Error Approximation

Let us begin by some general remarks. First off, note that the orthogonal projection onto the kernel of D_A^˚_,t{x along the image ofDA,t{x (where we drop the subscript) is given by

π“1´DpD^˚Dq^´1D^˚. (7.64) This implies immediately that the image ofD is in the kernel ofπ, indeed,

πpD^ψq “D^ψ´DpD^˚Dq^´1D^˚D^ψ“D^ψ´D^ψ“0. (7.65) Both the Bogomolny equations and the Nahm equations are vector bundle en-domorphism valued. To show that they are satisfied, we pick an arbitrary element ψPkerD^˚, apply the equation and show that the result is zero. Plugging in the def-initions of the transformed data, we see that the last step of the equation is applying the projectionπ, so we use two tricks to drop terms. First, anything in the image ofD vanishes, because of the calculation above, and second whenever D^˚ gets applied to ψ, the term vanishes.

It then turns out that we merely have to calculate some commutators, to apply the facts above. The final step uses the Weitzenb ¨ock formula that we have shown for the generalized LaplacianD^˚D. The important part about these is that they commute with the Clifford multiplications cli.

7.3.1 Nahm to Bogomolny

Theorem 7.20. If pT₀,T₁,T₂,T₃q satisfy the Nahm equations, then the transformed datum p^dA,^φqsatisfies the Bogomolny Equations.

Proof. Recall the Bogomolny equations

F^A“ ‹dAφ. (7.66) In coordinates we may write this as

F^A₂₃ “∇^A₁φ, F^A₃₁ “∇^A₂φ, F^A₁₂ “∇^A₃φ. (7.67) LetψPkerDx^˚. Then we can apply the equations toψto get

p∇^A₁φqψ´ r∇^A₂,∇^A₃sψ“0 (7.68) p∇^A₂φqψ´ r∇^A₃,∇^A₁sψ“0 (7.69) p∇^A₃φqψ´ r∇^A₁,∇^A₂sψ“0. (7.70) Plugging in the definitions and using

p∇^A₁φqψ“∇^A₁pφpψqq ´φp∇^A₁ψq, (7.71)

we arrive to the equivalent form of the first equations

πpB1πpitψqq ´πpitπpB1ψqq ´πpB2πpB3ψqq `πpB3πpB2ψqq “0. (7.72) It suffices to show that

B1πpitψq ´itπpB1ψq ´ B2πpB3ψq ` B3πpB2ψq Pkerπ. (7.73) Using the fact thatπ“¹´DxpDx^˚Dxq^´1Dx^˚, we end up with

´ B1DxpDx^˚Dxq^´1Dx^˚pitψq `itDxpDx^˚Dxq^´1Dx^˚pB1ψq (7.74)

` B2DxpDx^˚Dxq^´1Dx^˚pB3ψq ´ B3DxpDx^˚Dxq^´1Dx^˚pB2ψq. (7.75) Now permutingDx with the other operators, we calculate

rDx,Bjs “cljbi, rDx,its “i. (7.76) Using these brackets we arrive at

cl1bipDx^˚Dxq^´1Dx^˚pitψq ´ipDx^˚Dxq^´1Dx^˚pB1ψq

´cl2bipDx^˚Dxq^´1Dx^˚pB3ψq `cl3bipDx^˚Dxq^´1Dx^˚pB2ψq modimDx. (7.77) Next up we use the commutators withD^˚

rDx^˚,Bjs “cljbi, rDx^˚,its “ ´i, (7.78) to get modulo kerDx^˚

pcl1b1VqpDx^˚Dxq^´1ψ` pDx^˚Dxq^´1pcl1b1Vqψ (7.79)

` pcl2b1VqpDx^˚Dxq^´1pcl3b1Vqψ´ pcl3b1VqpDx^˚Dxq^´1pcl2b1Vqψ. (7.80) Note that we have notused that p^T0,T₁,T₂,T₃q satisfy the Nahm equations yet. Only now do we need the fact, that

pD^˚Dq^´1“1Sb^G, (7.81) whereGis the Greens Operator of the right rand side of the Weitzenb ¨ock formula7.5 for vanishing errorη. We arrive at

pcl1`cl1`cl2cl3´cl3cl2q

| {z }

2cl1´cl1´cl1“0

b1V

pDx^˚Dxq^´1ψ, (7.82)

which finishes the proof. To prove the other equations, cyclically permute1,2and3in

this proof.

