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Lectures on the Physical Approach to Donaldson and Seiberg-Witten Invariants of Four-Manifolds

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Lectures on the Physical Approach to Donaldson and Seiberg-Witten Invariants of Four-Manifolds Preprint typeset in JHEP style - HYPER VERSION Lectures On The Physical Approach To Donaldson And Seiberg-Witten Invariants Of Four-Manifolds Gregory W. Moore NHETC and Department of Physics and Astronomy, Rutgers University [email protected] Abstract: These are lecture notes for lectures at the Simons Center for Geometry and Physics scheduled for March 22-24, 2017. A truncated version was delivered at the Pre- String-Math school in Hamburg, July 17-18, 2017. THESE NOTES ARE UNDER CON- STRUCTION. They are available at http://www.physics.rutgers.edu/∼gmoore/SCGP- FourManifoldsNotes-2017.pdf COMMENTS WELCOME. August 25, 2017 Contents 1. Some Background History And The Plan Of The Lectures 3 1.1 Fundamental Group 3 1.2 Intersection Form 4 1.3 Whitehead Theorem 5 1.4 Serre's Theorem 5 1.5 Freedman's Theorem: Homeomorphism Type 5 1.6 Donaldson's Theorems: Diffeomorphism type 6 2. Plan For The Rest Of The Lectures 8 2.1 Acknowledgements 8 3. A Brief Review Of Cohomological TFT Path Integrals 9 3.1 A Nice Integral 9 3.2 Supersymmetric Representation Of The Nice Integral 10 3.2.1 Superspace 10 3.2.2 Rewriting The Integral 11 3.2.3 A Generalization: The Case Of \Nonzero Index" 13 3.3 Thom Isomorphism Theorem 14 3.4 The Localization Formula 14 3.5 Equivariance 16 3.6 The Fields, Equations, Symmetries Paradigm 18 4. Twisted N=2 SYM In Mathai-Quillen Form 19 4.1 Fields, Equations, Symmetries 19 4.2 Relation To Twisted N = 2 Field Theories 20 4.2.1 N = 2 Vectormultiplets 21 4.2.2 Topological Twisting 22 4.3 A Little Bit About The Moduli Space Of Instantons 24 4.3.1 Deformation Complex And Virtual Dimension 25 4.3.2 The Virtual Dimension Is The Ghost Number Anomaly 26 4.3.3 Some Essential Mathematical Properties Of Instanton Moduli Space 26 4.3.4 Instanton Moduli Space As The Q-Fixed-Point Locus 26 4.4 Observables 27 4.5 The Donaldson Polynomials 29 4.6 The Donaldson-Witten Partition Function 31 4.7 Generalizations: Lagrangian N=2 Theories 33 4.7.1 Hypermultiplets 33 4.7.2 Charge One Hypermultiplets In A G = U(1) Theory: Spin-c Struc- tures And The Seiberg-Witten Equations 34 { 1 { 4.7.3 Generalizing The Donaldson-Witten Partition Function 35 4.8 So, What Good Is It? 36 5. Mapping The UV Theory To The IR Theory 37 5.1 Motivation For Studying Vacuum Structure 37 5.2 The Classical Vacua And Spontaneous Symmetry Breaking 38 5.3 Quantum Vacua And The LEET 40 5.3.1 The General N=2 Supersymmetric LEET Describing Fluctuations 4 On The Coulomb Branch For Theory On R 42 5.4 Seiberg And Witten's LEET For G = SU(2) 44 5.4.1 An Alternative Description 46 5.4.2 An Important Symmetry 47 5.4.3 Seiberg-Witten Curves In General 47 5.5 BPS States 47 5.6 The Low Energy Theory Near u = ±Λ2 50 5.7 Putting The Twisted LEET On A General Four-Manifold X 51 5.7.1 Gravitational Couplings On The Coulomb Branch 52 5.7.2 Gravitational Couplings On The Higgs Branch 55 5.8 Mapping Operators From UV To IR 56 5.8.1 Mapping Operators On The Coulomb Branch 56 5.8.2 Mapping Operators On The Higgs Branch 56 5.8.3 Contact Terms 56 6. General Form Of The Higgs Branch Contribution 57 7. The Coulomb Branch Contribution aka The u-Plane integral 59 7.1 The Integral Over Fermions 59 7.2 The Photon Path Integral 60 7.3 Final Expression For The u-plane Integral 60 7.4 Metric Dependence: Wall-crossing 63 8. Derivation Of The Relation Between SW And Donaldson Invariants. 65 9. Simple Type And Witten's Conjecture 66 10. Applications Of The Physical Approach: Postdictions 67 10.1 SW=GW For Symplectic Manifolds 67 10.2 The Blowup Formula 67 10.3 Meng-Taubes: SW Invariants And Reidemeister-Milnor Torsion 68 11. Applications Of The Physical Approach: Predictions 68 11.1 New Formulae For Class Numbers 68 11.2 Donaldson Invariants For Other Simple Gauge Groups 68 11.3 The Geography Problem 69 { 2 { 11.3.1 Superconformal Singularities 69 11.4 Superconformal Simple Type And The Generalized Noether Inequality 70 12. Possible Future Directions. 