Edinburgh Lectures on Geometry, Analysis and Physics

Sir Michael Atiyah

Notes by Thomas K¨oppe Contents

Preface iii

1 From Euclidean 3-space to complex matrices1 1.1 Introduction...... 1 1.2 Euclidean geometry and projective space...... 2 1.3 From points to polynomials...... 2 1.4 Some physics: hyperbolic geometry...... 4 1.5 The hyperbolic conjecture...... 5 1.6 The Minkowski space conjecture...... 6 1.7 The normalised determinant...... 8 1.8 Relation to analysis and physics...... 10 1.9 Mysterious links with physics...... 12 List of conjectures...... 13 Bibliography...... 13

2 Vector bundles over algebraic curves and counting rational points 14 2.1 Introduction...... 14 2.2 Review of classical theory...... 14 2.3 Analogy with number theory...... 17 2.4 Relation between homology and counting rational points.... 17 2.5 The approach via Morse theory...... 20 2.6 Counting rational points...... 26 2.7 Comparison of equivariant Morse theory and counting rational points...... 31 2.8 Relation to physics...... 32 2.9 Finite-dimensional approximations...... 33 2.10 Relation of 휁-functions for ﬁnite ﬁelds and Riemann’s 휁-function 34 2.11 Arithmetic algebraic geometry (Arakelov theory)...... 35 2.12 Other question...... 36 Bibliography...... 36

ii Preface

These lecture notes are based on a set of six lectures that I gave in Edinburgh in 2008/2009 and they cover some topics in the interface between Geometry and Physics. They involve some unsolved problems and conjectures and I hope they may stimulate readers to investigate them. I am very grateful to Thomas K¨oppe for writing up and polishing the lectures, turning them into intelligible text, while keeping their informal nature. This involved a substantial eﬀort at times in competition with the demands of a Ph.D thesis. Unusually for such lecture notes I found little to alter in them. Michael Atiyah Edinburgh, September 2010

iii lecture series 1

From Euclidean 3-space to complex matrices

December 8 and 15, 2008

1.1 Introduction

3 We will formulate an elementary conjecture for 푛 distinct points in R , which is unsolved for 푛 ≥ 5, and for which we have computer evidence for 푛 ≤ 30. The conjecture would have been understood 200 years ago (by Gauss). What is the future for this conjecture?

∙ A counter-example may be found for large 푛.

∙ Someone (perhaps from the audience?) gives a proof.

∙ It remains a conjecture for 300 years (like Fermat).

To formulate the conjecture, we recall some basic concepts from Euclidean and hyperbolic geometry and from Special Relativity.

1 2

1.2 Euclidean geometry and projective space

2 {︀ 2 2 2 2 }︀ The two-dimensional sphere 푆 = (푥, 푦, 푧) ∈ R : 푥 + 푦 + 푧 = 1 is “the 1 same as” the complex projective line CP = C ⊔ {∞}, on which we have homogeneous coordinates [푢1 : 푢2]. Stereographic projection through a “north 2 2 pole” 푁 ∈ 푆 identiﬁes 푆 ∖ {푁} with C, and it extends to an identiﬁcation of 2 1 푆 with CP by sending 푁 to ∞.

Exercise 1.2.1. Suppose we have two stereographic projections from two ′ 1 1 “north poles” 푁 and 푁 . Show that these give a map CP → CP which is a complex linear transformation

푎푢 + 푏 푢′ = , where 푎, 푏, 푐, 푑 ∈ and 푎푑 − 푏푐 ̸= 0. 푐푢 + 푑 C

1 Hint: Start by considering stereographic projection from 푆 to R ﬁrst.

1.3 From points to polynomials

3 We will now associate to each set of 푛 distinct points in R a set of 푛 complex polynomials (deﬁned up to scaling).

3 The case 푛 = 2. Given two points 푥1, 푥2 ∈ R with 푥1 ̸= 푥2, deﬁne

푥2 − 푥1 2 푓(푥1, 푥2) := ∈ 푆 , ‖푥2 − 푥1‖ which gives a unit vector in the direction from 푥1 to 푥2. Under the identiﬁcation 2 ∼ 1 1 푆 = CP , 푓 associates to each pair (푥1, 푥2) a point in CP . Exchanging 푥1 2 and 푥2 is just the antipodal map 푥 ↦→ −푥 on 푆 .

3 The general case. Given 푛 (ordered) points 푥1, . . . , 푥푛 ∈ R , we obtain 1 푛(푛 − 1) points in CP by deﬁning

푥푗 − 푥푖 2 ∼ 1 푢푖푗 := ∈ 푆 = CP for all 푖 ̸= 푗. (1.1) ‖푥푗 − 푥푖‖

For each 푖 = 1, . . . , 푛 we deﬁne a polynomial 훽푖 ∈ C[푧] with roots 푢푖푗 (푗 ̸= 푖): ∏︁ 훽푖(푧) = (푧 − 푢푖푗) (1.2) 푗̸=푖

The polynomials 훽푖 are determined by their roots up to scaling. We make th the convention that if for some 푗 we have 푢푖푗 = ∞, then we omit the 푗 factor, so that 훽푖 drops one degree. In fact, a more invariant picture arises if

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instead we consider the associated homogeneous polynomials 퐵푖 ∈ C[푍0, 푍1] ∏︀ (︀ )︀ given by 퐵푖(푍0, 푍1) = 푗 푉푖푗푍0 − 푈푖푗푍1 , where [푈푖푗 : 푉푖푗] = [푢푖푗 : 1], so 훽푖(푧) = 퐵푖(푧, 1). We are now ready to state the simplest version of the conjecture:

3 Conjecture 1.3.1 (Euclidean conjecture). For all sets (푥1, . . . , 푥푛) ⊂ R of 푛 distinct points, the 푛 polynomials 훽1(푧), . . . , 훽푛(푧) are linearly independent over C.

Remark 1.3.2. The condition of linear independence of the polynomials 훽푖 is independent of the choice of stereographic projection in Equation (1.1) by Exercise 1.2.1.

3 Example (푛 = 3). Suppose 푥1, 푥2, 푥3 are distinct points in R . They are 2 3 automatically co-planar, so that 푥1, 푥2, 푥3 ∈ R ⊂ R . So the points 푢푖푗 lie in 1 2 ∼ 1 some great circle 푆 ⊂ 푆 = CP . We can choose the north pole 푁 for the stereographic projection in Equation

(1.1) either such that all 푢푖푗 lie in the equator, in which case |푢푖푗| = 1 and 1 푢푗푖 = −푢푖푗, or such that all 푢푖푗 lie on a meridian, in which case 푢푖푗 ∈ RP and  푢푗푖 = −1 푢푖푗. Let us stick with the ﬁrst convention, so that all 푢푖푗 lie on the equator and we have |푢푖푗| = 1 and 푢푗푖 = −푢푖푗. This deﬁnes three quadratics

훽1(푧) = (푧 − 푢12)(푧 − 푢13) = (푧 − 푢12)(푧 − 푢13)

훽2(푧) = (푧 − 푢21)(푧 − 푢23) = (푧 + 푢12)(푧 − 푢23)

훽3(푧) = (푧 − 푢31)(푧 − 푢32) = (푧 + 푢13)(푧 + 푢23)

In this case we can prove Conjecture 1.3.1 in two ways:

∙ By geometric methods: Represent quadratics by lines in a plane, then

linear dependence of the 훽푖 is the same as concurrence.

∙ By algebraic methods: Compute the determinant of the (3 × 3)-matrix

of coeﬃcients of the 훽푖 and show that it has non-vanishing determinant.

For the case 푛 = 4, there exists a proof using computer algebra. For 푛 ≥ 5, no proof is known, even for co-planar points (i.e. real polynomials). A proof will be rewarded with a bottle of champagne or equivalent. The easiest point of departure is to consider four points in a plane.

3 4

1.4 Some physics: hyperbolic geometry

2 3 Consider again the 2-sphere 푆 ⊂ R , and add a fourth variable 푡 (for “time”):

푥2 + 푦2 + 푧2 − 푅2푡2 = 0 (1.3)

This is the metric of Minkowski space-time. Here 푅 is the speed of light, and Equation (1.3) defines a light cone. Our original 2-sphere is the base of the light cone, the “celestial sphere” of an observer. + 2 ∼ 1 The (proper, orthochronous) Lorentz group 푆푂 (3, 1) acts on 푆 = CP  as a group of complex projective transformations 푆퐿(2; C) ±1 = 푃 푆퐿(2; C). The Euclidean version of this picture is the following: The rotation group 3  ∼ 2 ∼ 1 of R , 푆푂(3), acts as 푆푈(2) ±1 =: 푃 푆푈(2) = 푃 푈(2) on 푆 = CP preserving the metric given by Equation (1.3) (“rigid motion”). We can also see this as the 2 projectivisation of the action of 푆푈(2) or 푈(2) on C , and the projectivisation map ∼ 푆푈(2) 푃 푆푈(2) = 푆푂(3) is a double cover. This map is the restriction to the maximal compact subgroup ∼ + of the double cover 푆퐿(2; C) 푃 푆퐿(2; C) = 푆푂 (3, 1). ∼ + We have two different representations of 푆퐿(2; C) = 푆푂](3, 1) (double 3,1 cover): It acts on real 4-dimensional space-time R by proper, orthochronous 2 Lorentz transformations, and it acts on complex 2-dimensional space C (whose elements we call spinors). The fundamental link between these two representa- 2  × ∼ 1 tions is via projective spinors: A (projectivised) point in (C ∖ {0}) C = CP corresponds to a point on the base of the light cone, 푆2. Consider the hyperboloid given by 푥2 + 푦2 + 푧2 − 푅2푡2 = −푚2. Denote the interior of the base of the light cone by 퐻푚. The metric induced on 퐻푚 has constant negative curvature, and indeed it turns 퐻푚 into a model of hyperbolic 3-space with curvature −1/푚2. The Lorentz group 푆푂+(3, 1) acts transitively on hyperbolic 3-space 퐻3  by isometries, and it acts by 푆퐿(2; C) ±1 on the 2-sphere at infinity.

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1.5 The hyperbolic conjecture.

