Blowup relations on C 2 / Z 2 from Nakajima-Yoshioka blowup relations
BBlowup relations on C / Z from Nakajima-Yoshioka blowup relations Anton Shchechkin
Abstract
We obtain bilinear relations on Nekrasov partition functions, arising from study of tau func-tions of quantum q -Painlev´e equations, from Nakajima-Yoshioka blowup relations by an elementaryalgebraic approach.Additionaly, using this approach, we prove certain relations on Nekrasov partition functionsmodified by Chern-Simons term. Contents C / Z from blowup relations on C a r X i v : . [ m a t h - ph ] J un Introduction
Background and main results.
This paper is motivated by studies of so-called Painlev´e/gaugetheory (or Isomonodromy/CFT) correspondence, starting with the work [GIL12], where Painlev´e VItau function was written as a Fourier series of SU (2) Nekrasov partition function of instantons on C with four matters and (cid:15) + (cid:15) = 0 τ ( σ, s | z ) = (cid:88) n ∈ Z s n Z ( a + 2 n(cid:15) ; − (cid:15) , (cid:15) | z ) . (1.1)Then plenty generalizations of this formula appeared, in particular for the tau functions of Painlev´eV, III’s equations [GIL13], for the tau functions of q -Painlev´e equations [BS16q], [JNS17],[MN18]as well as for isomonodromic problems, more sophisticated than those corresponding to Painlev´eequations ([G15], [ILT14] etc.) The main idea of this generalization (and of Painlev´e/gauge theorycorrespondence) is that for each tau function on the Painlev´e side we should relate certain instantonpartition function on gauge side, such that tau function will be given by the Fourier series (1.1).Particularly, for the Painlev´e III( D (1)8 ) equation it is pure gauge SU (2) Nekrasov instanton partitionfunction on C , for q -Painlev´e equations one should take 5d instanton partion functions, adding onecompact dimension of radius R = − log q , for isomonodromic problems of rank N we should take SU ( N ) gauge group etc.It turns out that Painlev´e (differential and q -difference) equations and, presumably, more sophis-ticated isomonodromic problems are written as bilinear equations on these tau functions. Accordingto (1.1), such equations are equivalent to certain bilinear relations on Nekrasov partition functions,which have form (cid:88) n ∈ Z + j/ D (cid:16) Z ( a − n(cid:15) ; − (cid:15) , (cid:15) | z ) , Z ( a + 2 n(cid:15) ; − (cid:15) , (cid:15) | z ) (cid:17) = 0 , j = 0 , , (1.2)where D is certain differential or q -difference operator. One of the approaches to the proof ofPainlev´e/gauge theory correspondence in particular cases is to find such relations on appropriatepartition functions. For differential Painlev´e equations that was done from the CFT side of AGTrelation, using representation theory of Super Virasoro algebra ([BS14],[BS16b]). On the gauge the-ory side bilinear relations on Nekrasov partition functions appear from Nakajima-Yoshioka blowuprelations (proved in [NY05]) β dj ( q , q | z ) Z ( u ; q , q | z ) = (cid:88) n ∈ Z + j/ (cid:16) Z ( uq n ; q , q q − | q d z ) Z ( uq n ; q q − , q | q d z ) (cid:17) , q i = e R(cid:15) i , (1.3)namely, by excluding partition function in the l.h.s. from two such relations. However, in r.h.s. ofthese relations Ω-background parameters differ from that in (1.5).Appropriate relations from the gauge theory side of AGT possibly could be obtained from theblowup relations on C / Z (possibly modified by 5th compact dimension). Namely, in [BMT11] (seealso [BPSS13]) 4d blowup formula was proved Z X ( a, (cid:15) , (cid:15) | z ) = (cid:88) n ∈ Z (cid:16) Z ( a + 2 n(cid:15) ; 2 (cid:15) , − (cid:15) + (cid:15) | z ) , Z ( a + 2 n(cid:15) ; (cid:15) − (cid:15) , (cid:15) | z ) (cid:17) , (1.4)where X is minimal resolution of C / Z . However, 5d modification of these blowup relations seemto be missing in the literature. Reason for such two names for one correspondence is that Painlev´e equations arise from particular cases of isomon-odromic problems on Riemann surfaces with punctures, and that instanton partition functions of supersymmetric gaugetheories equal to certain CFT conformal blocks according to