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?
Best Answer
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 (eg An SO(3)-monopole cobordism formula relating Donaldson and Seiberg-Witten invariants, published as https://doi.org/10.1090/memo/1226), and some of their partial progress was used e.g. in the work of Kronheimer-Mrowka resolving Property (P).