The Cobordism hypothesis

Freed, Daniel

doi:10.1090/S0273-0979-2012-01393-9

The Cobordism hypothesis
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Bull. Amer. Math. Soc. 50 (2013), 57-92 Request permission

Abstract:

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

References

M. F. Atiyah, Bordism and cobordism, Proc. Cambridge Philos. Soc. 57 (1961), 200–208. MR 126856, DOI 10.1017/s0305004100035064
Michael Atiyah, Topological quantum field theories, Inst. Hautes Études Sci. Publ. Math. 68 (1988), 175–186 (1989). MR 1001453, DOI 10.1007/BF02698547
Lowell Abrams, Two-dimensional topological quantum field theories and Frobenius algebras, J. Knot Theory Ramifications 5 (1996), no. 5, 569–587. MR 1414088, DOI 10.1142/S0218216596000333
Dror Bar-Natan, On Khovanov’s categorification of the Jones polynomial, Algebr. Geom. Topol. 2 (2002), 337–370. MR 1917056, DOI 10.2140/agt.2002.2.337
Clark Barwick, $(\infty , n)$-Cat as a closed model category, ProQuest LLC, Ann Arbor, MI, 2005. Thesis (Ph.D.)–University of Pennsylvania. MR 2706984
Martin Betz and Ralph L. Cohen, Graph moduli spaces and cohomology operations, Turkish J. Math. 18 (1994), no. 1, 23–41. MR 1270436
Andrew J. Blumberg, Ralph L. Cohen, and Constantin Teleman, Open-closed field theories, string topology, and Hochschild homology, Alpine perspectives on algebraic topology, Contemp. Math., vol. 504, Amer. Math. Soc., Providence, RI, 2009, pp. 53–76. MR 2581905, DOI 10.1090/conm/504/09875
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, DOI 10.1063/1.531236
Arthur Bartels, Christopher L. Douglas, and André G. Henriques, Conformal nets and local field theory, 0912.5307.
Julia E. Bergner, A survey of $(\infty ,1)$-categories, Towards higher categories, IMA Vol. Math. Appl., vol. 152, Springer, New York, 2010, pp. 69–83. MR 2664620, DOI 10.1007/978-1-4419-1524-5_{2}
Alexander Beilinson and Vladimir Drinfeld, Chiral algebras, American Mathematical Society Colloquium Publications, vol. 51, American Mathematical Society, Providence, RI, 2004. MR 2058353, DOI 10.1090/coll/051
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, DOI 10.1090/S0894-0347-10-00669-7
David Ben-Zvi and David Nadler, The Character Theory of a Complex Group, 0904.1247. preprint.
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, DOI 10.1073/pnas.83.10.3068
Clark Barwick and Christopher Schommer-Pries, On the Unicity of the Homotopy Theory of Higher Categories, 1112.0040.
Jean Cerf, La stratification naturelle des espaces de fonctions différentiables réelles et le théorème de la pseudo-isotopie, Inst. Hautes Études Sci. Publ. Math. 39 (1970), 5–173 (French). MR 292089
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, DOI 10.1063/1.530746
K. Costello and O. Gwilliam, Factorization algebras in perturbative quantum field theory. http://math.northwestern.edu/$\sim$costello/factorization_public.html. in preparation.
Kevin Costello, Topological conformal field theories and Calabi-Yau categories, Adv. Math. 210 (2007), no. 1, 165–214. MR 2298823, DOI 10.1016/j.aim.2006.06.004
Moira Chas and Dennis Sullivan, String Topology, math/9911159.
S. K. Donaldson, Polynomial invariants for smooth four-manifolds, Topology 29 (1990), no. 3, 257–315. MR 1066174, DOI 10.1016/0040-9383(90)90001-Z
O. Davidovich, State sums in $2$-dimensional fully extended topological field theories, Ph.D. thesis, University of Texas at Austin, 2011. http://repositories. lib.utexas.edu/handle/2152/ETD-UT-2011-05-3139.
