Диссертация (1137342), страница 22
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It might be interesting to mention the appearance of a possibly related structure in the study of topological stringpartition functions [GHM, BGT].Rank two caseFor N = 2, the elementary 3-point RHPs can be solved in terms of Gauss hypergeometric functions so that Fredholm determinant representation (4.44) becomes completely explicit. Being rewritten in Fourier components, the blocks of K may bereduced to single infinite Cauchy matrices acting in `2 (Z).
We are going to use this~~ k−1Y,Qobservation to calculate the building blocks Z Y~k−1T [k] of principal minors of K~,Qkkin terms of monodromy data, and derive thereby a multivariate series representationfor the isomonodromic tau function of the Garnier system.984.4. Rank two caseGauss and Cauchy in rank 2The form of the Fuchsian system (4.14) is preserved by the following non-constantscalar gauge transformation of the fundamental solution and coefficient matrices:Φ (z) 7→ Φ̂ (z)n−2Y(z − al )κl ,l=0Al 7→ Âl + κl 1,l = 0, .
. . , n − 2.Under this transformation, the monodromy matrices Ml are multiplied by e−2πiκl , andthe associated Jimbo-Miwa-Ueno tau function transforms asYτJMU (a) 7→ τ̂JMU (a)(al − ak )−N κk κl +κk Tr Θl +κl Tr Θk .0≤k<l≤n−2The freedom in the choice of κ0 , . . . , κn−2 allows to make the following assumption.Assumption 4.25.
One of the eigenvalues of each of the matrices Θ0 , . . . , Θn−2 isequal to 0.This involves no loss in generality and means in particular that the ranks r[k] of the[k]coefficient matrices A1 in the auxiliary 3-point Fuchsian systems (4.16) are at mostN − 1.I,Jk−1[k]For r[k] = 1, the factor Z Ik−1Tin (4.67) can be computed in explicit form.,Jk kIn this case the sums such as (4.59) or (4.61) contain only one term, and the index r[k]can therefore be omitted. The matrix A1 ∈ CN ×N may be written as[k]ak A1 = −u[k] ⊗ v [k] .The crucial observation is that the blocks (4.61) are now given by single Cauchymatrices conjugated by diagonal factors (instead of being a sum of such matrices). In[k] order to put this to a good use, let us introduce two complex sequences xı ı∈I tJ ,k−1k[k] y ∈J tI of the same finite length |Ik−1 | + |Ik |.
Their elements are defined byk−1kshifted particle/hole positions:(p + σk−1,α ,ı ≡ (p, α) ∈ Ik−1 ,x[k](4.69a)ı :=−p + σk,α ,ı ≡ (−p, α) ∈ Jk ,(−q + σk−1,β , ≡ (−q, β) ∈ Jk−1 ,y[k] :=(4.69b)q + σk,β , ≡ (q, β) ∈ Ik .I,Jk−1[k]Lemma 4.26. If r[k] = 1, then Z Ik−1Tcan be written as,Jk kYYYYI,Jk−1[k][k] p;α[k][k] −p;α[k]Z Ik−1T=±ψψ̄ϕϕ̄×k ,Jkp;α−p;α(p,α)∈Ik−1(−p,α)∈Jk−1Y×x[k]ı−x[k](−p,α)∈JkYı,∈Ik−1 tJk ;ı<y[k]ı,∈Jk−1 tIk ;ı<YY[k]x[k]ı − y(p,α)∈Ik−yı[k].ı∈Ik−1 tJk ∈Jk−1 tIk(4.70)994. Fredholm determinant and Nekrasov sum representations of isomonodromic tau functionsProof.
The diagonal factors in (4.61) produce the first line of (4.70). It remains tocompute the determinant det (p,α)∈Ik−11p + σk−1,α + q − σk−1,β1−p + σk,α + q − σk−1,β(−p,α)∈Jk(p,α)∈Ik−11p + σk−1,α − q − σk,β (q,β)∈Ik(−p,α)∈Jk1−p + σk,α − q − σk,β (q,β)∈Ik(−q,β)∈Jk−1(−q,β)∈Jk−1,(4.71)which already includes the sign (−1) in (4.63a). The ± sign in (4.70) depends onthe ordering of rows and columns of the determinant (4.63a).
