Диссертация (1137342), страница 21
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. . , N }.It is useful to combine Fourier and color indices into one multi-index ı = (p, α) ∈ N :=Z0 × {1, . . . , N }. Unordered sets {ı1 , . . . , ım } ∈ 2N of such multi-indices are denotedby capital Roman letters I or J. Given a matrix M ∈ CN×N , we denote by MIJ its|I| × |J| restriction to rows I and columns J.~ J~ , where I~ = (I1 , . . . , In−3 ),Principal submatrices of K may be labeled by pairs I,J~ = (J1 , . . . , Jn−3 ) and I1...n−3 , J1...n−3 ∈ 2N . Namely, define0Jd[1]KI,~ J~:= 001I1Ia[2] 1Ib[2] 1000J0JJ1c[2]2J1I2d[2]2I2000··000[3] I2a0[3] I2b0··000··00000[4] I3a··00····0···J2I30000000Jc[3] 3················00000··00J0J000J200Jd[3]3I3·J3·940c[n−3]n−3Jn−4b[n−3]d[n−3]·In−20In−3n−3In−30Ia[n−2]0n−3Jn−34.3.
Fourier basis and combinatorics~ J~ will be referred to asFor reasons that will become apparent below, the pairs I,configurations. It is useful to keep in mind that the lower index in Ik , Jk correspondsto the annulus Ak , and the blocks of K are acting between spaces of holomorphicfunctions on the appropriate annuli.×2(n−3)~~Definition 4.16. A configuration I, J ∈ 2Nis called• balanced if |Ik | = |Jk | for k = 1, . .
. , n − 3;• proper if all elements of Ik (and Jk ) have positive (resp. negative) Fourierindices for k = 1, . . . , n − 3.The sets of all balanced and proper balanced configurations will be denoted by Confand Conf + , respectively.~ J~ ∈ Conf, defineDefinition 4.17. For I,I,Jk−1Z Ik−1k ,Jka[k]Ik−1b[k]Jk−1[k] Jkc Jk−1T [k] := (−1)|Ik | det Ik−1 Ik[k] Jkd Ik,k = 1, .
. . , n − 2.(4.63a)In order to have uniform notation, here we set I0 = J0 = In−2 = Jn−2 ≡ ∅, so thatIn−3 ,Jn−3|I1 |∅[1][1] J1[n−2][n−2] In−3ZI∅,T=(−1)detd,ZT=deta.∅, ∅1 ,J1I1Jn−3(4.63b)~~Proposition 4.18. The principal minor DI,:=detKvanishesunlessI,J∈~ J~~ J~I,Conf + , in which case it factorizes into a product of n − 2 finite (|Ik−1 | + |Ik |) ×(|Ik−1 | + |Ik |) determinants asDI,~ J~ =n−2YI,Jk−1Z Ik−1k ,JkT [k] .(4.64)k=1Proof. For k = 1, . . . , n − 3, exchange the (2k − 1)-th and 2k-th block row of thematrix KI,~ J~ . As such permutation can only change the sign of the determinant, theproposition for balanced configurations follows immediately from the block structureof the resulting matrix. The sign change is taken into account by the factor (−1)|Ik |in (4.63a).The only non-zero Fourierof a[k] , b[k] , c[k] , d[k] are given by (4.57).
coefficients~ J~ ∈ Conf is not proper, then at least one of theTherefore, if a configuration I,factors on the right of (4.64) vanishes due to the presence of zero rows or columns inthe relevant matrices.Corollary 4.19. Fredholm determinant τ (a) is given byτ (a) =Xn−2YI,Jk−1Z Ik−1k ,Jk~ J~)∈Conf + k=1(I,95T [k] .(4.65)4. Fredholm determinant and Nekrasov sum representations of isomonodromic tau functionsProof. Another useful consequence of the block structure of the operator K is thatTr K 2m+1 = 0 for m ∈ Z≥0 . This implies that det (1 − K) = det (1 + K).
It now suffices to combine this symmetry with von Koch’s formula (4.62) and Proposition 4.18.Let us now give a combinatorial description of the set Conf + of proper balancedconfigurations in terms of Maya diagrams and charged partitions.Definition 4.20. A Maya diagram is a map m : Z0 → {−1, 1} subject to the conditionthat m (p) = ±1 for all but finitely many p ∈ Z0± . The set of all Maya diagrams willbe denoted by M.A convenient graphical representation of m ∈ M is obtained by replacing −1’s and1’s by white and black circles located at the sites of half-integer lattice, see bottom partof Fig.
4.12 for an example. The white circles in Z0+ and black circles in Z0− are referredto as particles and holes in the Dirac sea, which itself corresponds to the diagram m0defined by m0 Z0± = ±1. An arbitrary diagram is completely determined by asequence p (m) = (p1 , . . . , pr ) of strictly decreasing positive half-integers p1 > . . . > prgiving the positions of particles, and a sequence h (m) = (−q1 , . . . , −qs ) of strictlyincreasing negative half-integers −q1 < . . .
< −qs corresponding to the positions ofholes. The integer Q (m) := |p (m)| − |h (m)| is called the charge of m.~ J~ ∈ Conf + , consider a pair of its multi-indices (Ik , Jk )Given a configuration I,associated to the annulus Ak . Recall that the Fourier indices of elements of Ik (andJk ) are positive (resp. negative). They can therefore be interpreted as positions ofparticles and holes of N different colors.
