Asymptotic Behaviour of

4.2. ASYMPTOTIC BEHAVIOUR OF𝑓 NEAR+1 75 The asymptotic𝑓_𝑎D can be chosen such that𝑓_𝑎D|_𝑢0 ≃ 𝑐₀⋅ (𝑢 − 𝑢₀)for some constant𝑐₀, which will be proven in Theorem 4.13. Hence only the asymptotic𝑓_𝑎then remains to be found, this is done in Theorem 4.16.

Thedifferential equationbetween the components of𝑓 is now found using the perturbation theoretic beta function, 𝛽, of a supersymmetric Lagrangian of the so-calledsuperQED, which is a variation on quantum electrodynamics, in Conjecture 4.14. Thisdifferential equation is then

ℱ_D(3)(𝑎_D) = −i

𝜋𝑎1D (Conjecture 4.15), which uses dual variables as described in Definition 3.34.

Hence the asymptotic 𝑓 is a solution of that differential equation using𝑓_𝑎 = −ℱ_D(1)(𝑓_𝑎D) (The- orem 4.16).

Then the respectivemonodromy matrixis derived in Corollary 4.17, after which thegroup of monodromy matrices as a subgroup of𝐺_𝒳 is defined in Theorem 4.18, making it a corollary to the Seiberg–Witten theorem (Theorem 3.48).

First the relatively simple asymptotic behaviour of𝑓_𝑎.

Theorem 4.13 (First component of the section 𝑓_𝑎∈ Γ 𝒳 near𝑓_𝑎

D|𝑢0 ≃ 0) Near any 𝑢0 ∈

ℳ, satisfying lim_𝑢→𝑢

0𝑓𝑎D|𝑢 = 0

2 _{(Theorem 3.48), the following asymptotic behaviour of𝑓}

𝑎D can

be achieved in the appropriate chart, for some𝑢₀-dependent𝑐₀∈ ℂ ⧵ {0}:

𝑓_𝑎D|_𝑢≃ 𝑐₀⋅ (𝑢 − 𝑢₀).

Note that𝑐₀ need not be uniquely defined by the choice of𝑢₀, and hence the double cover property of ℳ need not be ‘visible’ in the formula.

Proof This is implied by [8, 9], but not rigorously proven.

Part of the proof seems to be based on ℂ ⊗ 𝔤_eff ⧵ {0} being a double cover of ℳ (The- orem 3.24), and hence such that is locally looks like ℂ ⊗ 𝔤_eff in at least several different ways, whilelim_𝑢→𝑢

0𝑓𝑎D|𝑢= 0exists and is finite (Theorem 3.48).

It also seems to be based on the fact that all invertible entire holomorphic functionsℂ → ℂ

are of the form𝑧 ↦ 𝑎𝑧 + 𝑏for some constants𝑎, 𝑏 ∈ ℂ[10, p. 180]. ∎

The first step to find the differential equation relating the components of𝑓is from superQED’s beta function𝛽.

Conjecture 4.14 (SuperQED’s beta function, 𝛽) The effective Lagrangian in terms of dual variables and ℱ_D, from Legendre transformation Definition 3.34 and Equation 3.25, can be expanded near the singularity𝑎_D≈ 0(as fromlim_𝑢→𝑢0𝑓_𝑎D|_𝑢= 0) to get a generalisation of the so- called supersymmetric version of quantum electrodynamics,superQEDaccording to [9, pp. 20– 21].

From there superQED’s beta function, 𝛽, can be derived through quantum field theoretic perturbation theory [3], which implies the following differential equation near 𝑢₀:

𝛽(𝑔_effD) = 𝑎_Dd𝑔effD(𝑎D) d𝑎_D = 𝑔_effD(𝑎_D)3 8𝜋2 ,where ℱ_D(2)(𝑎_D) =𝜃effD(𝑎D) 2𝜋 + i 4𝜋 𝑔_effD(𝑎_D)2;

whereℱ_D(1)(𝑎_D) = −𝑎(Definition 3.34), and𝜃_effD(𝑎_D), 𝑔_effD(𝑎_D) ∈ ℝ. Note that𝜃_effD(𝑎_D) = 0in the case of superQED.

