nLab Kronheimer-Mrowka basic class

Idea

The Kronheimer-Mrowka basic classes are cohomology classes of the second cohomology of a simply connected 4-manifold generating its Donaldson polynomials in Donaldson theory, introduced by Peter Kronheimer and Tomasz Mrowka.

Description

For a 4-manifold $M$, its Donaldson invariants are an integer $\gamma_0(M)\in\mathbb{Z}$ and maps $\gamma_d(M)\colon H_2(M,\mathbb{Z})\rightarrow\mathbb{Z}[1/2]$ (into half-integers), which combine into the Donaldson polynomial:

$$\mathcal{D}_M\colon H_2(M,\mathbb{Z})\rightarrow\mathbb{R}, \mathcal{D}_M(x) =\sum_{d=0}^\infty\frac{\gamma_d(M)(x)}{d!}.$$

(Kronheimer & Mrowka 94, p. 3, Naber 11, p. 399)

Peter Kronheimer and Tomasz Mrowka introduced a condition known as Kronheimer–Mrowka simple type (KM symple type), which is sufficient to obtain the separate Donaldson invariants from their common Donaldson polynomial. For a KM-simple manifold $M$ there are cohomology classes $K_1,\ldots,K_s\in H^2(M,\mathbb{Z})$, called Kronheimer–Mrowka basic classes (short KM basic classes), as well as rational numbers $a_1,\ldots,a_s\in\mathbb{Q}$, called Kronheimer–Mrowka coefficients (short KM coefficients), so that:

$$\mathcal{D}_M(x) =\exp(Q_M(x,x)/2)\sum_{r=1}^s a_r\exp(K_r(x))$$

for all $x\in H_2(M,\mathbb{Z})$. Furthermore $w_2(M)=K_r\operatorname{mod}2\in H^2(M,\mathbb{Z}_2)$ for all KM basic classes.

(Kronheimer & Mrowka 94, Prop. 3, Kronheimer & Mrowka 95, Thrm. 1.7, Naber 11, Theorem A.5.1)

Although this reduction of the infinite sum of the Donaldson polynomial to a finite sum in early 1994 brought a significant simplification to Donaldson theory, it was overhauled just a few months later in late 1994 by the development of Seiberg-Witten theory. Edward Witten, presented in a lecture at MIT, used a purely physical argument to conjecture that Kronheimer-Mrowka basic classes are exactly the support of the Seiberg-Witten invariants $\operatorname{SW}\colon \operatorname{Spin}^\mathrm{c}(M)\rightarrow\mathbb{Z}$(hence the first Chern class $c_1\colon \operatorname{Spin}^\mathrm{c}(M)\rightarrow H^2(M,\mathbb{Z})$ of spinᶜ structures with a non-vanishing Seiberg-Witten invariant) and their Kronheimer-Mrowka coefficients are up to a topological factor exactly their Seiberg-Witten invariants. More concretely, it claims that a compact connected simply connected orientable smooth 4-manifold $M$ with $b_2^+(M)\geq 2$ odd is of Kronheimer–Mrowka simple type if and only if is of Seiberg–Witten simple type (meaning 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). In this case the Donaldson polynomial is given by:

$$\mathcal{D}_M(x) =\exp(Q_M(x,x)/2)\sum_{\mathfrak{s}\in\operatorname{Spin}^\mathrm{c}(M), \dim\mathcal{M}_{\mathfrak{s}}^\mathrm{SW}=0}2^{2+\frac{1}{4}(7\chi(M)+11\sigma(M))}\operatorname{SW}(M,\mathfrak{s})\exp(c_1(\mathfrak{s})(x)).$$

(Naber 11, p. 400)

References

• Peter Kronheimer, Tomasz Mrowka, Recurrence relations and asymptotics for four-manifold invariants, Bulletin of the American Mathematical Society, New Series, 30 (2): 215–221, arXiv:math/9404232, doi:10.1090/S0273-0979-1994-00492-6, ISSN 0002-9904, MR 1246469

• Peter Kronheimer, Tomasz Mrowka, Embedded surfaces and the structure of Donaldson’s polynomial invariants, Journal of Differential Geometry, 41 (3): 573–734, doi:10.4310/jdg/1214456482, ISSN 0022-040X, MR 1338483

• Gregory L. Naber, Topology, Geometry and Gauge fields – Foundations, Texts in Applied Mathematics 25 (2011) [doi:10.1007/978-1-4419-7254-5]

See also

• Wikipedia , Kronheimer–Mrowka basic class