Диссертация (1137342), страница 24
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Theexpression (4.101a) for M1 is then most easily deduced from the diagonal form of theproduct M1 M∞ = e−2πiS .Remark 4.33. The 2 F1 kernel is related to the so-called ZW -measures [BO05] arising inthe representation theory of the infinite-dimensional unitary group U (∞). It producesvarious other classical integrable kernels (such as sine and Whittaker) as limiting cases.The first part of Theorem 4.32, namely the Painlevé VI equation for D (t), was provedby Borodin and Deift in [BD]. Monodromy data for the associated Fuchsian system1094. Fredholm determinant and Nekrasov sum representations of isomonodromic tau functionshave been identified in [Lis]. To facilitate the comparison, let us note that indroducinginstead of λ and κ a new parameter σ̄ defined byQsin π (σ̄ − θ1 ) sin π (σ̄ + θ1 ) ,0 =± Γ (1 + σ + θ1 + 0 θ∞ ),λ=π2Γ (1 + 2σ) Γ (2 + 2σ)we have in particular that Tr M∞ M0 = 2e−2πi(σ+θ∞ ) cos 2πσ̄ and Tr Mt M1 = 2e2πi(σ+θ∞ ) cos 2πσ̄.The relation between parameters z, z 0 , w, w0 of [BD] and ours is(z, z 0 , w, w0 )[BD] = (σ̄ + θ1 , σ̄ − θ1 , σ − σ̄ + θ∞ , σ − σ̄ − θ∞ ) .AppendixRelation to Nekrasov functionsHere we demonstrate that the formula (4.88) can be rewritten in terms of Nekrasovfunctions.
This rewrite is conceptually important for identification of isomonodromictau functions with dual partition functions of quiver gauge theories [NO]. It is alsouseful from a computational point of view: naively, the formula (4.88) may producepoles in the tau function expansion coefficients when θk ± σk ±0 σk−1 ∈ Z. Ourcalculation shows that these poles actually cancel.The statement we are going to prove3 is the relation~00~ 0 ,m0 )+lsgn(Y~ ,m),mZY~Y ,m(T ) = (−1)lsgn(Y~ 0 ~0ẐY~Y ,Q,~Q (T ) ,(4.104)where~ 0 ~0ẐY~Y ,Q,~Q (TQNC (σα0 − σβ | Q0α , Qβ )) = QN×000 0α<β C σα − σ β Qα , Qβ C σα − σβ |Qα , Qβα,β0~0~eiδ~η ·Q +iδ~η·Q×Q (4.105) 1 1 ×N Yα , Yα 2 Z bif 0Yα0 , Yα0 2Z0bifαQN000α,β Z bif (σα + Qα − σβ − Qβ | Yα , Yβ ).× QNZ bif σα0 + Q0α − σ 0 − Q0 Yα0 , Y 0 Z bif (σα + Qα − σβ − Qβ | Yα , Yβ )α<ββββThe notation used in these formulas means the following:~ = (m, −m), Q~ 0 = (m0 , −m0 ), though the right side of (4.105) is defined even• Qwithout this specialization.• Y 0 and Y are identified, respectively, with Yk−1 and Yk in (4.88).
Similar con0ventions will be used for all other quantities. We denote, however, σ±= ±σk−1[k]and σ± = −θk ± σk ; T stands for T .3In the present chapter we do it only for N = 2 but the generalization is relatively straightforward.1104.5. Relation to Nekrasov functions• lsgn Y~ , m ∈ Z/2Z means the “logarithmic sign”, XX11~+p−,i +.lsgn Y , m := |q+ | · |p+ | +q+,i +22ii(4.106)Here, for example, |p+ | denotes the number of coordinates p+,i of particles in theMaya diagram corresponding to the charged partition (Y+ , m).