2

So𝑢₀∉ ℳ ⧵ 𝖭ℳ, since otherwise𝗀_ℳ would not be nondegenerate at𝑢₀, it would be zero, which cannot be true for anyKähler manifoldby its definition (Definition 3.33).

Proof The proof will not be stated here, but𝛽 is described in [8, 9].

Note that the natural scale of the theory, often called 𝜇, is proportional to𝑎_D according to [8, 9], and the considered varied coupling constant is𝑔_effD.

The Lagrangian that corresponds with superQED is described in [6, pp. 121–128], which is

supposed to be renormalisable. ∎

Thedifferential equationcan now be found when one generalises the result of Conjecture 4.14. Thereby findingℱ_D(3)near 𝑢₀= +1.

Conjecture 4.15 (Differential equation between𝑓’s components, ℱ_D(3)) Generalise the differential equation from Conjecture 4.14 with the addition of:

𝑎_Dd𝜃effD(𝑎D) d𝑎_D = 0.

This then gives the following differential equation:

𝑎_Dℱ_D(3)(𝑎_D) = −i 𝜋.

which should be true near𝑢 ≈ 𝑢₀= +1such that lim_𝑢→𝑢

0𝑓𝑎D|𝑢= 0 according to [8, 9, 11].

Proof Using the relation between𝜃_effD,𝑔_effD, andℱ_D(2)from Conjecture 4.14, one can find the equivalence with𝑎_Dℱ_D(3)(𝑎_D) = −i 𝜋: 𝑎_D d d𝑎_D i Im(ℱD (2)_(𝑎 D)) = 𝑎D d d𝑎_D 4𝜋i 𝑔_effD(𝑎_D)2 , = 4𝜋i 𝑔_effD(𝑎_D)3(−2) ⋅ 𝑎D d𝑔_effD(𝑎_D) d𝑎_D , = 4𝜋i 𝑔_effD(𝑎_D)3(−2) ⋅ 𝑔_effD(𝑎_D)3 8𝜋2 , = −i 𝜋; 𝑎_D d d𝑎_D Re(ℱD (2)_(𝑎 D)) = 𝑎D_d𝑎d D 𝜃_effD(𝑎_D) 2𝜋 , = 0 2𝜋 = 0.

Hence the differential equation is proved.

That these equations are true for𝑢 ≈ 𝑢₀ = +1, seems to be related to superQED not being ‘asymptotically free’ [3], which is implied by𝛽(𝑔_effD) > 0when𝑔_effD> 0. ∎

From the previous differential equation (Conjecture 4.15), with𝑓_𝑎= −ℱ_D(1)(𝑓_𝑎D), one can find theasymptotic behaviour of𝑓 near𝑢₀.

Theorem 4.16 (Asymptotic𝑓 ∈ Γ 𝒳 for 𝑢 ≈ 𝑢₀= +1) Near𝑢 ≃ 𝑢₀= +1, such thatlim_𝑢→𝑢0𝑓_𝑎D|_𝑢= 0,𝑓 is asymptotically: 𝑓|_𝑢= 𝑓𝑎D|𝑢 𝑓_𝑎|_𝑢 ≃ 𝑐₀⋅ (𝑢 − 𝑢₀) i 𝜋𝑓𝑎D|𝑢(ln(𝑓𝑎D|𝑢) − 1) − 𝐶𝑓𝑎D|𝑢+ 𝑎0 , ≃ _i 𝑐0⋅ (𝑢 − 𝑢0) 𝜋𝑓𝑎D|𝑢ln 𝑓_𝑎D|𝑢 𝑐0 + 𝐶0𝑓𝑎D|𝑢+ 𝑎0 ;

where𝐶,𝐶₀, and𝑎₀ are integration constants, and𝑐₀∈ ℂ ⧵ {0}. The second approximation of𝑓

4.2. ASYMPTOTIC BEHAVIOUR OF𝑓 NEAR+1 77 Proof The proof of𝑓_𝑎Dis given in Theorem 4.13, and from Conjecture 4.15 with𝑓_𝑎= −ℱ_D(1)(𝑓_𝑎D)

and𝑓_𝑎D|_𝑢≠ 0, one gets𝑓_𝑎:

