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The Cobordism hypothesis


Author:Daniel S. Freed
Journal: Bull. Amer. Math. Soc. 50 (2013), 57-92
MSC (2010): Primary 57R56
DOI: https://doi.org/10.1090/S0273-0979-2012-01393-9
Published electronically: October 11, 2012
Previous version:Original version posted October 11, 2012
MathSciNet review:2994995
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Abstract | References | Similar Articles | Additional Information

Abstract: In this expository paper we introduce extended topological quantum field theories and the cobordism hypothesis.


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[A1]
M. F. Atiyah, Bordism and cobordism, Proc. Cambridge Philos. Soc. 57 (1961), 200-208. MR 0126856 (23:A4150)
[A2]
-, Topological quantum field theories, Inst. Hautes Études Sci. Publ. Math. (1988), no. 68, 175-186 (1989). MR 1001453 (90e:57059)
[Ab]
Lowell Abrams, Two-dimensional topological quantum field theories and Frobenius algebras, J. Knot Theory Ramifications 5 (1996), no. 5, 569-587. MR 1414088 (97j:81292)
[B-N]
Dror Bar-Natan, On Khovanov's categorification of the Jones polynomial, Algebr. Geom. Topol. 2 (2002), 337-370 (electronic). MR 1917056 (2003h:57014)
[Ba]
Clark Barwick, (, n)-Cat as a closed model category, ProQuest LLC, Ann Arbor, MI, 2005. Thesis (Ph.D.)-University of Pennsylvania. MR 2706984
[BC]
Martin Betz and Ralph L. Cohen, Graph moduli spaces and cohomology operations, Turkish J. Math. 18 (1994), no. 1, 23-41. MR 1270436 (95i:58037)
[BCT]
Andrew J. Blumberg, Ralph L. Cohen, and Constantin Teleman, Open-closed field theories, string topology, and Hochschild homology, . MR 2581905 (2011f:55020)
[BD]
John C. Baez and James Dolan, Higher-dimensional algebra and topological quantum field theory, J. Math. Phys. 36 (1995), no. 11, 6073-6105. MR 1355899 (97f:18003)
[BDH]
Arthur Bartels, Christopher L. Douglas, and André G. Henriques, Conformal nets and local field theory, .
[Be]
Julia E. Bergner, A survey of -categories, Towards higher categories, IMA Vol. Math. Appl., vol. 152, Springer, New York, 2010, pp. 69-83. MR 2664620 (2011e:18001)
[BeDr]
Alexander Beilinson and Vladimir Drinfeld, Chiral algebras, American Mathematical Society Colloquium Publications, vol. 51, American Mathematical Society, Providence, RI, 2004. MR 2058353 (2005d:17007)
[BFN]
David Ben-Zvi, John Francis, and David Nadler, Integral transforms and Drinfeld centers in derived algebraic geometry, J. Amer. Math. Soc. 23 (2010), no. 4, 909-966. MR 2669705 (2011j:14023)
[BN]
David Ben-Zvi and David Nadler, The Character Theory of a Complex Group, . preprint.
[Bo]
Richard E. Borcherds, Vertex algebras, Kac-Moody algebras, and the Monster, Proc. Nat. Acad. Sci. U.S.A. 83 (1986), no. 10, 3068-3071. MR 843307 (87m:17033)
[BS]
Clark Barwick and Christopher Schommer-Pries, On the Unicity of the Homotopy Theory of Higher Categories, .
[C]
Jean Cerf, La stratification naturelle des espaces de fonctions différentiables réelles et le théorème de la pseudo-isotopie, Inst. Hautes Études Sci. Publ. Math. (1970), no. 39, 5-173. MR 0292089 (45:1176)
[CF]
Louis Crane and Igor B. Frenkel, Four-dimensional topological quantum field theory, Hopf categories, and the canonical bases, J. Math. Phys. 35 (1994), no. 10, 5136-5154. Topology and physics. MR 1295461 (96d:57019)
[CG]
K. Costello and O. Gwilliam, Factorization algebras in perturbative quantum field theory. costello/factorization_public.html">. in preparation.
[Co]
Kevin Costello, Topological conformal field theories and Calabi-Yau categories, Adv. Math. 210 (2007), no. 1, 165-214. MR 2298823 (2008f:14071)
[CS]
Moira Chas and Dennis Sullivan, String Topology, .
[D]
S. K. Donaldson, Polynomial invariants for smooth four-manifolds, Topology 29 (1990), no. 3, 257-315. MR 1066174 (92a:57035)
[Da]
O. Davidovich, State sums in -dimensional fully extended topological field theories, Ph.D. thesis, University of Texas at Austin, 2011. .
.
[DHS]
C. Douglas, A. Henriques, and C. Schommer-Pries, The -category of tensor categories. in preparation.
[Di]
R. Dijkgraaf, A geometrical approach to two-dimensional conformal field theory, Ph.D. thesis, University of Utrecht, 1989. .
.
[DW]
Robbert Dijkgraaf and Edward Witten, Topological gauge theories and group cohomology, Comm. Math. Phys. 129 (1990), no. 2, 393-429. MR 1048699 (91g:81133)
[EM]
Yakov M. Eliashberg and Nikolai M. Mishachev, The space of framed functions is contractible, .
[ENO]
Pavel Etingof, Dmitri Nikshych, and Viktor Ostrik, On fusion categories, Ann. of Math. (2) 162 (2005), no. 2, 581-642. MR 2183279 (2006m:16051)
[F]
Daniel S. Freed, Higher algebraic structures and quantization, Comm. Math. Phys. 159 (1994), no. 2, 343-398, . MR 1256993 (95c:58034)
[Fa]
L. Faddeev, Elementary introduction to quantum field theory, Quantum fields and strings: a course for mathematicians. Vol. 1 (P. Deligne et. al., ed.), American Mathematical Society, 1999, pp. 513-550. Notes by L. Jeffrey. MR 1701606 (2002a:81154)