The logarithmicQ~k−1 ,mk−1Ywhich appears in the representationsigns cancel in the product n−2k=1 Z Y~k ,mk(4.87) for the Garnier tau function.• δ~η and δ~η 0 are some explicit functions which are computed below. They justshift Fourier transformation parameters and their relevant combinations areexplicitly given by00eiδη+ −iδη− =eiδη+ −iδη−=1 (θk + σk−1 )2 − σk2,2σk−1 (θk − σk−1 )2 − σk2(4.107)22−1 (θk + σk ) − σk−1.22 σk (θk − σk )2 − σk−1• Z bif (ν|Y 0 , Y ) is the Nekrasov bifundamental contributionYYZ bif (ν|Y 0 , Y ) :=ν + 1 + aY 0 () + lY ()ν − 1 − aY () − lY 0 () .∈Y 0∈Y(4.108)12In particular, we have |Z bif (0|Y, Y )| =Q∈YhY ().• The three-point function C (ν|Q0 , Q) is defined byC (ν|Q0 , Q) ≡ C (ν|Q0 − Q) =G (1 + ν + Q0 − Q)0G (1 + ν) Γ (1 + ν)Q −Q,(4.109)where G (x) is the Barnes G-function. The only property of this function essential for our purposes is the recurrence relation G (x + 1) = Γ (x) G (x).• Using the formula (4.88), we assume a concrete ordering: p0+ , p0− , q+ , q− , p1 >p2 > .
. ., and in (4.105) we suppose that + < −.An important feature of the product (4.105) is that the combinatorial part in the 2ndline depends only on combinations such as σα + Qα , σα0 + Q0α . This is most crucial forthe Fourier transform structure of the full aswer for the tau function τGarnier (a).Let us now present the plan of the proof, which will be divided into several selfcontained parts. Most computations will be done up to an overall sign, and sometimeswe will omit to indicate this. In the end we will consider the limit θk → +∞,σk , σk−1 θk , σk , σk−1 → +∞ to recover the correct sign.1. First we will rewrite the formula (4.88) as0 ~0~ 0 ,m0~ ˆ~ 0 ~0ZY~Y ,m(T ) = ±eiδ1 η~ ·Q +iδ1 η~·Q ẐY~Y ,Q,~Q (T ) ,1114.
Fredholm determinant and Nekrasov sum representations of isomonodromic tau functionsˆ~ 0 ~0where ẐY~Y ,Q,~Q (T ) is expressed in terms of yet another function Z̃ bif ν Q0 , Y 0 ; Q, Y ,NY 1 1 ˆ Y~ 0 ,Q~ 0Z̃bif 0Qα , Yα , Qα , Yα − 2 Z̃bif 0Q0α , Yα0 , Q0α , Yα0 − 2 ×ẐY~ ,Q~ (T ) =αQN× QNα<βZ̃bif σα0 − σβ Q0α , Yα0 ; Qβ , Yβ,σα0 − σβ0 Q0α , Yα0 ; Q0β , Yβ0 Z̃ bif σα − σβ Qα , Yα ; Qβ , Yβα,βZ̃ bif(4.110)which is defined asYYY YZ̃ bif ν Q0 , Y 0 ; Q, Y =(−ν)q0 + 1(ν + 1)qi − 1(−ν)pi + 1(ν + 1)p0 − 1 ×i22i2iiQiQ00(ν + pi + qj )i,j (ν − qi − pj )Q i,j×Q.00i,j (ν − qi + qj )i,j (ν + pi − pj )2i(4.111)2. At the second step, we prove that Z̃ bif ν 0, Y 0 ; 0, Y ≡ Z̃ bif ν Y 0 , Y = ±Z bif ν Y 0 , Y .3.
Next it will be shown thatZ̃ bif ν Q0 , Y 0 ; Q, Y = C ν Q0 , Q Z bif ν + Q0 − QY 0 , Y .(4.112)4. Finally, we check the overall sign and compute extra contribution to ~η to absorbit.A realization of this plan is presented below.Step 1It is useful to decompose the product (4.88) into two different parts: a “diagonal”one, containing the products of functions of one particle/hole coordinate, and a “nondiagonal” part containing the products of pairwise sums/differences. Careful comparison of the formulas (4.88) and (4.110) shows that their non-diagonal parts actuallycoincide. Further analysis of (4.110) shows that its diagonal part is given byYYYYϕq,ϕ̄p, .ψp0 ,ψ̄q0 ,(p0 ,)∈I 0(−q 0 ,)∈J 0(−q,)∈J(p,)∈Iwith(1 + σk−1 + θk − σk )p0 − 1 (1 + σk−1 + θk + σk )p0 − 122,ψp0 , = = + : (1 + 2σk−1 )p0 − 1 ; = − : (−2σk−1 )p0 + 1 p0 − 12 !22ψ̄q0 ,(−σk−1 − θk + σk )q0 + 1 (−σk−1 − θk − σk )q0 + 12 2,= = + : (−2σk−1 )q0 + 1 ; = − : (1 + 2σk−1 )q0 − 1 q 0 − 12 !ϕq,(σk−1 + θk − σk + 1)q− 1 (−σk−1 + θk − σk + 1)q− 122 ,= = + : (−2σk )q+ 1 ; = − : (1 + 2σk )q− 1 q − 12 !ϕ̄p,(−σk−1 − θk + σk )p+ 1 (σk−1 − θk + σk )p+ 122.= = + : (1 + 2σk )p− 1 ; = − : (−2σk )p+ 1 p − 12 !222222112(4.113)4.5.