3 Given 푛 distinct, ordered points in 퐻 , define the point 푢푖푗 as the intersection 2 of the oriented geodesic joining 푥푖 to 푥푗 with the 푆 at infinity. We define 푛 polynomials 훽1, . . . , 훽푛, where 훽푖 has roots 푢푖푗, as before in Equation (1.2) (but note that in hyperbolic space we no longer have a notion of “antipodal points”). This brings us to the second, stronger version of the conjecture:

3 Conjecture 1.5.1 (Hyperbolic conjecture). For all sets (푥1, . . . , 푥푛) ⊂ 퐻 of 푛 distinct points, the 푛 polynomials 훽1(푧), . . . , 훽푛(푧) are linearly independent over C.

Remarks 1.5.2.

∙ There is good numerical evidence for the hyperbolic conjecture.

∙ The conjecture uses only the intrinsic geometry of hyperbolic 3-space, so it is invariant under the group of isometries (i.e. the Lorentz group).

3 3 3 ∙ A model for 퐻 is the open ball 퐵 ⊂ R . We can actually forget about 3 the geometry of 퐻 and just consider the points 푥1, . . . , 푥푛 to lie in 3 3 3 퐵 ⊂ R . Letting the radius of the ball 퐵 grow (which is equivalent to letting the curvature of the hyperbolic space go to zero) exhibits the Euclidean conjecture as a limiting case of the hyperbolic conjecture.

Remark 1.5.3 (The ball of radius 푅). As we said in Remark 1.5.2 (3), we can view the hyperbolic conjecture as a statement about points inside the unit ball 3 3 퐵 , and more generally inside any ball 퐵푅 of radius 푅 ≥ 0 – this corresponds to hyperbolic space of constant curvature −1/푅2.

We might expect that if the conjecture is false, then a counter-example would be given by a rather special conﬁguration of the 푛 points 푥1, . . . , 푥푛. The following example treats the most special conﬁguration, namely the collinear one.

3 Example. Let 푥1, . . . , 푥푛 be collinear in 퐵푅, and choose complex coordinates 2 on the boundary 푆 such that all the roots of 푝1 are at inﬁnity, so that 푝1(푧) = 1. 2 푛−1 But then 푝2(푧) = 푧, 푝3(푧) = 푧 ,..., 푝푛(푧) = 푧 , and these are clearly linearly independent.

5 6

1.6 The Minkowski space conjecture

3,1 Consider two world lines 휉1, 휉2 in R representing world-like motion of two “stars”. Consider the two points 푥1, 푥2 on 휉1, 휉2, respectively, representing events when an “observer” looks up into the sky and “sees” the other star on his celestial sphere, and denote the points on the respective celestial spheres 2 ∼ 1 푆 = CP by 푢12 and 푢21. We make this more precise and more general:

Given 푛 moving stars (i.e. non-intersecting world lines) 휉1, . . . , 휉푛 and 푛 events 푥푖 ∈ 휉푖, let 푢푖푗 be the point in the celestial sphere of 푥푖 at which the past light cone at 푥푖 intersects the world line 휉푗. In other words, 푥푖 “sees” 푛 − 1 other stars at points 푢푖푗 in its own celestial sphere. Since in (ﬂat) Minkowski space all celestial spheres can be identiﬁed by parallel translations, we may 1 consider all the points 푢푖푗 to live in the space CP . Again we form the polynomials 훽푖 from the roots 푢푖푗 as in Equation (1.2) and come to the third and strongest version of the conjecture.

3,1 Conjecture 1.6.1 (Minkowski space conjecture). Let 휉1, . . . , 휉푛 ⊂ R be 푛 non-intersecting world lines in Minkowski space and {푥1, . . . , 푥푛} a set of 푛 (distinct) events such that 푥푖 ∈ 휉푖 for all 푖. Then the polynomials 훽1(푧), . . . , 훽푛(푧) are linearly independent over C.

Remarks 1.6.2.

1. Since the Lorentz group is essentially 푆퐿(2; C), the Minkowski space conjecture is “physical”, i.e. Lorentz-invariant.

2. If all stars emerge from a “big bang”, i.e. if all world lines meet in a point in the past, then the Minkowski space conjecture reduces to the hyperbolic conjecture.

3. If stars are “static”, the Minkowski conjecture reduces to the Euclidean conjecture.

4. The Minkowski conjecture is true for 푛 = 2 (푢12 ̸= 푢21). There is no other evidence!

5. See [2] and [4] for details.

Challenge. Prove or disprove the Minkowski space conjecture for 푛 = 3.

Remarks.

1. Conjecture 1.6.1 refers to world lines. These can be interpreted as world lines of particles or “stars” in uniform motion and this gives one version

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of the conjecture. A stronger version arises if we allow all “physical motion” (i.e. not exceeding the velocity of light). In [2] I produced what purported to be an elementary counterexample for 푛 = 3. However, on closer inspection this involves motion faster than light, so the general conjecture is still open.

2. It is even tempting to consider motion on a curved space-time background but since we now have to worry about parallel transport it is not clear how to formulate a conjecture.

7 8

1.7 The normalised determinant

We begin by recalling some basic results from linear algebra. Consider the decomposition

2 2 ∼ 4 ∼ 2(︀ 2)︀ 2(︀ 2)︀ ∼ 3 1 R ⊗ R = R = Sym R ⊕ Λ R = R ⊕ R .

We can view the sum on the right-hand side as the decomposition of real (2×2)- matrices into symmetric and skew-symmetric parts, and we may think of the 2(︀ 2)︀ symmetric part Sym R as a space of symmetric polynomials (of degree 2) 2(︀ 2)︀ and of the alternating part Λ R as the “area” or “determinant”. The linear group 퐺퐿(2; R) acts on both summands and preserves this decomposition, and it acts on the area by multiplication by the determinant. 푆퐿(2; R) acts trivially 1 on the R -summand. The complex analogue of this picture is the following: The group 푆퐿(2; C) 2(︀ 2)︀ ∼ 푛 푛 ∼ 푛 ∼ 푛−1(︀ 2)︀ acts trivially on Λ C = C and on Λ (C ) = C. (Note: C = Sym C .) 2 The group action preserves the standard symplectic form on C . Now suppose we have 푛 distinct points 푥1, . . . , 푥푛 inside a ball of radius 1 푅, and the numbers 푢푖푗 ∈ CP are defined as in Equation 1.1. Lift the 푢푖푗 to 2  any 푣푖푗 ∈ C , i.e. pick a vector 푣푖푗 = (푧1, 푧2) such that 푧1 푧2 = 푢푖푗. Using the 2 2 ∨ standard symplectic form, we identify C with its dual (C ) , and using this identification we consider the 푣푖푗 as one-forms. Since 푢푖푗 ≠ 푢푗푖, 푣푖푗 ∧ 푣푗푖 ̸= 0. Now fix the constant multiplier by setting

∏︁ 푛−1(︀ 2 ∨)︀ ∼ 푛 ∨ 푝푖 = 푣푖푗 ∈ Sym (C ) = (C ) , 푗̸=푖 and deﬁne 푝1 ∧ 푝2 ∧ · · · ∧ 푝푛 퐷푅(푥1, . . . , 푥푛) = ∏︀ (︀ )︀ . (1.4) 푖<푗 푣푖푗 ∧ 푣푗푖

푛 푛 ∼ Remarks 1.7.1. Here the numerator is an element of Λ (C ) = C, concretely given by the determinant of the (푛 × 푛)-matrix of the coeﬃcients of the 2(︀ 2)︀ ∼ polynomials 푝푖. The denominator is a product of elements of Λ C = C. Changing the choice of 푉푖푗 by a factor 휆푖푗 multiplies both numerator and ∏︀ denominator by the same factor 푖̸=푗 휆푖푗, so 퐷푅 depends only on the points 푥1, . . . , 푥푛. Permuting the points 푥1, . . . , 푥푛 produces the same sign change in numerator and denominator, so 퐷푅 is invariant under permutations.

Definition 1.7.2 (Normalised determinant). For 푛 distinct points 푥1, . . . , 푥푛 3 in R inside a ball of radius 푅, we deﬁne the normalised determinant 퐷푅 to be as in Equation (1.4). (This normalization gives 퐷푅 = 1 for collinear points.)

8 Edinburgh Lectures on Geometry, Analysis and Physics 9

Computation of 퐷푅. Given 푛 distinct points 푥1, . . . , 푥푛 inside a ball of radius 푅, choose for each pair 푖 < 푗 lifts 푣푖푗, 푣푗푖 such that 푉푖푗 ∧ 푉푗푖 = 휔2. 푛−1−푖 푖 Write each 푝푖 in terms of the monomials 푡0 푡1, where {푡0, 푡1} is a basis 2 for C satisfying 푡0 ∧ 푡1 = 휔2. If we denote by 푃 the (푛 × 푛)-matrix whose (푖, 푗)-entry is the 푗th coeﬃcient of 푝푖, then 퐷푅(푥1, . . . , 푥푛) = det 푃 (hence the name “normalised determinant”).

Properties of the normalised determinant.

1. 퐷푅(푥1, . . . , 푥푛) is invariant under the 푆퐿(2; R)-action (i.e. the isometries 3 of 퐻푅) on the points 푥1, . . . , 푥푛, and it is continuous in (푥1, . . . , 푥푛).

2. The limit 퐷∞ := lim푅→∞ 퐷푅 exists and is invariant under the group of 3 Euclidean motions (translations and rotations of R ).

3. 퐷푅(푥1, . . . , 푥푛) = 1 for collinear points.

3 4. 퐷푅 → 퐷푅 under reﬂection of R (so 퐷푅 is real for coplanar points).

5. For 푛 = 3, 3 1 ∑︁ 퐷 = cos2(︀ 퐴푖 )︀ , ∞ 2 2 푖=1

where 퐴푖 are the angles of a triangle, varying between 1 for collinear and 9/8 for equilateral conﬁgurations. For 푛 ≥ 4, 퐷푅 is complex-valued in general.

6. 퐷∞ is scale-invariant: 퐷∞(휆푥1, . . . , 휆푥푛) = 퐷∞(푥1, . . . , 푥푛) for 휆 > 0.

7. In the hyperbolic case, 퐷푅(푥1, . . . , 푥푛) → 퐷푅(푥1, . . . , 푥푛−1) as |푥푛| → 푅. (This generalises to the so-called “cluster decomposition”: If the

points 푥1, . . . , 푥푛 fall into two “clusters” at great distance, then 퐷푅 is approximately the product of the 퐷푅’s of the clusters.)

The formalism of the normalised determinant allows us to rephrase our conjectures, and assuming normalisation we can actually state stronger forms:

∙ The Euclidean conjecture 1.3.1. Weak form: 퐷∞ ̸= 0. Strong form: |퐷∞| ≥ 1 after normalisation.