the AGT relation [AGT09]. q -Painlev´e equations, is (cid:88) n ∈ Z Z ( uq n ; q , q q − | q z ) Z ( uq n ; q q − , q | q z ) = (1 − ( q q ) / z / ) (cid:88) n ∈ Z Z ( uq n ; q , q q − | z ) Z ( uq n ; q q − , q | z ) . (1.5)It was proposed in [BS16q] (see (B.5) in loc. cit.) and proved in [BS18] for Painlev´e equations case q q = 1. It was proved in an elementary way, using Nakajima-Yoshioka blowup relations, but itseems that used approach cannot be generalized for arbitrary q , q . This and analogous relations areimportant for study of the quantum Painlev´e equations, namely they appear in Conjecture 4.2 from[BGM17] and also as in [BGM18] for Nekrasov partition function modified by Chern-Simons term. Inthis paper we find elementary way to obtain such relations for arbitrary q , q from Nakajima-Yoshiokablowup relations.Namely, results are as follows • We proved relations from Conjecture 4.2 from [BGM17]. These relations are above mentioned(1.5), and (3.18), (3.21), (3.22) from the main text. • We proved relation (3.23) on level 1 Chern-Simons-modified Nekrasov partition functions. • Using our approach, we prove certain relations on Chern-Simons modified Nekrasov partitionfunctions, namely Z [2] inst ( u ; q , q | z ) = ( z ; q , q ) ∞ Z [0] inst ( u ; q , q | z ) , (1.6)and (2.9) in the main text. Here the number in the square brackets indicates the Chern-Simonslevel.All these relations are relations on 5d SU (2) pure gauge Nekrasov partition functions and we provedthem for arbitrary q , q . As we mentioned above, (3.19) as well as (3.23), were proved in [BS18] for q q = 1. Relation (1.6) was proved in [BS18] for q q = 1, q q = 1, q q = 1 and relation (2.9) wasproved in [GNY06, Prop. 1.38] for q q = 1. It is easy to take standard limit to bilinear relations on4d Nekrasov partition functions, we do not discuss this. Method.
The method is based on the fact, that blowing up C twice in a way, represented onFig. 1 (where filed circle is point of an actual blowup) we get certain − . Note that presence or absence of additional 5th compact dimension does not affect onblowup geometry.As mentioned above, we are interested in bilinear relations, which could be obtained from blowupof C / Z . Its exceptional divisor is − C blowup, where it is − (cid:15) , (cid:15) this is represented by the value I = (cid:15) (1)1 /(cid:15) (1)2 + (cid:15) (2)2 /(cid:15) (2)1 , where (cid:15) ( η )1 , , η = 1 , −
1, and for X blowup relation (1.4) it equalsto −
2. Therefore we will call bilinear relations on Nekrasov partition functions with I = − − − (cid:98) Z ( u ; q , q ) = 0 and convolute it with another Nekrasovpartition function (here we omit dependence on z for simplicity) (cid:88) m ∈ Z Z ( u ( q q ) m ; q q , q − ) (cid:98) Z ( uq m ; q , q ) . (1.7) Strictly speaking, proof is done only for the case (cid:15) /(cid:15) ∈ Q ≤ , because only in this case we can guarantee convergenceof appropriate Nekrasov partition functions, see Subsection 2.1 for details. There is no such problem in 4d case. We are grateful to Hiraku Nakajima, who suggested to use this observation to study of bilinear relations on Nekrasovpartition functions, arising from Painlev´e equations. (cid:15) (cid:15) (cid:15) (cid:15) − (cid:15) (cid:15) − (cid:15) (cid:15) (cid:15) (cid:15) − (cid:15) (cid:15) − (cid:15) (cid:15) − (cid:15) (cid:15) − (cid:15) Figure 1: Blowup scheme of the approach2. Substituting (cid:98) Z ( q , q ) = (cid:80) terms (cid:80) n ∈ Z Z ( q , q q − ) Z ( q q − , q ) and using Nakajima-Yoshiokablowup relations (1.3) twice, we obtain linear combination of Nekrasov partition functions (cid:88) terms (cid:88) m,n ∈ Z Z ( q q , q − ) Z ( q , q q − ) Z ( q q − , q ) = (cid:88) terms (cid:88) m ∈ Z Z ( q q , q q − ) Z ( q q − , q ) = (cid:88) terms Z ( q q , q ) , (1.8)where we also omitted dependence on u , which is shifted appropriately. First sum representsseveral terms in the initial bilinear relation.3. Suppose that this linear combination is zero. Finally we prove, that initial − z -powers of the initial relations equalzero.Note that connection between − − Content.
