CFT approach to the q-painlevé VI equation

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1 Journal of Integrable Systems 07 7 doi: 0.093/integr/xyx009 CFT approach to the q-painlevé VI equation M. Jimbo Department of Mathematics Rikkyo University Toshima-ku Tokyo Japan H. Nagoya School of Mathematics and Physics Kanazawa University Kanazawa Ishikawa 90-9 Japan Corresponding author. nagoya@se.kanazawa-u.ac.jp and H. Sakai Graduate School of Mathematical Sciences University of Tokyo Meguro-ku Tokyo Japan Communicated by: Masatoshi Noumi Received on 6 June 07; editorial decision on 8 August 07; accepted on 30 August 07] Iorgov Lisovyy and Teschner established a connection between isomonodromic deformation of linear differential equations and Liouville conformal field theory at c =. In this article we present a q analogue of their construction. We show that the general solution of the q-painlevé VI equation is a ratio of four tau functions each of which is given by a combinatorial series arising in the AGT correspondence. We also propose conjectural bilinear equations for the tau functions. Keywords: isomonodromy deformation; q Painleve equation; conformal field theory. Dedicated to Professor Kazuo Okamoto on the occasion of his seventieth birthday. Introduction Theory of isomonodromic deformation and Painlevé equations has long been a subject of intensive study with many applications both in mathematics and in physics. Recent discovery of the isomonodromy/cft correspondence 3] opened up a new perspective to this domain. In a nutshell it states that the Painlevé tau functions are Fourier transform of conformal blocks at c =. The AGT relation allows one to write a combinatorial formula for the latter thereby giving an explicit series representation for generic Painlevé tau functions. The initial conjectures of 3] concern the sixth Painlevé PVI equation and at present three different proofs are known. In the first approach 4] one constructs solutions to the linear problem out of conformal blocks with the insertion of a degenerate primary field. The second approach 5] also makes use of CFT ideas but leads directly to bilinear differential equations for tau functions. The third approach 6] is based on the Riemann Hilbert problem. It is shown that Fredholm determinant representation for tau functions reproduces the combinatorial series expansion without recourse to conformal blocks or The authors 07. Published by Oxford University Press. This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License which permits non-commercial re-use distribution and reproduction in any medium provided the original work is properly cited. For commercial re-use please contact journals.permissions@oup.com Downloaded from on 0 July 08

2 M. JIMBO ET AL. the AGT relation. Painlevé equations of other types are also studied by confluence ] in the framework of irregular conformal blocks 7 8] through the AGT correspondence 9] and using Fredholm determinants 0]. It is natural to ask whether a similar picture holds for the q difference analogue of Painlevé equations. This has been studied in ] for the qpiiid 8 equation. The aim of the present article is to give a q analogue of the construction of 4]. Our method is based on the work ] where the five-dimensional AGT correspondence was studied in the light of the quantum toroidal gl algebra also known as the Ding Iohara Miki algebra. Having the central charge c = corresponds to choosing t = q where t q are the parameters of the algebra. This is a sort of classical limit in the sense that the quantum algebra reduces to the enveloping algebra of a Lie algebra see Section.. We use the trivalent vertex operators of ] to introduce a q analogue of primary fields and conformal blocks. As in the continuous case a key property is the braiding relation in the presence of the degenerate field. Following the scheme of 4] we then write the solution Yx of the Riemann problem as Fourier transform of appropriate conformal block functions. For the sake of concreteness we restrict our discussion to the setting relevant to the qpvi equation. Also we do not discuss the issue of convergence of these series. The above construction allows us to relate the unknown functions y z of qpvi to tau functions. More specifically we express y z in terms of four tau functions which arise from the expansions at x = 0or x = ; see below. On the other hand in view of the work 3 4] we expect that there are four more tau functions which naturally enter the picture; these are the ones related to the singular points other than x = 0. However for the lack of the notion of fusion it is not clear to us how to related them to y z. The absence of fusion is the main obstacle in the q analogue of isomonodromy theory. We present conjectural bilinear relations satisfied by these eight tau functions see Assuming the conjecture we give the final expression for y z in 3.4. The plan of the article is as follows. In Section we give a brief account of the results of ] restricting to the special case t = q. We introduce a q analogue of primary fields conformal blocks degenerate fields and their braiding. In Section 3 we apply them to the q analogue of the Riemann problem focusing attention to the setting of the qpvi equation. We derive a formula expressing the unknowns y z in terms of four tau functions. We then propose conjectural bilinear difference equations for eight tau functions. In appendix we give a direct combinatorial proof of the braiding relation used in the text. Notation. Throughout the article we fix q C such that q <. We set u] = q u / q a; q N = N j=0 aqj a... a k ; q = k j= a j; q and a; q q = jk=0 aqj+k. We use the q Gamma function q Barnes function and the theta function Ɣ q u = q; q q u G q u q u = qu ; q q q; q u ; q q; q q q u u / ϑu = q uu / q q u q x = x q/x q; q which satisfy Ɣ q = G q = and Ɣ q u + =u]ɣ q u G q u + = Ɣ q ug q u ϑu + = ϑu = ϑ u. Downloaded from on 0 July 08