∙ The hyperbolic conjecture 1.5.1. Weak form: |퐷푅| ̸= 0. Strong form |퐷푅| ≥ 1, after normalisation, with equality for collinear points.

∙ We also have a new conjecture, the monotonicity conjecture: |퐷푅| in- creases with 푅 (for ﬁxed 푥1, . . . , 푥푛).

9 10

Remarks 1.7.3. 퐷푅(푥) = 퐷휆푅(휆푥), so the hyperbolic conjecture is independent of 푅. So if it is true for ﬁnite 푅, then it is true for 푅 = ∞. The Minkowski space conjecture implies the hyperbolic conjecture: Shrink 2 2 ′ 푆푅 to 푆푅′ , where 푅 = |푥푛| = max푖 |푥푖|, then apply Property (7) inductively.

The normalised determinant 퐷푅 can be deﬁned for points inside any ellipsoid 푆, in which case we denote it by 퐷푆. This is because 푆 can be changed into a standard sphere by aﬃne linear transformations of 푅3 (which preserve straight lines). We can reduce 푥2 + 푦2 + 푧2 = 1 to 푥2/푎2 + 푦2/푏2 + 푧2/푐2 = 1 by choice of 푎, 푏, 푐 ≥ 1.

Remark 1.7.4 (Ellipsoid version). The Minkowski space conjecture can be ′ 3 stated in terms of ellipsoids: Suppose 푆 ⊇ 푆 are two ellipsoids in R containing 푛 distinct points (푥1, . . . , 푥푛). Then

|퐷푆′ (푥1, . . . , 푥푛)| ≥ |퐷푆(푥1, . . . , 푥푛)| .

To see this, consider the situation where 푆′ ⊇ 푆 are two light cones. Then ′ |퐷푆′ | ≥ |퐷푆|. A physical interpretation is that if 푆 is the vacuum light cone and 푆 the light cone in a medium, then |퐷med| ≤ |퐷vac|.

1.8 Relation to analysis and physics

3 The Dirac equation. Let 푠(푥) be a spinor ﬁeld in R . The Dirac equation in vacuum is 3 ∑︁ 휕푠 퐷푠 = 퐴 = 0 , 푗 휕푥 푗=1 푗

2 where 퐴푗 are (2 × 2)-matrices, 퐴푗 = −1, 퐴푖퐴푗 = −퐴푗퐴푖 = 퐴푘 (the Pauli matrices).

The point monopole. Given (푥1, . . . , 푥푛), consider these as locations of 푛 Dirac monopoles and take the Dirac equation 퐷푠 = 0 in the background

field. We need to impose suitable singular behaviour at 푥1, . . . , 푥푛 and decay at infinity. We expect an 푛-dimensional space of solutions. Examine the asymptotic behaviour at infinity: Can we find our polynomials 훽푖 in this (e.g. as a basis of the solutions)? Would this imply the Euclidean conjecture? In the hyperbolic case, the asymptotic behaviour may be exponential decay, with polynomial angular dependence. Would this imply the radius-푅 conjecture for finite 푅?

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The four-dimensional variant. Let 푀 4 be the Hawking-Gibbons 4-manifold, 4 3 which has an action of 푈(1). The quotient is 푀 푈(1) = R , and the 푈(1)- 3 action has 푛 fixed points, which determine 푛 points 푥1, . . . , 푥푛 in R . Reinterpret on 푀 4: The solutions of the four-dimensional Dirac equation on 푀 4 inherit an action of 푈(1). The invariant solutions on 푀 4 correspond 3 to the singular solutions on R . This disposes of the singular behaviour at 푥푖. We still require decay at infinity. Next step: The Dirac equation is conformally invariant, so we can form the conformal compactification 푀 (which has a mild singularity at infinity). This replaces asymptotic behaviour by local behaviour near infinity. In the final step, we form the twistor space of 푀 and use the complex methods of sheaf theory: Under the twistor transform, solutions of the Dirac equation correspond to sheaf cohomology. In particular, we expect a certain first cohomology to have dimension 푛. To relate this to polynomials and our conjectures, we must use real numbers and positivity. This is close to (real) algebraic geometry.

Hyperbolic analogue. Four-manifold 푁 4 with special metric and 푈(1)- 4 3 2 3 action with 푛 fixed points and 푁 푈(1) = 퐵 , the “inside” of 푆 in R with the hyperbolic metric. It admits a conformal compactification 푁, on which we have a 푈(1)-action with 푛 fixed points and a fixed 푆2 ⊂ 푁 ∖ 푁 4. Twistor methods still apply to this case, but is it better than the Euclidean case? This leads to the theory of LeBrun manifolds. See Atiyah-Witten, which includes a 3 problem about the existence of 퐺2 metrics on 7-manifolds which are R -bundles over 푁 4 and generalise the cases 푛 = 0 and 푛 = 1.

Lie group generalisation. The Euclidean conjecture implies the existence of a continuous map

(︀ 3)︀  × 푛  푛 푓푛 : 퐶푛 R → 퐺퐿(푛; C) (C ) → 푈(푛) 푇 compatible with the action of the symmetric group. Specifically, the value in 퐺퐿(푛; C) is the matrix of coefficients of the polynomials 푝푖, and the quotient × 푛 by (C ) accounts for the freedom of scale. The configuration space can be described as follows.

(︀ 3)︀ (︀ 푛)︀ 3 퐶푛 R = Lie 푇 ⊗ R ∖ 풮 ,

3 (︀ 푛)︀ where the R -factor contains the coordinates of the points, the factor Lie 푇 accounts for the 푛 points, and 풮 is the union of codimension-3 linear subspaces

풮훼, where 풮훼 is the kernel of the linear map (the root map)

(︀ 푛)︀ 3 3 훼 ⊗ idR3 : Lie 푇 ⊗ R → R

11 12 extending the roots 훼 of 푈(푛), accounting for the fact that the 푛 points are required to be distinct. (The roots of 푈(푛) are formed by elements 푥푖 − 푥푗. Note that Lie(︀푇 푛)︀ is the Cartan subalgebra of u(푛).) This leads us to a generalisation of our conjectures. Let 퐺 be a compact

Lie group (e.g. 푆푂(푛; R)) and 퐺C its complexiﬁcation (e.g. 푆푂(푛; C)). Let  푇 ≤ 퐺 be a maximal torus with complexiﬁcation 푇C, and let 푊 := 푁(푇 ) 푇 be the Weyl group of 퐺, which permutes the roots.

Conjecture 1.8.1 (Lie group conjecture). If 퐺 is a Lie group as above with rank 푛, then there exists a continuous map

3 퐺C 퐺 푓푛 : Lie(푇 ) ⊗ R = 풮 → → 푇C 푇 compatible with the action of the Weyl group 푊 .

In a joint paper with Roger Bielawski ([3]) we used Nahm’s equations

푑퐴 1 = [퐴 , 퐴 ] (and cyclic permutations), 푑푡 2 3 where 퐴푖 : (0, ∞) → Lie(퐺) are functions of 푡 subject to suitable boundary conditions 푡 → 0, 푡 → ∞, to prove the existence of a map to 퐺푇 . Problems:

1. For 퐺 = 푈(푛), is this the same as a map given by polynomials?

 2. Is there an explicit algebraic analogue for a map to 퐺C 푇C?.

3. Is there any generalisation of the hyperbolic conjecture from 퐺퐿(푛; C) to other Lie groups?

1.9 Mysterious links with physics

∙ Origin in Berry-Robbins on spin statistics.

∙ Link to Dirac equation?

∙ Generalisation to Minkowski space.

∙ Nahm’s equations and gauge theory.

∙ Link to Hawking-Gibbons metric?

∙ Twistor interpretation?

12 Edinburgh Lectures on Geometry, Analysis and Physics 13

1 Key fact of physics. The base of the light cone is CP . It is Penrose’s philosophy that this must be the origin of complex numbers in quantum theory, and it must lie behind any uniﬁcation of General Relativity and Quantum Mechanics.

What is the physical meaning of our conjectures?

List of conjectures

∙ Conjecture 1.3.1: The Euclidean conjecture (weak and strong).

∙ Conjecture 1.5.1: The hyperbolic conjecture (weak and strong).

∙ The monotonicity conjecture for the normalised determinant.

∙ Conjecture 1.6.1: The Minkowski space conjecture.

∙ Conjecture 1.8.1: The Lie group conjecture.

Bibliography

[1] M. Atiyah, Green’s functions for self-dual four-manifolds, in Math- ematical Analysis and Applications, Part A, 129–158, New York, 1981.

[2] M. Atiyah, Configurations of points, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci. 359 no. 1784, 1375–1387 (2001).

[3] M. Atiyah and R. Bielawski, Nahm’s equations, configuration spaces and flag manifolds, Bull. Braz. Math. Soc. (New Series) 33 no. 2, 157–176 (2002).

[4] M. Atiyah and P. Sutcliﬀe, The geometry of point particles, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 458 no. 2021, 1089–1115 (2002).

[5] M. Atiyah and E. Witten, 푀-theory dynamics on a manifold of

퐺2-holonomy, Adv. Theor. Math. Phys. 6 no. 1, 1–106 (2002).

[6] G. W. Gibbons and S. W. Hawking, Gravitational multi-instantons, Physics Letters B 78 no. 4, 430–432 (1978).

13 lecture series 2

Vector bundles over algebraic curves and counting rational points

February 9, 16, 23 and March 2, 2009

2.1 Introduction

There are two themes, both initiated by A. Weil:

1. Extension of classical ideas in algebraic geometry, number theory, physics from Abelian (scalars, 푈(1)) to non-Abelian (matrices, 푈(푛)) settings.

2. Connection between homology and counting rational points over ﬁnite ﬁelds.

2.2 Review of classical theory

(Abel, Jacobi, Riemann, . . . ) Consider complex projective space

푛−1 푛−1 푛  × CP ≡ P := (C ∖ {0}) C

푛−1 with homogeneous coordinates [푧1 : ... : 푧푛]. Rational functions on P are  fractions 푓(푧1, . . . , 푧푛) 푔(푧1, . . . , 푧푛), where 푓 and 푔 are homogeneous polynomials of the same degree. 푛−1 Note that a meromorphic function on P is determined up to scale by its zeros and poles (Liouville). On projective space, global complex analysis is just algebraic geometry (Serre). 푛−1 ∼ There is the standard line bundle 퐿 over P , i.e. 퐿 = 풪P푛−1 (1). Holo- morphic sections of its 푘th power 퐿푘 = 퐿 ⊗ 퐿 ⊗ · · · ⊗ 퐿 are just homogeneous polynomials of degree 푘.