Corollary7.21. For a generalized Nahm datumpT₀,T₁,T₂,T₃qwith the error term satisfying

G0is the Greens operator of 1_Hb

´clη, whereG0 (the Greens operator) commutes with the Clifford multiplication. We derive an expression of the Green operatorG ofDx^˚Dx in terms of the Green operatorG₀. . We see then, that where Gis the Greens operator ofDx^˚Dx andE is defined by the equation above. By looking at equations (7.79) and (7.80), we see that the error term’s first component is

Proof. Let us denote the operator (7.85) byX. Then XG₀ “1, and we see that ^kXψkě Cr²kψk by the definition of X for r large enough and some constant C (compare to argument for equation (7.126)). Using this forψ“G0ψshows that

Cr²kG0ψk²ďkXG0ψk²“kψk². (7.90) We conclude that kG0k “ Op^r´2q. This implies that ^E “ Op^r´4q, which implies the

claim (note thatηNahmis independent ofxPR³).

Corollary7.23. The first order term of the error mapηNÑηBis given by

η_Bp^ψq “ ´^4πp^G0η_NG₀ψq, (7.91) for a sectionψPΓpR³,Eq.

Proof. LetηN“εη, and compare coefficients by powers ofε. 7.3.2 Bogomolny to Nahm

Notation7.24. For brevity we denote the Bogomolny Dirac Operator by justD, instead ofDA,t in this section.

Theorem 7.25. Let pA,φq satisfy the Bogomolny equations. Then the transformed datum p∇t,T_jqsatisfies the Nahm equations.

Proof. Let us show the first Nahm equation.

∇tT1` rT2,T3s “0. (7.92) Take aψPkerD^˚ (i.e a section of the vector bundle) and calculate

p∇tT₁qpψq “∇pT₁ψq ´T₁∇tψ“π Btπpix₁ψq ´ix₁πpBtψq. (7.93) rT2,T3spψq “π ix2πpix3ψq ´ix3πpix2ψq. (7.94) So to show the equation, it suffices to show that

Btπp^ix1ψq ´^ix1πpBtψq ´^ix2πp^ix3ψq ´^ix3πp^ix2ψq Pkerπ. (7.95) We calculate

Btπpix1ψq ´ix1πpBtψq “ Btpix1ψ´DpD^˚Dq^´1D^˚qpix1ψq (7.96)

´ix1pBtψ´DpD^˚Dq^´1D^˚qpBtψq (7.97)

“ix1DpD^˚Dq^´1D^˚pBtψq ´ BtDpD^˚Dq^´1D^˚pix1ψq. (7.98)

Similarly

ix2πpix3ψq ´ix3πpix2ψq “ (7.99) ix3DpD^˚Dq^´1D^˚pix2ψq ´ix2DpD^˚Dq^´1D^˚pix3ψq. (7.100) By a calculation, we see that

∇^Apix1ψq “dpix1q ^ψ`ix1∇^Aψ (7.101) hence

DApix1ψq “icl1ψ`ix1DAψ, (7.102) so that

rDA,t,ix_js “ⁱclj, (7.103) and similarly

rDA,t,Bts “i. (7.104) Hence

ix₁DpD^˚Dq^´1D^˚pBtψq “ ´ⁱcl1pD^˚Dq^´1D^˚pBtψq mod imD. (7.105) Applying this to all leaves us modulo the image ofD with

`ipD^˚Dq^´1D^˚pix1ψq ´icl1pD^˚Dq^´1D^˚pBtψq (7.106)

`icl2pD^˚Dq^´1D^˚pix3ψq ´icl3pD^˚Dq^´1D^˚pix2ψq. (7.107) Then using the commutators with the adjoint

rD_A,t^˚ ,Bts “ ´i rD_A,t^˚ ,ixjs “iclj, (7.108) we can permuteD^˚ with the arguments to get by using thatψPkerD^˚,

´ pD^˚Dq^´1cl1ψ´cl1pD^˚Dq^´1ψ (7.109)

´cl2pD^˚Dq^´1cl3ψ`cl3pD^˚Dq^´1cl2ψ. (7.110) Note that we havenotused thatpA,φqsatisfy the Bogomolny equations yet. Only now do we need the fact, that

pD^˚Dq^´1“1Sb^G, (7.111) whereGis the Greens Operator of the right rand side of the Weitzenb ¨ock formula7.13 for vanishing error η. The import part is that pD^˚Dq^´1 commutes with the Clifford multiplication, so that

p´cl1´cl1´cl2cl3`cl3cl2qpD^˚Dq^´1ψ“0, (7.112) because cl2cl3“ ´cl1and cl3cl2“cl1. This shows the first Nahm equation. To prove the others, cyclically permute1,2and3in the proof above.