−ℱ(3)(𝑎_D) = +i 𝜋 1 𝑎_D, −ℱ(2)(𝑎_D) = i 𝜋ln 𝑎D− 𝐶, 𝑎 = −ℱ(1)(𝑎_D) = i 𝜋𝑎D(ln 𝑎D− 1) − 𝐶𝑎D+ 𝑎0

where𝑎₀ is an integration constant, just as𝐶is. Hence the first approximation of𝑓 is proved. From this first approximation, one can rewrite it to use𝐶₀ as in the second approximation, such that it is identical to the first. The second approximation is more like that shown in [11]. ∎

From that asymptotic behaviour of𝑓, one can find how different branches of𝑓 are related to each other near𝑢₀= +1. This can be done by considering the a closed path inℳ around𝑢₀ and how𝑓 transforms under that. This is similar to Corollary 4.12.

Corollary 4.17 (Monodromy of 𝑓 for 𝑢anticlockwise around 𝑢₀= +1, M₊₁) Moving𝑢 ∈ ℳ in an anticlockwise closed path around 𝑢₀ = +1, one finds the so-called monodromy matrix

M_𝑢 0 = M+1 for𝑓 3_: 𝑓 ↦ M_𝑢0(𝑓), M_𝑢0 = +1 0 −2 +1 , = 0 +1 −1 0 1 +2 0 1 0 −1 +1 0 , = S ∘ T₊₂∘ S−1∈ 𝐺_𝒳;

where𝐺_𝒳 is the fiber bundle 𝒳’s structure group, as fixed in Conjecture 3.45.

Proof The formula for𝑓in Theorem 4.16 can be used here, where one needs to substitute𝑢for the anticlockwisenearlyclosed path around𝑢₀= +1 𝛾: (−𝜋, +𝜋) → ℂ⧵{0}, with𝛾(𝑡) = 𝑢₀+𝑅(𝑡)ei𝑡for some chosen𝑅: (−𝜋, +𝜋) → (0, 1) that is sufficiently bounded above and satisfieslim_{𝑡→(−𝜋)}+𝑅(𝑡) =

lim_{𝑡→(+𝜋)}−𝑅(𝑡), to find 𝑓|_𝛾(𝑡) for some choice of branch cuts:

𝑓|_𝛾(𝑡)≃ 𝑐0⋅ (𝑅(𝑡)e i𝑡₎ i 𝜋𝑓𝑎D|𝛾(𝑡)(ln 𝑅(𝑡) + i𝑡) + 𝐶0𝑓𝑎D|𝛾(𝑡)+ 𝑎0 . Thenlim_{𝑡→(−𝜋)}+𝑓_𝑎

D|𝛾(𝑡) andlim𝑡→(+𝜋)−𝑓𝑎D|𝛾(𝑡)can be related to each other:

lim 𝑡→(+𝜋)−𝑓𝑎D|𝛾(𝑡)≃ 𝑐0⋅ _{𝑡→(+𝜋)}lim−𝑅(𝑡) (−1), = +𝑐₀⋅ lim 𝑡→(−𝜋)+𝑅(𝑡) (−1), ≃ + lim 𝑡→(−𝜋)+𝑓𝑎D|𝛾(𝑡);

which therefore implies𝑓_𝑎↦ +𝑓_𝑎 under the closed path, as had to be proven.

3

This actually relates branch cuts of𝑓 to each other, or more precisely it relates analytic continuations of𝑓 along that path (Theorem 3.48).

Now one can derive the monodromy of𝑓_𝑎:

lim

𝑡→(+𝜋)−𝑓𝑎|𝛾(𝑡)≃

i

𝜋 𝑡→(+𝜋)lim−𝑓𝑎D|𝛾(𝑡) ln _{𝑡→(+𝜋)}lim−𝑅(𝑡) + _{𝑡→(+𝜋)}lim−i𝑡 + 𝐶0 _{𝑡→(+𝜋)}lim−𝑓𝑎D|𝛾(𝑡) + 𝑎0,

= +i

𝜋 𝑡→(−𝜋)lim+𝑓𝑎D|𝛾(𝑡) ln _{𝑡→(−𝜋)}lim+𝑅(𝑡) + _{𝑡→(−𝜋)}lim+i𝑡 + i2𝜋 + 𝐶0 _{𝑡→(−𝜋)}lim+𝑓𝑎D|𝛾(𝑡) + 𝑎0,

≃ −2 lim

𝑡→(−𝜋)+𝑓𝑎D|𝛾(𝑡)+ _{𝑡→(−𝜋)}lim+𝑓𝑎|𝛾(𝑡) ;

hence𝑓 ↦ M_∞(𝑓)is proved, as had to be shown. ∎

Lastly the definition of themonodromy group𝛤(2), which relates all branch cuts of𝑓 to each other (Theorem 3.48).