C. Douglas, A. Henriques, and C. Schommer-Pries, The $3$-category of tensor categories. in preparation.
R. Dijkgraaf, A geometrical approach to two-dimensional conformal field theory, Ph.D. thesis, University of Utrecht, 1989. http://igitur-archive. library.uu.nl/dissertations/2011-0929-200347/UUindex.html.
Robbert Dijkgraaf and Edward Witten, Topological gauge theories and group cohomology, Comm. Math. Phys. 129 (1990), no. 2, 393–429. MR 1048699, DOI 10.1007/BF02096988
Yakov M. Eliashberg and Nikolai M. Mishachev, The space of framed functions is contractible, 1108.1000.
Pavel Etingof, Dmitri Nikshych, and Viktor Ostrik, On fusion categories, Ann. of Math. (2) 162 (2005), no. 2, 581–642. MR 2183279, DOI 10.4007/annals.2005.162.581
Daniel S. Freed, Higher algebraic structures and quantization, Comm. Math. Phys. 159 (1994), no. 2, 343–398. MR 1256993, DOI 10.1007/BF02102643
Ludwig Faddeev, Elementary introduction to quantum field theory, Quantum fields and strings: a course for mathematicians, Vol. 1, 2 (Princeton, NJ, 1996/1997) Amer. Math. Soc., Providence, RI, 1999, pp. 513–550. MR 1701606
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, DOI 10.1090/crmp/050/26
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, DOI 10.1093/acprof:oso/9780199534920.003.0019
Daniel S. Freed and Frank Quinn, Chern-Simons theory with finite gauge group, Comm. Math. Phys. 156 (1993), no. 3, 435–472. MR 1240583, DOI 10.1007/BF02096860
E. Getzler, Batalin-Vilkovisky algebras and two-dimensional topological field theories, Comm. Math. Phys. 159 (1994), no. 2, 265–285. MR 1256989, DOI 10.1007/BF02102639
Davide Gaiotto, $N=2$ dualities, JHEP 1208 (2011), 034, arXiv:0904.2715.
James Glimm and Arthur Jaffe, Quantum physics, 2nd ed., Springer-Verlag, New York, 1987. A functional integral point of view. MR 887102, DOI 10.1007/978-1-4612-4728-9
Davide Gaiotto, Gregory W. Moore, and Andrew Neitzke, Wall-crossing, Hitchin Systems, and the WKB Approximation, 0907.3987.
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, DOI 10.1007/s11511-009-0036-9
Sergei Gukov, Albert Schwarz, and Cumrun Vafa, Khovanov-Rozansky homology and topological strings, Lett. Math. Phys. 74 (2005), no. 1, 53–74. MR 2193547, DOI 10.1007/s11005-005-0008-8
Rudolf Haag, Local quantum physics, 2nd ed., Texts and Monographs in Physics, Springer-Verlag, Berlin, 1996. Fields, particles, algebras. MR 1405610, DOI 10.1007/978-3-642-61458-3
Kiyoshi Igusa, The space of framed functions, Trans. Amer. Math. Soc. 301 (1987), no. 2, 431–477. MR 882699, DOI 10.1090/S0002-9947-1987-0882699-7
Robion Kirby, A calculus for framed links in $S^{3}$, Invent. Math. 45 (1978), no. 1, 35–56. MR 467753, DOI 10.1007/BF01406222
Daniel M. Kan, Adjoint functors, Trans. Amer. Math. Soc. 87 (1958), 294–329. MR 131451, DOI 10.1090/S0002-9947-1958-0131451-0
Anton Kapustin, Topological field theory, higher categories, and their applications, Proceedings of the International Congress of Mathematicians. Volume III, Hindustan Book Agency, New Delhi, 2010, pp. 2021–2043. MR 2827874
Mikhail Khovanov, A categorification of the Jones polynomial, Duke Math. J. 101 (2000), no. 3, 359–426. MR 1740682, DOI 10.1215/S0012-7094-00-10131-7
M. Kontsevich and Yu. Manin, Gromov-Witten classes, quantum cohomology, and enumerative geometry, Comm. Math. Phys. 164 (1994), no. 3, 525–562. MR 1291244, DOI 10.1007/BF02101490
Joachim Kock, Frobenius algebras and 2D topological quantum field theories, London Mathematical Society Student Texts, vol. 59, Cambridge University Press, Cambridge, 2004. MR 2037238