14 Edinburgh Lectures on Geometry, Analysis and Physics 15

2 Algebraic curves. Let 푛 = 3, so we consider the projective plane P .A curve of degree 푘 is given as the locus of points 푧 such that 푓(푧1, 푧2, 푧3) = 0, where 푓 is a homogeneous polynomial of degree 푘. Non-singular curves are just compact Riemann surfaces, so topologically they are entirely determined by its genus 푔. A Riemann surface 푋 of genus 푔 has ﬁrst Betti number 푏1 = dim 퐻1(푋; Q) = 2푔. If 푋 is a curve of degree 푘 with double points, then removing these double points leaves a Riemann surface. We have a formula

1 푔 = (푘 − 1)(푘 − 2) − 훿 , 2 where 훿 is the number of double points. If 훿 = 0, then for 푘 = 1, 2 we find that 푋 is a rational curve, i.e. 푔 = 0; and for 푘 = 3 we get an elliptic curve with genus 푔 = 1. Another interpretation is that 푔 is the dimension of the space of holomorphic differentials (which look locally like 휑(푧) 푑푧, where 휑 is holomorphic). When 1 푔 = 0, the curve is the Riemann sphere CP = C ∪ {∞}, and the differential 푑푧 has a pole at infinity, so it is not holomorphic. When 푔 = 1, the curve is the  2 torus C Z , so the differential 푑푧 on C descends to a holomorphic differential on 푋.

1 Period matrices. Let 휔1, . . . , 휔푔 ∈ 퐻 (푋; C) be a basis of holomorphic diﬀerentials and 훼1, . . . , 훼2푔 ∈ 퐻1(푋; Z) a basis for the 1-cycles of a genus-푔 ∫︀ curve 푋. The (푔 × 2푔)-matrix with entries 휔푖 is called the period matrix of 훼푗 푋.

Divisors. We call a subvariety of codimension 1 a divisor. Since curves are 1-dimensional, divisors on curves are just points. The free Abelian group of all divisors of a variety 푋 is denoted by Div(푋), and so if 푋 is a curve, elements ∑︀푁 of Div(푋) are just formal sums 퐷 = 푖=1 푛푖푃푖, where 푃푖 ∈ 푋 are points. The ∑︀푁 degree of such a divisor 퐷 on a curve is deﬁned as deg 퐷 := 푖=1 푛푖.

Jacobians. The Jacobian of 푋, written 퐽(푋), is a complex torus of complex dimension 푔, given as

푔  퐽(푋) = C lattice = hol. diﬀerentials diﬀerentials with integer periods .

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The signiﬁcance of the Jacobian lies in the following observation. Let 휑 be a rational (meromorphic) function on 푋, and deﬁne the divisors

퐷0(휑) := set of zeros of 휑, with multiplicities,

퐷∞(휑) := set of poles of 휑, with multiplicities,

퐷(휑) := 퐷0(휑) − 퐷∞(휑) (the divisor of 휑).

Then deg 퐷0(휑) = deg 퐷∞(휑). This motivates the question for the converse: Given two divisors 퐷1 and 퐷2 of the same degree, when does there exist a function 휑 on 푋 with 퐷0(휑) = 퐷1 and 퐷∞(휑) = 퐷2? This is always true for 푔 = 0, but not otherwise. The “gap” between divisors of degree zero and divisors of meromorphic functions is measured precisely by the divisor class group Cl(푋). The degree-0 part of it is

divisors of degree 0 Cl0(푋) := . divisors of functions (Divisors of the form 퐷 = 퐷(휑) are also called principal divisors.) For 푔 = 0, the group Div0(푋) is trivial, but for 푔 = 1, the divisor class group is precisely the Jacobian (or its dual) – this is the content of the Abel-Jacobi Theorem. Moreover, the group Cl0(푋) is the group of isomorphism classes of holomorphic line bundles of degree 0, which are just given by elements of

(︀ )︀ Hom 휋1(푋), 푈(1)

(up to duality and complex structure). Differential geometry shows that a holomorphic line bundle of degree zero (i.e. first Chern class zero) has a unique flat unitary connection.

This is the beginning of the link with physics. Maxwell’s equations deal with the curvature of a line bundle on space-time.

16 Edinburgh Lectures on Geometry, Analysis and Physics 17

2.3 Analogy with number theory

Number theory Algebraic geometry

Ring of integers Z complex (affine) line, the ring C[푧] primes points factorisation of integers factorisation of polynomials 1 “infinite prime” point at infinity in P algebraic number field algebraic curve (covering of a line) lack of unique factorisation not all divisors come from functions ideal class group divisor class group

Galois group 휋1(푋)

The classical analogy is that between the ring of integers in number theory and the polynomial rings in geometry. A “half-way house” is an algebraic curve 푛 over a finite field. A finite field is a field F푞 with 푞 elements, where 푞 = 푝 for some prime 푝. We have “function field analogues” of geometric statements, e.g. a Riemann hypothesis (which is proved for finite fields; also for algebraic varieties of any dimension). The key fact for algebraic geometry over F푞 is the existence of the Frobenius map 푥 ↦→ 푥푞. (Recall that in characteristic 푝,(푥 + 푦)푝 = 푥푝 + 푦푝.) There is no such analogue in characteristic zero (but physics suggest rescaling the metric1).

2.4 Relation between homology and counting rational points

Definition 2.4.1 (Poincaréseries). For any topological space 푋 whose singular (︀ )︀ homology groups 퐻푘 푋; Q are finite-dimensional vector spaces, we define the Poincaréseries of 푋 to be the formal power series

∞ ∑︁ 푘 푃푋 (푡) := dim 퐻푘(푋; Q) 푡 . 푘=0

Proposition 2.4.2. If 푋 is a manifold or homotopy-equivalent to a manifold, then 푃푋 is in fact a polynomial.

푛−1 Example. Consider the space P . Over C, this has Poincar´eseries

푃 (푡) = 1 + 푡2 + 푡4 + ··· + 푡2푛−2 . (2.1)

1A quick explanation of this remark is in order: In differential geometry, a differential form scales with the power of its degree, so rescaling picks out the degree of the form. In characteristic 푝, the eigenvalues of the Frobenius map pick out the dimension of the cohomology.

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푛−1 Over F푞, the number of points in PF푞 is

푞푛 − 1 = 1 + 푞 + 푞2 + ··· + 푞푛−1 . 푞 − 1

This agrees with (2.1) if we put 푞 = 푡2. Note that since we can replace 푞 by 푞˜ = 푞푛, 푛 = 1, 2,..., we can think of 푞 as a variable like 푡. This extends to all algebraic varieties.

Exercise 2.4.3. Check that a similar relation between the Poincaréseries over C and the number of points over a finite field F푞 holds for the full flag variety 푈(푛)푇 푛. (Hint: Use successive fibrations by projective spaces.)

Generalisation from 푈(1) to 푈(푛). This corresponds to generalising from line bundles to vector bundles. In number theory, this corresponds to non- Abelian class ﬁeld theory. There are representations from the Galois group to 푈(푛), Langlands programme. . . In physics, this is related to non-Abelian gauge theories and Yang-Mills theory. Returning to algebraic geometry, we will focus on an algebraic curve 푋 (either over C or over F푞). The Jacobian is replaced by a “moduli space” of vector bundles over 푋. There are a few diﬃculties:

∙ There is no group structure (the tensor product does not preserve rank for ranks > 1).

∙ Bundles of rank 푛 can decompose into bundles of lower rank.

There is a moduli space 푀푠(푋, 푛, 푘) of holomorphic rank-푛 bundles of degree 푘 which are stable. Here 푘 is the degree of the determinant line bundle, 푛 푛 which is the ﬁrst Chern class; in symbols: deg 퐸 := deg Λ 퐸 ≡ 푐1(Λ 퐸). The space 푀푠(푋, 푛, 푘) is a compact algebraic variety if gcd(푛, 푘) = 1, e.g. if 푛 = 2, 푘 = 1.

For 푘 = 0, the space 푀푠(푋, 푛, 0) is the space of irreducible representations 휋1(푋) → 푈(푛). To see this, note that such a representation is a choice of 2푔 ∏︀푔 unitary matrices 퐴1, . . . , 퐴푔, 퐵1, . . . , 퐵푔 ∈ 푈(푛) such that 푖=1[퐴푖, 퐵푖] = 1, modulo conjugation by 푈(푛). ∏︀푔 For a general 푘, we replace this condition by 푖=1[퐴푖, 퐵푖] = 휁 id, where 휁 is a central element of 푈(푛) and 휁푘 = 1. (For example, for 푛 = 2, 푘 = 1, we ∏︀ have 푖[퐴푖, 퐵푖] = − id.) The general problem is to study 푀푠.

1. What does 푀푠 look like topologically?

(︀ )︀ 2. What are its Betti numbers 푏푖 = dim 퐻푖 푀푠; Q ?

18 Edinburgh Lectures on Geometry, Analysis and Physics 19

For a connected, oriented manifold 푋, we have the Poincar´epolynomial ∑︀dim 푋 푖 푃푋 (푡) = 푖=0 = 푏푖푡 . Note that for any compact manifold 푋, we have deg 푃푋 = dim 푋, and 푃푋 is palindromic (by Poincar´eduality). Furthermore, 푃푋×푌 (푡) = 푃푋 (푡)푃푌 (푡).

2 Example. If 푋 is a Riemann surface of genus 푔, then 푃푋 (푡) = 1 + 2 푔 푡 + 푡 , 2푔 and 푃퐽(푋)(푡) = (1 + 푡) .

What is 푃푀푠(푋,푛,푘)(푡)? What is it when gcd(푛, 푘) = 1? Let us consider the special case 푛 = 2, 푘 = 1 on a curve 푋 of genus 푔(푥) = 2. Then

(︀ 2 3 4 6)︀(︀ )︀4 (︀ 2 3 4 6)︀ 푃푀푠(푋,2,1)(푡) = 1+푡 +4푡 +푡 +푡 1+푡 = 1+푡 +4푡 +푡 +푡 푃퐽(푋)(푡).

Note. For 푛 = 2 and 푔 ≥ 2, we have dimC 푀푠(푋, 2, 푘) = (3푔 − 3) + 푔, and 푔 = dim 퐽(푋).