In Section 2 we recall necessary facts about Nekrasov partition functions and Nakajima-Yoshioka blowup relations. There we also prove convergence of 5d pure gauge SU (2) Nekrasov partitionfunction for (cid:15) /(cid:15) ∈ Q < .Section 3 describes our approach to obtain − − Acknowledgements.
We thank Hiraku Nakajima for telling the idea, that inspired writing thispaper, and pointing us to references [FS94], [B94], Mikhail Bershtein for interest to our work andstimulating discussions, Roman Gonin for discussion of Prop. 3.1.We are grateful to Pavlo Gavrylenko and Mykola Semenyakin for a careful reading of the Intro-duction.This work is partially supported by HSE University Basic Research Program and funded (par-tially) by the Russian Academic Excellence Project ’5-100’. Classification of − Nekrasov functions and Nakajima-Yoshioka blowup relations
We start from reviewing Nekrasov partition functions Z of pure SUSY SU (2) gauge theory on C extended by the 5th compact dimension and discuss their components.Full Nekrasov partition function Z splits into three factors (we follow conventions of [NY03L],[NY05]) Z = Z cl Z − loop Z inst . (2.1)In loc. cit. Z cl and Z − loop appear from the so-called ”perturbative” part. Nekrasov function dependson parameters of the Ω-background (cid:15) , (cid:15) , vacuum expectation values a , a with condition a + a = 0(we denote a = a − a ) and also on the radius R of the 5th compact dimension. In 5d case it isconvenient to use multiplicative parameters, connected with above by u i = e Ra i , q i = e R(cid:15) i , i = 1 , u u = 1 (we denote u = u /u ). To obtain pure SUSY SU (2) gauge theory on C oneshould tend R →
0, we will not discuss such limit in this paper. Sometimes we also want to modifyour pure SUSY SU (2) gauge theory by additional Chern-Simons theory of level l ∈ Z . We denotesuch instanton partition function (and related objects) by superscript [ l ].In this paper we will consider only region (cid:15) < < (cid:15) , which corresponds to central charge c ≤ (cid:15) + (cid:15) = 0 and2 (cid:15) + (cid:15) = 0 which are interesting in context of applications to Painlev´e equations, as we explainedin the Introduction. This region is also closed under Nakajima-Yoshioka blowup relations (see (2.21)below), which we discuss at the end of this Section. Instanton part of Nekrasov partition functions.
Instanton part of 5d Nekrasov function, mod-ified by Chern-Simons theory of level l is given by Nekrasov formula Z [ l ] inst ( u ; q , q | z ) = (cid:88) λ (1) ,λ (2) (cid:81) i =1 ( q q ) − l | λ ( i ) | T lλ ( i ) ( u i ; q , q ) (cid:81) i,j =1 N λ ( i ) ,λ ( j ) ( u i /u j ; q , q ) ( q − q − z ) | λ (1) | + | λ (2) | , (2.3)written in terms of combinatorial block N λ,µ ( u ; q , q ) = (cid:89) s ∈ λ (cid:16) − uq − a µ ( s ) − q l λ ( s )1 (cid:17) (cid:89) s ∈ µ (cid:16) − uq a λ ( s )2 q − l µ ( s ) − (cid:17) (2.4)and Chern-Simons term T λ ( u ; q , q ) = (cid:89) ( i,j ) ∈ λ u − q − i q − j . (2.5)Here λ (1) , λ (2) are partitions, | λ | = (cid:80) λ j and a λ ( s ) , l λ ( s ) denote lengths of arms and legs for the box s in the Young