ around the submanifold of vanishing loops. Our approach is to first compute the tubular metric of ${(ℳ^{2 N + 1})}_{C}$ around the diagonal submanifold, where ${(ℳ^{N})}_{C}$ is the Cartesian product of $N$ copies of $ℳ$ with a cyclic ordering. This gives an infinite sequence of tubular metrics such that the one relevant to $ℒ ℳ$ can be obtained by taking the limit $N \to \infty$ . Such metrics are computed by adopting an indirect method where the general tubular expansion theorem of [21] is crucially used. We discuss how the complete reparametrization isometry of loop space arises in the large- $N$ limit and verify that the corresponding Killing equation is satisfied to all orders in tubular expansion. These tubular metrics can alternatively be interpreted as some natural Riemannian metrics on certain bundles of tangent spaces of $ℳ$ which, for $ℳ \times ℳ$ , is the tangent bundle $T ℳ$ .

Keywords:

AMSC: 51P05, 81T30, 81T40, 46T05, 46T10, 53B20, 57N40, 57R70

References

1. C. G. Callan and L. Thorlacius, Sigma models and string theory, in Proc. Theoretical Advanced Study Institute in Elementary Particle Physics: Particles, Strings and Supernovae (Elsevier, 2003), pp. 795–878. Google Scholar
2. A. A. Tseytlin, Sigma model approach to string theory, Internat. J. Mod. Phys. A 4 (1989) 1257–1318. Link, Web of Science, ADS, Google Scholar
3. A. Kriegl and P. W. Michor, The Convenient Setting of Global Analysis, Mathematical Surveys and Monographs, Vol. 53 (American Mathematical Society, Providence, RI, 1997). Crossref, Google Scholar
4. A. Stacey, The differential topology of loop spaces, arXiv:math/0510097 [math.DG]]; See also: http://ncatlab.org/nlab/show/equivariant $+$ tubular $+$ neighborhoods. Google Scholar
5. E. Witten, Supersymmetry and Morse theory, J. Differential Geom. 17(4) (1982) 661–692. Crossref, Web of Science, Google Scholar
6. E. Witten, The index of the Dirac operator in loop space, Princeton preprint, PUPT-1050. Google Scholar
7. E. Frenkel, A. Losev and N. Nekrasov, Notes on instantons in topological field theory and beyond, Nucl. Phys. Proc. Suppl. 171 (2007) 215–230; hep-th/0702137. Crossref, ADS, Google Scholar
8. E. Frenkel, A. Losev and N. Nekrasov, Instantons beyond topological theory II, arXiv:0803.3302 [hep-th]. Google Scholar
9. P. Mukhopadhyay, On a coordinate independent description of string worldsheet theory, Ann. Henri Poincaré 15 (2014) 937–963; arXiv:0912.3987 [hep-th]. Crossref, Web of Science, ADS, Google Scholar
10. P. Mukhopadhyay, DeWitt–Virasoro construction in tensor representations, Adv. High Energy Phys. 2012 (2012) 415634; arXiv:1004.2396 [hep-th]. Crossref, Web of Science, Google Scholar
11. P. Mukhopadhyay, DeWitt–Virasoro construction, Pramana 76 (2011) 407–420; arXiv:1010.0930 [hep-th]. Crossref, Web of Science, ADS, Google Scholar
12. P. Mukhopadhyay, String worldsheet theory in Hamiltonian framework and background independence, arXiv:1003.6103 [hep-th]. Google Scholar
13. P. Mukhopadhyay, Towards a loop space description of non-linear sigma model, J. Phys. Conf. Ser. 410 (2013) 012132. Crossref, Google Scholar
14. P. Mukhopadhyay, On a semi-classical limit of loop space quantum mechanics, ISRN High Energy Phys. 2013 (2013) 398030; arXiv:1202.2735 [hep-th]. Crossref, Google Scholar
15. R. M. Wald, General Relativity (Chicago University Press, 1984). Crossref, Google Scholar
16. L. P. Eisenhart, Riemannian Geometry (Princeton University Press, Princeton, N.J., 1965). Google Scholar
17. M. Spivak, A Comprehensive Introduction to Differential Geometry, Vol. I, 2nd edn. (Publish or Perish, Inc., Wilmington, 1979). Google Scholar
18. G. E. Bredon, Topology and Geometry, Graduate Texts in Math., Vol. 139 (Springer, New York, 1993). Crossref, Google Scholar
19. P. S. Florides and J. L. Synge, Coordinate conditions in a Riemannian space for coordinates based on a subspace, Proc. Roy. Soc. London Ser. A 323 (1971) 1–10. Crossref, Web of Science, ADS, Google Scholar
20. A. Gray, Tubes, 2nd edn., Progress in Mathematics (Addison-Wesley Publishing Company, New York, 1990). Google Scholar
21. P. Mukhopadhyay, All order covariant tubular expansion, Rev. Math. Phys. 26(1) (2014) 1350019; arXiv:1203.1151 [gr-qc]. Link, Web of Science, Google Scholar
22. P. Mukhopadhyay, A cut-off tubular geometry of loop space II, in progress. Google Scholar
23. P. Petersen, Riemannian Geometry, 2nd edn., Graduate Texts in Mathematics, Vol. 171 (Springer, New York, 2006). Google Scholar
