Volume 8 (2006)

Download this article
For screen
For printing
Recent Volumes
Volume 1, 1998
Volume 2, 1999
Volume 3, 2000
Volume 4, 2002
Volume 5, 2002
Volume 6, 2003
Volume 7, 2004
Volume 8, 2006
Volume 9, 2006
Volume 10, 2007
Volume 11, 2007
Volume 12, 2007
Volume 13, 2008
Volume 14, 2008
Volume 15, 2008
Volume 16, 2009
Volume 17, 2011
Volume 18, 2012
Volume 19, 2015
The Series
About this Series
Ethics Statement
Purchase Printed Copies
Author Index
ISSN (electronic): 1464-8997
ISSN (print): 1464-8989
MSP Books and Monographs
Other MSP Publications

Introduction to the Gopakumar–Vafa Large N Duality

Dave Auckly and Sergiy Koshkin

Geometry & Topology Monographs 8 (2006) 195–456

DOI: 10.2140/gtm.2006.8.195

1 B Acharya, On realising N=1 super Yang–Mills in M–theory, arXiv:hep-th/0011089
2 V S Adamchik, Symbolic and numeric computations of the Barnes function, Comput. Phys. Comm. 157 (2004) 181–190 MR2105933
3 M Aganagic, A Klemm, M Mariño, C Vafa, The topological vertex, Comm. Math. Phys. 254 (2005) 425–478 MR2117633
4 M Aganagic, A Klemm, M Mariño, C Vafa, Matrix model as a mirror of Chern–Simons theory, J. High Energy Phys. (2004) MR2046799
5 M Aganagic, M Mariño, C Vafa, All loop topological string amplitudes from Chern–Simons theory, Comm. Math. Phys. 247 (2004) 467–512 MR2063269
6 M Aganagic, H Ooguri, N Saulina, C Vafa, Black holes, q–deformed 2d Yang–Mills, and non-perturbative topological strings, Nuclear Phys. B 715 (2005) 304–348 MR2135642
7 M Aganagic, C Vafa, G2–manifolds, mirror symmetry and geometric engineering, arXiv:hep-th/0110171
8 , Quantum cohomology at the Mittag–Leffler Institute (editor P Aluffi), Notes of Courses Given by Teachers at the School, Scuola Normale Superiore (1997) MR1664633
9 H H Andersen, Tensor products of quantized tilting modules, Comm. Math. Phys. 149 (1992) 149–159 MR1182414
10 H H Andersen, J Paradowski, Fusion categories arising from semisimple Lie algebras, Comm. Math. Phys. 169 (1995) 563–588 MR1328736
11 G E Andrews, R Askey, R Roy, Special functions, Encyclopedia of Mathematics and its Applications 71, Cambridge University Press (1999) MR1688958
12 T M Apostol, Modular functions and Dirichlet series in number theory, Graduate Texts in Mathematics 41, Springer (1990) MR1027834
13 V I Arnol'd, Mathematical methods of classical mechanics, Graduate Texts in Mathematics 60, Springer (1989) MR997295
14 M Atiyah, The geometry and physics of knots, Lezioni Lincee., Cambridge University Press (1990) MR1078014
15 M F Atiyah, R Bott, The moment map and equivariant cohomology, Topology 23 (1984) 1–28 MR721448
16 M Atiyah, J Maldacena, C Vafa, An M–theory flop as a large N duality, J. Math. Phys. 42 (2001) 3209–3220 MR1840340
17 M Audin, Torus actions on symplectic manifolds, Progress in Mathematics 93, Birkhäuser Verlag (2004) MR2091310
18 , Holomorphic curves in symplectic geometry (editors M Audin, J Lafontaine), Progress in Mathematics 117, Birkhäuser Verlag (1994) MR1274923
19 S Axelrod, S Della Pietra, E Witten, Geometric quantization of Chern–Simons gauge theory, J. Differential Geom. 33 (1991) 787–902 MR1100212
20 J Baez, J P Muniain, Gauge fields, knots and gravity, Series on Knots and Everything 4, World Scientific Publishing Co. (1994) MR1313910
21 B Bakalov, A Kirillov, Lectures on tensor categories and modular functors, University Lecture Series 21, American Mathematical Society (2001) MR1797619
22 D Bar-Natan, On the Vassiliev knot invariants, Topology 34 (1995) 423–472 MR1318886
23 D Bar-Natan, Perturbative Chern–Simons theory, J. Knot Theory Ramifications 4 (1995) 503–547 MR1361082