3 CFT APPROACH TO Q-PVI 3 A partition is a finite sequence of positive integers λ = λ... λ l such that λ λ l > 0. We set lλ = l. The conjugate partition λ = λ... λ l is defined by λ j = {i λ i j} l = λ. We denote by the set of all partitions. We regard a partition λ also as the subset {i j Z j λ i i } of Z and denote its cardinality by λ.for = i j Z >0 we set a λ = λ i j and l λ = λ j i. In the last formulas we set λ i = 0ifi > lλ resp. λ j = 0ifj > lλ.. q Analogue of conformal blocks In this section we collect background materials from ]. We shall restrict ourselves to the special case t = q where formulas of ] simplify considerably. We then introduce a q analogue of primary fields conformal blocks and the braiding relation for the degenerate fields analogue of operators.. Toroidal Lie algebra Let Z D be non-commuting variables satisfying ZD = qdz. The ring of Laurent polynomials C Z ± D ± in Z D is a Lie algebra with the Lie bracket given by the commutator. We consider its two-dimensional central extension which we denote by L. As a vector space L has C-basis Z k D l k l Z \{0 0} and central elements c c. The Lie bracket is defined by We set Z k D l Z k D l ]=q l k q l k Z k +k D l +l + δ k +k 0δ l +l 0q k l k c + l c. a r = Z r x + n = DZn x n = Zn D b r = D r y + n = ZD n y n = D n Z and introduce the generating series x ± z = n Z x± n z n y ± z = n Z y± n z n. We have two Heisenberg subalgebras of L a = r =0 Ca r Cc and b = r =0 Cb r Cc : a r a s ]=rδ r+s0 c b r b s ]= rδ r+s0 c. We say that an L-module has level l l if c acts as l id and c acts as l id. The Lie algebra L has automorphisms S T given by S : Z D D Z c c c c T : Z Z D DZ c c c c + c. They satisfy S 4 = ST 6 = id so that the group SL Z acts on L by automorphisms. We have Sb r = a r Sy ± z = x ± z.. Fock representations The most basic representations of L are the Fock representations of levels 0 and 0. Following ] we denote them by F u 0 and F u 0 u C respectively. The Fock representation F u 0 is irreducible under the Heisenberg subalgebra a. As a vector space we have F u 0 = Ca a...] vac where vac is a cyclic vector such that a r vac =0r > 0. The Downloaded from on 0 July 08

4 4 M. JIMBO ET AL. action of the generators x ± n is given by vertex operators x ± z q u± exp r q r a r z r exp ± r r q r a r z. r r The Fock representation F u 0 is the pullback of F u 0 by the automorphism S. It is irreducible with respect to the Heisenberg subalgebra b and y ± z act as vertex operators. On the other hand the action of a on F 0 u a basis of F 0 u is commutative. The joint eigenvectors { λ } λ of a are labelled by all partitions and constitute. In particular for the vector we have ur a r = q r x z = 0 x + z = δu/z where δx = n Z xn. Formulas for the action of a r and x ± z on a general λ can also be written explicitly. Since we do not use them in this article we refer the reader to 5]. Tensor products of N copies of F u 0 are closely related to representations of the W N algebra see e.g. 6 7]. We briefly explain this point in the case N =. Consider the tensor product of two Fock representations and its decomposition with respect to the Heisenberg subalgebra a F 0 u F 0 u = H u u where H is the Fock space of a and u u is the multiplicity space. The operators x ± z act on F 0 u F 0 u as a sum of two vertex operators. We can factor it into a product of two commuting operators q x ± z g ± z u u ±/ Tz; u /u ±/. The first factor represents the action of a on H g ± z =: exp ± r =0 q r a r + a r z r r where a r = a r id a r = id a r. The second factor acts on u u Tz; u = u + z + u z ± z =: exp ± q r a r a r z r. r r =0 and gives a free field realization of the deformed Virasoro algebra 8]. The space u u is a q analogue of the Virasoro Verma module with central charge c = and highest weight = where q = u /u. Downloaded from on 0 July 08

5 CFT APPROACH TO Q-PVI 5.3 Trivalent vertex operators For N Z we denote by F w N the pullback of F w 0 by the automorphism T N. The following intertwiners of L-modules are called trivalent vertex operators. ] N + wx = 0 w; N x = N x; 0 w N + wx : F 0 w ] : F N+ wx F N x F N+ wx F N x F 0 w. Such non-trivial intertwiners exist and are unique up to a scalar multiple ]. We introduce their components λ w 0 λ w relative to the basis { λ } of F w as follows. λ α = ˆtλ x w N λ wα α = λ ˆt λ x w N λ wα λ α F N x N+ α F wx. The numerical factors in front are ˆtλ x w N = qnλ c λ x λ f N λ w N ˆt λ x w N = qnλ f N λ c λ qw N λ x where nλ = l λ nλ = a λ λ λ f λ = λ q nλ nλ c λ = λ q l λ +a λ +. We normalize w w so that w vac = vac + w vac = vac +. Proposition. ] The operators λ w λ w are given by λ w =: + wη+ λ w λ w =: wη λ w lλ λ η ± λ w =: i η ± q j i i= j= where ± w =: exp q n n q a nw n n n =0 η ± w =: exp q n a n w n. n n =0 Downloaded from on 0 July 08

6 6 M. JIMBO ET AL. Of particular importance is their normal ordering rule. For a pair of partitions λ μ we introduce the Nekrasov factor by N λμ w = λ q l λ aμ w μ q a λ +lμ + w. Proposition. ] Let X Y be either or. Then the following normal ordering rule holds. { X λ zy μ w =: X λ zy μ w : Ĝq qw/zn μλ w/z for X = Y Ĝ q qw/zn μλ w/z for X = Y. where Ĝ q z = z; q q. It is useful to note the relations N λμ w = N μλ w w λ + μ f λ f μ. q nλ = f λq λ N λλ..3 c λ.4 Chiral primary field In what follows we set V w := F 0 q w F 0 q w. We fix its basis λ = λ + λ labelled by a pair λ = λ + λ of partitions. Consider the compositions of trivalent vertex operators : F 0 F F 0 F F 0 F 0 q /x q 3 /xw q 3 w /xw q 3 w q 3 w : F 0 F 0 F 0 F F 0 F q w q w /xw q w q /xw /x. By composing them we obtain an intertwiner of L-modules V w F 0 id V q w F V id /x /xw 3 q w F 0 /x V 3 q w..4 Taking further the vacuum-to-vacuum matrix coefficient of.4 we obtain a map V 3 ; w x : V w V 3 q w..5 Downloaded from on 0 July 08