diagram corresponding to the partition λ .The function Z [ l ] inst ( u ; q , q | z ) satisfies elementary symmetry properties: Z [ l ] inst ( u ; q , q | z ) = Z [ l ] inst ( u ; q , q | z ) = Z [ l ] inst ( u − ; q , q | z ) . (2.6)For l = 0 there is also elementary symmetry Z inst ( u ; q , q | z ) = Z inst ( u ; q − , q − | z ) . (2.7)Its proof is based on term by term coincidence of the power series, that’s why we have immediately Z [ − l ] inst ( u ; q , q | z ) = Z [ l ] inst ( u ; q − , q − | z ) (2.8)5n the case q q = 1 the symmetry q , q (cid:55)→ q − , q − is equivalent to the symmetry q ↔ q forarbitrary l .For general q , q , the situation with q , q (cid:55)→ q − , q − symmetry is more subtle. For l (cid:54) = 0 termby term comparison does not work. For l = 1, however, one has Z [1] inst ( u ; q , q | z ) = Z [1] inst ( u ; q − , q − | z ) . (2.9)The proof for the case q = q − , q = q case is given in [GNY06, Prop. 1.38]. We will prove thisequality in Subsection 4.2 for arbitrary q , q .For l = 2 it turns out that Z [2] inst ( u ; q , q | z ) = ( z ; q , q ) ∞ Z [0] inst ( u ; q , q | z ) , (2.10)so that Z [2] inst ( u ; q − , q − | z ) = 1 − z ( z ; q ) ∞ ( z ; q ) ∞ Z [2] inst ( u ; q , q | za ) . (2.11)We will discuss and prove equality (2.10) in Subsection 4.2. Convergence of Nekrasov partition functions.
Let us consider the convergence of the series(2.3). We proved that
Proposition 2.1.
Let q = q − m , q = q n , m, n ∈ Z ≥ , | q | (cid:54) = 1 and u (cid:54) = q k , k ∈ Z . Then series (2.3) for l = 0 converges uniformly and absolutely on every bounded subset of C . The proof of this Proposition is similar to the one in [ItsLTy14, Prop 1 (i)] and generalize Propo-sition [BS16q, Prop. 3.1.] to the case (cid:15) /(cid:15) ∈ Q < . Proof.
There exist constants L , L ∈ R > , such that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q k/ − q − k/ q / − q − / (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) > | k | L / , ∀ k ∈ Z (cid:54) =0 , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u / q k/ − u − / q − k/ q / − q − / (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) > L / , ∀ k ∈ Z . (2.12)Then we can bound (cid:81) i,j =1 N λ i ,λ j ( u i /u j ; q − m , q n ) as (cid:12)(cid:12)(cid:12) N λ ,λ (1; q − m , q n ) N λ ,λ (1; q − m , q n ) (cid:12)(cid:12)(cid:12) = (cid:89) s ∈ λ | q | m − n | q ( n ( a λ ( s )+1)+ ml λ ( s )) − q − ( n ( a λ ( s )+1)+ ml λ ( s )) || q ( na λ ( s )+ m ( l λ ( s )+1)) − q − ( na λ ( s )+ m ( l λ ( s )+1)) |· (cid:89) s ∈ λ ( λ ↔ λ ) > (cid:89) s ∈ λ | h λ | (cid:89) s ∈ λ | h λ | (cid:12)(cid:12)(cid:12) q m − n min ( m, n ) L ( q / − q − / ) (cid:12)(cid:12)(cid:12) | λ | + | λ | >> | λ | ! | λ | ! (dim λ dim λ ) (cid:12)(cid:12)(cid:12) q m − n min ( m, n ) L ( q / − q − / ) (cid:12)(cid:12)(cid:12) | λ | + | λ | , (2.13) (cid:12)(cid:12)(cid:12) N λ ,λ ( u ; q − m , q n ) N λ ,λ ( u − ; q − m , q n ) (cid:12)(cid:12)(cid:12) == (cid:89) s ∈ λ | q | m − n | u q ( n ( a λ ( s )+1)+ ml λ ( s )) − u − q − ( n ( a λ ( s )+1)+ ml λ ( s )) |· | u q ( na λ ( s )+ m ( l λ ( s )+1)) − u − q − ( na λ ( s )+ m ( l λ ( s )+1)) |· (cid:89) s ∈ λ ( λ ↔ λ , u ↔ u − ) > (cid:12)(cid:12)(cid:12) q m − n L ( q / − q − / ) (cid:12)(cid:12)(cid:12) | λ | + | λ | , (2.14)6here we used hook length formula for dim λ . Since (cid:80) | λ | = n (dim λ ) = n ! we have Z inst ( u ; q − , q | z ) < exp (cid:12)(cid:12)(cid:12)(cid:12) zq m − n min ( m, n ) L L ( q / − q − / ) (cid:12)(cid:12)(cid:12)(cid:12) .For (cid:15) /(cid:15) / ∈ Q arguments of above proof do not work. In this case poles in u of sum (2.3) are denseand it seems that the series diverges. However, we do not have any proof. For l (cid:54) = 0 we also don’tknow any proof, however numerical experiments suggest that it diverges when | l | >
2. According tosymmetry (2.8) below we will restrict ourselves to the levels l = 0 , ,
2. Such restriction is also naturalfrom the cluster point of view [BGM18]. We will consider case l = 2 only to discuss relation (2.10).Note also that results on convergence of Nekrasov instanton partition functions in other sector(namely (cid:15) , (cid:15) > (cid:15) , (cid:15) ) were obtained in paper[FML17]. Classical and 1-loop part of Nekrasov partition functions.
Classical and 1-loop parts of 5dNekrasov function are given by Z cl ( u ; q , q | z ) = ( q − q − z ) − log2 u q q , (2.15) Z − loop ( u ; q , q ) = ( u ; q , q ) ∞ ( u − ; q , q ) ∞ , (2.16)where q -Pochhammer symbol defined by( z ; q , . . . q N ) ∞ = ∞ (cid:89) i ,...i N =0 (cid:32) − z N (cid:89) k =1 q i k k (cid:33) (2.17)satisfy q -shift relations( z ; q , . . . q N ) ∞ / ( zq ; q , . . . q N ) ∞ = ( z ; q , . . . q N ) ∞ , ( z ; q ) ∞ / ( zq ; q ) ∞ = 1 − z. (2.18)These parts do not depend on l (however, Z cl becomes depend on l in case of SU ( r ), r > q ↔ q , u (cid:55)→ u − are also satisfied by classical and 1-loop partsof the full Nekrasov function Z . However, for the symmetry q , q (cid:55)→ q − , q − we have Z cl ( u ; q − , q − | z ) = ( q q ) − log2 u q q Z cl ( u ; q , q | z ) , (2.19)and Z − loop ( u ; q − , q − ) = ( uq ; q ) − ∞ ( u − ; q ) − ∞ ( u ; q ) − ∞ ( q u − ; q ) − ∞ Z − loop ( u ; q , q ) (2.20)(where we used properties ( z ; q − , q , . . . q N ) ∞ = ( zq ; q , . . . q N ) − ∞ and (2.18) successively), so sym-metry is broken for all cases except q q = 1. Functions Z [ l ] ( u ; q , q | z ) are known to satisfy Nakajima-Yoshioka blowup relations [NY05], [GNY06] β dj ( q , q | z ) Z [ l ] ( u ; q , q | z ) = (cid:88) n ∈ Z + j/ (cid:16) Z [ l ] ( uq n ; q , q q − | q d + l ( j − z ) Z [ l ] ( uq n ; q q − , q | q d + l ( j − z ) (cid:17) , (2.21)for j = 0 , l ∈ Z / Z , d = − , ,
1. Such coefficcients β dj turned out to be independent from l , they aregiven in the table β dj d = − d = 0 d = 1 j = 0 1 1 1 j = 1 ( q − q − z ) / − ( q q z ) / l = 0 and in Theorem 2.11 in [NY09] forthe case l = 1 , j = 0 and case l = 1 , j = 1, d = 0. We have not found cases l = 1, j = 1, d = ± | d | > d = ±
2) in Subsection 4.1.Note that, in fact, Nakajima-Yoshioka blowup relations are relations on Z [ l ] inst , and Z cl and Z − loop give ( l dn ) − z n , where z -independent coefficcient l dn is called blowup factor. Remark 2.1.