Theorem 4.18 (Monodromy group of 𝑓, 𝛤(2)) Define the monodromy group 𝛤(2) as the group that is generated by themonodromy matricesM_∞ andM₊₁:

M_∞= −1 +2

0 −1 , M+1=

+1 0 −2 +1 ;

which is a subgroup of𝐺_𝒳.

For each pair of branch cuts of𝑓: 𝑓₁ and𝑓₂; the function that satisfies 𝑓₁ ↦ 𝑓₂ has to be an element of the groups of monodromy matrices𝛤(2)(Theorem 3.48).

Proof The proof that𝛤(2)is a subgroup of𝐺_𝒳 is trivial, since bothM_∞andM₊₁are elements of𝐺_𝒳 as proved in Corollary 4.12 and in Corollary 4.17.

Now the only remaining proof is that all branch cuts of 𝑓 are related to each other with exactly only functions of𝛤(2). This is proved in the remainder of this proof.

For any pair of branch cuts 𝑓₁ ∈ Γ 𝒳 and 𝑓₂ ∈ Γ 𝒳, there is a monodromy relating them to each other: 𝑓₁|_𝑝 ↦ 𝑓₂|_𝑝; for each point in the underlying manifold 𝑝 ∈ ℳ ⧵ 𝖭ℳ. For each

𝑝 ∈ ℳ ⧵ 𝖭ℳ, and each sufficiently small open set containing𝑝, thismonodromycan be described using aclosed path: 𝛾: [0, 1] → ℂ ⧵ {−1, +1}, where𝛾(0) = 𝛾(1) = 𝑝; as theanalytic continuation of

𝑓₁ along𝛾 [10].

One could abusively use the following notation, component-wise:

𝑓₂|_𝑝= 𝑓₁|_𝑝+

𝛾

∂_𝑢(𝑓)|_𝑢d𝑢;

where the complex contour integral in the equation should be interpreted sufficiently4_{. Then,}

since each component of𝑓 is aholomorphic function(Remark 3.40), for each chosenclosed path

𝛾, one can rewrite the complex contour integral to sums of contour integrals over paths that move around only∞ or 𝑢 = +1 [10] (the paths can becontinuously deformed to those used in Corollary 4.12 and in Corollary 4.17).

Hence 𝛤(2)is sufficient in relating each possible branch cut of 𝑓, which is what had to be

shown. ∎

4_{For example for branches of complex logarithm, with the closed path𝛾: [0, 2] → ℂ ⧵ {0}such that: 𝛾(𝑡) = 𝛾}

1(𝑡)for

𝑡 ∈ [0, 1], and𝛾(𝑡) = 𝛾₁(𝑡 − 1)for𝑡 ∈ [1, 2], where𝛾₁(𝑡) = ei2𝜋⋅𝑡and𝛾(0) = 𝛾₁(0) = 𝛾₁(1) = 𝛾₁(1 − 1) = 𝛾₁(2 − 1) = 𝛾(2) = 1. Chose the principle branchln: ℂ ⧵ (−∞, 0] → ℂ: whereln(𝑅ei𝜑) = ln 𝑅 + 2𝜋i𝜑for𝑅 ∈ (0, ∞)and𝜑 ∈ (−𝜋, +𝜋); asln₁= ln, then ∂_𝑢(ln)|_𝑢= 𝑢−1. Thenln₂= ln₁+2𝜋i ⋅ 2, since∮_𝛾𝑢−1d𝑢 = ∮_𝛾

1𝑢

−1_{d𝑢 + ∮} 𝛾1𝑢

Chapter 5

Period Mapping

Here the last part of theSeiberg–Witten theorem(Theorem 3.48) is proved: anytrivialisationof a branch cut of𝑓 ∈ Γ 𝒳 under therestrictionofIm(ℱ(2)(𝑓)|_𝑢) > 0is found as in section 5.3.

