Two-point tau function and Painlevé V

In order to write the tau function as a Fredholm determinant, we will consider free Dirac equation in another gauge. Set the potential of the uniform magnetic field to be

A^共B兲= −Bydx, 共4.30兲 so that the corresponding Dirac Hamiltonian

Hˆ

tr共0兲=

冉

⁻⳵x−i^m⳵y+iBy ^⳵x−i−^⳵ym^−iBy

冊

commutes with thex-momentum operator Pˆ

x= −i⳵x. The eigenspace of Pˆ

x with momentum p is spanned by the functionsg共p,y兲e^ipx. Let us look at the solutions of the partial Dirac equation

共Hˆ_p−E兲g共p,y兲= 0, Hˆ_p=

冉

⁻i共⳵y+^mp−By兲 ^−i共⳵y−⁻m^p+By兲

冊

^{. 共4.31兲}

As above, we assume thatEis real and兩E兩⬍m. It is convenient to choose two linearly indepen-dent solutions of共4.31兲as follows:

B⬎0: ⌽^共⫾兲共p,y兲=

冋 ^冑

²¹␲^⌫

冉

^{2B␭2 + 1}

冊 册

^1/2

冢

^{⫾C+−1iCD+−共␭D−2共␭/2B2/兲−12B兲}

^{冉 冉}

^⫾⫾

^{冑 冑}

^2B2B

^{冉 冉}

^yy−−BBpp

^{冊冊 冊冊} 冣

^{, 共4.32兲}

B⬍0: ⌽^共⫾兲共p,y兲=

冋 ^冑

^21␲⌫

^冉

^{2兩B兩␭2 + 1}

^冊 册

^1/2

冢

^{⫾CiC+−1+DD−共␭−共␭22/2兩/2兩BB兩兲兩兲−1}

^冉

^⫾

^冉

^⫾

^冑

^2兩B兩

^冑

^2兩B

^冉

^y兩

^冉

^{+y+兩B兩p兩Bp}

^冊冊

^兩

^冊冊 冣

^,

共4.33兲

B= 0: ⌽^共⫾兲共p,y兲=e^⫿冑␭²+p²y

冑

2

冢

^⫾CiC+−1

^冉

⁺

^冉

^11⫾⫿

^{冑 冑}

^{␭␭2p2+p+pp22}

^{冊 冊}

^1/21/2

冣

^{. 共4.34兲}

Here,D_␣共s兲denotes the parabolic cylinder function and the constantC₊in共4.32兲–共4.34兲is deter-mined by 共4.6兲, 共4.25兲, and 共4.28兲, correspondingly. Note that ⌽^共+兲共p,y兲关⌽^共−兲共p,y兲兴 is square integrable asy→⬁关y→−⬁兴. These two solutions satisfy symmetry relations analogous to共3.39兲

⌽^共+兲共p,y兲=␴z⌽^共−兲共−p,−y兲, ⌽^共⫾兲共p,y兲=␴z⌽^共⫾兲共p,y兲. 共4.35兲 The normalization in共4.32兲–共4.34兲was chosen so that in all three cases

det共⌽^共+兲共p,y兲,⌽^共−兲共p,y兲兲= −i. 共4.36兲 We now adapt the reasoning of Sec. III C 1 to flat space. Consider a line Ly^共0兲=兵共x,y兲 苸R²兩y=y^共0兲其and an arbitraryC²-valued functiongy共0兲共x兲苸H^1/2共Ly共0兲兲, written as Fourier integral g_y共0兲共x兲=

冕

_−⬁^{⬁ dp g共p,y共0兲兲eipx. 共4.37兲}

Decompose Fourier transformg共p,y^共0兲兲as follows:

g共p,y^共0兲兲=˜g₊共p,y^共0兲兲⌽^共+兲共p,y^共0兲兲+˜g₋共p,y^共0兲兲⌽^共−兲共p,y^共0兲兲,

where ⌽^共⫾兲共p,y^共0兲兲 denote the functions defined by 共4.32兲, 共4.33兲, or 共4.34兲, depending on the value ofB. The formula共4.36兲and symmetry relations共4.35兲imply that

˜g_⫾共p,y^共0兲兲= ⫿i共⌽^共⫿兲共p,y^共0兲兲兲^†␴xg共p,y^共0兲兲. 共4.38兲 Recall that˜g₊共p,y^共0兲兲 and˜g₋共p,y^共0兲兲can be thought of as coordinates in the spaces of boundary values of solutions of the free Dirac equation 共Hˆ

tr

共0兲−E兲␺= 0 in the half planes y⬎y^共0兲 and y

⬍y^共0兲.

It is now straightforward to write down the analogs of Propositions 3.8 and 3.9.