푛 Let det: 푀푠(푋, 푛, 0) → 퐽(푋) be the determinant map 퐸 ↦→ det 퐸 ≡ Λ 퐸, and denote by 푀 0 the ﬁbre of det over some point. We have a general result.

Theorem 2.4.4 (Formula for general 푔 ≥ 2).

(1 + 푡3)2푔 푡2푔(1 + 푡)2푔 푃 0 (푡) = − (2.2) 푀 (1 − 푡2)(1 − 푡4) (1 − 푡2)(1 − 푡4)

Exercise 2.4.5. This should be a palindromic polynomial of degree 6푔 − 6, all of whose coeﬃcients are non-negative. Prove this.

Let us write in short 푀푔(푛, 푘) for 푀푠(푋, 푛, 푘), the moduli space of stable vector bundles of rank 푛 and degree 푘 on a smooth curve 푋 of genus

푔. What can we say for 푛 ≥ 2? For gcd(푛, 푘) = 1, 푀푔(푛, 푘) is a complex manifold of dimension (3푔 − 3) + 푔. Topologically, 푀푔(푛, 푘) is given by ∏︀푔 2휋푖/푛 퐴1, . . . , 퐴푔, 퐵1, . . . , 퐵푔 ∈ 푈(푛) such that 푖=1[퐴푖, 퐵푖] = 휎, where 휎 = 푒 , modulo conjugation by 푈(푛).

Specific question. What is the homology of 푀푔(푛, 푘)? What is its Poincar´e polynomial? Recall:

푁 ∑︁ 푖(︀ )︀ 푖 푃푀 (푡) := dim 퐻 푀푔(푛, 푘); Q 푡 푖=0

2푔 Here 푁 = 8푔 − 6. For 푛 = 1, 푃푀푔(1,푘)(푡) = (1 + 푡) , independent of 푘. 0 For 푛 = 2, 푘 = 1, the moduli space decomposes as 푀푔(2, 1) = 푀푔 (2, 1) × 0 퐽(푋), and the Poincar´epolynomial of 푀푔 (2, 1) is given by Equation (2.2).

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2.5 The approach via Morse theory

2.5.1 Basic Morse theory

Let 푌 be an 푛-dimensional manifold and 푓 : 푌 → R a function; the points 푥 ∈ 푌 where 푑푓(푥) = 0 are called the critical points of 푌 . The Hessian, which we write brieﬂy as “푑2푓”, is a quadratic form, and we call 푓 a Morse function if 푑2푓 is non-degenerate at all critical points of 푓. By the Morse Lemma, there exist near every critical point 푝 local coordinates {푥푖} in which 푓 takes the form 2 2 2 2 2 푓(푝 + 푥) = 푓(푝) − 푥1 − 푥2 − · · · − 푥푟 + 푥푟+1 + ··· + 푥푛 . The integer 푟 is called the Morse index of the critical point. If 푟 = 0, 푓 has a minimum; if 푟 = 푛, 푓 has a maximum, and if 0 < 푟 < 푛, 푓 has a saddle point. If 푓 is a Morse function on 푌 , the Morse polynomial is

∑︁ 훾(푄) 푀푌,푓 (푡) = 푡 , 푄 the sum over all non-degenerate critical points 푄, and 훾(푄) is the Morse index of 푄. It can be shown that

푀푌,푓 (푡) ≥ 푃푌 (푡), with equality in “good cases”.

Examples.

1 ∙ Let 푌 = 푆 and 푓 : 푌 → R the height function. Then 푀푌,푓 (푡) = 푃푌 (푡) = 1 + 푡; this is a “good case”.

∙ Let 푌 = 푆1, but “pinched”, and 푓 again the height function. Then

푀푌,푓 (푡) = 2 + 2푡, a “bad case”.

1 1 ∙ Let 푌 = 푆 × 푆 be the torus and 푓 the height function. Then 푀푌,푓 (푡) = 2 2 1 + 2푡 + 푡 = (1 + 푡) = 푃푌 (푡), another “good case”.

푛−1 ∙ Let 푌 = CP and ∑︀푛 휆 |푧 |2 푓(푧) = 푖=1 푖 푖 with 휆 < ··· < 휆 . ∑︀푛 2 1 푛 푖=1 |푧푖|

Then the critical points of 푓 are 푄푗 where 푧푗 = 1 and 푧푖 = 0 for 푖 ̸= 푗, 2 2푛−2 with indices 훾(푄푗) = 2푗 −2. Hence 푀푌,푓 (푡) = 1+푡 +···+푡 = 푃푌 (푡), and we have another “good case”.

20 Edinburgh Lectures on Geometry, Analysis and Physics 21

We generalise the notion of non-degeneracy to allow critical submanifolds. 푄 ⊆ 푌 is a critical submanifold if 푑푓 = 0 along 푄 and 푑2푓 is non-degenerate in normal directions. The Morse index of 푄, written again as 훾(푄), is the number of linearly independent negative normal directions. Such a function will be called a Morse-Bott function.

Definition 2.5.1. If 푓 : 푌 → R is a Morse-Bott function, the Morse polynomial of 푓 is ∑︁ 훾(푄) 푀푌,푓 (푡) = 푡 푃푄(푡), 푄 where the sum is taken over all non-degenerate critical submanifolds 푄 ⊂ 푌 .

Again we have the Morse inequality 푀푌,푓 (푡) ≥ 푃푌 (푡), with equality in good cases.

Examples.

푛−1 ∙ Let 푌 = CP and ∑︀푛 휆 |푧 |2 푓(푧) = 푖=1 푖 푖 with 휆 ≤ · · · ≤ 휆 , 휆 ̸= 휆 . ∑︀푛 2 1 푛 1 푛 푖=1 |푧푖|

푛−2 If for example 휆1 = 휆2 = ··· = 휆푛−1 < 휆푛, then 푄min = CP and 푄max = {pt.} = [0 : ... : 0 : 1], and so

2푛−2 푀 (푡) = 푃 푛−2 (푡) + 푡 = 푃 (푡). 푌,푓 CP 푌

More generally, if 휆1 = ··· = 휆푟 < 휆푟+1 < ··· 휆푛, then

2푟 2푛−2 푀 (푡) = 푃 푟−2 (푡) + 푡 + ··· + 푡 = 푃 (푡). 푌,푓 CP 푌

∞ 2 1 ∙ Now take 푛 = ∞ in the last example. Then 푃CP (푡) = 1+푡 +··· = 1−푡2 . But we still have

2푟 2푟+2 푃 ∞ (푡) = 푃 푟−1 (푡) + 푡 + 푡 + ··· , CP CP

1 ∑︀∞ 2푟 so we conclude that 푃 푟−1 (푡) = − 푡 . CP 1−푡2 푘=푟

The last example is the prototype of the method to compute 푃푄min (푡) of some critical manifold 푄min in terms of the (possibly inﬁnite-dimensional) total space and higher critical points. We will use this method again later to compute the Poincar´eseries of the moduli space of 푈(2)-bundles over a curve of genus 푔.

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2.5.2 Equivariant cohomology, or The effect of symmetry

Let 퐺 be a compact Lie group (for instance 푈(1) or 푈(푛)) and suppose 퐺 acts on a manifold 푌 . If the action is free, then 푌/퐺 is a manifold and has nice cohomology and Poincar´eseries. If the action is not free, 푌/퐺 has singularities. What to do?

* * Definition 2.5.2 (Equivariant cohomology). We deﬁne 퐻퐺(푌 ) := 퐻 (푌퐺) to be the 퐺-equivariant cohomology of 푌 , where 푌퐺 is given by the Borel construction  푌퐺 := (퐸퐺 × 푌 ) 퐺 , where 퐸퐺 is a contractible space with a free 퐺-action, and the action of 퐺 on 퐸퐺 × 푌 is 푔.(푒, 푦) = (푔.푒, 푔.푦). (In fact, 퐸퐺 is the total space of the classifying ﬁbration 퐺 ˓→ 퐸퐺 퐵퐺.)

Example. Let 퐺 = 푈(1) and 퐸퐺 = ∞ ∖ {0} = lim(︀ 푁 ∖ {0})︀. Then C −→ C  ∞ 퐵퐺 := 퐸퐺 퐺 = CP . We compute:

1 퐻* (pt.) = 퐻*(︀ ∞)︀ and 푃 (푡) = 1 + 푡2 + ··· = . 퐺 CP 1 − 푡2

Note that the projection

  푌퐺 = (퐸퐺 × 푌 ) 퐺 → 퐸퐺 퐺 =: 퐵퐺 ≃ {pt.}퐺 gives a homomorphism

* (︀ )︀ *(︀ )︀ * (︀ )︀ 퐻퐺 {pt.} = 퐻 퐵퐺 −→ 퐻퐺 푌 ,

* (︀ )︀ which turns 퐻퐺 푌 into a graded module over the graded cohomology ring * 퐻퐺(pt.). We saw from the example that for 퐺 = 푈(1), the equivariant * *(︀ ∞)︀ cohomology 퐻퐺(pt.) = 퐻 CP is a polynomial ring in one variable 푢 of degree 2, and we may take 푢 to be the Chern class of the tautological line ∞ bundle on CP . More generally, for 퐺 = 푈(푛) the equivariant cohomology * *(︀ )︀ 퐻퐺(pt.) = 퐻 퐵푈(푛) is a polynomial ring in 푛 variables 푢1, . . . , 푢푛 of degrees 2, 4,..., 2푛, and again the 푢푖 may be interpreted as the Chern classes of the (︀ ∞)︀ tautological 푛-plane bundle over 퐵푈(푛) = Gr푛 C .

Definition 2.5.3. Let 푌 be a manifold with an action of a compact Lie group 퐺 as above. The equivariant Poincar´eseries of 푌 is

∞ 퐺 ∑︁ 푘 (︀ )︀ 푘 푃푌 (푡) = dim 퐻퐺 푌 푡 . 푘=0

22 Edinburgh Lectures on Geometry, Analysis and Physics 23

∼   Remark 2.5.4. If the action of 퐺 on 푌 is free, then 푌퐺 = (퐸퐺×푌 ) 퐺 ≃ 푌 퐺, 퐺 퐺 and so 푃푌 (푡) = 푃푌/퐺(푡) is a polynomial. In general, however, 푃푌 (푡) is only a power series which is the expansion of a rational function. If 푌 is contractible, * (︀ )︀ *(︀ )︀ then 푌퐺 ≃ 퐵퐺, and so 퐻퐺 푌 = 퐻 퐵퐺 .