24 D Bar-Natan, From astrology to topology via Feynman diagrams and Lie algebras, from: "Proceedings of the 19th Winter School “Geometry and Physics” (Srni 1999)", Rend. Circ. Mat. Palermo (2) (2000) 11–16 MR1758073
25 D Bar-Natan, S Garoufalidis, L Rozansky, D P Thurston, Wheels, wheeling, and the Kontsevich integral of the unknot, Israel J. Math. 119 (2000) 217–237 MR1802655
26 D Bar-Natan, S Garoufalidis, L Rozansky, D P Thurston, The Århus integral of rational homology 3–spheres I. A highly non trivial flat connection on S³, Selecta Math. (N.S.) 8 (2002) 315–339 MR1931167
27 D Bar-Natan, R Lawrence, A rational surgery formula for the LMO invariant, Israel J. Math. 140 (2004) 29–60 MR2054838
28 K Behrend, Algebraic Gromov–Witten invariants, from: "New trends in algebraic geometry (Warwick, 1996)", London Math. Soc. Lecture Note Ser. 264, Cambridge Univ. Press (1999) 19–70 MR1714820
29 K Behrend, Localization and Gromov–Witten invariants, from: "Quantum cohomology (Cetraro, 1997)", Lecture Notes in Math. 1776, Springer (2002) 3–38 MR1911301
30 R Bott, L W Tu, Differential forms in algebraic topology, Graduate Texts in Mathematics 82, Springer (1982) MR658304
31 V Bouchard, B Florea, M Mariño, Counting higher genus curves with crosscaps in Calabi–Yau orientifolds, arXiv:hep-th/0405083
32 V Bouchard, B Florea, M Mariño, Topological open string amplitudes on orientifolds, arXiv:hep-th/0411227
33 G E Bredon, Introduction to compact transformation groups, Pure and Appliced Mathematics 46, Academic Press (1972) MR0413144
34 K S Brown, Cohomology of groups, Graduate Texts in Mathematics 87, Springer (1982) MR672956
35 A Bruguières, Catégories prémodulaires, modularisations et invariants des variétés de dimension 3, Math. Ann. 316 (2000) 215–236 MR1741269
36 J Bryan, R Pandharipande, Curves in Calabi–Yau threefolds and topological quantum field theory, Duke Math. J. 126 (2005) 369–396 MR2115262
37 F Cachazo, K Intriligator, C Vafa, A large N duality via a geometric transition, Nuclear Phys. B 603 (2001) 3–41 MR1838691
38 F Cachazo, S Katz, C Vafa, Geometric transitions and N=1 quiver theories, arXiv:hep-th/0108120
39 P Candelas, X C de la Ossa, Comments on conifolds, Nuclear Phys. B 342 (1990) 246–268 MR1068113
40 V Chari, A Pressley, A guide to quantum groups, Cambridge University Press (1994) MR1300632
41 C H Clemens, Double solids, Adv. in Math. 47 (1983) 107–230 MR690465
42 H Cohen, L Lewin, D Zagier, A sixteenth-order polylogarithm ladder, Experiment. Math. 1 (1992) 25–34 MR1181084
43 D A Cox, S Katz, Mirror symmetry and algebraic geometry, Mathematical Surveys and Monographs 68, American Mathematical Society (1999) MR1677117
44 , Quantum fields and strings: a course for mathematicians. Vol 1, 2 (editors P Deligne, P Etingof, D S Freed, L C Jeffrey, D Kazhdan, J W Morgan, D R Morrison, E Witten), American Mathematical Society (1999) 727–1501 MR1701618
45 P Di Francesco, P Mathieu, D Sénéchal, Conformal field theory, Graduate Texts in Contemporary Physics, Springer (1997) MR1424041
46 D E Diaconescu, B Florea, A Grassi, Geometric transitions, del Pezzo surfaces and open string instantons, Adv. Theor. Math. Phys. 6 (2002) 643–702 MR1969655
47 D E Diaconescu, B Florea, A Misra, Orientifolds, unoriented instantons and localization, arXiv:hep-th/0305021
48 D E Diaconescu, B Florea, N Saulina, A vertex formalism for local ruled surfaces, Comm. Math. Phys. 265 (2006) 201–226 MR2217303
49 L Dixon, J Harvey, C Vafa, E Witten, Strings on orbifolds, Nuclear Phys. B 261 (1985) 678–686 MR0851703
50 S K Donaldson, R P Thomas, Gauge theory in higher dimensions, from: "The geometric universe (Oxford, 1996)", Oxford Univ. Press (1998) 31–47 MR1634503