In section 5.1 an invertible function 𝐿 is found that maps each ∂_𝑢(𝑓)|_𝑢 up to 𝐺_𝒳 (under the restriction of Im(ℱ(2)(𝑓)|_𝑢) > 0 as from Theorem 3.47 (Definition 5.1)) to a lattice from ℂ

(Theorem 5.2 and Theorem 5.3). Hence one can use such a lattice without loss of generality, since eachtrivialisationof∂_𝑢(𝑓)differsby a function in𝐺_𝒳.

Then in section 5.2, for each𝑢 ∈ ℂ ⧵ {−1, +1}, a manifold 𝐸_𝑢 is made from a cubic curve (Definition 5.6). This manifold is thenisomorphicto the respective lattice (Theorem 5.8). From that one can find∂_𝑢(𝑓)in Theorem 5.9, up to some remainingdegree of freedom: 𝑢 ↦ 𝑔_𝑢.

This freedom𝑢 ↦ 𝑔_𝑢will then befixedto get∂_𝑢(𝑓). Then this isintegratedto get𝑓 ∈ Γ 𝒳 (up to𝐺_𝒳, and under the restriction ofIm(ℱ(2)(𝑓)|_𝑢) > 0), in section 5.3 (Conjecture 5.10). In order to prove that this𝑓 describes the actualeffective Lagrangian: (𝒞_eff, ℒ_eff) ≃ (𝒞_𝑓ℳ

𝑎, ℒ

ℳ

𝑓_𝑎D); it has to

comply with chapter 4, this is done in Conjecture 5.11 and Conjecture 5.12. Therebyfinishing the proof of theSeiberg–Witten theorem(Theorem 3.48).

5.1 Lattice Map

This section describes each ∂_𝑢(𝑓)|_𝑢, as from Definition 3.39, under the additional restriction of

Im(ℱ(2)(𝑓_𝑎)|_𝑢) > 0(Definition 5.1), in relation to the lattice𝐿(∂_𝑢(𝑓)|_𝑢) = ℂ/(∂_𝑢(𝑓)|_𝑢⋅ ℤ + ∂_𝑢(𝑓)|_𝑢⋅ ℤ)

(Theorem 5.2). This relation is invertible up to trivialisation of the fiber bundle 𝒳, which is equivalent to invertibility up to𝒳’s structure group𝐺_𝒳 (Theorem 5.3).

After that a trivial concept ofintegrationon𝐿(∂_𝑢(𝑓)|_𝑢) is described (Theorem 5.4), which is only meant as anintroductionto the similar concept in thenext section in Theorem 5.8.

Then in Theorem 5.5, the proof is given that one canuniquelyfind any trivialisationof any branch cut of𝑓 ∈ Γ 𝒳 from: each∂_𝑢(𝑓)|_𝑢, and an integration constant1.

First the restriction caused by the assumptionIm(ℱ(2)(𝑓_𝑎))|_𝑢> 0(Theorem 3.47).

Definition 5.1 (Restriction of ∂_𝑢(𝑓), 𝕃_𝒳) Define the following subset of the vectorspaceℂ2:

𝕃_𝒳= 𝜋𝑎D

𝜋_𝑎 ∈ (ℂ

2_{) Im} 𝜋𝑎D

𝜋_𝑎 > 0 .

1_{In Conjecture 5.10, the proof is given that the integration constant is unique actually.}

From𝑓 ∈ Γ 𝒳, under the restrictions of Definition 3.39,addthe following restriction for each

𝑢 ∈ ℳ ⧵ 𝖭ℳ for the remainderof thischapter:

∂_𝑢(𝑓)|_𝑢= ∂𝑢(𝑓𝑎D)|𝑢

∂_𝑢(𝑓_𝑎)|_𝑢 ∈ 𝕃𝒳,

which is equivalent toIm(ℱ(2)(𝑓_𝑎))|_𝑢> 0(Theorem 3.47); as stated in Theorem 3.48. From any element in𝕃_𝒳, one can define a lattice using𝐿.

Theorem 5.2 (Lattice mapping,𝐿) Consideringℂ ≃ ℝ2, alattice can be made fromℂ and two complex numbers,𝜋_𝑎

D, 𝜋𝑎 that are linearly independent inℝ

2_{, which is equivalent toIm(}𝜋𝑎D

𝜋𝑎 ) ≠

0.