24. S. Sasaki, On the differential geometry of tangent bundles of Riemannian manifolds, Tohoku Math. J. 10(3) (1958) 338–354. Crossref, Google Scholar
25. O. Kowalski and M. Sekizawa, Natural transformations of Riemannian metrics on manifolds to metrics on tangent bundles — A classification, Bull. Tokyo Gakugei Univ. (4) 40 (1988) 1–29. Google Scholar
26. E. Musso and F. Tricerri, Riemannian metrics on tangent bundles, Ann. Mat. Pura Appl. (4) 150 (1988) 1–20. Crossref, Web of Science, Google Scholar
27. S. Gudmundsson and E. Kappos, On the geometry of tangent bundles, Expo. Math. 20(1) (2002) 1–41. Crossref, Google Scholar
28. M. T. K. Abbassi and M. Sarih, On natural metrics on tangent bundles of Riemannian manifolds, Arch. Math. (Brno) 41(1) (2005) 71–92. Google Scholar
29. F. J. Chinea, Symmetries in tetrad theories, Class. Quantum Grav. 5 (1988) 135–145. Crossref, Web of Science, ADS, Google Scholar
30. S. Kobayashi, Fixed points of isometries, Nagoya Math. J. 13 (1958) 63–68. Crossref, Google Scholar
31. S. Kobayashi and K. Nomizu, Foundations of Differential Geometry II (Wiley, New York, 1969). Google Scholar
32. R. Giles and C. B. Thorn, A lattice approach to string theory, Phys. Rev. D 16 (1977) 366–386. Crossref, Web of Science, ADS, Google Scholar
33. C. B. Thorn, The theory of interacting relativistic strings, Nucl. Phys. B 263 (1986) 493–556. Crossref, Web of Science, ADS, Google Scholar
34. C. B. Thorn, Reformulating string theory with the 1/ $N$ expansion, in Proc. Sakharov Memorial Lectures in Physics, Vol. 1, Moscow, Russia (1991), pp. 447–453; Florida Univ. Gainesville, UFIFT-HEP-91-23 (91, rec. Dec.) 9 pp. (200590); hep-th/9405069. Google Scholar
35. O. Bergman and C. B. Thorn, String bit models for superstring, Phys. Rev. D 52 (1995) 5980–5996; hep-th/9506125. Crossref, Web of Science, ADS, Google Scholar
36. M. Srednicki and R. P. Woodard, A world sheet regularization for Witten’s string field theory, Phys. Lett. B 196 (1987) 55–59. Crossref, Web of Science, ADS, Google Scholar
37. V. D. Gershun and A. I. Pashnev, Relativistic system of interacting points as a discrete string, Theor. Math. Phys. 73 (1987) 1227–1232; Teor. Mat. Fiz. 73 (1987) 294–301. Google Scholar
38. J. Bordes and F. Lizzi, Computation of amplitudes in the discretized approach to string field theory, Phys. Rev. Lett. 61 (1988) 278–281. Crossref, Web of Science, ADS, Google Scholar
39. A. T. Filippov and A. P. Isaev, Gauge models of discrete strings, Mod. Phys. Lett. A 4 (1989) 2167–2176. Link, Web of Science, ADS, Google Scholar
40. A. T. Filippov and A. P. Isaev, Gauge models of fermionic discrete ‘strings’, Mod. Phys. Lett. A 5 (1990) 325–335. Link, Web of Science, ADS, Google Scholar
41. A. T. Filippov, D. Gangopadhyay and A. P. Isaev, Discrete ‘strings’ and gauge model of three confined particles, Int. J. Mod. Phys. A 7 (1992) 3639–3663. Link, Web of Science, ADS, Google Scholar
42. H. L. Verlinde, Bits, matrices and 1/ $N$ , JHEP 0312 (2003) 052; hep-th/0206059. Crossref, ADS, Google Scholar
43. J.-G. Zhou, PP wave string interactions from string bit model, Phys. Rev. D 67 (2003) 026010; hep-th/0208232. Crossref, Web of Science, ADS, Google Scholar
44. D. Vaman and H. L. Verlinde, Bit strings from $N = 4$ gauge theory, JHEP 0311 (2003) 041; hep-th/0209215. Crossref, ADS, Google Scholar
45. U. Danielsson, F. Kristiansson, M. Lubcke and K. Zarembo, String bits without doubling, JHEP 0310 (2003) 026; hep-th/0306147. Crossref, ADS, Google Scholar
46. A. Karch, Light cone quantization of string theory duals of free field theories, hep-th/0212041. Google Scholar
47. A. Dhar, G. Mandal and S. R. Wadia, String bits in small radius AdS and weakly coupled $N = 4$ super Yang–Mills theory. 1, hep-th/0304062. Google Scholar
48. A. Clark, A. Karch, P. Kovtun and D. Yamada, Construction of bosonic string theory on infinitely curved anti-de Sitter space, Phys. Rev. D 68 (2003) 066011; hepth/0304107. Crossref, Web of Science, ADS, Google Scholar
49. L. F. Alday, J. R. David, E. Gava and K. S. Narain, Towards a string bit formulation of $N = 4$ super Yang–Mills, JHEP 0604 (2006) 014; hep-th/0510264. Crossref, ADS, Google Scholar
50. L. Alvarez-Gaume, D. Z. Freedman and S. Mukhi, The background field method and the ultraviolet structure of the supersymmetric nonlinear sigma model, Ann. Phys. 134 (1981) 85–109. Crossref, Web of Science, ADS, Google Scholar
51. U. Muller, C. Schubert, A. M. E. van de Ven, A closed formula for the Riemann normal coordinate expansion, Gen. Rel. Grav. 31 (1999) 1759–1768; gr-qc/9712092. Crossref, Web of Science, ADS, Google Scholar