51 V G Drinfel'd, Almost cocommutative Hopf algebras, Algebra i Analiz 1 (1989) 30–46 MR1025154
52 J J Duistermaat, G J Heckman, On the variation in the cohomology of the symplectic form of the reduced phase space, Invent. Math. 69 (1982) 259–268 MR674406
53 Y Eliashberg, A Givental, H Hofer, Introduction to symplectic field theory, from: "GAFA 2000 (Tel Aviv, 1999)", Geom. Funct. Anal. Special Volume, Part II (2000) 560–673 MR1826267
54 P Etingof, Geometry and quantum field theory, Feynman calculus, MIT Open Courseware, Mathematics 18.238 (2002)
55 C Faber, R Pandharipande, Hodge integrals and Gromov–Witten theory, Invent. Math. 139 (2000) 173–199 MR1728879
56 R Fenn, C Rourke, On Kirby's calculus of links, Topology 18 (1979) 1–15 MR528232
57 D Fiorenza, R Murri, Matrix integrals and Feynman diagrams in the Kontsevich model, Adv. Theor. Math. Phys. 7 (2003) 525–576 MR2030059
58 O Forster, Lectures on Riemann surfaces, Graduate Texts in Mathematics 81, Springer (1991) MR1185074
59 P J Freyd, D N Yetter, Braided compact closed categories with applications to low-dimensional topology, Adv. Math. 77 (1989) 156–182 MR1020583
60 P Freyd, D Yetter, J Hoste, W B R Lickorish, K Millett, A Ocneanu, A new polynomial invariant of knots and links, Bull. Amer. Math. Soc. (N.S.) 12 (1985) 239–246 MR776477
61 K Fukaya, Morse homotopy and Chern–Simons perturbation theory, Comm. Math. Phys. 181 (1996) 37–90 MR1410567
62 W Fulton, J Harris, Representation theory: a first course, Graduate Texts in Mathematics 129, Springer (1991) MR1153249
63 S Garoufalidis, M Mariño, On Chern–Simons matrix models, arXiv:math.GT/0601390
64 A Giveon, A Kehagias, H Partouche, Geometric transitions, brane dynamics and gauge theories, J. High Energy Phys. (2001) MR1878692
65 R Gopakumar, C Vafa, On the gauge theory/geometry correspondence, Adv. Theor. Math. Phys. 3 (1999) 1415–1443 MR1796682
66 T Graber, R Pandharipande, Localization of virtual classes, Invent. Math. 135 (1999) 487–518 MR1666787
67 A Grassi, M Rossi, Large N dualities and transitions in geometry, from: "Geometry and physics of branes (Como, 2001)”", Ser. High Energy Phys. Cosmol. Gravit., IOP (2003) MR1950959
68 P Griffiths, J Harris, Principles of algebraic geometry, Pure and Applied Mathematics, Wiley-Interscience (1978) MR507725
69 M Gromov, Pseudoholomorphic curves in symplectic manifolds, Invent. Math. 82 (1985) 307–347 MR809718
70 N Halmagyi, T Okuda, V Yasnov, Large–N duality, lens spaces and the Chern–Simons matrix model, J. High Energy Phys. (2004) MR2080896
71 S K Hansen, T Takata, Reshetikhin–Turaev invariants of Seifert 3–manifolds for classical simple Lie algebras, J. Knot Theory Ramifications 13 (2004) 617–668 MR2080126
72 G Hardy, E Wright, An introduction to the theory of numbers, Oxford University Press (1979) MR0568909
73 J Harris, Algebraic geometry, a first course, Graduate Texts in Mathematics 133, Springer (1992) MR1182558
74 J Harris, I Morrison, Moduli of curves, Graduate Texts in Mathematics 187, Springer (1998) MR1631825
75 R Hartshorne, Algebraic geometry, Graduate Texts in Mathematics 52, Springer (1977) MR0463157
76 K Hori, S Katz, A Klemm, R Pandharipande, R Thomas, C Vafa, R Vakil, E Zaslow, Mirror symmetry, Clay Mathematics Monographs 1, American Mathematical Society (2003) MR2003030
77 S Hu, Lecture notes on Chern–Simons–Witten theory, World Scientific Publishing Co. (2001) MR1852998
78 J E Humphreys, Introduction to Lie algebras and representation theory, Graduate Texts in Mathematics 9, Springer (1978) MR499562
79 A Iqbal, All genus topological string amplitudes and 5–brane webs as Feynman diagrams, arXiv:hep-th/0207114