Proposition 4.2: Let us consider a strip S=兵共x,y兲苸R²兩y^共L兲⬍y⬍y^共R兲其. Suppose that ␺ 苸H^1/2共⳵S兲can be continued toS as a solution of the free Dirac equation共Hˆ

tr

共0兲−E兲␺^{= 0. Then,}

冉

^␺˜␺^˜R,+^L,−共共p兲^p兲

冊

⁼

^冉

^{0 11 0}

^冊 冉

^␺˜␺^˜L,+R,−^共共p兲^p兲

冊

^{. 共4.39兲}

Proposition 4.3: Assume that the strip S contains one branching point a₀ (i.e., y^共L兲⬍a_0y

⬍y^共R兲) and introduce a horizontal branch cut ᐉ=共−⬁+ia_0y,a_0x+ia_0y兴. Suppose that ␺ 苸H^1/2共⳵S兲 is the boundary value of a multivalued solution of the free Dirac equation on S\ᐉ, which is characterized by the monodromy e^2␲i␯at the point a₀. Then,

冉

^␺˜␺^˜L,−R,+^共共p兲^p兲

冊

⁼

冉

^␣␥ˆˆS^S共共a^a00兲^{兲 ␤}␦^ˆˆSS共a^共a0⁰兲^兲

冊冉

^␺˜␺^˜L,+R,−^共共p兲^p兲

冊

^,

where

共␣^ˆ_S共a0兲␺^˜L,+兲共p兲=

冕

−⬁

⬁

⌬˙

−

共a₀,␯兲共p,q兲␺^˜L,+共q兲dq, 共4.40兲

共␤^ˆ_S共a0兲␺^˜R,−兲共p兲=

冕

_−⬁^{⬁ G˙+共a0,␯兲共p,q兲␺˜R,−共q兲dq, 共4.41兲}

共␥^ˆ_S共a₀兲␺^˜L,+兲共p兲=

冕

_−⬁^{⬁ G˙−共a0,␯兲共p,q兲␺˜L,+共q兲dq, 共4.42兲}

共␦^ˆS共a0兲␺^˜R,−兲共p兲=

冕

−⬁

⬁

⌬˙

+

共a₀,␯兲共p,q兲␺^˜R,−共q兲dq, 共4.43兲

and

The definition of the tau function of the Dirac Hamiltonian on the plane with two branch pointsa₁ anda₂is also completely analogous to the disk case. Repeating the arguments of Sec.

III A, one ends up with the following Fredholm determinant representation:

␶共a兲= det共1 −␣^ˆ共a₂兲␦^ˆ共a₁兲兲, 共4.46兲 where␣^ˆ共a2兲 and␦^ˆ共a1兲 are given by 共4.40兲 and共4.43兲. As above, the fact that the tau function depends only on the distance between the pointsa₁ anda₂ allows us to choosea₁= 0,a₂=t+i0 共t苸R⁺兲, and the invariance ofHˆ

tr

共0兲with respect to x-translations reduces the problem of calcula-tion of ␶共a兲 to finding the form factors ⌬˙

⫾共0,␯兲共p,q兲. Finally, the symmetry of the free Dirac

HamiltonianHˆ

tr

共0兲 combined with the relations共4.35兲implies that

⌬˙

⫾共0,␯兲共p,q兲are determined by the relation共4.44兲or by the equivalent formula

1

Here, the first factor corresponds to a singular gauge transformation removing the AB field, and the second one comes from the change of the vector potential of the uniform magnetic field from 共4.1兲to共4.30兲.

Also note thatyandy⬘^{in共4.44兲and共4.48兲}can be chosen arbitrarily. Analogous observation in the disk case allowed us to obtain a more explicit representation for⌬˙

⫾共0,␯兲共p,q兲by analyzing the

asymptotics of a relation similar to 共4.48兲 near the disk boundary. For B⫽0 the asymptotic analysis of the left hand side of共4.48兲asy,y⬘^→⫾⬁becomes rather complicated and we have not managed to repeat the above trick in this case. However, when both p and q are positive or negative, one can chooseyandy⬘in such a way that the arguments of parabolic cylinder functions in the right hand side of one of the relations共4.48兲are equal to zero. This leads to a simpler关than 共4.48兲for general y,y⬘^兴representation of ⌬˙

⫾共0,␯兲共p,q兲.

Example:Assume thatB⬎0,p⬎0, and⌰= −␲/2. Then, setting in 共4.48兲y=p/B,y⬘^=q/B, and taking into account共4.49兲, one finds a triple integral representation for⌬˙

+ 共0,␯兲共p,q兲

⌬˙

Much simpler results can be obtained for B= 0. In this case, it is convenient to introduce instead of the momentump a rapidity variable ␪pdefined by

p=␭sinh␪p,

冑

␭²+p²=␭cosh␪p. Partial waves共4.34兲can then be written as

⌽^共⫾兲共p,y兲= e^{⫿␭ycosh␪p}

冑

2 cosh␪p

冉

⫾^CiC^+−1e₊^⫾␪e^⫿␪p/p2/2

冊

^.

Set⌰= −␲/2 and substitute the representation共4.29兲 for ⌬共z,z⬘兲 共recall that it is valid for y,y⬘

⬎0兲 into 共4.48兲 and共4.49兲. After elementary integration over x andx⬘ in the left hand side of 共4.48兲we find

It should be mentioned that the formulas equivalent to 共4.50兲and 共4.51兲 were first obtained by Schroer and Truong.²⁷ In a context similar to ours, they were rediscovered by Palmer in Ref.4.

We finally comment on the limiting form of Eq.共1.1兲in flat space. Introduces=R⁻²tand let R→⬁. Then, settingw共s兲= 1 −y共t兲 and using asymptotic behavior of the parameters␤ ^and ␦ ⁱⁿ 共1.3兲

␤⁻␦⯝R²␥⬘^, ␦⯝R⁴␦⬘^,

␥⬘^{=m2−E2+B共1 +␯1+␯2兲}

2 , ␦⬘^{= −B2}

8 , 共4.52兲

it is straightforward to check that y共t兲 satisfies Painlevé V equation d²y

Dans le document On Painlevé VI transcendents related to the Dirac operator on the hyperbolic disk (Page 35-38)