Seiberg-Witten Invariants and Superconducting Vortices

Using a reformulation of topological N = 2 QFT’s in M-theory setup, where QFT is realized via M5 branes wrapping co-associative cycles in a G2 manifold constructed from the space of self-dual 2-forms over a four-fold X , we show that superconducting vortices are mapped to M2 branes stretched between M5 branes. This setup provides a physical explanation of Taubes’ construction of the Seiberg-Witten invariants when X is symplectic and the superconducting vortices are realized as pseudo-holomorphic curves. This setup is general enough to realize topological QFT’s arising from N = 2 QFT’s from all Gaiotto theories on arbitrary 4-manifolds. December 25, 2020 e-mail: cecotti@sissa.it e-mail: cgerig@math.harvard.edu e-mail: vafa@g.harvard.edu

[1]  Chris Gerig.  Seiberg–Witten and Gromov invariants for self-dual harmonic 2–forms , 2018, Geometry & Topology.

[2]  Chris Gerig.  Taming the pseudoholomorphic beasts in R × ( S 1 × S 2 ) , 2018 .

[3]  S. Gukov.  Trisecting non-Lagrangian theories , 2017, 1707.01515.

[4]  D. Gaiotto.  Preprint Typeset in Jhep Style -hyper Version N = 2 Dualities , 2022 .

[5]  C. Vafa,et al.  Classification of Complete N = 2 Supersymmetric Theories in 4 Dimensions , 2011, 1103.5832.

[6]  G. ’t Hooft,et al.  Computation of the quantum effects due to a four-dimensional pseudoparticle , 2011 .

[7]  E. Witten.  From superconductors and four-manifolds to weak interactions , 2007 .

[8]  D. joyce.  Riemannian Holonomy Groups And Calibrated Geometry , 2007 .

[9]  C. Taubes.  A proof of a theorem of Luttinger and Simpson about the number of vanishing circles of a near-symplectic form on a 4-dimensional manifold , 2006 .

[10]  C. Taubes,et al.  Seiberg-Witten and Gromov invariants for symplectic 4-manifolds , 2005 .

[11]  C. Vafa,et al.  Khovanov-Rozansky Homology and Topological Strings , 2004, hep-th/0412243.

[12]  C. Vafa,et al.  Topological M-theory as unification of form theories of gravity , 2004, hep-th/0411073.

[13]  K. Honda.  Transversality Theorems for Harmonic Forms , 2004 .

[14]  C. Taubes.  Seiberg-Witten invariants, self-dual harmonic 2-forms and Hofer-Wysocki-Zehnder formalism , 2002 .

[15]  J. Morgan.  Seiberg-Witten invariants , 2001 .

[16]  M. Mariño.  Topological quantum field theory and four manifolds , 2000, hep-th/0008100.

[17]  Dominic Joyce,et al.  Compact Manifolds with Special Holonomy , 2000 .

[18]  R. Bryant.  Calibrated Embeddings in the Special Lagrangian and Coassociative Cases , 1999, math/9912246.

[19]  H. Ooguri,et al.  Knot Invariants and Topological Strings , 1999, hep-th/9912123.

[20]  C. Taubes.  Seiberg{Witten Invariants and Pseudo-Holomorphic Subvarieties for Self-Dual, Harmonic 2{Forms , 1999, math/9907199.

[21]  C. Taubes.  A compendium of pseudoholomorphic beasts in R × ( S 1 × S 2 ) , 1999 .

[22]  R. C. Mclean.  Deformations of calibrated submanifolds , 1998 .

[23]  W. Nahm.  On electric-magnetic duality , 1997 .

[24]  E. Witten,et al.  Integration over the u-plane in Donaldson theory , 1997, hep-th/9709193.

[25]  M. Hutchings,et al.  An introduction to the Seiberg-Witten equations on symplectic manifolds , 1997 .

[26]  C. Lebrun,et al.  Yamabe Constants and the Perturbed Seiberg-Witten Equations , 1996, dg-ga/9605009.

[27]  C. Vafa,et al.  Self-dual strings and N = 2 supersymmetric field theory , 1996, hep-th/9604034.

[28]  C. Vafa,et al.  D-branes and topological field theories , 1995, hep-th/9511222.

[29]  M. Douglas,et al.  New Phenomena in SU(3) Supersymmetric Gauge Theory , 1995, hep-th/9505062.

[30]  C. Taubes.  The Seiberg-Witten and Gromov invariants , 1995 .

[31]  E. Witten.  Monopoles and four-manifolds , 1994, hep-th/9411102.

[32]  E. Witten,et al.  Electric - magnetic duality, monopole condensation, and confinement in N=2 supersymmetric Yang-Mills theory , 1994, hep-th/9407087.

[33]  C. Taubes.  THE SEIBERG-WITTEN INVARIANTS AND SYMPLECTIC FORMS , 1994 .

[34]  E. Witten.  Chern-Simons gauge theory as a string theory , 1992, hep-th/9207094.

[35]  Claude Viterbo,et al.  An introduction to symplectic topology , 1991 .

[36]  Edward Witten,et al.  Quantum field theory and the Jones polynomial , 1989 .

[37]  Edward Witten,et al.  Topological quantum field theory , 1988 .

[38]  M. Gromov.  Pseudo holomorphic curves in symplectic manifolds , 1985 .