According to symmetries (2.8) , (2.19) , (2.20) coefficients β d, [ l ] j for arbitrary d satisfy β d, [ l ] j ( q − , q − | z ) = ( − j β − d, [ − l ] j ( q , q | z ) , (2.22) where we restored β d, [ l ] j dependence on l for arbitrary d . C / Z from blowup relations on C There are also blowup relations on Nekrasov partition functions on C / Z which have form Z [ l ] X ( u ; q , q | z ) = (cid:88) n ∈ Z + j/ D (cid:16) Z [ l ] ( uq n ; q , q q − | z ) , Z [ l ] ( uq n ; q q − , q | z ) (cid:17) , j = 0 , Z [ l ] X is certain instanton partition function on X , which is minimal resolution of C / Z . Explicittype of this partition function depends on bilinear q -difference in z operator D in r.h.s.As already mentioned in Introduction, such relations can be used to prove formula (1.1) for taufunctions of q -difference Painlev´e equations. However, to do this one needs only bilinear relations onNekrasov partition functions, which can be obtained from the above relations by exluding Z [ l ] X . Theserelations are given by sum of following terms (cid:98) Z [ l ] d ( u ; q , q | z ) = (cid:88) n ∈ Z + j/ (cid:15) n Z [ l ] ( uq n ; q , q q − | q d z ) Z [ l ] ( uq n , q q − , q | q d z ) , d ∈ Z , (3.2)with coefficcients, independent from u . We will see below that sign (cid:15) = ± j = 1.We call such bilinear relations ” − (cid:15) (1)1 /(cid:15) (1)2 + (cid:15) (2)2 /(cid:15) (2)1 = − −
1. Ouraim is to find approach to derive − l = 0. Generalization for l = 1will be given in Subsection 3.4.Let us make a certain convolution of Z ( u ; q q , q − | z ) with (3.2) (cid:88) m ∈ Z + j/ (cid:15) m Z ( u ( q q ) m ; q q , q − | ( q q ) d + d z ) (cid:98) Z d ( uq m ; q , q | q d + d z ) (3.3)or, explicitly writing − (cid:88) m,n ∈ Z + j/ (cid:15) m + n Z ( uq m q m ; q q , q − | ( q q ) d + d z ) ×Z ( uq n q m ; q , q q − | q d − d q d + d z ) Z ( uq m + n )2 , q q − , q | q d − d q d + d z ) , (3.4)8here we introduced d and d , such that d = d − d . Let us make substitution m = m (cid:48) + n (cid:48) , n = m (cid:48) − n (cid:48) in this expression (cid:15) j (cid:88) m (cid:48) ∈ Z + j/ ,n (cid:48) ∈ Z + (cid:15) (cid:88) m (cid:48) ∈ Z + j/ ,n (cid:48) ∈ Z + Z ( u ( q q ) m (cid:48) + n (cid:48) ) ; q q , q − | ( q q ) d + d z ) Z ( u ( q q ) m (cid:48) ( q − q ) n (cid:48) ; q , q q − | q d − d q d + d z ) × Z ( uq m (cid:48) , q q − , q | q d z ) (3.5)Using Nakajima-Yoshioka blowup relations (2.21) for the first pair of Nekrasov partition functions(summing up over n (cid:48) ), we obtain (cid:88) m (cid:48) ∈ Z (cid:15) m (cid:48) β d m (cid:48) + j mod 2 ( q q , q q − | ( q q ) d z ) Z ( u ( q q ) m (cid:48) ; q q , q q − | ( q q ) d z ) Z ( uq m (cid:48) , q q − , q | q d z ) , (3.6)which could be summed up again, using Nakajima-Yoshioka blowup relations (2.21), to (cid:88) i =0 , (cid:15) i β d i + j mod 2 ( q q , q q − | ( q q ) d z ) β d i ( q q , q | z ) Z ( u ; q q , q | z ) (3.7)Let us take sum of terms (3.2) and make convolution (3.3) with the whole sum. Assume that aftersuch convolution we obtain Z ( u ; q q , q | z ) with zero coefficient. Note that to make such convolution,uniform for all terms, we should take terms in the sum with the same shift d + d .Below we prove that the initial sum is also zero in case j = 0. Case j = 1 is more subtle: we provethat initial sum vanishes, when there are two different convolutions of the initial sum. In principle,we have such possibility, because we could take different pairs d and d , s.t. d − d = d . Below forsimplicity we denote d = d + d . Proposition 3.1.
