Fivebranes and 4-Manifolds

  • Abhijit Gadde
  • Sergei Gukov
  • Pavel Putrov
Part of the Progress in Mathematics book series (PM, volume 319)


We describe rules for building 2d theories labeled by 4-manifolds. Using the proposed dictionary between building blocks of 4-manifolds and 2d \(\mathcal{N} = (0,2)\) theories, we obtain a number of results, which include new 3d \(\mathcal{N} = 2\) theories T[M3] associated with rational homology spheres and new results for Vafa–Witten partition functions on 4-manifolds. In particular, we point out that the gluing measure for the latter is precisely the superconformal index of 2d (0, 2) vector multiplet and relate the basic building blocks with coset branching functions. We also offer a new look at the fusion of defect lines/walls, and a physical interpretation of the 4d and 3d Kirby calculus as dualities of 2d \(\mathcal{N} = (0,2)\) theories and 3d \(\mathcal{N} = 2\) theories, respectively.



We thank F. Quinn, D. Roggenkamp, C. Schweigert, A. Stipsicz, and P. Teichner for patient and extremely helpful explanations. We also thank T. Dimofte, Y. Eliashberg, A. Kapustin, T. Mrowka, W. Neumann, T. Okazaki, E. Sharpe, C. Vafa, J. Walcher, and E. Witten, among others, for a wide variety of helpful comments. The work of A.G. is supported in part by the John A. McCone fellowship and by DOE Grant DE-FG02-92-ER40701. The work of S.G. is supported in part by DOE Grant DE-FG03-92-ER40701FG-02 and in part by NSF Grant PHY-0757647. The work of P.P. is supported in part by the Sherman Fairchild scholarship and by NSF Grant PHY-1050729. Opinions and conclusions expressed here are those of the authors and do not necessarily reflect the views of funding agencies.


  1. [Aus90]
    D.M. Austin, SO(3)-instantons on L(p, q) ×R. J. Differ. Geom. 32 (2), 383–413 (1990)MathSciNetzbMATHGoogle Scholar
  2. [Ass96]
    T. Asselmeyer, Generation of source terms in general relativity by differential structures. Classical Quantum Gravity 14, 749–758 (1997). [gr-qc/9610009]
  3. [Akb12]
    S. Akbulut, 4-Manifolds. Oxford Graduate Texts in Mathematics, vol. 25 (Oxford University Press, Oxford, 2016)Google Scholar
  4. [AG]
    A. Adams, D. Guarrera, Heterotic flux Vacua from hybrid linear models (2009) [arXiv:0902.4440]
  5. [AG04]
    B.S. Acharya, S. Gukov, M theory and singularities of exceptional holonomy manifolds. Phys. Rep. 392, 121–189 (2004). [hep-th/0409191]
  6. [AV01]
    B.S. Acharya, C. Vafa, On domain walls of N=1 supersymmetric Yang-Mills in four-dimensions (2001). [hep-th/0103011]
  7. [AW77]
    M.F. Atiyah, R.S. Ward, Instantons and algebraic geometry. Commun. Math. Phys. 55 (2), 117–124 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  8. [ABT10]
    L.F. Alday, F. Benini, Y. Tachikawa, Liouville/Toda central charges from M5-branes. Phys. Rev. Lett. 105, 141601 (2010). [arXiv:0909.4776]
  9. [AGT10]
    L.F. Alday, D. Gaiotto, Y. Tachikawa, Liouville correlation functions from four-dimensional Gauge theories. Lett. Math. Phys. 91, 167–197 (2010). [arXiv:0906.3219]
  10. [APS73]