80 V F R Jones, A polynomial invariant for knots via von Neumann algebras, Bull. Amer. Math. Soc. (N.S.) 12 (1985) 103–111 MR766964
81 V G Kac, Infinite-dimensional Lie algebras, Cambridge University Press (1990) MR1104219
82 C Kassel, Quantum groups, Graduate Texts in Mathematics 155, Springer (1995) MR1321145
83 C Kassel, M Rosso, V G Turaev, Quantum groups and knot invariants, Panoramas et Synthèses 5, Société Mathématique de France (1997) MR1470954
84 R Kirby, A calculus for framed links in S³, Invent. Math. 45 (1978) 35–56 MR0467753
85 A A Kirillov Jr., On an inner product in modular tensor categories, J. Amer. Math. Soc. 9 (1996) 1135–1169 MR1358983
86 T Kohno, Conformal field theory and topology, Translations of Mathematical Monographs 210, American Mathematical Society (2002) MR1905659
87 M Kontsevich, Intersection theory on the moduli space of curves and the matrix Airy function, Comm. Math. Phys. 147 (1992) 1–23 MR1171758
88 M Kontsevich, Feynman diagrams and low-dimensional topology, from: "First European Congress of Mathematics, Vol. II (Paris, 1992)", Progr. Math. 120, Birkhäuser (1994) 97–121 MR1341841
89 M Kontsevich, Enumeration of rational curves via torus actions, from: "The moduli space of curves (Texel Island, 1994)", Progr. Math. 129, Birkhäuser (1995) 335–368 MR1363062
90 M Kontsevich, Vassiliev's knot invariants, Adv. Soviet Math. 16/2, American Mathematical Society (2003) 137–150 MR1237836
91 S Koshkin, Conormal bundles to knots and the Gopakumar–Vafa conjecture, arXiv:math.DG/0503248
92 J M F Labastida, M Mariño, C Vafa, Knots, links and branes at large N, arXiv:hep-th/0010102
93 T T Q Le, J Murakami, T Ohtsuki, On a universal perturbative invariant of 3–manifolds, Topology 37 (1998) 539–574 MR1604883
94 J Li, C C M Liu, K Liu, J Zhou, A mathematical theory of the topological vertex, arXiv:math.AG/0408426
95 J Li, Y S Song, Open string instantons and relative stable morphisms, Adv. Theor. Math. Phys. 5 (2001) 67–91 MR1894338
96 J Li, G Tian, Virtual moduli cycles and Gromov–Witten invariants of algebraic varieties, J. Amer. Math. Soc. 11 (1998) 119–174 MR1467172
97 J Li, G Tian, Virtual moduli cycles and Gromov–Witten invariants of general symplectic manifolds, from: "Topics in symplectic 4–manifolds (Irvine, CA, 1996)", First Int. Press Lect. Ser., I, Int. Press, Cambridge, MA (1998) 47–83 MR1635695
98 C C M Liu, Moduli of J–holomorphic curves with Lagrangian boundary conditions and open Gromov–Witten invariants for an S¹–equivariant pair, PhD thesis arXiv:math.SG/0210257
99 G Lusztig, Quantum groups at roots of 1, Geom. Dedicata 35 (1990) 89–113 MR1066560
100 S Mac Lane, Categories for the working mathematician, Graduate Texts in Mathematics 5, Springer (1998) MR1712872
101 S Majid, A quantum groups primer, London Mathematical Society Lecture Note Series 292, Cambridge University Press (2002) MR1904789
102 M Mariño, Enumerative geometry and knot invariants, from: "Infinite dimensional groups and manifolds", IRMA Lect. Math. Theor. Phys. 5, de Gruyter MR2104354
103 M Mariño, Chern–Simons theory and topological strings, Rev. Modern Phys. 77 (2005) 675–720 MR2168778
104 M Mariño, Chern–Simons theory, matrix integrals, and perturbative three-manifold invariants, Comm. Math. Phys. 253 (2005) 25–49 MR2105636
105 M Mariño, Chern–Simons theory, matrix models and topological strings, International Series of Monographs on Physics 131, The Clarendon Press Oxford University Press (2005) MR2177747
106 I Maulik, N Nekrasov, A Okounkov, R Pandharipande, Gromov–Witten theory and Donaldson–Thomas theory I, arXiv:math.AG/0312059
107 I Maulik, N Nekrasov, A Okounkov, R Pandharipande, Gromov–Witten theory and Donaldson–Thomas theory II, arXiv:math.AG/0406092
