Relation to Painlevé VI

Let us briefly describe the relation of the present paper to the PBT work.¹⁴ Recall that the HamiltonianHˆ共0兲 of a Dirac particle in the absence of the AB fluxes is given by the formula共2.3兲 with

K=R⁻¹关2共1 −兩z兩²兲⳵z+共1 + 2b兲z¯兴, K^ⴱ= −R⁻¹关2共1 −兩z兩²兲⳵¯z+共1 − 2b兲z兴.

Consider the operator

Aˆ=U共Hˆ共0兲−E兲U␴z, U= diag

冉冉

^mm+−EE

冊

^1/4,

冉

^mm+−EE

冊

^−1/4

冊

^.

It is straightforward to check thatAˆ coincides with the operatorm−Dkstudied by PBT关see, e.g., the formulas共1.14兲–共1.16兲in Ref.14兴if we identify

m_PBT=

冑

m²−E², k_PBT= −b.

In the presence of branch points, one should only replaceHˆ共0兲in the definition ofAˆ by the operator Hˆ共a,␯兲, introduced in the beginning of this section 共recall that Hˆ共a,␯兲 is obtained from the Dirac Hamiltonian with AB field by a singular gauge transformation兲. Thus there is a unique correspon-dence between the multivalued solutions of the Dirac equation considered in Ref. 14 and the solutions of共Hˆ共a,␯兲−E兲␺= 0. Using this correspondence, we now reformulate the key steps of the PBT analysis in the context of the present work.

1. Symmetries and elementary solutions

The Hamiltonian Hˆ共0兲 transforms covariantly under the action of the isometry group of the Poincaré disk. In particular, if ␺共z兲 satisfies the equation 共Hˆ共0兲−E兲␺= 0, then for any g=

共

^␣_␤¯␤

␣

¯

兲

苸SU共1 , 1兲 the function␺g共z兲defined by

␺g共z兲=

冉

^{v共g,z兲0共1−2b兲/2} v共g,z兲^{−0关共1+2b兲/2兴}

冊

^{␺共zg−1共z兲兲, v共g,z兲=␣¯}_␣⁻₋^␤_{␤¯ z}^{¯z, 共3.69兲}

is a solution of the same equation. A basis in the Lie algebra of the corresponding complexified infinitesimal symmetries can be chosen in the following way:

M₊= −z²⳵z+⳵¯z+bz− z 2␴z,

M₋=⳵z−¯z²⳵¯z−bz¯+¯z 2␴z, M₃=z⳵z−¯z⳵¯z−b+ ¹₂␴z=Lˆ−b. These generators satisfysl共2兲commutation relations

关M3,M_⫾兴= ⫾M_⫾, 关M+,M₋兴= 2M₃

and, therefore, one may introduce two families of solutions兵w_l其,兵w_l^ⴱ其of the Dirac equation onD with one branch point atz= 0, on which the symmetries act as follows:

M₃wl=共l−b兲wl, M₃w_l^ⴱ= −共l+b兲wlⴱ, M₊wl=m₊共l,b兲wl+1, M₊w_l^ⴱ=w_l−1^ⴱ ,

M₋w_l=w_l−1, M₋w_l^ⴱ=m₋共l,b兲w_l+1^ⴱ . 共3.70兲 These relations fix兵wl其and兵wlⴱ其up to overall normalization. Since兵wl其and兵wlⴱ其diagonalize both the Hamiltonian and angular momentum, they are related to the radial wave functions introduced in Sec. II. It is convenient to choose

wl共z兲=e^−i␲l elemen-tary solutions with one branch point atz=a,

w_l共z,a兲=V共T关a兴,z兲w_l共T关−a兴z兲, w_l^ⴱ共z,a兲=V共T关a兴,z兲w_l^ⴱ共T关−a兴z兲, where

T关−a兴z= z−a

1 −¯za , V共T关a兴,z兲= diag

冉冉

^{1 −1 −¯zaza¯}

冊

^{共1−2b兲/2,}

冉

^{1 −1 −¯zaza¯}

冊

^{−关共1+2b兲/2兴}

冊

^.

2. Local expansions and deformation equations

Now consider multivalued solutions of the Dirac equation, which are branched at two points a₁,a₂苸D with fixed monodromies e^2␲i␯1,2. In a sufficiently small finite neighborhood of each branch point, any such solution can be represented by an expansion of the form

␺关aj兴=_n

兺

mono-dromy which are square integrable共with the measured␮兲as 兩z兩→1.

• Wj共z,^˜␯兲andW_j^ⴱ共z,^˜␯兲have local expansions of the form uniquely. Therefore, the expansions共3.71兲and共3.72兲can be thought of as defining the coefficients a_n,j^k ,b_n,j^k ,c_n,j^k , and d_n,j^k as functions ofa and␯^˜.