Corollary7.26. For a generalized Bogomolny datumpA,φqwith error term satisfyingkclηkă kG0k(in the operatornorm fromW^1,2toW^1,2), the transformed error term is given by

ψÞÑπ˝ G0is the Greens operator of

1_Hb p∇^Aq^˚∇^A´ p^φ´^it1q² (7.115) and for a sectionψPΓpI,Vq.

Proof. The proof is identical to the Nahm case.

Corollary7.27. IfkclηkăkG0k^´1 (in the operatornorm fromW^1,2 toW^1,2), then the trans-formed error termη_Nahmis bounded onI.

Proof. This follows from the same argument as in the Nahm case,kG₀ψkěCt²kψkso

thatkG0kď _Ct¹2, henceEis bounded.

The idea of the proof of the boundary conditions is identical for both directions. First an approximation operator is introduced that has the same boundary behavior as the Dirac operator. Then it is shown that the kernel of this new operator is close to the original one, in the sense that the boundary behavior of the elements in the kernel is sufficiently the same. Finally explicit solutions in the kernel of the approximating operator are given and it is shown that those have the correct boundary behavior.

We have to modify these proofs in two steps. We define the same approximating operator but have to show that the additional term clη in the Weitzenb ¨ock formulas still allow to show that the elements in the kernel are close. For this only need the asymptotic of the Weitzenb ¨ock formulas, which is unchanged by the modifications.

This is justified in equations (7.126) and (7.147).

The second step is that we need to make sure that the solutions of the approxi-mated operator still have the correct boundary behavior (this uses satisfaction of the equations). Again it turns out that we merely need an asymptotic satisfaction of the equations, i.e. the transformed error has to beηPOp^r´2qin the Bogomolny equations andηPOp_t`1¹ qfortÑ ´1andηPOp_t´1¹ q fortÑ1in the Nahm equations.

Since these poofs use a vast amount of details, we will not reproduce them com-pletely here but rather give a overview of the important steps and show where adjust-ments have to be made.

7.4.1 Nahm to Bogomolny

We closely follow [Hit83] in this section.

Theorem7.29. Letp^T0,T₁,T₂,T₃qbe an approximate solution to the Nahm equations of charge kPN, so that the transformed error term satisfiesη˜ POp^r´2q. Then the transformed datum pA,φqsatisfies the boundary conditionsBCk.

Proof. We useCas a generic constant in this proof. AsSUp2q-representations we have the following decomposition. decomposes the representations^Skand^Sk´2 into one-dimensional weight spaces,

S^k“Vk‘Vk´2‘ ¨ ¨ ¨ ‘V´k (7.122) S^k´2“^Vk´2‘^Vk´4‘ ¨ ¨ ¨ ‘^V´k`2 (7.123)

Now letv_˘be the highest (lowest) weight ofS^k, which implies that it is fixed inHbV the Weitzenb ¨ock formula implies that forkxkěr,

hDxψ,Dxψiě p^r2´^kclηkq^kψk2ě^Cr2kψk2, (7.126) approximated to orderr^´1by elements in the kernel ofD₀^˚.

[Hit83] now goes on to shows that the solutions of D0^˚, which can be expressed byg_˘, satisfy the boundary conditions which implies the satisfaction of the boundary conditions of the elements in the kernel of D^˚. Notice that our D0 is identical¹ to the one in [Hit83]. The only part where the equations come into play is shortly after equation [Hit83, Equation (2.13)], wherekF^AkPOp^r´2qneeds to implykdAφkPOp^r´2q, which is true because of the assumption ˜ηPOp^r´2qon the transformed error term.

Corollary7.30. IfkclηkăkG0k^´1andDx^˚Dx is a positive operator, then the extended Nahm transform from Nahm to Bogomolny is well defined.

Proof. This follows by Theorem7.29and Corollary7.21.

1Dxis called ∆pxq in Hitchin’s notation, Dx⁰ is denoted by ∆₀pxq and defined shortly after [Hit83, Theorem2.8]. The relation between our definitions isiDx“∆pxq. Note also that Hitchin defines∆pxqfor xPR⁴, but restricts it to^xPR³after [Hit83, Equation (2.10)]. Finally note that Hitchin uses the interval I“ p0,2q.

7.4.2 Bogomolny to Nahm

We follow closely the description in Nakajima’s paper [Nak91]. Similarly to this work Cis used as a generic constant.

Theorem7.31. LetpA,φqbe an approximate solution to the Nahm equations of chargekPN and let the transformed error term satisfyη˜ POp_t`1¹ qfort Ñ ´1andη˜ POp_t´1¹ qfor tÑ1. Then the transformed datum satisfies the boundary conditionsNCk.