    M.F. Atiyah, V. Patodi, I. Singer, Spectral asymmetry and Riemannian geometry. Bull. Lond. Math. Soc 5 (2), 229–234 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  11. [AOSV05]
    M. Aganagic, H. Ooguri, N. Saulina, C. Vafa, Black holes, q-deformed 2d Yang-Mills, and non-perturbative topological strings. Nucl. Phys. B715, 304–348 (2005). [hep-th/0411280]
  12. [BB13]
    F. Benini, N. Bobev, Two-dimensional SCFTs from wrapped Branes and c-extremization. J. High Energy Phys. 1306, 005 (2013). [arXiv:1302.4451]
  13. [BM09]
    C. Bachas, S. Monnier, Defect loops in gauged Wess-Zumino-Witten models. J. High Energy Phys. 1002, 003 (2010). [arXiv:0911.1562]
  14. [BR07]
    I. Brunner, D. Roggenkamp, B-type defects in Landau-Ginzburg models. J. High Energy Phys. 0708, 093 (2007). [arXiv:0707.0922]
  15. [BT96]
    M. Blau, G. Thompson, Aspects of N(T) ≥ 2 topological gauge theories and D-branes. Nucl. Phys. B492, 545–590 (1997). [hep-th/9612143]
  16. [BT97]
    M. Blau, G. Thompson, Euclidean SYM theories by time reduction and special holonomy manifolds. Phys. Lett. B415, 242–252 (1997). [hep-th/9706225]
  17. [BDP]
    C. Beem, T. Dimofte, S. Pasquetti, Holomorphic blocks in three dimensions. J. High Energy Phys. 2014 (12), Article 177, 118 pp. (2014)Google Scholar
  18. [BJR08]
    I. Brunner, H. Jockers, D. Roggenkamp, Defects and D-Brane monodromies. Adv. Theor. Math. Phys. 13, 1077–1135 (2009). [arXiv:0806.4734]
  19. [BVS95]
    M. Bershadsky, C. Vafa, V. Sadov, D-branes and topological field theories. Nucl. Phys. B463, 420–434 (1996). [hep-th/9511222]
  20. [BdDO02]
    C. Bachas, J. de Boer, R. Dijkgraaf, H. Ooguri, Permeable conformal walls and holography. J. High Energy Phys. 0206, 027 (2002). [hep-th/0111210]
  21. [BEHT13]
    F. Benini, R. Eager, K. Hori, Y. Tachikawa, Elliptic genera of two-dimensional N = 2 gauge theories with rank-one gauge groups. Lett. Math. Phys. 104 (4), 465–493 (2014)Google Scholar
  22. [BHKK99]
    O. Bergman, A. Hanany, A. Karch, B. Kol, Branes and supersymmetry breaking in three-dimensional gauge theories. J. High Energy Phys. 9910, 036 (1999). [hep-th/9908075]
  23. [BJKZ96]
    P. Berglund, C.V. Johnson, S. Kachru, P. Zaugg, Heterotic coset models and (0,2) string vacua. Nucl. Phys. B460, 252–298 (1996). [hep-th/9509170]
  24. [CH85]
    C.G. Callan, J.A. Harvey, Anomalies and fermion zero modes on strings and domain walls. Nucl. Phys. B250, 427 (1985)MathSciNetCrossRefGoogle Scholar
  25. [CR10]
    N. Carqueville, I. Runkel, Rigidity and defect actions in Landau-Ginzburg models. Commun. Math. Phys. 310, 135–179 (2012). [arXiv:1006.5609]
  26. [CCV]
    S. Cecotti, C. Cordova, C. Vafa, Braids, walls, and mirrors (2011). [arXiv:1110.2115]
  27. [Don83]
    S.K. Donaldson, An application of gauge theory to four-dimensional topology. J. Differ. Geom. 18, 279–315 (1983)MathSciNetzbMATHGoogle Scholar
  28. [DS08]
    R. Dijkgraaf, P. Sulkowski, Instantons on ALE spaces and orbifold partitions. J. High Energy Phys. 0803, 013 (2008). [arXiv:0712.1427]
  29. [DS10]
    J. Distler, E. Sharpe, Heterotic compactifications with principal bundles for general groups and general levels. Adv. Theor. Math. Phys. 14, 335–398 (2010). [hep-th/0701244]
