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


  • [BHMV95]. 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.
  • [Bla10]. 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.
  • [BN05]. 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.
  • [BS11a]. 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.
  • [BS11b]. 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.
  • [CK14]. 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.
  • [ES16a]. 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.
  • [ES16b]. 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.
  • [EST16]. M. Ehrig, C. Stroppel, and D. Tubbenhauer. Generic \(\mathfrak{gl}_2\)-foams, web and arc algebras. 2016. arxiv 1601.08010.
  • [EST17]. 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.
  • [ETW18]. 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.
  • [Kho02]. 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.
  • [Kho04]. 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.
  • [KK99]. 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.
  • [Koc04]. 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.
  • [Kup96]. 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.
  • [MPT14]. 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.
  • [SW12]. 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.
  • [Tub14a]. 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.
  • [Tub14b]. D. Tubbenhauer. \( \mathfrak{sl}_n\)-webs, categorification and Khovanov-Rozansky homologies. 2014. arxiv 1404.5752.
  • [TVW17]. 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.
  • [Wil18]. 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 für Mathematik, Universität Zürich, Winterthurerstrasse 190, Campus Irchel, CH-8057 Zürich, Switzerland
Wilbert, Arik
Department of Mathematics, University of Georgia, Athens, GA 30602, 526 Boyd GSRC, USA