108 D McDuff, D Salamon, J–holomorphic curves and symplectic topology, American Mathematical Society Colloquium Publications 52, American Mathematical Society (2004) MR2045629
109 D Metzler, Topological and smooth stacks, arXiv:math.DG/0306176
110 J W Milnor, J D Stasheff, Characteristic classes, Annals of Mathematics Studies 76, Princeton University Press (1974) MR0440554
111 M Müger, On the structure of modular categories, Proc. London Math. Soc. (3) 87 (2003) 291–308 MR1990929
112 D Mumford, Geometric invariant theory, Ergebnisse der Mathematik und ihrer Grenzgebiete 34, Springer (1965) MR0214602
113 D Mumford, Picard groups of moduli problems, from: "Arithmetical Algebraic Geometry (Proc. Conf. Purdue Univ., 1963)", Harper & Row (1965) 33–81 MR0201443
114 D Mumford, Curves and their Jacobians, The University of Michigan Press, Ann Arbor, Mich. (1975) MR0419430
115 D Mumford, Towards an enumerative geometry of the moduli space of curves, from: "Arithmetic and geometry, Vol II", Progr. Math. 36, Birkhauser (1983) 271–328 MR0717614
116 L Ng, Conormal bundles, contact homology and knot invariants, from: "The interaction of finite-type and Gromov–Witten invariants (BIRS 2003)", Geom. Topol. Monogr. 8 (2006) 129–144
117 T Ohtsuki, Finite type invariants of integral homology 3–spheres, J. Knot Theory Ramifications 5 (1996) 101–115 MR1373813
118 A Okounkov, R Pandharipande, Hodge integrals and invariants of the unknot, Geom. Topol. 8 (2004) 675–699 MR2057777
119 H Ooguri, A Strominger, C Vafa, Black hole attractors and the topological string, Phys. Rev. D (3) 70 (2004) MR2123156
120 H Ooguri, C Vafa, Knot invariants and topological strings, Nuclear Phys. B 577 (2000) 419–438 MR1765411
121 H Ooguri, C Vafa, Worldsheet derivation of a large N duality, Nuclear Phys. B 641 (2002) 3–34 MR1928179
122 F Perroni, Orbifold cohomology of ADE–singularities, arXiv:math.AG/0510528
123 M Polyak, Feynman diagrams for pedestrians and mathematicians, from: "Graphs and patterns in mathematics and theoretical physics", Proc. Sympos. Pure Math. 73, Amer. Math. Soc. (2005) 15–42 MR2131010
124 V V Prasolov, A B Sossinsky, Knots, links, braids and 3–manifolds, Translations of Mathematical Monographs 154, American Mathematical Society (1997) MR1414898
125 J H Przytycki, P Traczyk, Invariants of links of Conway type, Kobe J. Math. 4 (1988) 115–139 MR945888
126 J G Ratcliffe, Foundations of hyperbolic manifolds, Graduate Texts in Mathematics 149, Springer (1994) MR1299730
127 M Reid, The moduli space of 3–folds with K=0 may nevertheless be irreducible, Math. Ann. 278 (1987) 329–334 MR909231
128 N Y Reshetikhin, V G Turaev, Ribbon graphs and their invariants derived from quantum groups, Comm. Math. Phys. 127 (1990) 1–26 MR1036112
129 N Reshetikhin, V G Turaev, Invariants of 3–manifolds via link polynomials and quantum groups, Invent. Math. 103 (1991) 547–597 MR1091619
130 D Rolfsen, Knots and links, Mathematics Lecture Series 7, Publish or Perish (1990) MR1277811
131 M Rosso, Quantum groups at a root of 1 and tangle invariants, Internat. J. Modern Phys. B 7 (1993) 3715–3726 MR1241466
132 L Rozansky, A large k asymptotics of Witten's invariant of Seifert manifolds, Comm. Math. Phys. 171 (1995) 279–322 MR1344728
133 D Salamon, Lectures on Floer homology, from: "Symplectic geometry and topology (Park City, UT, 1997)", IAS/Park City Math. Ser. 7, Amer. Math. Soc. (1999) 143–229 MR1702944
134 H Samelson, Notes on Lie algebras, Universitext, Springer (1990) MR1056083
135 S Sawin, Quantum groups at roots of unity and modularity, arXiv:math.QA/0308281
