# What else is Seiberg-Witten Theory equal to?

In low-dimensional topology there have been a bunch of invariants defined, and Seiberg-Witten Theory seems to make its appearance in [a lot of] them:
1) Heegaard Floer homology = SW Floer homology (Kutluhan, Lee, Taubes)
2) Embedded Contact homology = SW Floer homology (Taubes)
3) Gromov-Witten invariant = 4-dimensional SW-invariant (Taubes)
4) Turaev torsion = 3-dimensional SW-invariant (Turaev)
5) Milnor torsion (hence Alexander invariant) = 3-dimensional SW-invariant (Meng, Taubes)
6) Donaldson-Smith standard surface count = 4-dimensional SW-invariant (Usher)
7) Casson invariant (hence integral Theta divisor) = 3-dimensional SW-invariant (Lim)
8) Poincare Invariant = SW-invariant for algebraic surfaces (Okonek, et al.)

Conjectured:
8) Heegaard Floer closed 4-manifold invariant = SW-invariant (Ozsvath, Szabo)
*Analog of (1) above in dimension 4
9) Lagrangian matching invariant = SW-invariant (Perutz)
*Analog of (6) above for broken Lefschetz fibrations
10) Near-symplectic Gromov-Witten count = SW-invariant (Taubes)
*Analog of (4) above for near-symplectic manifolds, counting holomorphic curves in the complement of the degenerate circles of the near-symplectic form -- but this invariant hasn't really been defined yet

Does/should it stop there? Are there constructions out there that Seiberg-Witten Theory could possibly have a link with?

• This should probably be community wiki, since there may be multiple useful answers. Check out: ams.org/mathscinet-getitem?mr=2405163 – Ian Agol Jun 13 '12 at 17:07
• Also: the title suggests that you're looking for things which are different from SW, but the last paragraph seems to ask for more things to add to your list of "things SW can do". What's the question you're actually asking? – Marco Golla Jun 13 '12 at 17:18
• They're supposed to complement each other -- I am wondering about the "why" aspect of SW-theory appearing all over the place. – Chris Gerig Jun 13 '12 at 17:44
• 5) is joint work of Taubes and Meng. – John Klein Jun 14 '12 at 1:49
• 1) is also addressed by Colin, Ghiggini and Honda (I think they're still in the phase of writing, but there's already something online: arxiv.org/abs/1008.2734 ) – Marco Golla Jun 14 '12 at 7:46

There is a conjectured categorification of the HOMFLY polynomial, which the conjectured property that it recovers knot Floer homology (which is the analogue of Heegaard Floer for knots). Maybe once there are categorified versions of the Reshetikhin-Turaev invariants and generalizations, there may also be a connection of these invariants with Heegaard Floer homology.

There is also an important conjecture of Witten which is motivated by certain gauge theory dualities relating Seiberg-Witten and Donaldson invariants of 4-manifolds. Feehan and Leness used to work hard on this, and some of their partial progress was used e.g. in the work of Kronheimer-Mrowka resolving Property (P).

• So is this related to SW or not related to SW (for us in the peanut stalls). – David Roberts Jun 14 '12 at 3:57
• @ David: Conjecturally, knot Floer homology should be equivalent to a flavor of SW knot Floer homology. In fact, knot Floer homology is a special case of Juhasz's sutured Floer homology, which should be equivalent to Kronheimer-Mrowka's "monopole Floer homology" (I think monopole refers to Seiberg-Witten theory). However, I'm not sure this equivalence has been proved yet. front.math.ucdavis.edu/0807.4891 – Ian Agol Jun 14 '12 at 4:24
• @Agol: Monopole Floer homology is another name for SW Floer homology, and in any case the two sutured Floer homologies are in fact isomorphic as abelian groups (i.e. maybe without a decomposition with respect to spin^c structures). This follows from work of Lekili (arxiv.org/abs/0903.1773) together with the equivalence of Heegaard Floer and monopole Floer homologies proved by either Kutluhan-Lee-Taubes or Taubes + Colin-Ghiggini-Honda. – Steven Sivek Jun 14 '12 at 13:46
• @Chris: I think this paper describes the challenges left in completing the program. ams.org/mathscinet-getitem?mr=2189928 – Ian Agol Jun 14 '12 at 22:55
• In arxiv.org/abs/1001.5024 I together with Goettsche and Yoshioka proved SW = Donaldson for complex projective surfaces. It is still open for arbitrary $4$-manifolds, but we reduce it to another conjecture involving Nekrasov's partition function. – Hiraku Nakajima Jun 15 '12 at 21:44

Khovanov homology could probably fit into one of the two categories (although I'm not experienced enough to judge/guess whether it belongs to the "SW does it all" or to the "SW can't do it" category).

At a first glance, the two objects look quite different: Khovanov homology is a theory for knots and links in $S^3$ (and cobordisms between them), that categorifies the Jones polynomial, detects the unknot. It can be defined combinatorially in terms of knot diagrams, but I think that the original definition had a flavour of category theory and representation theory. However, Witten has recently proposed a gauge-theoretic approach to it, so maybe there's a deep connection after all.

Moreover, it's been known for some time that Khovanov homology and Heegaard Floer homology are related. Take a knot $K\subset S^3$: there's a spectral sequence, whose $E_2$-page is defined in terms of (a suitable variant of) $KhH(K)$, that converges to $\widehat{HF}(\Sigma(K))$, the Floer homology of the double cover of $S^3$, branched over $K$.