This ambiguity doesnot play any role as the relevant sign appears twice in the full product (4.64).On the other hand, the notation introduced above allows to rewrite (4.71) as a(|Ik−1 | + |Ik |) × (|Ik−1 | + |Ik |) Cauchy determinant!ı∈Ik−1 tJk1,det[k][k]xı − y ∈Jk−1 tIk|Ik |and the factorized expression (4.70) easily follows.[1]We now restrict ourselves to the case N = 2, where the condition r = . . . =[n−2]r= 1 does not lead to restrictions on monodromy. Let us start by preparing asuitable notation.• The color indices will take values in the set {+, −} and will be denoted by , 0 .[k]• Recall that the spectrum of A1 coincides with that of Θk . According to Assumption 4.25, the diagonal matrix Θk has a zero eigenvalue for k = 0, .
. . , n − 2.Its second eigenvalue will be denoted by −2θk . Obviously, there is a relation2θk ak = v [k] · u[k] ,k = 1, . . . , n − 2,where v · u = v+ u+ + v− u− is the standard bilinear form on C2 . The eigenvaluesof the remaining local monodromy exponent Θn−1 may be parameterized asθn−1, =n−2Xθk + θn−1 , = ±.k=0[k][k][k][k]• Also, the spectra of A0 and A∞ = −AP0 −A1 coincide with the spectra of Sk−1and −Sk . Since furthermore Tr Sk = kj=0 Tr Θj , we may write the eigenvaluesof Sk asσk, = −kX = ±,θj + σk ,j=0where σ0 ≡ θ0 and σn−2 ≡ −θn−1 .100k = 0, . . . , n − 2,(4.72)4.4.
Rank two caseThe non-resonancy of monodromy exponents and Assumption 4.3 imply that2θk ∈/ Z\ {0} ,11|<σk | ≤ , σk 6= ± ,22k = 0, . . . , n − 1,k = 1, . . . , n − 3.To simplify the exposition, we add to this extra genericity conditions.Assumption 4.27. For k = 1, . . . , n − 2, we haveσk−1 + σk ± θk ∈/ Z,σk−1 − σk ± θk ∈/ Z.It is also assumed that σk 6= 0 for k = 0, .
. . , n − 2.Let us introduce the spacenMΘ = [M0 , . . . , Mn−1 ] ∈ (GL (N, C)) / ∼ M0 . . . Mn−1 = 1, Mk ∈ [e2πiΘk ] for k = 0, . . . , n − 1of conjugacy classes of monodromy representations of the fundamental group withfixed local exponents. The parameters σ1 , . . . , σn−3 are associated to annuli A1 , . . . , An−3and provide n−3 local coordinates on MΘ (that is, exactly one half of dim MΘ = 2n − 6).The remaining n − 3 coordinates will be defined below.Our task is now to find the 3-point solution Ψ[k] explicitly. The freedom in thechoiceits normalization allows to pick any representative in the conjugacy class [k] of[k] A0 , A1 for the construction of the 3-point Fuchsian system (4.16).
An importantfeature of the N = 2 case is that this conjugacy class is completely fixed by localmonodromy exponents Sk−1 , Θk and −Sk . We can set in particular[k]A0 = diag {σk−1,+ , σk−1,− } ,[k]ak A1 = −u[k] ⊗ v [k] ,with σk−1,± parameterized as in (4.72) and[k]u± =(σk−1 ± θk )2 − σk2ak ,2σk−1[k]v± = ±1.As in Subsection 4.2.5, one may first construct the solution Φ̃[k] of the rescaledsystem![k][k]A0A∂z Φ̃[k] = Φ̃[k]+ 1,(4.73)zz−1having the same monodromy around 0, 1, ∞ as the solution Φ[k] of the originalsystem (4.16) has around 0, ak and ∞. To write it explicitly in terms of the Gausshypergeometric function 2 F1 a,c b ; z , we introduce a convenient notation,θ1 + θ2 + θ3 , θ1 + θ2 − θ3χ; z := 2 F1;z ,θ1 θ32θ1θ32 − (θ1 + θ2 )21 + θ1 + θ2 + θ3 , 1 + θ1 + θ2 − θ3θ2z 2 F1φ; z :=;z .θ1 θ32 + 2θ12θ1 (1 + 2θ1 )(4.74)θ21014.
