By Katz N.M.

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Using the group action ˜ over the pull-back π ∗ (M ) of M to N∞ , which we construct a bundle E 34 S. DONALDSON AND E. SEGAL can also be viewed as a bundle over P with ﬁbre S 4 × M . The connections ˜ A section s of M on T P, U and E induce a natural connection A0 on E. ∗ induces a section s of π (M ) and we have a connection As = s∗ (A0 ) on ˜ over N∞ . The stated properties of the connection on the a bundle s∗ (E) universal bundle imply that As is isomorphic to A over the section at inﬁnity and satisﬁes the condition F2 = 0, simply because the connection on the universal bundle is anti-self-dual on each R4 slice in M × R4 .

In other words, N2 −→ Kv1 σ ∩ Z2 , (α, β) −→ α · (1, 0) + β · (0, 1) = (α, β) AN INVITATION TO TORIC DEGENERATIONS 45 and N2 −→ Kv3 σ ∩ Z2 , (α, β) −→ α · (0, −1) + β · (1, −a) = (β, −α − βa) are isomorphisms of additive monoids. This shows C[Kvi σ ∩ Z2 ] C[x, y], i = 1, 3, as abstract rings. For i = 2 the integral generators (−1, a), (−1, 0) of the extremal −1 rays of Kv2 σ generate a proper sublattice of Z2 of index a = det −1 a 0 . Thus (−1, 0), (−1, a) also do not suﬃce to generate Kv2 σ ∩ Z2 as a monoid, for a > 1.

In this section we discuss the converse question. Let σ be a G2 structure on Y and A be a GAUGE THEORY IN HIGHER DIMENSIONS, II 31 G2 -instanton on a bundle E over Y . Let k be a positive integer and P be an associative submanifold in Y . When does the triple (A, P, k) appear as the limit of smooth G2 -instantons with respect to a sequence of deformations σi of σ? In this subsection we will explain that there is a natural candidate criterion for “bubbling” question. In particular when the gauge group is SU (2) and when k = 1 we will argue that this occurs if for some spin structure on P the coupled Dirac operator on E|P , deﬁned by the restriction of the connection A, has a nontrivial kernel.

### A conjecture in arithmetic theory of differential equations by Katz N.M.

by David

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