M. Kontsevich and Y. Soibelman, Notes on $A_\infty$-algebras, $A_\infty$-categories and non-commutative geometry, Homological mirror symmetry, Lecture Notes in Phys., vol. 757, Springer, Berlin, 2009, pp. 153–219. MR 2596638
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, DOI 10.4310/CNTP.2007.v1.n1.a1
Jacob Lurie, On the classification of topological field theories, Current developments in mathematics, 2008, Int. Press, Somerville, MA, 2009, pp. 129–280. MR 2555928
—, Higher Algebra. http://math.harvard.edu/\~lurie/papers/higheralgebra.pdf. draft version.
—, Lectures on the cobordism hypothesis. http://www.ma.utexas.edu/video/perspectives.html. Perspectives in Geometry lectures at University of Texas, January, 2009.
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, DOI 10.1007/BF02102111
George W. Mackey, The mathematical foundations of quantum mechanics, Mathematical Physics Monograph Series, Benjamin/Cummings Publishing Co., Inc., Advanced Book Program, Reading, Mass., 1980. A lecture-note volume; Reprint of the 1963 original. MR 676642
J. P. May, Stable algebraic topology, 1945–1966, History of topology, North-Holland, Amsterdam, 1999, pp. 665–723.
Saunders MacLane, Categories for the working mathematician, Graduate Texts in Mathematics, Vol. 5, Springer-Verlag, New York-Berlin, 1971. MR 0354798
John W. Milnor, Topology from the differentiable viewpoint, University Press of Virginia, Charlottesville, Va., 1965. Based on notes by David W. Weaver. MR 0226651
J. Milnor, Morse theory, Annals of Mathematics Studies, No. 51, Princeton University Press, Princeton, N.J., 1963. Based on lecture notes by M. Spivak and R. Wells. MR 0163331, DOI 10.1515/9781400881802
John Milnor, Lectures on the $h$-cobordism theorem, Princeton University Press, Princeton, N.J., 1965. Notes by L. Siebenmann and J. Sondow. MR 0190942, DOI 10.1515/9781400878055
Gregory W. Moore and Yuji Tachikawa, On $2d$ TQFTs whose values are holomorphic symplectic varieties, 1106.5698.
Scott Morrison and Kevin Walker, Higher categories, colimits, and the blob complex, Proc. Natl. Acad. Sci. USA 108 (2011), no. 20, 8139–8145. MR 2806651, DOI 10.1073/pnas.1018168108
Paul S. Aspinwall, Tom Bridgeland, Alastair Craw, Michael R. Douglas, Mark Gross, Anton Kapustin, Gregory W. Moore, Graeme Segal, Balázs Szendrői, and P. M. H. Wilson, Dirichlet branes and mirror symmetry, Clay Mathematics Monographs, vol. 4, American Mathematical Society, Providence, RI; Clay Mathematics Institute, Cambridge, MA, 2009. MR 2567952
Gregory Moore and Nathan Seiberg, Classical and quantum conformal field theory, Comm. Math. Phys. 123 (1989), no. 2, 177–254. MR 1002038, DOI 10.1007/BF01238857
Ib Madsen and Ulrike Tillmann, The stable mapping class group and $Q(\Bbb C \mathrm P^\infty _+)$, Invent. Math. 145 (2001), no. 3, 509–544. MR 1856399, DOI 10.1007/PL00005807
Ib Madsen and Michael Weiss, The stable moduli space of Riemann surfaces: Mumford’s conjecture, Ann. of Math. (2) 165 (2007), no. 3, 843–941. MR 2335797, DOI 10.4007/annals.2007.165.843
David Mumford, Towards an enumerative geometry of the moduli space of curves, Arithmetic and geometry, Vol. II, Progr. Math., vol. 36, Birkhäuser Boston, Boston, MA, 1983, pp. 271–328. MR 717614
L. S. Pontryagin, Smooth manifolds and their applications in homotopy theory, American Mathematical Society Translations, Ser. 2, Vol. 11, American Mathematical Society, Providence, R.I., 1959, pp. 1–114. MR 0115178