  30. [DGG1]
    T. Dimofte, D. Gaiotto, S. Gukov, Gauge theories labelled by three-manifolds. Commun. Math. Phys. 325 (2), 367–419 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  31. [DGG2]
    T. Dimofte, D. Gaiotto, S. Gukov, 3-Manifolds and 3d Indices. [arXiv:1112.5179]
  32. [DGG13]
    T. Dimofte, M. Gabella, A.B. Goncharov, K-Decompositions and 3d gauge theories (2013). [arXiv:1301.0192]
  33. [DGH11]
    T. Dimofte, S. Gukov, L. Hollands, Vortex counting and Lagrangian 3-manifolds. Lett. Math. Phys. 98, 225–287 (2011). [arXiv:1006.0977]
  34. [DVV02]
    R. Dijkgraaf, E.P. Verlinde, M. Vonk, On the partition sum of the NS five-brane (2002). [hep-th/0205281]
  35. [DHSV07]
    R. Dijkgraaf, L. Hollands, P. Sulkowski, C. Vafa, Supersymmetric gauge theories, intersecting Branes and free fermions. J. High Energy Phys. 0802, 106 (2008). [arXiv:0709.4446]
  36. [dDHKM02]
    J. de Boer, R. Dijkgraaf, K. Hori, A. Keurentjes, J. Morgan, et al., Triples, fluxes, and strings. Adv. Theor. Math. Phys. 4, 995–1186 (2002). [hep-th/0103170]
  37. [Fre82]
    M. Freedman, The topology of four dimensional manifolds. J. Differ. Geom. 17, 357–453 (1982)MathSciNetzbMATHGoogle Scholar
  38. [FH90]
    M. Furuta, Y. Hashimoto, Invariant instantons on S 4. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 37 (3), 585–600 (1990)MathSciNetzbMATHGoogle Scholar
  39. [FW99]
    D.S. Freed, E. Witten, Anomalies in string theory with D-branes. Asian J. Math. 3, 819 (1999). [hep-th/9907189]
  40. [FGP13]
    H. Fuji, S. Gukov, P. Sulkowski, Super-a-polynomial for knots and BPS states. Nucl. Phys. B867, 506–546 (2013). [arXiv:1205.1515]
  41. [FSV12]
    J. Fuchs, C. Schweigert, A. Velentino, Bicategories for boundary conditions and for surface defects in 3-d TFT. Commun. Math. Phys. 321 (2), 543–575 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  42. [FGSA]
    H. Fuji, S. Gukov, P. Sulkowski, H. Awata, Volume conjecture: refined and categorified. Adv. Theor. Math. Phys. 16 (2), 1669–1777 (2012)MathSciNetzbMATHGoogle Scholar
  43. [FGSS]
    H. Fuji, S. Gukov, M. Stos̆ić, P. Sulkowski, 3d analogs of Argyres-Douglas theories and knot homologies. J. High Energy Phys. 2013, 175 (2003)Google Scholar
  44. [Gan96]
    O.J. Ganor, Compactification of tensionless string theories (1996) [hep-th/9607092]
  45. [Guk05]
    S. Gukov, Three-dimensional quantum gravity, Chern-Simons theory, and the A-polynomial. Commun. Math. Phys. 255 (3), 577–627 (2005). [hep-th/0306165]
  46. [Guk07]
    S. Gukov, Gauge theory and knot homologies. Fortschr. Phys. 55, 473–490 (2007). [arXiv:0706.2369]
  47. [Gai12]
    D. Gaiotto, N=2 dualities. J. High Energy Phys. 1208, 034 (2012). [arXiv:0904.2715]
  48. [GL91]
    T. Gannon, C. Lam, Gluing and shifting lattice constructions and rational equivalence. Rev. Math. Phys. 3 (03), 331–369 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  49. [GL92]
    T. Gannon, C. Lam, Lattices and \(\Theta\)-function identities. I: Theta constants. J. Math. Phys. 33, 854 (1992)MathSciNetzbMATHGoogle Scholar