[FHLT]
Daniel S. Freed, Michael J. Hopkins, Jacob Lurie, and Constantin Teleman, Topological quantum field theories from compact Lie groups, A celebration of the mathematical legacy of Raoul Bott, CRM Proc. Lecture Notes, vol. 50, Amer. Math. Soc., Providence, RI, 2010, pp. 367-403. . MR 2648901 (2011i:57040)
[FHT]
Daniel S. Freed, Michael J. Hopkins, and Constantin Teleman, Consistent
orientation of moduli spaces, The many facets of geometry, Oxford Univ. Press, Oxford, 2010, pp. 395-419. . MR 2681705 (2011h:57043)
[FQ]
Daniel S. Freed and Frank Quinn, Chern-Simons theory with finite gauge group, Comm. Math. Phys. 156 (1993), no. 3, 435-472, . MR 1240583 (94k:58023)
[G]
E. Getzler, Batalin-Vilkovisky algebras and two-dimensional topological field theories, Comm. Math. Phys. 159 (1994), no. 2, 265-285. MR 1256989 (95h:81099)
[Ga]
Davide Gaiotto, dualities, JHEP 1208 (2011), 034, .
[GJ]
James Glimm and Arthur Jaffe, Quantum physics, second ed., Springer-Verlag, New York, 1987. A functional integral point of view. MR 887102 (89k:81001)
[GMN]
Davide Gaiotto, Gregory W. Moore, and Andrew Neitzke, Wall-crossing, Hitchin Systems, and the WKB Approximation, .
[GMTW]
Søren Galatius, Ulrike Tillmann, Ib Madsen, and Michael Weiss, The homotopy type of the cobordism category, Acta Math. 202 (2009), no. 2, 195-239, . MR 2506750 (2011c:55022)
[GSV]
Sergei Gukov, Albert Schwarz, and Cumrun Vafa, Khovanov-Rozansky homology and topological strings, Lett. Math. Phys. 74 (2005), no. 1, 53-74. MR 2193547 (2007a:57014)
[H]
Rudolf Haag, Local quantum physics, second ed., Texts and Monographs in Physics, Springer-Verlag, Berlin, 1996. Fields, particles, algebras. MR 1405610 (98b:81001)
[I]
Kiyoshi Igusa, The space of framed functions, Trans. Amer. Math. Soc. 301 (1987), no. 2, 431-477. MR 882699 (88g:57034)
[K]
Robion Kirby, A calculus for framed links in , Invent. Math. 45 (1978), no. 1, 35-56. MR 0467753 (57:7605)
[Ka]
Daniel M. Kan, Adjoint functors, Trans. Amer. Math. Soc. 87 (1958), 294-329. MR 0131451 (24:A1301)
[Kap]
Anton Kapustin, Topological field theory, higher categories, and their applications, Proceedings of the International Congress of Mathematicians. Volume III (New Delhi), Hindustan Book Agency, 2010, pp. 2021-2043. . MR 2827874
[Kh]
Mikhail Khovanov, A categorification of the Jones polynomial, Duke Math. J. 101 (2000), no. 3, 359-426. MR 1740682 (2002j:57025)
[KM]
M. Kontsevich and Yu. Manin, Gromov-Witten classes, quantum cohomology, and enumerative geometry, Comm. Math. Phys. 164 (1994), no. 3, 525-562, . MR 1291244 (95i:14049)
[Ko]
Joachim Kock, Frobenius algebras and D topological quantum field theories, London Mathematical Society Student Texts, vol. 59, Cambridge University Press, Cambridge, 2004. MR 2037238 (2005a:57028)
[KS]
M. Kontsevich and Y. Soibelman, Notes on -algebras, -categories and non-commutative geometry, Homological mirror symmetry, Lecture Notes in Phys., vol. 757, Springer, Berlin, 2009, pp. 153-219. . MR 2596638 (2011f:53183)
[KW]
Anton Kapustin and Edward Witten, Electric-magnetic duality and the geometric Langlands program, Commun. Number Theory Phys. 1 (2007), no. 1, 1-236, . MR 2306566 (2008g:14018)
[L1]
Jacob Lurie, On the classification of topological field theories, Current developments in mathematics, 2008, Int. Press, Somerville, MA, 2009, pp. 129-280. . MR 2555928 (2010k:57064)
[L2]
-, Higher Algebra. . draft version.
[L3]
-, Lectures on the cobordism hypothesis. . Perspectives in Geometry lectures at University of Texas, January, 2009.
[LZ]
Bong H. Lian and Gregg J. Zuckerman, New perspectives on the BRST-algebraic structure of string theory, Comm. Math. Phys. 154 (1993), no. 3, 613-646. MR 1224094 (94e:81333)
[Ma]
George W. Mackey, The mathematical foundations of quantum mechanics: A lecture-note volume, W., A. Benjamin, Inc., New York-Amsterdam, 1963. MR 676642 (84g:81003)
[May]
J. P. May, Stable algebraic topology, 1945-1966, History of topology, North-Holland, Amsterdam, 1999, pp. 665-723.
[Mc]
Saunders Mac Lane, Categories for the working mathematician, second ed., Graduate Texts in Mathematics, vol. 5, Springer-Verlag, New York, 1998. MR 0354798 (50:7275)
[Mi1]
John W. Milnor, Topology from the differentiable viewpoint, Based on notes by David W. Weaver, The University Press of Virginia, Charlottesville, Va., 1965. MR 0226651 (37:2239)
[Mi2]
-, Morse theory, Based on lecture notes by M. Spivak and R. Wells. Annals of Mathematics Studies, No. 51, Princeton University Press, Princeton, N.J., 1963. MR 0163331 (29:634)
[Mi3]
-, Lectures on the -cobordism theorem, Notes by L. Siebenmann and J. Sondow, Princeton University Press, Princeton, N.J., 1965. MR 0190942 (32:8352)
[MoT]
Gregory W. Moore and Yuji Tachikawa, On TQFTs whose values are holomorphic symplectic varieties, .
[MoW]
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Additional Information