Richard S. Palais, Seminar on the Atiyah-Singer index theorem, Annals of Mathematics Studies, No. 57, Princeton University Press, Princeton, N.J., 1965. With contributions by M. F. Atiyah, A. Borel, E. E. Floyd, R. T. Seeley, W. Shih and R. Solovay. MR 0198494, DOI 10.1515/9781400882045
Frank Quinn, Lectures on axiomatic topological quantum field theory, Geometry and quantum field theory (Park City, UT, 1991) IAS/Park City Math. Ser., vol. 1, Amer. Math. Soc., Providence, RI, 1995, pp. 323–453. MR 1338394, DOI 10.1090/pcms/001/05
Charles Rezk, A Cartesian presentation of weak $n$-categories, Geom. Topol. 14 (2010), no. 1, 521–571. MR 2578310, DOI 10.2140/gt.2010.14.521
N. Yu. Reshetikhin and V. G. Turaev, Ribbon graphs and their invariants derived from quantum groups, Comm. Math. Phys. 127 (1990), no. 1, 1–26. MR 1036112, DOI 10.1007/BF02096491
N. Reshetikhin and V. G. Turaev, Invariants of $3$-manifolds via link polynomials and quantum groups, Invent. Math. 103 (1991), no. 3, 547–597. MR 1091619, DOI 10.1007/BF01239527
G. B. Segal, Felix Klein Lectures 2011. http://www.mpim-bonn.mpg.de/node/3372/abstracts.
Graeme Segal, The definition of conformal field theory, Topology, geometry and quantum field theory, London Math. Soc. Lecture Note Ser., vol. 308, Cambridge Univ. Press, Cambridge, 2004, pp. 421–577. MR 2079383
—, String algebras. http://www.cgtp.duke.edu/ITP99/segal/stanford/ lect5.pdf.
S. Smale, On the structure of manifolds, Amer. J. Math. 84 (1962), 387–399. MR 153022, DOI 10.2307/2372978
Edwin H. Spanier, Algebraic topology, Springer-Verlag, New York-Berlin, 1981. Corrected reprint. MR 666554, DOI 10.1007/978-1-4684-9322-1
R. F. Streater and A. S. Wightman, PCT, spin and statistics, and all that, Princeton Landmarks in Physics, Princeton University Press, Princeton, NJ, 2000. Corrected third printing of the 1978 edition. MR 1884336
René Thom, Quelques propriétés globales des variétés différentiables, Comment. Math. Helv. 28 (1954), 17–86 (French). MR 61823, DOI 10.1007/BF02566923
Constantin Teleman, Topological field theories in 2 dimensions, European Congress of Mathematics, Eur. Math. Soc., Zürich, 2010, pp. 197–210. MR 2648326, DOI 10.4171/077-1/9
—, Lie group actions in categories and Langlands duality. http://www.ma.utexas.edu/video/perspectives.html. Perspectives in Geometry lectures at University of Texas, October, 2011.
Ulrike Tillmann, The classifying space of the $(1+1)$-dimensional cobordism category, J. Reine Angew. Math. 479 (1996), 67–75. MR 1414388, DOI 10.1515/crll.1996.479.67
Ulrike Tillmann, On the homotopy of the stable mapping class group, Invent. Math. 130 (1997), no. 2, 257–275. MR 1474157, DOI 10.1007/s002220050184
V. G. Turaev, Quantum invariants of knots and 3-manifolds, De Gruyter Studies in Mathematics, vol. 18, Walter de Gruyter & Co., Berlin, 1994. MR 1292673, DOI 10.1515/9783110883275
Edward Witten, Topological quantum field theory, Comm. Math. Phys. 117 (1988), no. 3, 353–386. MR 953828, DOI 10.1007/BF01223371
Edward Witten, Topological sigma models, Comm. Math. Phys. 118 (1988), no. 3, 411–449. MR 958805, DOI 10.1007/BF01466725
Edward Witten, Quantum field theory and the Jones polynomial, Comm. Math. Phys. 121 (1989), no. 3, 351–399. MR 990772, DOI 10.1007/BF01217730
Edward Witten, Fivebranes and knots, Quantum Topol. 3 (2012), no. 1, 1–137. MR 2852941, DOI 10.4171/QT/26
K. Walker, On Witten’s $3$-manifold invariants. http://canyon23.net/math/1991TQFTNotes.pdf.