Define𝐿 to describe such a lattice from a vector in 𝕃_𝒳:

𝐿: 𝕃_𝒳→ ℂ 𝜋_𝑎 D⋅ ℤ + 𝜋𝑎⋅ ℤ 𝜋_𝑎D 𝜋_𝑎 ∈ 𝕃𝒳 , ∂_𝑢(𝑓)|_𝑢↦ ℂ ∂_𝑢(𝑓)|_𝑢⋅ ℤ + ∂_𝑢(𝑓)|_𝑢⋅ ℤ , = 𝐴 ⊆ ℂ ∀𝑎 ∈ 𝐴 ∀𝑧₁, 𝑧₂∈ ℤ: 𝑎 + ∂_𝑢(𝑓)|_𝑢⋅ 𝑧₁+ ∂_𝑢(𝑓)|_𝑢⋅ 𝑧₂∈ 𝐴 .

Furthermore note that any lattice can be written in terms of𝐿’s codomain. Proof The proof is divided in several steps.

The equivalence of ℝ-linear independence of 𝜋_𝑎

D and 𝜋𝑎, with Im(

𝜋_𝑎D

𝜋𝑎 ) ≠ 0 is proved in two

steps: first that theℝ-linear independenceimpliesIm(𝜋𝑎D

𝜋𝑎 ) ≠ 0, and then itsconverse.

So first theimplication: assume𝜋_𝑎

D and𝜋𝑎 areℝ-linear independent complex numbers. Then

𝜋_𝑎

D, 𝜋𝑎∈ ℂ ⧵ {0}, so

𝜋_𝑎D

𝜋𝑎 ∈ ℂ ⧵ {0}exists. From the linear independence, one also finds that there

does not exist any𝑎 ∈ ℝ ⧵ {0}such that𝑎𝜋_𝑎− 𝜋_𝑎

D = 0. Hence there does not exist any𝑎 ∈ ℝsuch

that𝑎 =𝜋𝑎D

𝜋𝑎 , which impliesIm(

𝜋_𝑎D

𝜋𝑎 ) ≠ 0. Hence this implication has be proven.

Now the converse: assume Im(𝜋𝑎D

𝜋𝑎 ) ≠ 0 for some 𝜋𝑎D, 𝜋𝑎 ∈ ℂ such that it exists. From that

existence, 𝜋𝑎D

𝜋𝑎 has to exist in addition to being non-zero, so 𝜋𝑎D, 𝜋𝑎 ∈ ℂ ⧵ {0}. By the complex

numbers, there exists an unique𝑎 ∈ ℂ ⧵ {0}such that 𝑎𝜋_𝑎− 𝜋_𝑎D = 0, which is𝑎 = 𝜋𝑎D

𝜋_𝑎 . But that

unique number𝑎, is not a real number, since by assumption Im(𝑎) = Im(𝜋𝑎D

𝜋𝑎 ) ≠ 0. Hence there

does not exist any𝑎 ∈ ℝ ⧵ {0}such that𝑎 =𝜋𝑎D

𝜋𝑎 , which is equivalent to theℝ-linear independence

of𝜋_𝑎

D and𝜋𝑎. Hence the equivalence is proved.

So the codomain of the function𝐿 is a non-strict subset of the set of lattices from ℂ. The only remaining part of the proof is that the restriction ofIm(𝜋𝑎D

𝜋𝑎 ) ≠ 0toIm(

𝜋_𝑎D

𝜋𝑎 ) > 0can be done

without loss of generality.

One can rephrase this as: for any (𝜋₁, 𝜋₂) ∈ ℂ2 that satisfiesIm(𝜋1

𝜋2) < 0, there exists some

(𝜋_𝑎D, 𝜋_𝑎) ∈ ℂ2 that satisfiesIm(𝜋𝑎D

𝜋_𝑎 ) > 0, such that𝜋1⋅ ℤ + 𝜋2⋅ ℤ = 𝜋𝑎D⋅ ℤ + 𝜋𝑎⋅ ℤ. This is relatively

simple to prove, since it is implied by the special case: (𝜋_𝑎

D, 𝜋𝑎) = (−𝜋1, 𝜋2); since−1 ∈ ℤ.