  50. [GL92]
    T. Gannon, C. Lam, Lattices and θ-function identities. II: Theta series. J. Math. Phys. 33, 871 (1992)MathSciNetzbMATHGoogle Scholar
  51. [GK02]
    J.P. Gauntlett, N. Kim, M five-branes wrapped on supersymmetric cycles. 2.. Phys. Rev. D65, 086003 (2002). [hep-th/0109039]
  52. [GK09]
    A. Giveon, D. Kutasov, Seiberg duality in Chern-Simons theory. Nucl. Phys. B812, 1–11 (2009). [arXiv:0808.0360]
  53. [GP15]
    S. Gukov, D. Pei, Equivariant Verlinde formula from fivebranes and vortices (2015). [arXiv:1501.0131]
  54. [GS99]
    R.E. Gompf, A.I. Stipsicz, 4-manifolds and Kirby calculus. Graduate Studies in Mathematics, vol. 20 (American Mathematical Society, Providence, RI, 1999)Google Scholar
  55. [GW09]
    D. Gaiotto, E. Witten, Supersymmetric boundary conditions in N=4 super Yang-Mills theory. J. Stat. Phys. 135, 789–855 (2009). [arXiv:0804.2902]
  56. [GGP13]
    A. Gadde, S. Gukov, P.J. Putrov, Walls, lines, and spectral dualities in 3d Gauge theories. J. High Energy Phys. 2014, 47 (2014)CrossRefGoogle Scholar
  57. [GKW00]
    J.P. Gauntlett, N. Kim, D. Waldram, M Five-branes wrapped on supersymmetric cycles. Phys. Rev. D63, 126001 (2001). [hep-th/0012195]
  58. [GMN10]
    D. Gaiotto, G.W. Moore, A. Neitzke, Four-dimensional wall-crossing via three-dimensional field theory. Commun. Math. Phys. 299, 163–224 (2010). [arXiv:0807.4723]
  59. [GPS93]
    S.B. Giddings, J. Polchinski, A. Strominger, Four-dimensional black holes in string theory. Phys. Rev. D48, 5784–5797 (1993). [hep-th/9305083]
  60. [GST02]
    S. Gukov, J. Sparks, D. Tong, Conifold transitions and five-brane condensation in M theory on spin(7) manifolds. Classical Quantum Gravity 20, 665–706 (2003). [hep-th/0207244]
  61. [GSW87]
    M.B. Green, J. Schwarz, E. Witten, Superstring Theory. vol. 1: Introduction, 1st edn. (Cambridge, New York, 1987)Google Scholar
  62. [GVW00]
    S. Gukov, C. Vafa, E. Witten, CFT’s from Calabi-Yau four folds. Nucl. Phys. B584, 69–108 (2000). [hep-th/9906070]
  63. [GRRY11]
    A. Gadde, L. Rastelli, S.S. Razamat, W. Yan, The 4d superconformal index from q-deformed 2d Yang-Mills. Phys. Rev. Lett. 106, 241602 (2011). [arXiv:1104.3850]
  64. [Har79]
    J.L. Harer, Pencils of Curves on 4-Manifolds (ProQuest LLC, Ann Arbor, MI, 1979). Thesis (Ph.D.)-University of California, BerkeleyGoogle Scholar
  65. [HW97]
    A. Hanany, E. Witten, Type IIB superstrings, BPS monopoles, and three-dimensional gauge dynamics. Nucl. Phys. B492, 152–190 (1997). [hep-th/9611230]
  66. [HW04]
    K. Hori, J. Walcher, D-branes from matrix factorizations. C. R. Phys. 5, 1061–1070 (2004). [hep-th/0409204]
  67. [Ito93]
    M. Itoh, Moduli of half conformally flat structures. Math. Ann. 296 (4), 687–708 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  68. [Joh95]
    C.V. Johnson, Heterotic coset models. Mod. Phys. Lett. A10, 549–560 (1995). [hep-th/9409062]
  69. [KP]
    V.G. Kac, D.H. Petersen, Infinite-dimensional Lie algebras, theta functions and modular forms. Adv. Math. 53 (2), 125–264 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  70. [KS10]