Equivariant Morse theory. Suppose 퐺 acts on 푌 and 푓 : 푌 → R is a 퐺- invariant Morse-Bott function, i.e. 푓 is a Morse-Bott function and 푓(푔.푦) = 푓(푦)  for all 푔 ∈ 퐺. If 퐺 acts freely on 푌 , then 푓 induces a function 푓퐺 : 푌 퐺 → R, and we can apply Morse theory to 푓퐺. Otherwise, consider 푓 on 푌 , but remember the 퐺-action and use 퐻퐺, that is, consider 푓 as a Morse function on 푌퐺.

Example. Let 푌 = 푆2 and 퐺 = 푈(1), acting by a simple rotation with two ﬁxed points, and let 푓 be the height function. Then

1 푡2 1 + 푡2 푀 퐺 (푡) = + = . 푌,푓 1 − 푡2 1 − 푡2 1 − 푡2 ⏟ ⏞ ⏟ ⏞ min. max

퐺 2  2 This is a “good case”, since we also have 푃푌 (푡) = (1 + 푡 ) (1 − 푡 ).

Some criteria for a good Morse(-Bott) function. The following conditions allow us to conclude that a Morse polynomial (or power series) is “good”, i.e. equal to the Poincar´eseries.

푛−1 ∙ If all Morse indices and all Betti numbers are even. (E.g. for CP .)

∙ In the equivariant case: If each critical submanifold is point-wise ﬁxed by a some 푈(1) ⊂ 퐺 which has no ﬁxed vectors in the negative normal bundle.

We will use these criteria in gauge-theoretical computations in the following section.

2.5.3 Application to infinite dimensions (gauge theory)

Let 푋 be a surface of genus 푔 ≥ 2 and 퐴 a 퐺-connection for a vector bundle 푛 of rank 푛 over 푋, where 퐺 = 푈(푛). For the trivial bundle 푋 × C ,

2 ∑︁ 퐴 = 퐴푖(푥) 푑푥푖 , 푖=1 where (푥1, 푥2) are local coordinates on 푋 and 퐴푖 ∈ u(푛), the Lie algebra of skew-Hermitian (푛 × 푛)-matrices. The curvature of the connection is (locally,

23 24 or globally in the case of the trivial bundle)

2(︀ )︀ 퐹퐴 = 푑퐴 + 퐴 ∧ 퐴 ∈ Ω 푋; u(푛) .

The Lie algebra u(푛) admits an invariant inner product, so we can deﬁne a norm ‖−‖ on it. The Yang-Mills functional of the connection 퐴 is ∫︁ 2 휑(퐴) := ‖퐹퐴‖ 푑 Vol . 푋 The key idea is to apply Morse theory to 휑.

1. The function 휑 is a function on the inﬁnite-dimensional space 풜 of all connections. This is an aﬃne-linear space, hence contractible.

2. The function 휑 is invariant under the inﬁnite-dimensional symmetry group of all bundle automorphisms 풢 = Map(푋, 퐺), the so-called group of gauge transformations.

3. Inside 풢 we have the subgroup 풢0 ⊂ 풢 of based maps 푋 → 퐺, which is the kernel of ev: 풢 → 퐺, the evaluation at a base point 푥0 ∈ 푋 given by ev(푓) = 푓(푥0). That is, 풢0 consists of all those gauge transformations which are the identity at 푥0.

The restricted group 풢0 acts freely on 풜, and so we can reduce to a  퐺-action on 풜 풢0. Moreover, 풢-equivariant cohomology on 풜 becomes  퐺-equivariant cohomology on 풜 풢0.

4. We will apply 풢-equivariant Morse theory to the Yang-Mills functional 휑 on the space 풜.

The critical connections for 휑 are the those for which the curvature 퐹퐴 is covariantly constant. The absolute minimum appears when 퐹퐴 = 0, i.e. when 퐴 is flat (or more generally central harmonic). For higher critical points, 퐴 decomposes.

Example. Let us consider the simplest case, 푛 = 2. That is, we consider rank-2 bundles, or 푈(2)-bundles, on a Riemann surface 푋. The determinant 2 line bundle det 퐸 of a rank-2 bundle 퐸 has degree 푘 = 푐1(퐸) = 푐1(Λ 퐸), and 퐸 is topologically non-trivial whenever 푘 ̸= 0. Let us assume 푘 = 1; so we are in a diﬀerent component of the moduli space than for 푘 = 0.

At the absolute minimum, 풢 acts freely. The moduli space 푀푔(2, 1) is a manifold and contributes 푃푀푔(2,1)(푡). At higher critical points, the bundle is ∼ a direct sum of line bundles, 퐸 = 퐿1 ⊕ 퐿2, and deg 퐿1 + deg 퐿2 = 1. Assume

24 Edinburgh Lectures on Geometry, Analysis and Physics 25

without loss of generality that deg 퐿2 > deg 퐿1. Now 풢 acts with isotropy subgroup 푈(1) and contributes

(1 + 푡)4푔 푃 푈(1) = . 퐽(푋)×퐽(푋) 1 − 푡2

*(︀ )︀ What is the contribution of the total space 풜? We know that 퐻풢 풜 = 퐻*(︀퐵풢)︀, but how do we calculate this? Following Atiyah and Bott [5, S2] we have:

1. 퐵풢 = Map(︀푋, 퐵풢)︀.

∞ 2. For 퐺 = 푈(1), we have 퐵퐺 = CP . So

(︀ ∞)︀ ∏︁ 1 ∞ Map 푋, CP = Z × 푆 × CP , 2푔

and 2푔 2 푃퐵풢(푡) = (1 + 푡) (1 − 푡 ) .

3. For 퐺 = 푈(푛), we have 푈(푛) ∼ 푈(1) × 푆3 × · · · × 푆2푛−1, so

∏︀푛 (︀ 2푖−1)︀2푔 푖=1 1 + 푡 푃퐵풢(푡) = . (2.3) (︁∏︀푛−1(︀ 2푖)︀)︁ (︀ 2푛)︀ 푖=1 1 − 푡 1 − 푡

All of these are “good cases”. We ﬁnish with a computation to prove Theorem 2.4.4. ∞ 푡2푔(1 + 푡)2푔 1 ∑︁ = 푡2푔+4푖(1 + 푡)2푔 (2.4) (1 − 푡2)(1 − 푡4) 1 − 푡2 푖=1

2 −1 ∞ On the right-hand side we recognise the factors (1 − 푡 ) = 푃CP (푡) and 2푔 (1 + 푡) = 푃퐽(푋)(푡). We obtain one big equation

{︀minimum}︀ + {︀higher critical points}︀ = {︀total space}︀ , where

minimum = 푃 0 , the series of the space of interest, 푀푔 (2,1) higher points = the expression (2.4), and total space = (1 + 푡3)2푔(1 − 푡2)(1 − 푡4) from Equation (2.3) with 푛 = 2, for the Yang-Mills functional 휑 on the space of all connections on 푈(2)-bundles with ﬁxed degree 1. The contribution from the higher critical points is given by the 퐿1 ⊕ 퐿2 (with ﬁxed total degree), which is the origin of the Jacobian

25 26

factor 푃퐽(푋)(푡).

Remark 2.5.5. For 푛 ≥ 3, even if we only want to deal with the co-prime case gcd(푛, 푘) = 1, the inductive step will need a general case (e.g. 푛 = 3, 푘 = 1 can decompose into 퐸2 ⊕ 퐸1 with rk 퐸푖 = 푖 and deg 퐸2 = 0, deg 퐸1 = 1). But Morse theory still works to give induction if we use equivariant cohomology and equivariant Poincar´eseries. (The Poincar´eseries 푃푀 (푡) will not be a polynomial).

2.6 Counting rational points

2.6.1 Finite fields

Fields with finitely many elements are either the integers modulo some prime  푝, written F푝 := Z 푝Z, or some algebraic extension thereof, written F푞 with 푞 = 푝푛 for some 푛 ≥ 1. Note that every field is a vector space over its prime subfield F푝, and the characteristic is in each case the prime 푝. We can consider an algebraic variety 푉 defined over any field, in particular over F푞 – for example by considering as the defining equations of 푉 polynomials with integer coefficients and reducing modulo 푝.

(︀ 푛)︀ (︀ 푛 )︀ × Example (Projective spaces). Let 푉 := P F푞 = F푞 ∖ {0} F푞 . The number of points in 푉 is

푞푛 − 1 푁 (푉 ) = = 1 + 푞 + 푞2 + ··· + 푞푛−1 . 푞 푞 − 1

Observe:

1. Over the ﬁeld F푞푚 , the number of points is

푚 2푚 푚(푛−1) 푁푞푚 (푉 ) = 1 + 푞 + 푞 + ··· + 푞 ,

so varying 푚 determines a polynomial in 푞 via 푚 ↦→ 푁푞푚 (푉 ) ∈ Z[푞].

2 푛 푛−1 2. Setting 푞 = 푡 gives the Poincarépolynomial of P(C ) = CP . This indicates a relation between counting rational points over finite fields and Betti numbers of complex varieties.

3. Replacing 푞 by 푞−1 gives

푞푛(1 − 푞−푛) 푁 (푉 ) = = 푞푛−1(︀1 + 푞−1 + ··· 푞−(푛−1))︀ , 푞 푞(1 − 푞−1)

26 Edinburgh Lectures on Geometry, Analysis and Physics 27

and 푁 (푉 ) 푞 = 1 + 푞−1 + 푞−2 + ··· + 푞−(푛−1) (Poincar´eDuality). 푞푛−1

4. Let 푛 → ∞. We get 1(︀1−푞−1)︀, and putting 푞 = 푡−2 we get 1(︀1−푡2)︀ =

∞ 푃CP (푡).

Zeta functions. The 휁-function of an algebraic variety 푉 over F푞 is

(︃ ∞ )︃ ∑︁ 푡푚 푍 (푡) = exp 푁 푚 (푉 ) , 푉 푞 푚 푚=1 where 푁푞푚 (푉 ) is the number of points of 푉 over the finite field F푞푚 . We define further −푠 휁푉 (푠) := 푍푉 (푞 ), which is the analogue of the Riemann 휁-function. Note that |푞−푠| = 푞−ℜ(푠). In 1 the special case where 푉 is a single point, 푍푉 (푡) = 1−푡 .

2.6.2 The Weil conjectures

(The Weil conjectures were proved by A. Grothendieck and P. Deligne.)

