nLab simple type conjecture

Idea

In Seiberg-Witten theory, the simple type conjecture is a conjecture about vanishing Seiberg-Witten invariants. A Riemannian 4-manifold, for which all Seiberg-Witten invariants of positive-dimensional Seiberg-Witten moduli spaces vanish, is said to be of Seiberg-Witten simple type (simple type if context is clear). In this case, non-vanishing Seiberg-Witten invariants only come from zero-dimensional Seiberg-Witten moduli spaces by counting its points with a sign determined by their orientation. Now the simple type conjecture states:

Every simply connected Riemannian 4-manifold $M$ with $b_2^+(M)\geq 2$ is of simple type.

As the Seiberg-Witten invariants of the second complex projective plane $M=\mathbb{C}P^2$ with $b_2^+(M)=1$ show, the conjecture doesn’t hold under this condition. So far, the simple type conjecture is known to hold for symplectic manifolds and under reduction $mod 2$ after Kato & Makamura & Yasui 20.

Related concepts

Articles about Seiberg-Witten theory:

• Ordinary theory:

  • Seiberg-Witten equations

  • linearized Seiberg-Witten equations

  • Seiberg-Witten invariant

  • simple type conjecture

  • Seiberg-Witten moduli space

  • Seiberg-Witten flow

• Adapted theory:

  • D=6 Seiberg-Witten theory

  • D=7 Seiberg-Witten theory (over G₂ manifolds)

  • D=8 Seiberg-Witten theory (over Spin(7) manifolds)

  • quaternionic Seiberg-Witten theory

  • pseudo-Riemannian Seiberg-Witten theory

References

• Paul Feehan, Thomas Leness, Witten’s conjecture for many four-manifolds of simple type, Journal of the European Mathematical Society 17 4, pp. 899-923 (2015) [arXiv:/math/0609530 doi:10.4171/JEMS/521]

• Tsuyoshi Kato, Nobuhiro Nakamura, Kouichi Yasui, The simple type conjecture for mod 2 Seiberg-Witten invariants (2020) [arXiv:2009.06791]