7 CFT APPROACH TO Q-PVI 7 The matrix coefficients of.5 are given by μ V ; w x λ 3 = C qnλ + c λ+ q nλ c λ q nμ + c μ+ q nμ c μ f λ f μ + f μ q λ μ μ x μ λ vac λ+ q w λ q w μ + q 3 w μ q 3 w vac.6 where we set λ = λ + + λ for λ = λ + λ and C is a scalar factor independent of λ μ. We choose the normalization V q 3 x 3 N 3 3 ; w x = N 3 = ɛɛ =± G q + ɛ 3 ɛ..7 G q + 3 G q Here and after stands for the matrix coefficient. We regard.5asaq analogue of the chiral primary field of conformal dimension =..5 Conformal block function The m + point conformal block function is the expectation value of a product of chiral primary fields see Fig. m m F ; x m... x m+ σ m σ m σ 0 m m = V ; w m x m V ; w m x m V ; w x σ m σ m σ 0 m+ σ m where w p = q p w. The right-hand side is actually independent of w. Using formula.6 applying the normal ordering rule. and further using..3 for simplification we obtain the following explicit expression. m m F ; x m... x m+ σ m σ m σ 0 = m N p= p σ p σ p λ...λ m p= q pσ p x p+ m p= x σ p p σ p p m q p λ p m x p= p m p= ɛɛ =± N λ p ɛ λ p q ɛσp p ɛ σp ɛ..8 ɛɛ =± N λ p ɛ λ p q ɛσp ɛ σp ɛ We have set σ 0 = 0 σ m = m+ λ 0 = λ m = and the sum is taken over all λ p = λ p + λ p p =... m. Downloaded from on 0 July 08

8 8 M. JIMBO ET AL. Fig.. q-conformal block function. Formula.8 is the q version of the AGT formula for conformal blocks 9] specialized to the case t = q. The derivation given above from the quantum algebra viewpoint is due to ]..6 Degenerate field and braiding The normalization factor.7 has a zero when = / and 3 =±/. In this case we redefine the primary field by N = lim ± ε 0 G q ε N ε ± V ; w x = lim ± ε 0 G q ε V ε ; w x. ± We consider the conformal block function when one of the primary fields is V. We keep using the same letter F to denote them. By choosing m = and specializing parameters in.8 the four point conformal blocks can be evaluated in terms of Heine s basic hypergeometric series α β F γ ; x = Ɣ qαɣ q β Ɣ q γ q α q β φ ; q x q γ as follows. Here we have set F ; x + ε x = N q q x 0 x F ε +/4 + ε ε 0 + ε ; qx qx F ; x 0 + ε x = N q 0 0 F x ε0 +/4 + + ε 0 + ε 0 + ε 0 ; qx x N = q x /4 x 0 x. μμ =± G q + μ + μ 0. G q + G q 0 The following braiding relation holds. Downloaded from on 0 July 08

9 CFT APPROACH TO Q-PVI 9 Theorem.3 We have V ; x + ε V ; x + ε 0 = V ; x ε=± 0 + ε V ; x 0 + ε 0 B εε ] x q / x.9 0 x x where the braiding matrix is given by B εε ] x = ε ϑ 0 + ε + ε 0 ϑ 0 ϑ + ε + + ε 0 + u ϑ + u.0 with x = q u. For the vacuum matrix element the relation.9 is a consequence of the known connection formula for basic hypergeometric functions. At this writing it is not clear to us whether the general case can be deduced from this and the intertwining relation. In Appendix we give a direct combinatorial proof of the braiding relation.9. The braiding matrix.0 has the following properties. B εε ] qx ] = B εε x. 0 0 B 0 x ] ] = B q x. 0 ] x det B = ϑ ϑ u 0 ϑ 0 ϑ u + ] B x is -periodic in q-painlevé VI equation 3. Riemann problem This article deals with a non-linear non-autonomous q-difference equation called q-painlevé VIqPVI equation. It is derived from deformation theory of linear q-difference equations as a discrete analogue of isomonodromic deformation 0]. A remark is in order concerning the name of the equation. Discrete Painlevé equations are classified by rational surface theory and sometimes called by the name of the surface ]. From that viewpoint qpvi is an equation corresponding to the surface of type A 3. Besides this surface has symmetry by the Weyl group D 5. Accordingly qpvi is also referred to as the q-painlevé equation of type A 3 or the q-painlevé equation with D 5 -symmetry. Regarding discrete Painlevé equations pioneering researches by Ramani Grammaticos and their coworkers are well-known. The qpvi equation in the special case when an extra symmetry condition is fulfilled has been studied earlier by them under the name of discrete Painlevé III equation ]. Downloaded from on 0 July 08

10 0 M. JIMBO ET AL. In this section we review the content of 0] concerning the generalized Riemann problem associated with the qpvi equation. Let 0 t C.Weset R = max q tq tq t R = min q q tq tq t and assume that R < R. Fix also σ C and s C. Suppose we are given matrices Y x t Y 0t x t Y 0 x t with the following properties. i These matrices are holomorphic in the domains Y x t : R < x Y 0t x t : R < x < R Y 0 x t : x < R. 3. ii They are related to each other by Y x t = Y 0t x tb x Y 0t x t = Y 0 x tb x 3. ] B x = B + q x ] t B σ x = B q σ + t t 0 s x 0 iii At x or x 0 they have the behaviour Y x t = I + Y tx + Ox x 0 x x x Y x t = GtI + Ox 0 x x t x It follows that Yx t = Y x t is a solution of linear q-difference equations of the form We have Yqx t = Ax tyx t Ax t = A x + A tx + A 0 t x q x tq 3.6 Yx qt = Bx tyx t Bx t = xi + B 0t x tq. 3.7 t A = diagq q A 0 t = tq 3 t Gt diagq 0 q 0 Gt det Ax t = x q x tq t. x q x tq Though the argument is quite standard we outline the derivation for completeness. Downloaded from on 0 July 08