71 A. Orientation, Spin, Spinc, P in±, And Pinc Structures On Manifolds 72 A.1 Reduction Of Structure Group: General Discussion 72 A.2 Obstructions To Spin And Pin Structures 73 A.3 Spinc Structures On Four-Manifolds 75 A.4 't Hooft Flux 76 B. The 11/8 Conjecture 76 C. Theorems On SW Moduli Space 77 D. Constraints On π1 Of Three-Manifolds 77 1. Some Background History And The Plan Of The Lectures Let us summarize some of the standard results on 4-manifolds. See the texbooks [8, 16, 17, 48] for details. In these lectures X will always denote a compact, connected, orientable four-manifold without boundary. 1.1 Fundamental Group ∼ First, if π is any finitely presented group then there is a compact 4-fold X with π1(X) = π. 1 Since the word problem for groups is undecidable this means we cannot hope to classify all compact 4-manifolds. But we can still hope to understand simply connected 4-folds. 1To prove this theorem of Markov one takes a connected sum of S1 ×S3, one summand for each generator in the finite presentation of π. Choose a basepoint in the connected sum and a representative of each of the generators of the fundamental group. Now in the presentation of π each relation corresponds to a word in these generators, and hence corresponds to an embedded circle in the connected sum. Draw a tubular neighborhood around each of the embedded circles so that the neighborhood is D × S1. Displace these so that they do not intersect. Next do surgery to replace by neighborhoods of the form D~ × S1 where now the longitude becomes contractible in the disk D~. Finally, using the Seifert-van Kampen theorem one can prove that the resulting manifold has π1(X) =∼ π. Four is the first dimension in which this happens: See Appendix D below. { 3 { 1.2 Intersection Form There is another interesting topological invariant, the intersection number. The intersection number of an oriented compact four-manifold gives an invariant: 2 2 H (X; Z) × H (X; Z) ! Z (1.1) It is an invariant of the homotopy type of X. Poincar´eduality says that on free group 2 2 H (X; Z)=Tors(H (X; Z)) of rank b2(X) it is a perfect pairing, and therefore corresponds to a symmetric integral unimodular bilinear form QX . If α is a cohomology class Poincar´edual to S(α) then we the oriented intersection number can be written in several different ways: Z Z S(α) · S(β) = αβ = β (1.2) X S(α) 2 2 The way the group H (X; Z)=Tors(H (X; Z)) will enter in our considerations is as a lattice in the vector space of DeRham cohomology classes ¯ 2 2 H (X) ⊂ HDR(X) (1.3) defined as the set of classes with integral periods. Of course this only makes sense when X has a differentiable structure. In that case 2 ∼ 2 2 H¯ (X) = H (X; Z)=Tors(H (X; Z)) (1.4) and Z QX (!1;!2) := !1 ^ !2 (1.5) X b+ b− The intersection form QX has signature (+1 2 ; −1 2 ). That is, if we consider it as a 2 quadratic form on the real vector space H (X; Z) ⊗ R then, after a suitable choice of basis + it can be brought to this diagonal form. More invariantly, b2 is the rank of the maximal 2 sublattices in H (X; Z) on which the restriction of QX is positive definite. When X is oriented and has a Riemannian metric (or just a conformal structure) we can define a Hodge dual ∗ :Ω2(X) ! Ω2(X). It satisfies ∗2 = +1 and so we can speak of self-dual forms: ∗! = ! (1.6) and anti-self-dual forms: ∗! = −! (1.7) 2 The Hodge theorem allows us to identify H (X; R) with the space of harmonic two-forms 2 + H (X) and ∗ preserves this space. Then we can interpret b2 as the dimension of the vector − space of harmonic self-dual two-forms, and b2 as the dimension of the vector space of harmonic anti-self-dual two-forms. 2 2 Remark: Later on we will be working with various torsors of H¯ (X) inside H (X; R). For example, given an integral liftw ¯2(X) of w2(X) we will form 1 Γ := w¯ (X) + H¯ 2(X) ⊂ H2(X; ) (1.8) w 2 2 R Similar torsors associated to integral lifts of other mod-two classes will also play a role. { 4 { 1.3 Whitehead Theorem In 1949 J.H.C. Whitehead introduced the notion of CW decomposition of manifolds to classify homotopy type. In [35] Milnor observed that an interesting consequence is that two ∼ simply connected oriented four-manifolds X1;X2 are homotopy equivalent iff QX1 = QX2 . To prove this one notes that the cell-decomposition of a simply connected four-fold is _ 2 4 Si _ D (1.9) i 3 W 2 and is determined by the homotopy class of a map f : S ! i Si , which can be related to the intersection matrix. 2 1.4 Serre's Theorem Thus we come to the classification of integral unimodular forms.
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