Hence the proof is complete. ∎

Now one can describe theinvertibilityof𝐿. This is where the equality𝐺_𝒳 = SL(2, ℤ)(Conjec- ture 3.45) becomes significant in the derivation of𝑓. Note that the equality actually comes from equating two subsets ofℂ, that are of the form𝜋₁⋅ ℤ + 𝜋₂⋅ ℤ(with the restriction of 𝜋1

5.1. LATTICE MAP 81 Theorem 5.3 (Invertibility, 𝐿) The group SL(2, ℤ) is described in Conjecture 3.45, and is equal to 𝒳’s structure group 𝐺_𝒳. By Theorem 3.42, 𝕃_𝒳/𝐺_𝒳 ⊆ 𝐹/𝐺_𝒳 makes sense, and can be made into a fiber bundleoverℳ ⧵ 𝖭ℳ like𝒳.

Extendthe definition of𝐿 to the invertiblefunction:

𝐿: 𝕃_𝒳/𝐺_𝒳 → ℂ 𝜋_𝑎 D⋅ ℤ + 𝜋𝑎⋅ ℤ 𝜋_𝑎D 𝜋_𝑎 ∈ 𝕃𝒳 , 𝐺_𝒳(∂_𝑢(𝑓)|_𝑢) ↦ ℂ ∂_𝑢(𝑓)|_𝑢⋅ ℤ + ∂_𝑢(𝑓)|_𝑢⋅ ℤ ; where𝐺_𝒳(∂_𝑢(𝑓)|_𝑢) = {M(∂_𝑢(𝑓)|_𝑢) ∈ 𝕃_𝒳|M ∈ 𝐺_𝒳}.

Proof The proof can be found in [4, pp. 329–330], here only sketches will be given.

The proof that any M ∈ 𝐺_𝒳 can be restricted to a function M: 𝕃_𝒳 → 𝕃_𝒳 is based on the following. Assume 𝜋𝑎D 𝜋_𝑎 ∈ 𝕃𝒳, and 𝜋_𝑎 D 𝜋_𝑎 = 𝛼 𝛽 𝛾 𝛿 𝜋_𝑎 D 𝜋_𝑎 where 𝛼 𝛽 𝛾 𝛿 ∈ 𝐺𝒳, so𝛼, 𝛽, 𝛾, 𝛿 ∈ ℤ and 𝛼𝛿 − 𝛽𝛾 = +1 > 0 by the definition of SL(2, ℤ) = 𝐺_𝒳 [4]. Then the following proves

𝜋_𝑎 D 𝜋_𝑎 ∈ 𝕃𝒳: Im 𝜋𝑎D 𝜋𝑎 = Im ⎛ ⎜ ⎜ ⎜ ⎝ 𝛼𝜋𝑎D 𝜋𝑎 + 𝛽 𝛾 𝜋_𝑎D 𝜋𝑎 + 𝛿 ∗ 𝛾𝜋𝑎D 𝜋_𝑎 + 𝛿 2 ⎞ ⎟ ⎟ ⎟ ⎠ = 𝛼𝛿 − 𝛽𝛾 𝛾𝜋𝑎D 𝜋_𝑎 + 𝛿 2Im 𝜋_𝑎 D 𝜋_𝑎 > 0

In [4, pp. 329–330] the proof of the equivalence of: 𝐿(𝑎) = 𝐿(𝑏), for any 𝑎, 𝑏 ∈ 𝕃_𝒳; with: the existence of a M ∈ 𝐺_𝒳 such that 𝑎 = M(𝑏); is given. This means that: 𝐿(𝑎) = 𝐿(𝑏) implies

𝑎, 𝑏 ∈ 𝐺_𝒳(𝑎); which is equivalent to: 𝐿(𝐺_𝒳(𝑎)) = 𝐿(𝐺_𝒳(𝑏))implies𝐺_𝒳(𝑎) = 𝐺_𝒳(𝑏); and hence the

invertibilityof the new 𝐿2. ∎

For similarity with the integration that will be defined later in Theorem 5.7, define an integra- tion on𝐿(∂_𝑢(𝑓)|_𝑢). This integration can then be used to find∂_𝑢(𝑓)|_𝑢up to𝐺_𝒳, as in Theorem 5.3.