Proof. Define

DA,t“DA` pφ´itq: ΓpSbEq ÑΓpSbEq (7.131) and use the index theorem of Callias (see Theorem7.17) to see that this operator has indexk. The Weitzenb ¨ock formula shows that

D_A,t^˚ DA,t“∇^˚∇´1Sb pφ´it1q²´clη, (7.132) which is a positive operator ifkclηkis small enough in the operator norm fromW^1,2to L². Hence kerL²DA^˚,t isk-dimensional. This defines a subbundle of the trivial vector bundle

VĂ^L2pR³,Sb^Eq Ñ^I (7.133) by letting Vt “ ker_L2D_A,t^˚ . Nakajima shows [Nak91] that the boundary conditions assure that we can find an assymptotic gauge that the bundle E decomposes into L‘L^˚ the sum of two line bundles (the eigenspaces of φ for eigenvalues i and ´i) and using this for the operator implies that

DA^˚,t “ ^{B1 0}

is isomorphic to the Spin bundle ofS², which decomposes intoS^` andS^´ andD_A^˘₈ are the Dirac operators of S² twisted by A8. A8 is the connection induced by A on the line bundleL. The matrix acts on the decomposition

SbE“L^˚bS^{`</sup>‘L<sup>˚</sup>bS<sup>´</sup>‘LbS<sup>`}‘L‘S^´ (7.139)

“Op´^k´1q ‘Op´^k`1q ‘Op<sup>k</sup>´1q ‘Op<sup>k</sup>`1q. (7.140) This time we won’t be able to give a explicit basis in kerpD_A,t⁰ q^˚, but we can give a approximation in the following sense. Focusing on the boundary conditions attÑ ´1, we define ak-dimensional space of functions close to kerpD_A,t⁰ q^˚ by

ψpr,θq “χprqe^´pt`1qrr^k´22 forCindependent ofRandt. In this sense it is an approximate solution to kerppD_A,t⁰ q^˚q for largeR. A solutionϕof the equation

D_A,t^˚ DA,tϕ“1Sb

∇^˚_A∇A´(φ´it)²´clη

ϕ“D_A,t^{˚ ψ} (7.143) then gives an element in kerD_A,t^˚ as ψ´DA,tϕ. We will show, that the boundary behavior of this element is dominated by ψ, but first we argue that such a ϕ exists (uniquely). The equation

1Sb

∇^˚_A∇_A´(φ´it)²´clη

ϕ´D_A,t^{˚ ψ}“0 (7.144) is the Euler Lagrange equation of the functional

Spϕq “ ¹

2k∇Aϕk²_L2`¹

2kpφ´itqϕk²_L2´hϕ, clηϕi_L2´ϕ,D_A,t^{˚ ψ}_L2. (7.145) This is a strictly convex function, which is also differentiable and coercive (see [JT80, Proposition IV.4.1]). By methods of variational calculus (see e.g. [JT80, Proposition VI.8.5]), it has a unique minimumSpϕq ďSp0q “0. Therefore

kDA,tϕk²_L2 “k∇Aϕk²_L2`kpφ´itqϕk²_L2´2hϕ, clηϕi_L2 ď2

ϕ,D_A,t^{˚ ψ}_L2. (7.146)

For t sufficiently close to ´1, we can estimate all these terms from below against the L²norm ofϕ, i.e.

kp1`tqϕk<sup>2</sup><sub>L</sub>2 ďC k∇Aϕk<sup>2</sup><sub>L</sub>2`kpφ´itqϕk²_L2´2hϕ, clηϕi

“CkDA,tϕk²_L2, (7.147) for details on how to estimate the first two terms see [Nak91, Equations (2.4) and (2.5)]. But then we can restricttto a smaller neighborhood of´¹, so that it is satisfied also with our error term hclηϕ,ϕi ď kclηkkϕk² (independent of ϕ). Once we have established this equations, we can apply this to equation (7.146), to get

kDA,tϕk⁴_L2 correct boundary conditions. This is shown in [Nak91, Lemma2.6] where Nakajima shows that the action of Ti on fPH⁰pCP¹,Op^k´1qqis a multiple of a k-dimensional irreducible sup²q representation. The multiplicity is then fixed by satisfaction of the Nahm equations, where we need that the transformed error ˜ηis inOp_t´1¹ q fort Ñ 1 Nahm transform form Bogomolny to Nahm is well defined.

Proof. This follows by Theorem7.31and Corollary7.26.

Im Dokument Generalized Seiberg-Witten and the Nahm Transform (Seite 103-113)