Ehrig, Michael; Tubbenhauer, Daniel; Wilbert, Arik

Singular TQFTs, Foams and Type D Arc Algebras

Doc. Math. 24, 1585-1655 (2019)
DOI: 10.25537/dm.2019v24.1585-1655
Communicated by Dan Ciubotaru


We combinatorially describe the 2-category of singular cobordisms, called (rank one) foams, which governs the functorial version of Khovanov homology. As an application we topologically realize the type D arc algebra using this singular cobordism construction.

Mathematics Subject Classification

16G10, 17B10, 17B37, 57M27, 81R50


singular TQFTs, cobordism/foam categories, functorial Khovanov homology, web/arc algebras, Lie type D


  • 1. C. Blanchet, N. Habegger, G. Masbaum, and P. Vogel. Topological quantum field theories derived from the Kauffman bracket. Topology, 34(4):883-927, 1995. DOI 10.1016/0040-9383(94)00051-4; zbl 0887.57009; MR1362791.
  • 2. C. Blanchet. An oriented model for Khovanov homology. J. Knot Theory Ramifications, 19(2):291-312, 2010. DOI 10.1142/S0218216510007863; zbl 1195.57024; MR2647055; arxiv 1405.7246.
  • 3. D. Bar-Natan. Khovanov's homology for tangles and cobordisms. Geom. Topol., 9:1443-1499, 2005. DOI 10.2140/gt.2005.9.1443; zbl 1084.57011; MR2174270; arxiv math/0410495.
  • 4. J. Brundan and C. Stroppel. Highest weight categories arising from Khovanov's diagram algebra I: cellularity. Mosc. Math. J., 11(4):685-722, 821-822, 2011. zbl 1275.17012; MR2918294; arxiv 0806.1532.
  • 5. J. Brundan and C. Stroppel. Highest weight categories arising from Khovanov's diagram algebra III: category \(\mathcal{O} \). Represent. Theory, 15:170-243, 2011. DOI 10.1090/S1088-4165-2011-00389-7; zbl 1261.17006; MR2781018; arxiv 0812.1090.
  • 6. Y. Chen and M. Khovanov. An invariant of tangle cobordisms via subquotients of arc rings. Fund. Math., 225(1):23-44, 2014. DOI 10.4064/fm225-1-2; zbl 1321.57031; MR3205563; arxiv math/0610054.
  • 7. M. Ehrig and C. Stroppel. \(2\)-row Springer fibres and Khovanov diagram algebras for type D. Canad. J. Math., 68(6):1285-1333, 2016. DOI 10.4153/CJM-2015-051-4; zbl 06663416; MR3563723; arxiv 1209.4998.
  • 8. M. Ehrig and C. Stroppel. Diagrammatic description for the categories of perverse sheaves on isotropic Grassmannians. Selecta Math. (N.S.), 22(3):1455-1536, 2016. DOI 10.1007/s00029-015-0215-9; zbl 1343.05162; MR3518556; arxiv 1511.04111.
  • 9. M. Ehrig, C. Stroppel, and D. Tubbenhauer. Generic \(\mathfrak{gl}_2\)-foams, web and arc algebras. 2016. arxiv 1601.08010.
  • 10. M. Ehrig, C. Stroppel, and D. Tubbenhauer. The Blanchet-Khovanov algebras. In Categorification and higher representation theory, volume 683 of Contemp. Math., pages 183-226. Amer. Math. Soc., Providence, RI, 2017. DOI 10.1090/conm/683/13707; zbl 06708136; MR3611714; arxiv 1510.04884.
  • 11. M. Ehrig, D. Tubbenhauer, and P. Wedrich. Functoriality of colored link homologies. Proc. Lond. Math. Soc., 117(5):996-1040, 2018. DOI 10.1112/plms.12154; zbl 06991332; MR3877770; arxiv 1703.06691.
  • 12. M. Khovanov. A functor-valued invariant of tangles. Algebr. Geom. Topol., 2:665-741, 2002. DOI 10.2140/agt.2002.2.665; zbl 1002.57006; MR1928174; arxiv math/0103190.
  • 13. M. Khovanov. \( \mathfrak{sl}(3)\) link homology. Algebr. Geom. Topol., 4:1045-1081, 2004. DOI 10.2140/agt.2004.4.1045; zbl 1159.57300; MR2100691; arxiv math/0304375.
  • 14. M. Khovanov and G. Kuperberg. Web bases for \(\mathfrak{sl}(3)\) are not dual canonical. Pacific J. Math., 188(1):129-153, 1999. DOI 10.2140/pjm.1999.188.129; zbl 0929.17012; MR1680395; arxiv q-alg/9712046.
  • 15. J. Kock. Frobenius algebras and \(2\) D topological quantum field theories, volume 59 of London Mathematical Society Student Texts. Cambridge University Press, Cambridge, 2004. zbl 1046.57001; MR2037238.
  • 16. G. Kuperberg. Spiders for rank \(2\) Lie algebras. Comm. Math. Phys., 180(1):109-151, 1996. DOI 10.1007/BF02101184; zbl 0870.17005; MR1403861; arxiv q-alg/9712003.
  • 17. M. Mackaay, W. Pan, and D. Tubbenhauer. The \(\mathfrak{sl}_3\)-web algebra. Math. Z., 277(1-2):401-479, 2014. DOI 10.1007/s00209-013-1262-6; zbl 1321.17010; MR3205780; arxiv 1206.2118.
  • 18. C. Stroppel and B. Webster. 2-block Springer fibers: convolution algebras and coherent sheaves. Comment. Math. Helv., 87(2):477-520, 2012. DOI 10.4171/CMH/261; zbl 1241.14009; MR2914857; arxiv 0802.1943.
  • 19. D. Tubbenhauer. \( \mathfrak{sl}_3\)-web bases, intermediate crystal bases and categorification. J. Algebraic Combin., 40(4):1001-1076, 2014. DOI 10.1007/s10801-014-0518-5; zbl 1322.17010; MR3273401; arxiv 1310.2779.
  • 20. D. Tubbenhauer. \( \mathfrak{sl}_n\)-webs, categorification and Khovanov-Rozansky homologies. 2014. arxiv 1404.5752.
  • 21. D. Tubbenhauer, P. Vaz, and P. Wedrich. Super \(q\)-Howe duality and web categories. Algebr. Geom. Topol., 17(6):3703-3749, 2017. DOI 10.2140/agt.2017.17.3703; zbl 06791660; MR3709658; arxiv 1504.05069.
  • 22. A. Wilbert. Topology of two-row Springer fibers for the even orthogonal and symplectic group. Trans. Amer. Math. Soc., 370(4):2707-2737, 2018. DOI 10.1090/tran/7194; zbl 06830286; MR3748583; arxiv 1511.01961.


Ehrig, Michael
Beijing Institute of Technology, School of Mathematics and Statistics, Liangxiang Campus of Beijing Institute of Technology, Fangshan District, 100488 Beijing, China
Tubbenhauer, Daniel
Institut f\``ur Mathematik, Universit\''at Z\``urich, Winterthurerstrasse 190, Campus Irchel, CH-8057 Z\''urich, Switzerland
Wilbert, Arik
Department of Mathematics, University of Georgia, Athens, GA 30602, 526 Boyd GSRC, USA