11 CFT APPROACH TO Q-PVI We take the determinant of the relations and rewrite them as /x tq /x; q q /x tq t /x; q det Y x t = ϑ q + x q t/x; q q + x det Y 0t x t ϑσ qx tq t /x; q = s ϑ q + x q t+ + x/t; q q t +t+ + q x ϑ 0 qx q + x/t; q t t x det Y 0 x t. Both sides of this expression are single valued and holomorphic on C. In view of the behaviour we conclude that det Y x t = q /x tq t /x; q /x tq /x; q. In particular the matrices Y x t Y 0t x t and Y 0 x t are generically invertible. Rewriting and using the periodicity of the braiding matrices we obtain Ax t := Y qx ty x t = Y 0t qx ty 0t x t = Y 0 qx ty 0 x t. A similar argument shows that Ax t is a rational function of x with the only poles at x = q q t. Using once again we find that Ax t has the form stated above. The derivation of 3.7 is quite similar. Introduce y = yt z = zt and w = wt by Ax t + = q wx y x q x tq Ay t ++ = y tq t qzy q where we use ± to indicate components of a matrix. Notation being as above the compatibility Ax qtbx t = Bqx tax t 3.8 of 3.6 and 3.7 leads to the qpvi equation. Here and after we write f t = f qt f t = f q t. Proposition 3. 0] The functions y z solve the qpvi system with the parameters yy = z tb z tb a 3 a 4 z b 3 z b 4 zz = y ta y ta b 3 b 4 y a 3 y a 4 a = q a = q t a 3 = q a 4 = q b = q 0 t b = q 0 t b 3 = q b 4 = q. Downloaded from on 0 July 08

12 M. JIMBO ET AL. Remark 3. From the compatibility condition 3.8 we obtain the expression of the matrix Bx t in terms of y and z etc. In particular the + element of the matrix B 0 q t is written as B 0 q t + = q+ zw q z CFT construction We are now in a position to construct solutions of the generalized Riemann problem in terms of conformal block functions. Following the method of 4] we consider the following sums of 5 point conformal blocks Y εε x t = k ε Y 0t εε x t = x k 0t ε = x k 0t ε Y 0 εε x t = x t k 0 ε s n t F ; q t+x q t t 3.0 n Z ε ε ε σ + n 0 s n t F ; q t q t+x t 3. n Z ε σ + n + ε σ + n 0 n+ ε t s F ; q t q t+x t n Z ε σ + n + σ + n + ε 0 s n t F ; q t t q t+x 3. n Z ε σ + n ε 0 where k ε = q ε/ ε t+ N ε ˆτ k 0t ε = k εq + / k 0 ε = k 0t εq t +t/+ t t t and ˆτ = n Z s n F t ; q t t. σ + n 0 We assume that these series converge in the domains 3.. The asymptotic behaviour are simple consequences of the expansion.8 of conformal blocks. The crucial point is the validity of the connection formulas due to the periodicity property.3 of the braiding matrix. Therefore formulas solve the generalized Riemann problem. Define the tau function by ] τ t s σ t = q t+ +t Gq + G q 0 ˆτ. 0 Downloaded from on 0 July 08

13 CFT APPROACH TO Q-PVI 3 Explicitly it has the AGT series representation ] τ t s σ t = ] ] s n t σ +n t 0 C t σ + n Z t σ + n t n Z with the definition ] C t σ εε = =± G q + ε + ε σg q + εσ t + ε 0 0 G q + σg q σ ] Z t σ t = t λ εε =± N λ ε q ε ε σ N λ ε q εσ t ε 0 0 λ=λ + λ εε =± N. λελ ε q ε ε σ Comparing the asymptotic expansion at x = 0 and x = we can express the + element of the matrix Ax t and Bx t in terms of tau functions. Theorem 3.3 The following expressions hold for y z and w in terms of tau functions: y =q t τ3τ 4 τ τ 3.4 τ τ τ τ z = q τ τ q τ τ 3.5 w =q Ɣ q q Ɣ q τ. τ Here we put ] ] τ = τ t s σ t τ = τ t s σ t 0 0 ] ] τ 3 = τ t 0 + s σ + t τ 4 = τ t 0 s σ t. Proof. From the explicit expression of the 5-point conformal block function.8 we have F 3 ; x 3 x x = N 4 σ σ 0 σ 0 = N 3 4 σ q σ σ x 0 3 F q 3 4 x 4 3 σ 3 F ; x 3 x 4 σ σ ; x x σ σ 0 + Ox + O. x 3 Downloaded from on 0 July 08