Theorem 2.6.1. Let 푉 be a non-singular projective algebraic variety over a finite field F푞. Then

1. 푍푉 (푡) is a rational function of 푡.

2. If 푛 = dim 푉 , then

푝1(푡) 푝3(푡) ··· 푝2푛−1(푡) 푍푉 (푡) = , 푝0(푡) 푝2(푡) ··· 푝2푛(푡)

−푖/2 where each root 휔 of 푝푖 has |휔| = 푞 .

3. The roots of 푝푖 are interchanged with the roots of 푝2푛−푖 under the substi- tution 푡 → 1푞푛 푡.

4. If 푉 is the reduction of an algebraic variety over a subfield of C, then the Betti numbers 푏푖 of the variety 푉 (C) are 푏푖 = deg 푝푖.

Remark 2.6.2. Part (2) of Theorem 2.6.1 is the Riemann hypothesis for function ﬁelds. Part (3) is the functional equation for 휁(푠).

Steps in the proof.

27 28

푖(︀ )︀ 푖(︀ )︀ 1. Deﬁne cohomology groups 퐻 푉 which are the analogues to 퐻 푉 (C) . (Done by Grothendieck.)

2. Use the Frobenius map 휑: 푉 → 푉 , 푥 ↦→ 푥푞. This maps preserves both multiplication and addition. The ﬁxed points of 휑푚 are the points of 푉 (F푞푚 ), and there are 푁푞푚 (푉 ) of them.

3. Apply the Lefschetz ﬁxed point theorem: The number of ﬁxed points of a map 푓 : 푋 → 푋 is

dim 푋 ∑︁ 푖 (︀ * 푖 푖 )︀ (−1) tr 푓 : 퐻 (푋; Z) → 퐻 (푋; Z) . 푖=0

Take 푋 = 푉 , 푓 = 휑 and 퐻푖 to be Grothendieck cohomology:

∑︁ (︀ 푚 * 푖 푖 )︀ ∑︁ 푖 ∑︁ 푚 푁푞푚 (푉 ) = tr (휑 ) : 퐻 (푉 ) → 퐻 (푉 ) = (−1) 휔푖푗 , 푖 푖 푗

where the 휔푖푗 are the eigenvalues of 휑* acting on 퐻푖(푉 ).

4. Now compute:

(︃ ∞ )︃ ∑︁ 푡푚 푍 (푡) = exp 푁 푚 (푉 ) 푉 푞 푚 푚=1 (︁∑︁ 푖 ∑︁ )︁ ∏︁  ∏︁ = exp (−1) − log(1 − 휔푖푗 푡) = 푝푖(푡) 푝푖(푡), 푖 푗 푖 odd 푖 even ∏︀ where 푝푖(푡) = 푗(1 − 휔푖푗 푡). This proves the theorem subject to

5. Poincar´eduality, and

푖/2 6. the Riemann hypothesis: |휔푖푗| = 푞 for all 푖, 푗 (done by Deligne).

Example. Let 푉 = 푋푔 be an algebraic curve of genus 푔. Then

∏︀2푔 (︀ )︀ 푗=1 1 − 휔푗푡 푍푉 (푡) = (︀1 − 푡)︀(︀1 − 푞푡)︀ and ∏︀2푔 (︀ −푠)︀ 푗=1 1 − 휔푗푞 휁푉 (푠) = . (︀1 − 푞−푠)︀(︀1 − 푞−푠+1)︀

Example. Let 푉 = 푀푔(푛, 푘) with gcd(푛, 푘) = 1 be the moduli of stable vector bundles over 푋푔 of rank 푛 and degree 푘. If we can compute 푁푞푚 (푉 ) for all 푚, then Theorem 2.6.1 gives the Betti numbers of 푉 (C), i.e. the Poincar´e polynomial of 푀푔(푛, 푘) over C.

28 Edinburgh Lectures on Geometry, Analysis and Physics 29

How do we compute the number of points of 푀푔(푛, 푘) over F푞? We use two key ideas:

1. All bundles are trivial if we allow poles (of all orders), i.e. if we work

with the ﬁeld of rational functions on 푋푔.

2. The vector space 퐴 of power series over F푞 of the form

∞ ∑︁ 푗 푎푗푡 ∈ F푞[[푡]] (2.5) 푗=0

is inﬁnite-dimensional but compact, since it is a product of ﬁnite (hence compact) sets.

The space 퐴 has a natural measure 휇, which is normalised such that

휇(퐴) = 1. Let 퐴푟 ≤ 퐴 be the linear subspace of power series of the form (2.5)  푟 which satisfy 푎0 = 푎1 = ··· = 푎푟−1 = 0. Then the quotient space 퐴 퐴푟 has 푞 −푟 points, so 휇(퐴푟) = 푞 . We deﬁne the inﬁnite projective space over F푞 to be

∞ ∞  × P(F푞 ) ≡ F푞P := (퐴 ∖ {0}) F푞 .

Since {0} has measure zero,

(︀ ∞)︀ ⃒ ×⃒ 1 휇 푞푃 = 휇(퐴) ⃒ ⃒ = . F F푞 푞 − 1

∞ 1 (Compare this with the Poincaréseries 푃CP (푡) = 1−푡 .) The way in which we just dealt with infinite dimensions and computed measures is our inspiration for counting points in moduli spaces over a finite field F푞: Allowing poles and using measures we can compute the number of points as ratios of measures.

Example. The group of isomorphism classes of line bundles over 푋푔 is iso- morphic to the divisor class group Cl(푋푔) of 푋푔, which is

(︀ )︀ Cl(푋푔) := Div(푋푔) 퐷 ∼ 퐷 + (푓) .

∑︀ A divisor 퐷 is a formal ﬁnite sum 퐷 = 푗 푘푗푄푗, where the 푄푗 ∈ 푋푔 are points and 푘푗 ∈ Z. Now pick a local coordinate 푢 near a point 푄 and let 푓 be a local power series ∞ ∑︁ 푘 푓(푢) = 푎푘 푢 , with 푎−푁 ̸= 0 . 푘=−푁 −푁 Multiplication by elements of a compact group 풦푄 reduces this to 푓(푢) = 푢 . (The group is the group of holomorphic power series around 푄 with non-

29 30

vanishing constant term, i.e. the invertible elements.) So the group Div(푋푔) of all divisors on 푋푔(F푞) is ∏︁ Div(푋푔) = 풦푥∖풜푥 = 풦∖풜 ,

푥∈푋푔

0 and the group of divisor classes of degree 0, written Cl (푋푔), is

0 × Cl (푋푔) = 풦∖풜/퐾 ,

(︀ ×)︀ where 퐾 = 퐾(푋푔) is the function ﬁeld of 푋푔. The measure 휇 풜/퐾 is ﬁnite, and counting points gives the answer 푞2푔.

Bundles of higher rank. To study the moduli space 푀푔(푛, 푘) for 푛 > 1, i.e. the moduli space of bundles of higher rank, we can use the same method, provided we ﬁx the determinant. We have

1 ∑︁ 1 = (푞 − 1) , 휇(풦) |Aut(퐸)| 퐸 where 휇 is the Tamagawa measure (with 푐 = 1). We have further

1 2 = 푞(푛 −1)(푔−1)휁 (2) ··· 휁 (푛). 휇(풦) 푋푔 푋푔

In particular, for 푛 = 2 and 푘 = 1 the sum over all bundles 퐸 splits into a sum over stable bundles and a sum over unstable bundles, where for a stable bundle 퐸 we have Aut(퐸) = {1}. Therefore

∞ ∑︁ 1 ⃒ 0 ⃒ ∑︁ 1 = ⃒푀 (2, 1)⃒ + , |Aut(퐸)| 푔 |Aut(퐸)| 퐸 푟=1 where the last sum is a geometric series running over all bundles 퐸 = 퐿푟 ⊕퐿1−푟 ⃒ 0 ⃒ and extensions. This gives an explicit formula for ⃒푀푔 (2, 1)⃒, and hence by the

Weil conjectures for 푃푀푔(2,1)(푡).

0 Computing measures. Let 훼 run over all points of 푀푔 (푛, 푘), i.e. orbits of 풦 acting on 풜*퐾×. Then

∑︁ (︀  )︀ (︀ * ×)︀ 휇 풦 퐾훼 = 휇 풜 퐾 = 퐶 , 훼 or ∑︁ 1 퐶 = . |풦 | 휇(풦) 훼 훼

30 Edinburgh Lectures on Geometry, Analysis and Physics 31

The only automorphisms of line bundles are scalars, so |풦훼| = 푞 − 1. Also,

∑︁ 1 |퐽(푋푔)| = . |풦 | 푞 − 1 훼 훼

We need to know the value of 퐶 and 휇(풦). Both depend on the precise  normalisation of 휇. If we choose 퐶 = 1, then we get 1 휇(풦) = |퐽(푋푔)|.

2.7 Comparison of equivariant Morse theory and counting rational points

We obtain the same formula for 푃푀 (푡) and agreement term by term in the method of the proof. This also works for all 푛, 푘 and other groups than 푈(푛). The key points are the following:

∙ The total space is “trivial”: The space of connections is aﬃne-linear, hence contractible, and the Tamagawa measure of 푆퐿(푛; C) is 1.

∙ Let 퐼 be the isotropy group. We can compute 푃퐵퐼 (푡) and divide by 휇(퐼).

(︀ )︀ ∼ With 풢 = Map 푋푔, 푈(푛) = 풦,

푛 푛−1 ∏︁ 2푘−1 2푔 2푛 ∏︁ 2푘 2 푃퐵풢(푡) = (1 + 푡 ) (1 − 푡 ) (1 − 푡 ) , 푘=1 푘=1 and 1 2 = 푞(푛 −1)(푔−1)휁 (2) ··· 휁 (푛). 휇(풦) 푋푔 푋푔 These agree using the formula

2푔 ∏︁ −푠  −푠 1−푠 휁푋푔 (푠) = (1 − 휔푖푞 ) (1 − 푞 )(1 − 푞 ). 푖=1

Questions.

1. Why do these two formulae agree? (“Quantum analogue of the Weil conjectures”)

2. Is there an extension of the Weil conjectures to inﬁnite dimensions?

3. Is computing measures on ad`elicspaces analogous to Feynman integration in gauge theories?

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2.8 Relation to physics

Does physics help us understand the questions we raised in the last section? Is there a relation to the original 휁-function? (This leads to arithmetic algebraic geometry (Arakelov theory) and further speculations.)