This yields a bijection between the set ofpairs (Ik , Jk ) verifying the balance condition |Ik | = |Jk | and the setMN0 =n XNm(1) , . . . , m(N ) ∈ MN α=1oQ m(α) = 0of N -tuples of Maya diagrams with vanishing total charge. We thereby obtain aone-to-one correspondence× . . . × MNConf + ∼= MN0 .{z}| 0n−3 factorsDefinition 4.21. A charged partition is a pair Ŷ = (Y, Q) ∈ Y × Z. The integer Qis called the charge of Ŷ .There is a well-known bijection between Maya diagrams and charged partitions,whose construction is illustrated in Fig.
4.12. Given a Maya diagram m ∈ M, we startfar on the north-west axis and draw a segment directed to the south-east above eachblack circle and a segment directed north-east above each white circle. The resultingpolygonal line defines the outer boundary of the Young diagram Y corresponding tom. The charge Q = Q (m) of Ŷ is the signed distance between Y and the north-eastaxis. In the case Q (m) = 0, the sequences p (m) and −h (m) give the Frobeniuscoordinates of Y .964.3.
Fourier basis and combinatoricsNENWQ......9 7 5 3 1 - 1 -3 - 5 - 7 - 92 2 2 2 2 2 2 2 2 2Figure4.12:ThecorrespondencebetweenMayadiagramsandchargedpartitions;here the charge Q (m) = 2 and the positions of particles and7 3 151,,,andh(m)=−,−.holes are given by p (m) = 132 2 2 222~ , withLet us write N -tuples Ŷ (1) , . . . , Ŷ (N ) of charged partitions as Y~ , Q~ = Q(1) , . . . , Q(N ) ∈ ZN . The set of such N Y~ = Y (1) , .
. . , Y (N ) ∈ YN and Q∼ Ntuples with zero total charge can be identified with MN0 = Y × QN , where QNdenotes the AN −1 root lattice: XNno~ ∈ ZN QN := QQ(α) = 0 .α=1This suggests to introduce an alternative notation for elementary finite determinantfactors in (4.65). For |Ik−1 | = |Jk−1 | and |Ik | = |Jk |, we define~Y~ k−1,QZ Y~k−1~,QkkI,Jk−1T [k] := Z Ik−1T [k] ,k ,Jk(4.66) ~ k−1 , Y~k , Q~ k ∈ YN × QN are associated to N -tuples of Maya diwhere Y~k−1 , Qagrams describing subconfigurations (Ik−1 , Jk−1 ), (Ik , Jk ).
In what follows, the twonotations are used interchangeably.The structure of the expansion of τ (a) may now be summarized as follows.Theorem 4.22. Fredholm determinant τ (a) giving the isomonodromic tau functionτJMU (a) can be written as a combinatorial seriesτ (a) =XXn−2Y~Y~ k−1,QZ Y~k−1~,QkT [k] ,(4.67)k~ 1 ,...Q~ n−3 ∈QN Y~1 ,...Y~n−3 ∈YN k=1Q~~ k−1Y,Qwhere Z Y~k−1T [k] are expressed by (4.66), (4.63) in terms of matrix elements of~,Qkk3-point Plemelj operators in the Fourier basis.Example 4.23.
Let us outline simplifications to the above scheme in the case N = 2,~ J~ ∈ Conf +n = 4 corresponding to the Painlevé VI equation. Here a configuration I,is given by a single pair (I, J) of multi-indices whose structure may be described asfollows: I (and J) encode the positions of particles (resp. holes) of two colors {+, −},974. Fredholm determinant and Nekrasov sum representations of isomonodromic tau functionsand the total number of particles in I coincides with the total number of holes inJ. Relative positions of particles and holes of each color are described by two Youngdiagrams Y+ , Y− ∈ Y. The vectors (Q+ , Q− ) ∈ Q2 of the charge lattice are labeled bya single integer n = Q+ = −Q− ∈ Z.
In the notation of Subsection 4.2.5, the series(4.65) can be rewritten asτ (t) ==XXn∈Zp+ ,p− ∈2 + ; h+ ,h− ∈2 −|p+ |−|h+ |=|h− |−|p− |=nXZ0X−(−1)|p+ |+|p− | det ah++ ,h−− det d hp++ ,h,p− =p ,pZ0ZY+ ,Y− ,n T(4.68)[L]ZY+ ,Y− ,nT[R],n∈Z Y+ ,Y− ∈Yp ,ph ,hwhere ZY+ ,Y− ,n T [L] = (−1)|p+ |+|p− | det d p++ ,p−− and Z Y+ ,Y− ,n T [R] = det ah++ ,h−− . Inthese equations, the particle/hole positions (p+ , h+ ) and (p− , h− ) for the 1st and 2ndcolor are identified with a pair of Maya diagrams, subsequently interpreted as chargedpartitions (Y+ , n) and (Y− , −n).Remark 4.24. Describing the elements of Conf + in terms of N -tuples of Young diagrams and vectors of the AN −1 root lattice is inspired by their appearance in thefour-dimensional N = 2 supersymmetric linear quiver gauge theories.
Combinatorialstructure of the dual partition functions of such theories [Nek, NO] coincides with thatof (4.67). These partition functions can in fact be obtained from our construction orits higher genus/irregular extensions by imposing additional spectral constraints onmonodromy.
It will shortly become clear that the multiple sum over QN is responsiblefor a Fourier transform structure of the isomonodromic tau functions. This structurewas discovered in [GIL12, ILT13] for Painlevé VI, understood for N = 2 and arbitrarynumber of punctures within the framework of Liouville conformal field theory [ILTe],and conjectured to appear in higher rank in [Gav].
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