Fredholm determinant and Nekrasov sum representations of isomonodromic tau functionsThe solution of (4.73) can then be written as[k]Φ̃[k] (z) = Sk−1 (−z)Sk−1 Ψ̃in (z) ,(4.75)where Sk−1 is a constant connection matrix encoding the monodromy (cf (4.15)), and[k]Ψ̃in is given byθk[k](z) = χΨ̃in;z ,±σk−1 σk±±(4.76)θk[k]Ψ̃in(z) = φ;z .±σk−1 σk±∓ It follows that Φ[k] (z) = Φ̃[k] azk and[k]Ψ+(z) =−Sak k−1[k]Ψ̃inzak[k]z ∈ Cin .,(4.77a)[k][k][k]Let us also note that det Φ̃[k] (z) = const · (−z)Tr A0 (1 − z)Tr A1 implies that det Ψ̃in (z) =(1 − z)−2θk , which in turn yields a simple representation for the inverse matrix [k][k]zz2θkΨ̃− Ψ̃inak−1z in −− ak Sk−1[k][k]+− Ψ+ (z) = 1 −z ∈ Cin . ak ,[k][k]zzak− Ψ̃inΨ̃inakak−+++(4.77b)The equations (4.75)–(4.76) are adapted for the description of local behavior of[k][k]Ψ (z) inside the disk around 0 bounded by the circle Cin , cf left part of Fig.
4.9. To[k][k]calculate Ψ+ (z) inside the disk around ∞ bounded by Cout , let us first rewrite (4.75)using the well-known 2 F1 transformation formulas. One can show that[k][k][k]Φ̃[k] (z) = Sk−1 C∞(−z)Sk Ψ̃out (z) G∞,where(4.78)θk−1(z) = χ;z,∓σk σk−1±±θk[k]−1Ψ̃out(z) = φ;z,∓σk σk−1±∓[k]Ψ̃out(4.79)andG[k]∞[k]C∞=1=2σk−θk + σk−1 + σk θk + σk−1 − σk−θk + σk−1 − σk θk + σk−1 + σkΓ (2σk−1 ) Γ (1 + 2σk )Γ (1 + σk−1 + σk − θk ) Γ (σk−1 + σk + θk )−,Γ (2σk−1 ) Γ (1 − 2σk )Γ (1 + σk−1 − σk − θk ) Γ (σk−1 − σk + θk )Γ (−2σk−1 ) Γ (1 − 2σk )Γ (1 − σk−1 − σk − θk ) Γ (θk − σk−1 − σk )Γ (−2σk−1 ) Γ (1 + 2σk )−Γ (1 − σk−1 + σk − θk ) Γ (θk − σk−1 + σk )(4.80).(4.81)As a consequence,[k]Ψ+(z) =[k] −SkD∞ak[k]Ψ̃outzak102G[k]∞,[k]z ∈ Cout ,(4.82a)4.4.
Rank two caseno[k][k][k]where D∞ = diag d∞,+ , d∞,− is a diagonal matrix expressed in terms of monodromyas[k][k].= Sk−1 Sk−1 C∞D∞[k]Analogously to (4.77b), it may be shown that for z ∈ Cout−1[k]Ψ+ (z) [k]z− Ψ̃outakak 2θk [k] −1 +−−−[k] −1k = 1−D∞.G∞ aSk[k][k]zzz− Ψ̃outΨ̃outakak[k]Ψ̃out zak−+++(4.82b)We now have at our disposal all quantities that are necessary to compute theexplicit form of the integral kernels of a[k] , b[k] , c[k] , d[k] in the Fredholm determinantrepresentation (4.44) of the Jimbo-Miwa-Ueno tau function, as well as of diagonalfactors ψ [k] , ϕ[k] , ψ̄ [k] , ϕ̄[k] in the building blocks (4.70) of its combinatorial expansion(4.67).Lemma 4.28.