    A. Kapustin, N. Saulina, Surface operators in 3d topological field theory and 2d rational conformal field theory, in Mathematical Foundations of Quantum Field Theory and Perturbative String Theory. Proceedings of Symposia in Pure Mathematics, vol. 83 (American Mathematical Society, Providence, 2011), pp. 175–198Google Scholar
  71. [KS11]
    A. Kapustin, N. Saulina, Topological boundary conditions in abelian Chern-Simons theory. Nucl. Phys. B845, 393–435 (2011). [arXiv:1008.0654]
  72. [KW07]
    A. Kapustin, E. Witten, Electric-magnetic duality and the geometric Langlands program. Commun. Num. Theor. Phys. 1, 1–236 (2007). [hep-th/0604151]
  73. [KW13]
    A. Kapustin, B. Willett, Wilson loops in supersymmetric Chern-Simons-matter theories and duality (2007). [arXiv:1302.2164]
  74. [KOO99]
    T. Kitao, K. Ohta, N. Ohta, Three-dimensional gauge dynamics from brane configurations with (p,q) - five-brane. Nucl. Phys. B539, 79–106 (1999). [hep-th/9808111]
  75. [Loc87]
    R. Lockhart, Fredholm, Hodge and Liouville theorems on noncompact manifolds. Trans. Am. Math. Soc. 301 (1), 1–35 (1987)MathSciNetCrossRefGoogle Scholar
  76. [LP72]
    F. Laudenbach, V. Poénaru, A note on 4-dimensional handlebodies. Bull. Soc. Math. Fr. 100, 337–344 (1972)MathSciNetzbMATHGoogle Scholar
  77. [Mar95]
    N. Marcus, The other topological twisting of N=4 Yang-Mills. Nucl. Phys. B452, 331–345 (1995). [hep-th/9506002]
  78. [Mac99]
    M. Mackaay, Spherical 2-categories and 4-manifold invariants. Adv. Math. 143 (2), 288–348 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  79. [MNVW98]
    J. Minahan, D. Nemeschansky, C. Vafa, N. Warner, E strings and N=4 topological Yang-Mills theories. Nucl. Phys. B527, 581–623 (1998). [hep-th/9802168]
  80. [MQSS12]
    I.V. Melnikov, C. Quigley, S. Sethi, M. Stern, Target spaces from Chiral gauge theories. J. High Energy Phys. 1302, 111 (2013). [arXiv:1212.1212]
  81. [Nor69]
    R.A. Norman, Dehn’s lemma for certain 4-manifolds. Invent. Math. 7, 143–147 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  82. [Nak94]
    H. Nakajima, Instantons on ALE spaces, quiver varieties, and Kac-Moody Algebras. Duke Math. 76, 365–416 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  83. [NRXS12]
    S. Nawata, P. Ramadevi, Zodinmawia, X. Sun, Super-A-polynomials for twist knots. J. High Energy Phys. 1211, 157 (2012). [arXiv:1209.1409]
  84. [Oht99]
    K. Ohta, Supersymmetric index and s rule for type IIB branes. J. High Energy Phys. 9910, 006 (1999). [hep-th/9908120]
  85. [OA97]
    M. Oshikawa, I. Affleck, Boundary conformal field theory approach to the critical two-dimensional Ising model with a defect line. Nucl. Phys. B495, 533–582 (1997). [cond-mat/9612187]
  86. [OY13]
    T. Okazaki, S. Yamaguchi, Supersymmetric boundary conditions in 3D N = 2 theories, in String-Math 2013. Proceedings of Symposia in Pure Mathematics, vol. 88 (American Mathematical Society, Providence, 2014), pp. 343–349Google Scholar
  87. [Pfe04]
    H. Pfeiffer, Quantum general relativity and the classification of smooth manifolds (2004). [gr-qc/0404088]