The lowest order coefficients satisfy a set of deformation equations ina共see Theorem 5.0 in Ref.14兲. If we introduce the 2⫻2 matrices with the elementsAjk=aj␦jk,A¯_jk=¯a_j␦jk,⌳jk=^˜␯j␦jk, and also

共a1兲jk=a_1/2,j^k /共1 −兩ak兩²兲, e=⌳−b1+关a1,A兴, 共b1兲jk=b_1/2,j^k /共1 −兩ak兩²兲, f=Ab₁A¯−b₁,

共c1兲jk=c_1/2,j^k /共1 −兩ak兩²兲, g=c₁−A¯c₁A,

共d₁兲jk=d₁^k_/2,j/共1 −兩ak兩²兲, h=⌳−b1−关d₁,A¯兴, 共3.73兲 these equations are given by

de=fG+Fg+关E,e兴, 共3.74兲

df=eF+Fh+Ef−fH, 共3.75兲

dg=hG+Ge+Hg−gE, 共3.76兲

dh=gF+Gf+关H,h兴, 共3.77兲

where

E=共⌳−共b+ 1/2兲1兲AdA¯ +A¯ dA

1 −兩A兩² +关dA,a₁兴,

F=dAb₁A¯+Ab₁dA¯, G=dA¯c₁A+A¯c₁dA,

H= −共⌳−共b− 1/2兲1兲AdA¯+A¯ dA

1 −兩A兩² +关dA¯,d₁兴.

One also has symmetry relations

ef−fh=ge−hg= 0, 共3.78兲

e²−fg=h²−gf=␮²1, 共3.79兲

关esin␲⌳共1 −兩A兩²兲兴^†=hsin␲⌳共1 −兩A兩²兲, 共3.80兲 关f sin␲⌳共1 −兩A兩²兲兴^†=fsin␲⌳共1 −兩A兩²兲, 共3.81兲 关gsin␲⌳共1 −兩A兩²兲兴^†=gsin␲⌳共1 −兩A兩²兲. 共3.82兲 In addition, the diagonal elementsa_1/2,j^j andd_1/2,j^j may be expressed as follows:

a_1/2,j^j =¯a_jm₊共␯^˜j− 1/2,b兲+

兺

_k

⫽j

ejka_1/2,k^j +

兺

_k ^{fjk¯akc1/2,kj , 共3.83兲}

d₁^j_/2,j=a_jm₋共−^˜␯j− 1/2,b兲−_k

兺

_⫽j^{hjkd1j/2,k−}

兺

_k ^{gjkakb1j/2,k. 共3.84兲}

It is known that the system共3.74兲–共3.77兲combined with the relations 共3.78兲–共3.82兲can be inte-grated in terms of a Painlevé VI transcendent. In particular, if we choosea₁= 0,a₂=

冑

s, and set

f₁₂f₂₁ f₁₁f₂₂=1 −w

1 −s, 共3.85兲

then for^˜␯1,2⬎0 the functionw共s兲satisfies PVI Eq. 共1.1兲with parameters共1.3兲共for the details of the proof, see Ref.14兲.

3. Tau function

The link between the deformation equations and the tau function considered above is provided by a formula for the derivative of the Green’s function ofHˆ共a,␯兲,

共1 −兩aj兩²兲⳵a_jG¨共a,␯兲共z,z⬘^{兲= 1} 2Rsin␲␯^˜j

W_j共z,^˜␯兲丢W_j^ⴱ共z⬘^,˜␯兲^†, 共3.86兲

共1 −兩aj兩²兲⳵a¯

jG¨共a,␯兲共z,z⬘兲= 1 2Rsin␲␯^˜j

W^ⴱ_j共z,^˜␯兲丢Wj共z⬘^,˜␯兲^†. 共3.87兲 Remarkable factorized form of these expressions follows from the fact that Green’s function G¨共a,␯兲共z,z⬘^兲 inverts the operator Hˆ共a,␯兲−E, and from the analysis of the local expansions of G¨共a,␯兲共z,z⬘兲 near the branch points. Numerical factors in 共3.86兲 and 共3.87兲 may be determined using a variant of Stokes theorem calculations from Sec. III A. Different boundary conditions for Hˆ共a,␯兲 are encoded into the choice of 兵␯^˜j其: the value ^˜␯j=␯j+ 1 共or ^˜␯j=␯j兲 corresponds to the functions whose upper共lower兲component is regular ataj.

Using 共3.86兲 and 共3.87兲 one can show 共see Theorem 6.3 in Ref. 14兲 that the logarithmic derivative of the␶-function 共3.6兲 may be written in terms of the lowest order expansion coeffi-cients ofWj共z,^˜␯兲andW^ⴱ_j共z,^˜␯兲,

dln␶共a,a⁰兲=

兺

j

1

1 −兩aj兩²兵a_1/2,j^j daj+d_1/2,j^j da¯_j其. 共3.88兲 Specializing this formula to the casea₁= 0,a₂=

冑

s, and^˜␯1,2⬎0 and using共3.83兲and共3.84兲, PBT have obtained the relation共1.2兲between the corresponding ␶-function and the Painlevé VI tran-scendent defined by 共3.85兲. Analogous result holds for arbitrary branch point positions, as the

␶-function actually depends only on the geodesic distance betweena₁anda₂. The proof of the last statement is given in the Appendix.

Dans le document On Painlevé VI transcendents related to the Dirac operator on the hyperbolic disk (Page 25-28)