14 4 M. JIMBO ET AL. By using this expression we calculate the asymptotic behaviour of Y x t Y 0 x t and get ] t τ Gt εε = q C Ɣ t t q ε ε s σ t ε Y Ɣ q + ε t + = Ɣ q 0 τ Ɣ q τ τ where C = 0 + t + ε + t + + ε see for Gt and Y t. We can write down the exponents of matrices Ax t and Bx t in these terms. In particular the expressions A 0 t + = q 3 t q 0 q 0 Gt ++Gt + A t + = q q Y t + det Gt give the formula for w and y. The expression of z is obtained from B 0 q t + = Y t + Y q t + and the relation 3.9. In 3] Mano constructed two parametric local solutions of the qpvi equation near the critical points t = 0. The above formula for y extends Mano s asymptotic expansion to all orders. We expect that the formula 3.5 for z can be simplified see 3.4 below. 3.3 Bilinear equations In 4] Tsuda and Masuda formulated the tau function of the qpvi equation to be a function on the weight lattice of D 5 with symmetries under the affine Weyl group. Motivated by 4] we consider the set of eight tau functions defined by the AGT series 3.3: ] ] τ = τ t s σ t + τ = τ t s σ t 0 0 ] ] τ 3 = τ t 0 + s σ + t τ 4 = τ t 0 s σ t ] τ 5 = τ t s + ] σ t τ 6 = τ t s σ t 0 0 τ 7 = τ t ] s σ + t τ 8 = τ 0 t + ] s σ t. 0 For convenience we shift the parameter in the previous section to + /. Downloaded from on 0 July 08

15 CFT APPROACH TO Q-PVI 5 On the basis of a computer run we put forward: Conjecture 3.4 For the eight tau functions defined above the following bilinear equations hold. τ τ q t τ 3 τ 4 q tτ 5 τ 6 = τ τ t τ 3 τ 4 q t tτ 5 τ 6 = τ τ τ 3 τ 4 + q tq t τ 7 τ 8 = τ τ q t τ 3 τ 4 + q t tq t τ 7 τ 8 = τ 5 τ 6 + q +t / t τ 7 τ 8 τ τ = τ 5 τ 6 + q + +t / t τ 7 τ 8 τ τ = 0 3. τ 5 τ 6 + q 0 +t τ 7 τ 8 q t τ 3 τ 4 = 0 3. τ 5 τ 6 + q 0 +t τ 7 τ 8 q t τ 3 τ 4 = The solution y z of qpvi are expressed as y = q t τ3τ 4 τ τ z = q t t τ7τ 8 τ 5 τ The bilinear equations were originally proved in 3] in the special case when the tau functions are given by Casorati determinants of basic hypergeometric functions. It would be interesting if one can prove them in the general case by a proper extension of the methods of 5] or6]. Remark 3.5 Formula 3.5 for z is obtained from the asymptotic expansion at x =. The expansion at x = 0 gives an alternative formula for the same quantity. Comparing these and shifting to + / we arrive at the following bilinear relation: τ τ τ τ = q/+ q / q t τ q 0 q 3 τ 4 τ 3 τ 4. 0 This is consistent with the conjectured relations Acknowledgments The authors would like to thank Kazuki Hiroe Oleg Lisovyy Toshiyuki Mano Yousuke Ohyama Yusuke Ohkubo Teruhisa Tsuda Yasuhiko Yamada and Shintaro Yanagida for discussions. Funding JSPS KAKENHI JP6K0583 to M.J. in part; JSPS KAKENHI JP5K7560 to H.N. in part; JSPS KAKENHI JP5K04894 to H.S. in part. Downloaded from on 0 July 08

16 6 M. JIMBO ET AL. Appendix A. Braiding relation In this appendix we give a direct proof of Theorem.3. We rewrite the braiding relation.9 into the form of matrix elements α β λ μ for any partitions λ μ α β. The matrix element α β V ; x + ε V + ε σ ; x λ μ can be read off from the coefficient of t λ + μ x α β 3 in the 6 point conformal block t F ; x 3 + ε 3 x x t. σ 0 Formula.8 tells that it is a sum over a pair of partitions. By using N λμ = δ λμ N λλ the sum can be reduced further to one over a single partition. We thus find that Theorem.3 is equivalent to the following proposition. Proposition A. For any partitions λ μ α and β wehave where q q ε +/4 x ε =± Ɣ q + ε + ε σ X ε Ɣ q + ε λμαβ σ ; x x x = ε=± B εε q σ qx x X + λμαβ σ ; x x εσ+/4 x σ x ε =± Ɣ q + ε + εσ Y ε λμαβ Ɣ q + εσ σ ; x x ] q / x A. x = N αβ q N βλ q / σ N βμ q / +σ λ + μ q η + β x qx α + β N αηq N ηλ q +/ σ N ημ q +/ +σ x η x N ηβ q + N ηη X λμαβ σ ; x x = X + λμβα σ ; x x Y + λμαβ σ ; x x = N λμ q σ N αλ q σ / N βλ q σ / λ + μ λ + η qx q x α + β N ημq N αη q +σ +/ N βη q +σ +/ x η x N λη q σ + N ηη Y λμαβ σ ; x x = Y + μλαβ σ ; x x. Downloaded from on 0 July 08

17 CFT APPROACH TO Q-PVI 7 In the following we show that the identity A. is reduced to the connection formula of basic hypergeometric functions. First we prepare several lemmas. Lemma A. For any λ μ we have N λ μ u = N μλu. Proof. Since a λ j i = l λ i j for all i j Z >0 wehave N λ μ u = q aλij lμij u q aμij+lλij+ u. ij λ Therefore to prove the Lemma it suffices to show that the identity q aλij lμij u = q aμij+lλij+ u A. ij λij/ μ ij μ ij λij/ μ holds for all λ μ. Note that the product consists of all entries i j such that λ i >μ i and μ i + j λ i.weshowa. by induction with respect to the length of λ. Consider a sub-sequence λ n+ λ n+... λ n+s in λ such that λ n μ n λ i >μ i i = n +... n + s λ n+s+ μ n+s+. Put λ = λ n+... λ n+s and μ = μ n+... μ n+s. Then we have for i =... s and μ i + j λ i l μ n + i j = l μ i j l λ n + i j = l λi j because λ n μ n and λ n+s+ μ n+s+. We further divide a partition λ into sub-sequences λ m+... λ m+r such that λ m+ μ m λ i > μ i i = m +... m + r λ m+r+ μ n+r. Put ˆλ = λ m+... λ m+r and ˆμ = μ m+... μ m+r. Then we have again for i =... r and ˆμ i + j ˆλ i l μ m + i j = l ˆμ i j l λm + i j = lˆλi j because λ m+ μ m and λ m+r+ μ m+r. Replacing λ μ by ˆλ ˆμ we may assume without loss of generality that λ >μ λ i >μ i i =... lλ. A.3 If lλ = it is easy to check A.. Suppose A. is true for lλ = l. Let λ satisfy lλ = l. Assuming A.3 we choose the number k such that μ l k+ >λ l μ l k. Downloaded from on 0 July 08