  88. [Qui79]
    F. Quinn, Ends of maps. I. Ann. Math. (2) 110 (2), 275–331 (1979)Google Scholar
  89. [Qui82]
    F. Quinn, Ends of maps. III. Dimensions 4 and 5. J. Differ. Geom. 17 (3), 503–521 (1982)Google Scholar
  90. [QS02]
    T. Quella, V. Schomerus, Symmetry breaking boundary states and defect lines. J. High Energy Phys. 0206, 028 (2002). [hep-th/0203161]
  91. [Roh89]
    R. Rohm, Topological defects and Differential structures. Ann. Phys. 189, 223 (1989)MathSciNetCrossRefGoogle Scholar
  92. [Sav02]
    N. Saveliev, Fukumoto-Furuta invariants of plumbed homology 3-spheres. Pac. J. Math. 205 (2), 465–490 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  93. [Sla09]
    J. Sladkowski, Exotic smoothness and astrophysics. Acta Physiol. Pol. B40, 3157–3163 (2009). [arXiv:0910.2828]
  94. [Smi10]
    A. Smilga, Witten index in supersymmetric 3d theories revisited. J. High Energy Phys. 1001, 086 (2010). [arXiv:0910.0803]
  95. [SW94]
    N. Seiberg, E. Witten, Electric - magnetic duality, monopole condensation, and confinement in N=2 supersymmetric Yang-Mills theory. Nucl. Phys. B426, 19–52 (1994). [hep-th/9407087]
  96. [VW94]
    C. Vafa, E. Witten, A strong coupling test of S duality. Nucl. Phys. B431, 3–77 (1994). [hep-th/9408074]
  97. [Wit87]
    E. Witten, Elliptic genera and quantum field theory. Commun. Math. Phys. 109, 525 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  98. [Wit88]
    E. Witten, Topological quantum field theory. Commun. Math. Phys. 117, 353 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  99. [Wit93]
    E. Witten, The Verlinde algebra and the cohomology of the Grassmannian (1993). [hep-th/9312104]
  100. [Witt93]
    E. Witten, Phases of N=2 theories in two-dimensions. Nucl. Phys. B403, 159–222 (1993). [hep-th/9301042]
  101. [Wit94]
    E. Witten, Monopoles and four manifolds. Math. Res. Lett. 1, 769–796 (1994). [hep-th/9411102]
  102. [War95]
    N. Warner, Supersymmetry in boundary integrable models. Nucl. Phys. B450, 663–694 (1995). [hep-th/9506064]
  103. [Wit96]
    E. Witten, Five-brane effective action in M theory. J. Geom. Phys. 22, 103–133 (1997). [hep-th/9610234]
  104. [Wit98]
    E. Witten, Toroidal compactification without vector structure. J. High Energy Phys. 9802, 006 (1998). [hep-th/9712028]
  105. [Wit99]
    E. Witten, Supersymmetric index of three-dimensional gauge theory, in The Many Faces of the Superworld (World Scientific, River Edge, 2000), pp. 156–184zbMATHGoogle Scholar
  106. [Wit03]
    E. Witten, SL(2,Z) action on three-dimensional conformal field theories with Abelian symmetry, in From Fields to Strings: Circumnavigating Theoretical Physics, vol. 2 (World Scientific, Singapore, 2005), pp. 1173–1200Google Scholar
  107. [WA94]
    E. Wong, I. Affleck, Tunneling in quantum wires: a boundary conformal field theory approach. Nucl. Phys. B417, 403–438 (1994)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Abhijit Gadde
    • 1
  • Sergei Gukov
    • 1
  • Pavel Putrov
    • 1
  1. 1.California Institute of TechnologyPasadenaUSA

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