Theorem 5.4 (Integration on any 𝐿(∂_𝑢(𝑓)|_𝑢)) For any∂_𝑢(𝑓)|_𝑢∈ 𝕃_𝒳, one can define an integ- ration for any 𝑧₀, 𝑧₁∈ 𝐿(∂_𝑢(𝑓)|_𝑢) and any holomorphic𝑔: 𝐿(∂_𝑢(𝑓)|_𝑢) → ℂ:

𝑧1

𝑧0

𝑔(𝑧)d𝑧;

such that it is equal to the ordinary complex contour integral ∫_𝛾𝑔(𝑧)d𝑧 (modulo implied terms of 0 described later) for any path 𝛾: [0, 1] → ℂ: such thatℂ ∋ 𝛾(𝑗) = 𝑧_𝑗 ∈ 𝐿(∂_𝑢(𝑓)|_𝑢)(modulo the expected terms) for each𝑗 ∈ {0, 1}.

Since the path𝛾 in the previous can be deformed, as in ordinary complex analysis [10], while keeping the integral ∫_𝛾 equal, the definition of the integration on 𝐿(∂_𝑢(𝑓)|_𝑢) is modulo (for any

𝑝 ∈ 𝐿(∂_𝑢(𝑓)|_𝑢)) ∫_𝑝𝑝= 0 + ℤ ∫_𝛾 𝑎D+ℤ ∫𝛾𝑎, where: 𝛾_𝑎 D: [0, 1] → 𝐿(∂𝑢(𝑓)|𝑢), 𝛾𝑎: [0, 1] → 𝐿(∂𝑢(𝑓)|𝑢), 𝛾_𝑎 D(𝑡) = 𝑡 ⋅ ∂𝑢(𝑓𝑎D)|𝑢, 𝛾𝑎(𝑡) = 𝑡 ⋅ ∂𝑢(𝑓𝑎)|𝑢.

2_{Note that anyfunction:}

𝐹: 𝐴 → 𝐵; can be uniquely described by agraph:𝐺_𝐹, which is a subset of𝐴 × 𝐵; that satisfies the following: ∀(𝑎₁, 𝑏₁), (𝑎₂, 𝑏₂) ∈ 𝐺_𝐹: (𝑎₁= 𝑎₂⇒ 𝑏₁= 𝑏₂). Then the following notation is defined: for any(𝑎, 𝑏) ∈ 𝐺_𝐹, then 𝐹: 𝑎 ↦ 𝑏 = 𝐹(𝑏).

From double periodicity of the implied𝑔: ℂ → ℂ[10, p. 324], the function𝑔 as before has to be a constant function: so use𝑔: 𝑧 ↦ 𝑔 ∈ ℂ; then the integral returns a value in a lattice3_:

𝑧1

𝑧0

𝑔(𝑧)d𝑧 = 𝑔 ⋅ (𝑧₁− 𝑧₀) ∈ 𝐿(𝑔 ⋅ ∂_𝑢(𝑓)|_𝑢).

Proof Most of the actual proofs that might be needed are actually already given in the the- orem. Furthermore, the proof that any closed path in𝐿(∂_𝑢(𝑓)|_𝑢)can be continuously deformed to a sum of𝛾_𝑎D and𝛾_𝑎 or to a point (of which the integral is always0), is intuitively understandable

when considering𝐿(∂_𝑢(𝑓)|_𝑢)a torus. ∎

Last in this section is the proof that any trivialisation of any branch cut of 𝑓 ∈ Γ 𝒳 can be found from𝑢 ↦ 𝐿(∂_𝑢(𝑓)|_𝑢)and an integration constant𝑓𝐶 ∈ 𝒳.

Theorem 5.5 (Constructing ∂_𝑢(𝑓) and 𝑓 from integration) Up to an overall 𝐺_𝒳, and an integration constant, one can find 𝑓 ∈ Γ 𝒳 uniquelyfrom 𝑢 ↦ 𝐿(∂_𝑢(𝑓)|_𝑢)(ignoring the choice of

𝖭ℳ).

This is done by first fixing a base point 𝑃 ∈ ℂ ⧵ {−1, +1}, and the integration constant

In document Seiberg-Witten Theory (Page 82-114)