For N = 2, the integral kernels (4.57) can be expressed asa[k] (z, z 0 ) =1−−Sk−1ak1−[k]b0z0ak(z, z ) = −−Sak k−1akz02θk K (z) K (z) K (z 0 ) −K (z 0 ) +++−−−+−−1K−+ (z) K−− (z)−K−+ (z 0 ) K++ (z 0 )Sak k−1 ,z − z02θk[k] −1G∞K++ (z) K+− (z)K−+ (z) K−− (z)z − z00(4.83a)0K̄−− (z ) −K̄+− (z )−K̄−+ (z 0 ) K̄++ (z 0 )[k]kaSk D∞−1(4.83b)[k]c0(z, z ) =1−[k] −SkD∞ak0zak2θk 00K−− (z ) −K+− (z )K̄++ (z) K̄+− (z)[k]G∞−K−+ (z 0 ) K++ (z 0 )K̄−+ (z) K̄−− (z)z − z0Sak k−1 ,(4.83c)d[k] (z, z 0 ) =[k] −SkD∞ak1− 1−ak 2θkz0K̄++ (z) K̄+− (z)K̄−+ (z) K̄−− (z)z − z000K̄−− (z ) −K̄+− (z )−K̄−+ (z 0 ) K̄++ (z 0 )[k]kaSk D∞−1(4.83d)where we introduced a shorthand notation K (z) =matrices[k]Ψ̃in,out(z) and[k]G∞[k]Ψ̃in zak, K̄ (z) =[k]Ψ̃out zak; theare defined by (4.76), (4.78) and (4.80).Proof.
Straightforward substitution.Lemma 4.29. Under genericity assumptions on parameters formulated above, the103,,4. Fredholm determinant and Nekrasov sum representations of isomonodromic tau functionsFourier coefficients which appear in (4.60) are given byQ0Pk−110 =± (θk + σk−1 + σk )p+ 12p;j=0 θk −σk−1 −(p− 2 )[k]=ψ(−) ,akp − 21 ! (2σk−1 )p+ 12Q1(1−θ−σ+0 σk )p− 1 − Pk−1kk−10 =±j=0 θk +σk−1 −(p+ 2 )[k]2ψ̄ p; =(−) ,akp − 21 ! (1 − 2σk−1 )p− 12Q0Pk1(θ+σ−σ)10kk−1k p+ =±j=0 θk −σk +(p+ 2 ) [k][k] −p;2=ϕad∞, ,kp − 12 ! (−2σk )p+ 12Q0(1−θ+σ+σk )p− 1 − Pkj=0 θk +σk +(p− 1 )0kk−1 =±−12[k]2ϕ̄ −p; =akd[k]∞, ,1p − 2 ! (1 + 2σk )p− 1(4.84a)(4.84b)(4.84c)(4.84d)2where = ± and (c)l :=Γ (c + l)denotes the Pochhammer symbol.Γ (c)[k]Proof.
From the first equation in (4.60a), the representation (4.77a) for Ψ+ (z) on[k]Cin , and hypergeometric contiguity relations such as(c − a) (c − b)a, ba + 1, b + 1a + 1, b + 1z 2 F1; z +(z − 1) 2 F1;z ,;z =2 F1cc+2c+1c (c + 1)it follows thatXp∈Z0+ψ[k] p1z − 2 +p (θ + σ )2 − σ2z1 + θk + σk−1 + σk , 1 + θk + σk−1 − σkkk−1k;2 F11 + 2σk−12σk−1ak −S= −ak k−1 .22(θk − σk−1 ) − σk2σk−12 F1z1 + θk − σk−1 + σk , 1 + θk − σk−1 − σk;1 − 2σk−1akThis in turn implies the equation (4.84a).
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