By Selman Akbulut
Within the spring of 1985, A. Casson introduced a fascinating invariant of homology 3-spheres through buildings on illustration areas. This invariant generalizes the Rohlin invariant and provides magnificent corollaries in low-dimensional topology. within the fall of that very same yr, Selman Akbulut and John McCarthy held a seminar in this invariant. those notes grew out of that seminar. The authors have attempted to stay on the subject of Casson's unique define and continue via giving wanted information, together with an exposition of Newstead's effects. they've got usually selected classical concrete techniques over basic tools. for instance, they didn't try and provide gauge concept reasons for the result of Newstead; as a substitute they his unique thoughts.
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Additional resources for Casson's Invariant for Oriented Homology Three-Spheres: An Exposition. (MN-36) (Mathematical Notes)
28) Since ρ acts freely, the fundamental group of the Enriques surface E is Z2 and the Euler number is χ (E) = χ (K3)/2 = 12. As ω2,0 (and ω¯ 2,0 = ω0,2 ) is anti-invariant, the cohomology groups have dimensions h 00 = h 22 = 1, h 10 = h 01 = h 20 = h 02 = 0 and h 11 = χ (E) − 2 = 10. e. K E⊗2 = O E , hence non-trivial: K E = O E . 29) Counting BPS States on the Enriques Calabi-Yau 41 and with Picard number 18, one can explicitly check  that the invariant part − anti-invariant part K3 of 3,19 under ρ ∗ are + K3 = 1,1 − K3 (2) ⊕ E 8 (−2), =[ 1,1 (2) ⊕ E 8 (−2)] ⊕ 1,1 g .
Klemm, M. 9), y 2 , is defined by y 2 = 2y + y − − y 2 . 11) We will denote by y2± , y2 the imaginary parts of these moduli. 12) 1 and p R = v · w/Y 2 ∈ C is given by pR = 1 Y 1/2 1 1 1 1 −n + my 2 + ζ J2 m 2 y + + ζ J2 n 2 y − + ζ J2 q · y . 13) With this parameterization, the resulting topological couplings will have good convergence properties in the region Im y → ∞. 14) therefore ν = 1. 4). 4), only the untwisted sector J = 1 contributes. The reason for that is that in the twisted sectors J = 2, 3, the lattice s1,1 where the reduction is performed has the shift β J = δ = z − z .
We then obtain for this point the massless spectrum of N = 4 supersymmetric gauge theory, which has a vanishing beta function and no Higgs branch. 1). As usual one gets a SU (2) gauge symmetry enhancement at level 1. In addition one gets four hypermultiplets in the fundamental representation of SU (2), one from each fixed point of the T2 . The resulting gauge theory is N = 2, SU (2) Yang-Mills theory with four 42 A. Klemm, M. Mariño massless hypermultiplets. This theory has a vanishing beta function and it is believed to be conformal .
Casson's Invariant for Oriented Homology Three-Spheres: An Exposition. (MN-36) (Mathematical Notes) by Selman Akbulut