Relation to Nekrasov functionsThe notation [ = + : X; = − : Y ] means that we should substitute this constructionby X when = + and by Y when = −. Comparing these expressions with (4.88),~ 0 ,m0 ˆ Y~ 0 ~0we may compute the ratios of diagonal factors which appear in Z YY~ ,mẐY~ ,Q,~Q :(σk−1 + θk − σk ) (σk−1 + θk + σk ),[ = + : 2σk−1 ; = − : 1](−σk−1 − θk + σk )−1 (−σk−1 − θk − σk )−1,δ ψ̄q0 , =[ = + : (−2σk−1 )−1 ; = − : 1](4.114)(σk−1 + θk − σk ) (−σk−1 + θk − σk )δϕq, =,[ = + : 1; = − : 2σk ](−σk−1 − θk + σk )−1 (σk−1 − θk + σk )−1δ ϕ̄p, =.[ = + : 1; = − : (−2σk )−1 ]Since |p± | − |q± | = Q± , these formulas allow to determine the corrections δ1 η± :δψp0 , =0eiδ1 η+ =0iδ1 η−e(θk + σk−1 )2 − σk2,2σk−12= (θk − σk−1 ) −σk2 ,2,e−iδ1 η+ = (θk − σk )2 − σk−1e−iδ1 η−2(θk + σk )2 − σk−1.=2σk(4.115)One could notice that some minus signs should also be taken into account, so that00 ~0~ 0 ,m0~ ˆ~ 0 ~0(T ) = (−1)|q+ |+|p− | eiδ1 η~ ·Q +iδ1 η~·Q ẐY~Y ,Q,~Q (T ) .ZY~Y ,mThis is however not essential, as these signs will be recovered at the last step.
Amore important thing to note is that in the reference limit described by θk → +∞,σk , σk−1 θ, σk , σk−1 → +∞ one has0 sgn eiδ1 η± = sgn eiδ1 η± = 1.Step 2Let us now formulate and prove combinatorialTheorem 4.34. Z̃ bif ν 0, Y 0 ; 0, Y ≡ Z̃ bif ν Y 0 , Y = ±Z bif ν Y 0 , Y .This statement follows from the following two lemmas.Lemma 4.35. Equality Z bif = ±Z̃ bif holds for the diagrams Y 0 , Y ∈ Y iff it holds forY 0 , Y with added one column of admissible height L.Proof. Let us denote the new value of Z bif by Z ∗bif 4 , thenQQ(1 + ν)L i L + p0i + 21 + ν (1 − ν)L i L + pi + 21 − ν∗QQZ bif .Z bif =110i L − qi + 2 + νi L − qi + 2 − ν(4.116)The extra factor comes only from the product over 2L new boxes.
To explain how itsexpression is obtained, we will use the conventions of Fig. 4.14.Everywhere in this appendix X ∗ denotes the value of a quantity X after appropriate transformation.41134. Fredholm determinant and Nekrasov sum representations of isomonodromic tau functionsFigure 4.14: A Young diagram Y 0∗ obtained from Y 0 ={6, 4, 4, 2, 2} by addition of a column of length L = 7.To compute the contribution from the red boxesjust to multiply theQ it is enough10corresponding shifted hook lengths, which yields i L + pi + 2 + ν . To compute thecontribution from the green boxes one has to first write down the product of numbersfrom ν + L to ν + 1 (i.e.
the Pochhammer symbol (1 + ν)L in the numerator), keepingin mind that each step down by one box decreases the leg-length of the box by atleast one. Then one has to take into account that some jumps in this sequence aregreater than one: this happens exactly when we meet some rows of the transposeddiagram. We mark with the green crosses the boxes whose contributions should becancelled from the initial product: they produce the denominator.Next let us check what happens with Z̃ bif .
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