136 J Sawon, Perturbative expansion of Chern–Simons theory, from: "The interaction of finite-type and Gromov–Witten invariants (BIRS 2003)", Geom. Topol. Monogr. 8 (2006) 145–166
137 E Schrödinger, Statistical thermodynamics, Cambridge University Press (1962) MR0149891
138 G B Segal, The definition of conformal field theory, from: "Differential geometrical methods in theoretical physics (Como, 1987)", NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 250, Kluwer Acad. Publ. (1988) 165–171 MR981378
139 G Segal, Two-dimensional conformal field theories and modular functors, from: "IXth International Congress on Mathematical Physics (Swansea, 1988)", Hilger (1989) 22–37 MR1033753
140 G Segal, The definition of conformal field theory, from: "Topology, geometry and quantum field theory", London Math. Soc. Lecture Note Ser. 308, Cambridge Univ. Press (2004) 421–577 MR2079383
141 B Siebert, Symplectic Gromov–Witten invariants, from: "New trends in algebraic geometry (Warwick, 1996)", London Math. Soc. Lecture Note Ser. 264, Cambridge Univ. Press (1999) 375–424 MR1714832
142 D J Simms, N M J Woodhouse, Lectures in geometric quantization, Lecture Notes in Physics 53, Springer (1976) MR0672639
143 B Simon, Representations of finite and compact groups, Graduate Studies in Mathematics 10, American Mathematical Society (1996) MR1363490
144 S Sinha, C Vafa, $\mathrmSO,
145 I Smith, R Thomas, Symplectic surgeries from singularities, Turkish J. Math. 27 (2003) 231–250 MR1975340
146 J Sondow, Analytic continuation of Riemann's zeta function and values at negative integers via Euler's transformation of series, Proc. Amer. Math. Soc. 120 (1994) 421–424 MR1172954
147 E H Spanier, Algebraic topology, Springer (1981) MR666554
148 G 't Hooft, A planar diagrams theory for strong interactions, Nuclear Phys. B 75 (1974) 461–470
149 C H Taubes, Lagrangians for the Gopakumar–Vafa conjecture, Adv. Theor. Math. Phys. 5 (2001) 139–163 MR1894340
150 A Tsuchiya, K Ueno, Y Yamada, Conformal field theory on universal family of stable curves with gauge symmetries, from: "Integrable systems in quantum field theory and statistical mechanics", Adv. Stud. Pure Math. 19, Academic Press (1989) 459–566 MR1048605
151 V G Turaev, The Conway and Kauffman modules of a solid torus, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 167 (1988) 79–89 MR964255
152 V G Turaev, Quantum invariants of knots and 3–manifolds, de Gruyter Studies in Mathematics 18, Walter de Gruyter & Co. (1994) MR1292673
153 V G Turaev, H Wenzl, Quantum invariants of 3–manifolds associated with classical simple Lie algebras, Internat. J. Math. 4 (1993) 323–358 MR1217386
154 R Vakil, The moduli space of curves and Gromov–Witten theory, arXiv:math.AG/0602347
155 A Vistoli, Intersection theory on algebraic stacks and on their moduli spaces, Invent. Math. 97 (1989) 613–670 MR1005008
156 C A Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Mathematics 38, Cambridge University Press (1994) MR1269324
157 A Weil, Elliptic functions according to Eisenstein and Kronecker, Classics in Mathematics, Springer (1999) MR1723749 Reprint of the 1976 original
158 H Wenzl, Braids and invariants of 3–manifolds, Invent. Math. 114 (1993) 235–275 MR1240638
159 E Witten, Quantum field theory and the Jones polynomial, Comm. Math. Phys. 121 (1989) 351–399 MR990772
160 E Witten, Two-dimensional gravity and intersection theory on moduli space, from: "Surveys in differential geometry (Cambridge, MA, 1990)", Lehigh Univ. (1991) 243–310 MR1144529
161 E Witten, Chern–Simons gauge theory as a string theory, from: "The Floer memorial volume", Progr. Math. 133, Birkhäuser (1995) 637–678 MR1362846
162 A Zee, Quantum field theory in a nutshell, Princeton University Press (2003) MR1978227