18 8 M. JIMBO ET AL. Writing λ = ν λ l we compare A. for λ and for ν. In the left hand side of A. the factors for i <lremain the same as for ν. The product of factors for i = l is expressed as λl μ l +k j=0 q k j u k j= q λ l +μ l j +j u. A.4 In the right hand side of A. the factors for j >λ l remain the same as for ν. The product of factors for i = l is equal to λ l μ l q j u. j=0 The products of factors for i = l k... l and j λ l are equal to λ l μ i q l i m u m= before adding the l-th row and after they become λ l μ i q l i m u. m=0 Namely we lose the factor l i=l k q λ l +μi+l i u and gain the factor k m= qm u. Therefore the outcome can be written as A.4. We introduce the following operations on partitions λ : λ =λ... λ lλ A.5 r n λ =λ +... λ n + λ n+... n Z 0. A.6 Here we identify a partition λ with a sequence λ... λ lλ Lemma A.3 Let λ η be partitions. Then N ηλ q = 0 if and only if η = r n λ for some n 0. Proof. It is easy to show the if part. To show the only if part assume that η satisfies N ηλ q = 0. Then we have l η i j + a λ i j + = 0 i j η A.7 l λ i j + a η i j = 0 i j λ. A.8 Downloaded from on 0 July 08

19 CFT APPROACH TO Q-PVI 9 Using them we show i If lλ < i lη then η i =. ii If i minlλ lη then η i λ i + iii If i minlλ lη and η i λ i then i < lλ and η i λ i+. iv If i minlλ lη then η i λ i+. To see i suppose there exists such an i that lλ < i lη and η i. Then η η l ηη = 0 and a λ η = in contradiction with A.7. Since η i > 0 for i lη we must have η i =. To see ii suppose there exists such an i minlλ lη that η i >λ i +. Take the largest such i and let j = λ i +. Then i j η l η i j = 0 and a λ i j = which contradicts A.7. Under the assumption of iii we have i η i λ and a η i η i = 0. Further if i = lλ or else i < lλ and λ i+ <η i then we have l λ i η i = 0 which contradicts A.8. To show iv first assume i lλ and η i <λ i+ η i+ = λ i+ hold. Then i η i + λ a η i η i + = and l λ i η i + = which contradicts A.8. Similarly if i = lλ and η lλ <λ lλ then the same conclusion takes place. Now we prove the Lemma. If η i = λ i + for all i lλ then together with i we have η = r lη λ. Otherwise from ii iii and iv there exists an n such that η i = λ i + i n and η i = λ i+ for all n < i lλ. Moreover if lη lλ then η lλ η lλ = λ lλ which contradicts iii. Therefore lη = lλ and η = r n λ. Lemma A.4 Let λ μ and n Z 0. Using the notation A.5 and A.6 we set Then we have N ηλ q N ηη l = lλ k = lμ η = r n λ γ = r n μ { η = η n l λ n l+ n l. = N η λq N η η q η λ j= A.9 μ l N μη u = N μ η q q j u l u q l i+aμi u A.0 q lμlj+j u N μλ u = N μ λq u μ l+ j= i= q j u q lμl+j+j u l q l i+aμi+ u A. i= N μλ u N μη qu = N μ λq u u A. N μ η u N μλ u N μη qu = N μλqu N μη q u q η λ + k u A.3 qu Downloaded from on 0 July 08

20 0 M. JIMBO ET AL. μ l N γ λ u = N γ λq u q l+ γ μ q j u l u q l i+aμi u. A.4 q lμlj+j u Proof. By the definition we have j= a λi j = a λ i j + i =... l j Z >0 l λi j = l λ i j + i j λ a η i j = a η i j + i =... minl η l j Z >0 { l η i j = l η i j + i j η n l orn l and j n + i i > 0 j = n l. Using these relations we show below A.9 A.4. Proof of A.9. First we consider the case n l. We have N ηλ q N η λq = = ij η ij η λl j= q lηij a λ ij q lηij+ a λ ij+ q l λ lj+ j l l q l+i λ i q l i+η i i= i= ij λ ij λi<l i= q l λ ij+aηij q l λ ij++aηij+ and N η η N ηη = = l i= ij η ij η q lηij+ aηij+ q lηij aηij ij η q l+i+ η i q l i+η i. ij η q lηij++aηij++ q lηij+aηij+ Since η = λ +... λ n + λ n+... λ l and η λ = l + n + λ n+ we obtain A.9 for n l. Second we consider the case n l.wehave N ηλ q n N η λq = i= q n i λ i l i= q l i λ i q i l i+λ l i= q n+ i λ i n+ i=l+ q n+ i λ i q l+i λ i q l i+λ i = l i= q n l Downloaded from on 0 July 08