The Yang-Mills functional came from physics over 4-dimensional space- time. It can be considered formally over a compact Riemannian manifold 푋 of any dimension 푑. In particular, ∙ if 푑 = 2, 푋 is a Riemann surface and we have many results about moduli spaces;

∙ if 푑 = 4 we have Donaldson theory.

Quantum field theory. 1. Hamiltonian approach: Consider space and time separately. We have a Hilbert space ℋ of states, and a self-adjoint “Hamiltonian” operator 퐻 acting on ℋ. The evolution is given by the unitary operator 푒푖푡퐻 on ℋ.

2. Lagrangian formulation (relativistically invariant): Let 퐿 be a functional on some space of functions 푓 on space-time, e.g. 퐿(푓) = ∫︀ |∇푓|2.

3. The Feynman integral is ∫︀ exp(︀ 푖 퐿(푓))︀, integrated over all functions 푓 on ~ 3 ⟨︀ 푖휏퐻 ⟩︀ R × [0, 휏] with 푓(0) = 푢 and 푓(휏) = 푣, determines the value 푢, 푒 푣 . This relates to the Hamiltonian approach. (Recall that the Lagrangian and Hamiltonian are related via the Legendre transform.)

Topological quantum field theories. For some special Lagrangians, we get 퐻 = 0, and so time evolution is just the identity. In this case, the Feynman integrals give topological information, and we call these cases topological quantum field theories. There are many interesting examples of topological QFTs in dimensions 2, 3 and 4. In four dimensions, we get Donaldson theory and Seiberg-Witten theory, but these have no parameters. In three dimensions, we get Chern-Simons theory, which does have an interesting parameter. Let 퐴 be a 퐺-connection over 푋, where 퐺 = 푈(푛). Let

2휋 ∫︁ 2 퐿 = 퐶푆(퐴) = tr(︀퐴 ∧ 푑퐴 + 퐴 ∧ 퐴 ∧ 퐴)︀ . 푘 푋 3 The Hilbert space is the space of holomorphic sections of a line bundle 푘 퐿 over 푀푛(푋푔), where 푋푔 is a Riemann surface. (This three-dimensional theory is related to two-dimensional conformal ﬁeld theory.) We get topological invariants of 3-manifolds and knots inside them (Jones, Witten).

32 Edinburgh Lectures on Geometry, Analysis and Physics 33

In two dimensions, there is also a Yang-Mills theory with Lagrangian 퐿(퐴) = ‖퐹 ‖2 = ∫︀ |퐹 |2. (This is the function on the space of connections 퐴 푋푔 퐴 to which we applied equivariant Morse theory.) This theory is physical and not just topological, but we can solve it exactly. A coupling constant 휖 is introduced and the Feynman integral is formally ∫︁ 1 (︁ 1 2)︁ 푍(휖) = exp − ‖퐹퐴‖ 푑퐴 . vol(풢) 풜 2휖

This has a non-trivial dependence on 휖 and can be used to compute the *(︀ )︀ multiplicative structure on the cohomology ring 퐻 푀(푋푔, 푛) (Witten). This quantum field theory looks promising, but does not give the Poincaré series of 푀(푋푔, 푛). Question: Is there an analogue over a finite field (where the Frobenius map is related to scaling 휖)? Another possibility is to use a 1 (super-symmetric) variant of Chern-Simons theory for a 3-manifold 푆 × 푋푔 (or more generally a circle bundle or a Seifert fibration). The Hilbert space is *(︀ )︀ Ω 푀(푋푔, 푛) , the space of all differential forms on 푀(푋푔, 푛), which comes equipped with a differential 푑 and its adjoint (with respect to the symplectic structure) 푑*. However, this seems to involve integration for functions on 1 푆 × 푋푔, while we want just functions on 푋푔 (for the analogy with finite fields). A possible idea is contained in Witten-Beasley for another theory of Chern- 1 Simons type, where integration is reduced to 푋푔 ⊂ 푆 × 푋푔 as the fixed-point set of a symmetry.

2.9 Finite-dimensional approximations

We can use approximations to link topology with ﬁnite ﬁelds and then pass to a limit. Let us consider approximations to 퐵퐺. For 퐺 = 푈(푛),

푈(푁) 퐵퐺 = lim = lim Gr ( 푁 ) = Gr ( ∞). −→ −→ 푛 C 푛 C 푁→∞ 푈(푛) × 푈(푁 − 푛) 푁→∞

For maps 푓 : 푋푔 → 퐵퐺, fix a degree deg(푓) = 푚 (and then let 푚 → ∞). For 푁 fixed 푁, 푚, the space of holomorphic maps 푓 : 푋푔 → Gr푛(C ) of degree 푚 forms a finite-dimensional algebraic variety 푉 (푁, 푚). The idea of finite-dimensional approximations is the following: Holomorphic maps are determined by their behaviour at “poles”, and the Graßmannians

퐺푟푛 can be embedded in projective space. We can study whether continuous maps can be approximated by holomorphic maps, apply the Weil conjectures to 푉 (푁, 푚) and take limits. This is a reasonable programme.

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2.10 Relation of 휁-functions for finite fields and Riemann’s 휁-function

The original Riemann 휁-function is

∞ ∑︁ 1 ∑︁ (︁ 1 )︁−1 휁(푠) = = 1 − , 푛푠 푝푠 푛=1 푝 prime where the last expression is also known as the Euler product, whose factors are so-called local factors. (They are called thus with reference to the closed points (푝) of the scheme Spec(Z).) The 휁-function ostensibly contains information about the set of primes. Now let 푉 be an algebraic variety over a finite field F푝. We want to define a 휁-function for 푉 . If 푉 = {*} = Spec(F푝) is a single point, let

(︀ −푠)︀−1 휁푉 (푠) := 1 − 푝 .

In general, if 푉 is any variety defined over Z, we define ∏︁ 휁푉 (푠) := 휁푉푝 (푠), 푝 where 푉푝 is the reduction of 푉 modulo 푝. We need to look out for special “bad” primes and add a term for the “infinite prime” (arising in valuation theory). −푠 By the Weil conjectures, 휁푉푝 (푠) is given by a rational function of 푡 = 푝 in terms of the Frobenius action on cohomology.

Example. Let 푉 be an elliptic curve (i.e. of genus 1) deﬁned over Z. The Weil formula gives (1 − 훼푡)(1 − 훽푡) 푍(푡) = , (1 − 푡)(1 − 푝푡) where 훼, 훽 are eigenvalues of the Frobenius map 휑 on 퐻1, and further we have −1/2 −1 (︀ * )︀ |훼| = |훽| = 푝 , 훽 = 훼 and 훼 + 훽 = 푎 = tr 휑 |퐻1(푉 ) . Put

(︁ −푠)︁ −푠 1−2푠 퐿푝(푠) = numerator of 푍(푡) with 푡 = 푝 = 1 − 푎푝푝 + 푝 , and ∏︁ 퐿푉 (푠) = 푐. 퐿푝(푠). 푝

Theorem 2.10.1 (Hasse-Weil Conjecture). With 푉 as above and with suitable choices for the infinite prime and for bad primes, the function 퐿푝(푠) extends holomorphically to all 푠 ∈ C, and 퐿푉 (푠) = ±퐿푉 (2푠).

The Hasse-Weil Conjecture has now been proved by Wiles, Taylor and

34 Edinburgh Lectures on Geometry, Analysis and Physics 35 others. Similar conjectures exist for all 푉 and all 퐻푖. (There is one 퐿-function for each 푖.)

Remark 2.10.2 (The ad`elicpicture for Q or number ﬁelds). This is a comment on the double coset space 풦∖퐺퐴/퐺퐾 used for an algebraic curve over F푝. For Q or Z and for 푆퐿(2) we have 푆푂(2; R)∖푆퐿(2; R), which is the upper-half plane (or hyperbolic plane). The double coset space is

ℳ :== 푆푂(2; R)∖푆퐿(2; R)/푆퐿(2; Z), the moduli space of elliptic curves. To compute the area of ℳ. we start with 푆퐿(2; R)/푆퐿(2; Z), which is a three-dimensional manifold with an invariant volume form. We decompose it into 푆푂(2; R)-orbits and integrate.

2.11 Arithmetic algebraic geometry (Arakelov theory)

Suppose we have an algebraic variety 푉 of dimension 푑 defined over the integers Z. We can either embed Z into C and consider 푉 (C) as a complex variety, or  we can form the residues Z → Z 푝 and get a corresponding variety 푉푝. So in fact we get a family 푉푝 over the primes in Z, and we include 푉∞ sitting over the infinite prime. This family, a scheme over Spec Z, is an algebraic variety of dimension 푑 + 1. If 푑 = 0 we get a number field, if 푑 = 1 we get a so-called arithmetic surface. “In the big picture, physics is at infinity, and number theory at the finite points.” We may try to extend theorems from surfaces to their arithmetic analogues.

Non-Abelian theories. For 푑 = 0 and 퐺 = 푆퐿(푛), we have the Langlands programme, also known as non-Abelian class-field theory. For 푑 = 1 we study the local theory at 푝. For 푝 = ∞, we have the geometric Langlands programme, which has been related by Witten to quantum field theories over 푉 (C). What is the ultimate goal? Perhaps quantum field theories over arithmetic varieties? One would start with the case 푑 = 1.

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2.12 Other question

Can we extend our results from curves to varieties of higher dimensions? Recall that for a curve 푋푔 and gauge group 퐺 = 푆푈(푛), we know the Poincar´eseries

푛 푛−1 ∏︁ 2푘−1 2푔 2푛 ∏︁ 2푘 2 푃Map(푋푔,퐵퐺)(푡) = (1 + 푡 ) (1 − 푡 ) (1 − 푡 ) . 푘=1 푘=1

Over F푞, −1 (푛2−1)(푔−1) vol(풦) = 푞 휁푋푔 (2) ··· 휁푋푔 (푛), where 2푔 ∏︁ −푠  −푠 1−푠 휁푋푔 (푠) = (1 − 휔푖푞 ) (1 − 푞 )(1 − 푞 ) 푖=1 and 풦 is the maximal compact subgroup of 퐺퐴푋 . Both formulae extend from curves to varieties 푉 of all dimensions and still appear to be closely related. We may study, for example, bundles over

2 ∙ P ,

2 1 ∙ (P , P ),

1 ∙ P × 푋푔 (here Morse theory is trickier),

∙ 푋푔 with gauge group Ω(퐺), this is related to the previous point,

∙ and also Weil theory for some inﬁnite-dimensional cases.

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