Daniel S. Freed
Affiliation: The University of Texas at Austin, Mathematics Department RLM 8.100, 2515 Speedway Stop C1200, Austin, Texas 78712-1202
Email: dafr@math.utexas.edu

DOI: https://doi.org/10.1090/S0273-0979-2012-01393-9
Received by editor(s): November 15, 2011
Received by editor(s) in revised form: September 14, 2012
Published electronically: October 11, 2012
Additional Notes: The work of this author was supported by the National Science Foundation under grant DMS-0603964
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

Davide Silvano Achille Gaiotto (born 11 March 1977) is an Italian theoretical physicist who deals with quantum field theories and string theory.

Biography[edit]

Gaiotto won 1996 the silver medal as Italian participants in the International Mathematical Olympiad and 1995 gold medal at the International Physics Olympiad in Canberra. He was an undergraduate student at Scuola Normale Superiore in Pisa from 1996 to 2000. From 2004 to 2007 he was a post-doctoral researcher at Harvard University and then to 2011 the Institute for Advanced Study. Since 2011 he has been working at the Perimeter Institute for Theoretical Physics in Waterloo (Ontario) .

He introduced new techniques in the study and design of four-dimensional (N = 2) supersymmetric conformal field theories. He constructed from M5-branes, which are wound around Riemann surfaces with punctures. This led to new insights into the dynamics of four-dimensional (supersymmetric) gauge theories. With Juan Maldacena he studied these gauge theories using the AdS/CFT correspondence. In 2010 he had with Yuji Tachikawa and Luis Alday, developed the AGT correspondence (named after the authors), a duality in the 6D (2,0) superconformal field theory with compactification on a surface to a conformal field theory on the surface (Liouville field theory). In 2012 he received the New Horizons in Physics Prize and 2011 the Gribov Medal.

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