21 CFT APPROACH TO Q-PVI and N η η N ηη = q n l q n l+ l i=. q l+i λ i q l i+λ i Since η λ =n + l we obtain A.9 for n l. Proof of A.0. First we consider the case n l. Then by l η l we obtain N μη u N μ η q u = = ij μi l μ l j= q lμij+j u l q l i+aμi+ u q lμij+j u q j u l q lμlj+j u i= i= q l i+aμi u. Second we consider the case n l. Then we have a η i + j = a η i j if l<i n and l η i = l η i + for i Z >0. Hence we obtain lμ μi N μη u N μ η q u = i=l+ j=μ i+ + q aηij u l i= ql i+aμi u μl+ j= q lμl+j+j 3 u q aμn+ u = = μl μ l j= j=μ n+ + qj u μ n+ j= qj u μl j= q lμl+j+j 3 u q j u l q lμlj+j u i= q l i+aμi u. l i= ql i+aμi u q μ q μ n+ l u u Proof of A.. A.. Proof of A.. Proof of A.3. In A.0 choose n = l then rename l by l + and λ i by λ i. We obtain This follows from A. and A.0. Using the relation. and A. we have N μλ un μη q u N μλ qun μη qu = q η λ k N λμ u N η μ q u N λ μ qu N ημ qu = q η λ k k i= λ k+ j= q j u η k+ q lηk+j+j 3 u q l λk+j+j u q j u j= q k i+a λi+ u q k i+aηi u. Downloaded from on 0 July 08

22 M. JIMBO ET AL. First we consider the case n l. By definition we have for k + n l λ k + j j =... λ n+ l η k + j = n k + j = λ n+ + l λ k + j j = λ n λ + and for k + n + l η k + j = l λ k + j j =... η k+. Hence we obtain for k + n N μλ un μη q λ u N μλ qun μη qu = k+ q η λ k j= q j u q l λk+j+j u λ n+ q k+ n+λ n+ u λk++ = q η λ k qk n+λ n+ u q u. j= j=λ n+ + λk+ + j= q l λ k+j+j u q l λ k+j +j 3 u q j u We also obtain for k + n + N μλ un μη q λ u N μλ qun μη qu = k+ q j u λ k+ q lλk+j+j u q η λ k q l λk+j+j u q j u j= j= k i=n+ q k i+a λi+ u q k i+a λi+ u = q η λ k qk n+λ n+ u q u. Since η λ =n λ n+ we obtain A.3 for n l. Second we consider the case n l. By the definition we have l η i j = l λ i j i j η j l η i = n i i Z >0. Downloaded from on 0 July 08

23 CFT APPROACH TO Q-PVI 3 Hence we obtain N μλ un μη q u N μλ qun μη qu = q η λ k k i=n+ λ k+ j= q j u η k+ q lλk+j +j 3 u qn+k u q l λk+j+j u q j u q u j= q k i u q k i u = q η λ k q n+k u q u. Since η λ =n we obtain A.3 for n l. Proof of A.4. By A. we have γ l+ N γ λ u = N γ λq q j u u q lγ l+j+j u j= l q l i+aγ i+ u. Recall that γ = s n μ. First we consider the case n μ. By the definition we have l γ i j = a γ i j = Hence for l + μ n+ we have N γ λ u = N γ λq u i= { l μ i j + i j γ j n l μ i j + i j γ j n + a μ i j i =... μ n+ j Z >0 n j i = μ n+ + j Z >0 a μ i j i j γ i μ n+ + j Z >0. n j= l q l i+aμi u i= μl+ j= q j u q lμl+j+j 3 u μ l+ j=n+ μ l = N γ λq q j u u q lμlj+j u q μ q μ l u j= l q l i+aμi u q μ l u. i= q lμl+j++j u n+ +l+n u Downloaded from on 0 July 08

24 4 M. JIMBO ET AL. Similarly for l + = μ n+ + wehave N γ λ u = N γ λq u n j= μ l = N γ λq u j= q j u q lμl+j+j 3 u q j u q lμlj+j u l q l i+aμi u q μ l u i= q n u l q μ l u i= q l i+aμi u q μ l u. Here we use the inequality μ l+ n. For l + μ n+ + wehave μ l N γ λ u = N γ λq u l i=μ n+ + j= q j u q lμlj+j u q l i +aμi u j= μ n+ i= μ l = N γ λq q j u l u q lμlj+j u i= q l i+aμi u q l μ n+ +n u q l i+aμi u q l μ n+ +n u. Here we use the inequality μ l n. Since γ μ =n μ n+ we obtain A.4 for n μ. Second we consider the case n μ. By the definition we have l γ i j = l μ i j i j γ i { n j i = j Z >0 a γ i j = a μ i j i j γ i j Z >0. Hence we obtain μ l N γ λ u = N γ λq q j u u q lμlj+j u j= l q l i+aμi + u q l +n u. i= Since γ μ =n we obtain A.4 for n μ. Lemma A.4 enables us to reduce λ to λ of X ε λμαβ and Y ε λμαβ. Denote by T the q-shift operator with respect to x : T f x... x n = f qx... x n. Downloaded from on 0 July 08

