Witten's conjecture relating Donaldson and Seiberg–Witten invariants

Let XX be a smooth, closed, oriented 4-manifold with b1(X)=0b_1(X)=0 and odd b2+(X)>1b_2^+(X)>1. Let XX have Seiberg–Witten simple type, and let K1,,KsK_1,\dots,K_s be the Donaldson basic classes of XX, with corresponding nonzero Donaldson coefficients β1,,βs\beta_1,\dots,\beta_s. A class KK is a Seiberg–Witten basic class when SWX(K)0SW_X(K)\ne 0.

Witten's conjecture. Then XX has simple type; the basic classes of XX are precisely the Seiberg–Witten basic classes of XX; and there is a nonzero constant c(X)c(X) depending only on XX such that

βr=c(X)SWX(Kr)\beta_r=c(X)\cdot SW_X(K_r)

for all r=1,,sr=1,\dots,s. This conjecture proposes a precise relationship between Donaldson and Seiberg–Witten invariants. The source introduces it after recalling the Donaldson structure theorem; no resolution status is supplied in the provided text.

Sources & referencesView supporting material

Primary source

John A. Baldwin and Steven Sivek, “Stein fillings and SU(2) representations”, arXiv:1611.05629 (2016).

Additional references

8 papers in this index state this conjecture (1995–2016). The statement above is taken from the most recent of them; the others are arXiv:1408.5085, arXiv:1408.5307, arXiv:math/0609530, arXiv:math/0106221, arXiv:math/9907107, arXiv:dg-ga/9509005, arXiv:alg-geom/9505018.

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