25 CFT APPROACH TO Q-PVI 5 Lemma A.5 For partitions λ μ α βwehave X ε λμαβ σ ; x x = C q ε σ / q ε σ / q lλ ε + +σ +/ T X + λμαβ + σ + ; x x Y + λμαβ σ ; x x = C q± σ / q σ + q lλ++σ T Y + λμαβ + σ + ; x x Y λμαβ σ ; x x = C q σ x x q lλ T Y λμαβ + σ + ; x x where C = α lλ j= β lλ j= q j+ σ / q lαlλj+j+ σ 3/ q j σ / q l β lλj+j σ 3/ q lλ α β x lλ. q lλ i+aαi+ σ +/ lλ i= q lλ i+a β i σ +/ Proof of Proposition A.. By induction. If all partitions λ μ α and β are equal to then the identity A. is the connection formula of basic hypergeometric functions. By definition we have the symmetries lλ X ε λμαβ σ ; x x = X ε μλαβ σ ; x x X λμαβ σ ; x x = X + λμβα σ ; x x Y λμαβ σ ; x x = Y + μλαβ σ ; x x Y ε λμαβ σ ; x x = Y ε λμβα σ ; x x B ε x ] = B σ +ε x ] B σ ε x ] = B σ ε+ x ] σ x and by Lemma A. we have x X ε λμαβ σ ; x x = Y ε β α μ λ σ ; qx q x. Recall also the relations. and. for the braiding matrix. Because of these relations it suffices to show that for any partitions λ μ α and β the connection formula A. is equivalent to the formula A. for λ μ α and β acting by a q-difference operator. i= x x Downloaded from on 0 July 08

26 6 M. JIMBO ET AL. By Lemma A.5 the left-hand side of A. is computed as follows: q q +/4 x ε =± Ɣ q + + ε σ X + λμαβ x Ɣ q + σ ; x x = q q + +/4 x q /4 ε =± Ɣ q ε σ + q ± σ / x Ɣ q + q Cq σ / q lλ + +σ +/ T X + λμαβ + σ + ; x x = Cq σ 3/4 q± σ / q q q lλ+ +σ +3/4 T q + x x ε =± Ɣ q ε σ + Ɣ q + +/4 X + λμαβ + σ. Similarly by Lemma A.5 the right-hand side of A. is computed as follows: εσ+/4 q σ qx ε =± Ɣ q + ε + εσ Y ε λμαβ x ε=± Ɣ q + εσ σ ; x x B ε+ x ] q / x σ x x εσ+/+/4 ε =± Ɣ q + ε + + εσ + q ± σ / Ɣ q + εσ + q = q σ qx +/ x ε=± + B ε+ σ + x ] q +/ +// x x x Cq lλ++εσ +/ T Y ε λμαβ + σ + = Cq σ 3/4 q± σ / q { q lλ+ +σ +3/4 T ε=± qx + B ε+ σ + x ] q +/ +// x x +/ q σ 3/4 q σ +/ ε =± Ɣ q + ε + + εσ + Ɣ q + εσ + εσ+/+/4 Y ε λμαβ + σ + x x +/ }. Therefore the identity A. with parameters σ for λ μ α β is reduced to the identity A. with parameters + / σ + / for λ μ α β. Downloaded from on 0 July 08

27 CFT APPROACH TO Q-PVI 7 References. Gamayun O. Iorgov N. & Lisovyy O. 0 Conformal field theory of Painlevé VI. JHEP p.. Gamayun O. Iorgov N. & Lisovyy O. 03 How instanton combinatorics solves Painlevé VIV and IIIs. J. Phys. A: Math. Theor Iorgov N. Lisovyy O. & Tykhyy Yu. 03 Painlevé VI connection problem and monodromy of c = conformal blocks. JHEP p. 4. Iorgov N. Lisovyy O. & Teschner J. 05 Isomonodromic tau-functions from Liouville conformal blocks. Commun. Math. Phys Bershtein M. & Shchechkin A. 05 Bilinear equations on Painlevé τ functions from CFT. Commun. Math. Phys Gavrylenko P. & Lisovyy O. Fredholm determinant and Nekrasov sum representations of isomonodromic tau functions. arxiv: Nagoya H. 05 Irregular conformal blocks with an application to the fifth and fourth Painleve equations. J. Math. Phys Bershtein M. & Shchechkin A. 07 Bäcklund transformation of Painlevé IIID8 τfunction. J. Phys. A: Math. Theor p. 9. Bonelli G. Lisovyy O. Maruyoshi K. Sciarappa A. & Tanzini A. On Painleve/gauge theory correspondence. arxiv: Gavrylenko P. & Lisovyy O. Pure SU gauge theory partition function and generalized Bessel kernel. arxiv: Bershtein M. & Shchechkin A. 07 q-deformed Painlevé τ function and q-deformed conformal blocks. J. Phys. A: Math. Theor p.. Awata H. Feigin B. & Shiraishi J. 0 Quantum algebraic approach to refined topological vertex. JHEP p. 3. Sakai H. 998 Casorati determinant solutions for the q-difference sixth Painlevé equation. Nonlinearity Tsuda T. & Masuda T. 006 q-painlevé VI equation arising from q-uc hierarchy. Commun. Math. Phys Feigin B. & Tsymbaliuk A. 0 Heisenberg action in the equivariant K-theory of Hilbert schemes via Shuffle Algebra. Kyoto J. Math Feigin B. Hashizume K. Hoshino A. Shiraishi J. & Yanagida S. 009 A commutative algebra on degenerate CP and Macdonald polynomials. J. Math. Phys Miki K. 007 A q γanalog of the W + algebra. J. Math. Phys Shiraishi J. Kubo H. Awata H. & Odake S. 996 A quantum deformation of the Virasoro algebra and the Macdonald symmetric functions. Lett. Math. Phys Awata H. & Yamada Y. 00 Five-dimensional AGT conjecture and the deformed Virasoro algebra. JHEP p. 0. Jimbo M. & Sakai H. 996 A q-analog of the sixth Painlevé equation. Lett. Math. Phys Sakai H. 00 Rational surfaces associated with affine root systmes and geometry of Painlevé equations. Commun. Math. Phys Ramani A. Grammaticos B. & Hietarinta J. 99 Discrete versions of the Painlevé equations. Phys. Rev. Lett Mano T. 00 Asymptotic behaviour around a boundary point of the q-painlevé VI equation and its connection problem. Nonlinearity Downloaded from on 0 July 08