Consider − bilinear relation on Nekrasov partition functions which is sum ofterms (3.2) .(i) Case j = 0 . If there is a vanishing convolution, given by (3.3) , then initial − bilinear relationis satisfied.(ii) Case j = 1 . If there is at least two vanishing convolutions, s.t. d (cid:54) = d (cid:48) , then initial − bilinear relation is satisfied.Proof. In terms of Z inst , summands (3.2) have the form( q − q − z ) − log2 u q q (cid:88) n ∈ Z + j/ (cid:15) n u nd ( q q ) n d ( q − q − z ) n l dn Z inst ( uq n ; q , q q − | q d z ) Z inst ( uq n , q q − , q | q d z ) , (3.8)where l dn are blowup coefficcients for C / Z blowup, resulting from Z − loop .We convolute relation from these summands with Z ( u ; q q , q − | z ) = z log2 u q q q + ∞ (cid:88) p =0 b p ( u ) z p , (3.9) b ( u ) = ( q q − ) log2 u q q q Z − loop ( u ; q q , q − ) (3.10)(i) In the case j = 0 the whole sum, as series in z , has form z − log2 u q q + ∞ (cid:88) k =0 c k ( u ) (cid:15) k z k/ , (3.11)9o we see that − z andsign (cid:15) signifies that.After convolution we have up to common factor (cid:88) m ∈ Z (cid:15) m u ( d ) m ( q q ) ( d ) m z m + ∞ (cid:88) p =0 b p ( uq m q m )(( q q ) ( d ) z ) p + ∞ (cid:88) k =0 c k ( uq m )( q d z ) k/ = 0 (3.12)Then, equating to zero coefficcients in front of power z n/ , n ∈ Z ≥ , due to b ( u ) (cid:54) = 0 we obtain c = 0 , c = 0 , c = 0 . . . successively.(ii) In the case j = 1 there is no such splitting z − log2 u q q + ∞ (cid:88) k =0 d k ( u ) z k . (3.13)Two vanishing convolutions with d (cid:54) = d (cid:48) give us up to common factor (cid:88) m ∈ Z +1 / (cid:15) m u d m ( q q ) d m z m + ∞ (cid:88) p =0 b p ( u ( q q ) m )(( q q ) d z ) p + ∞ (cid:88) k =0 d k ( uq m )( q d z ) k = 0 (3.14) (cid:88) m ∈ Z +1 / (cid:15) m u d (cid:48) m ( q q ) d (cid:48) m z m + ∞ (cid:88) p =0 b p ( u ( q q ) m )(( q q ) d (cid:48) z ) p + ∞ (cid:88) k =0 d k ( uq m )( q d (cid:48) z ) k = 0 (3.15)Equating to zero coefficcients in front of power z n +1 / , n ∈ Z ≥ , we will obtain 2 × d n ( uq ) and d n ( uq − ) with inhomogeneity that equals linear combination of d k , k < n . Fundamentalmatrix of this system is (cid:32) u d / ( q q ) d / b ( uq q ) q d p u − d / ( q q ) d / b ( uq − q − ) q d p u d (cid:48) / ( q q ) d (cid:48) / b ( uq q ) q d (cid:48) p u − d (cid:48) / ( q q ) d (cid:48) / b ( uq − q − ) q d (cid:48) p (cid:33) , (3.16)its determinant is equal to( q q ) ( d + d (cid:48) ) / b ( uq q ) b ( u ( q q ) − ) q ( d + d (cid:48) ) p ( u ( d − d (cid:48) ) / − u ( d (cid:48) − d ) / ) , (3.17)which is nonzero for general u . So we obtain d = 0 , d = 0 , d = 0 . . . successively.We see that in case j = 0 sign (cid:15) corresponds to the branch of square root z / , and we will omitit in next Subsection. Let us itemize different − − d + d We knowNakajima-Yoshioka blowup relations (2.21) only for cases d = − , ,
1, that’s our another restriction. • Case d + d = 2 . We have only one term with d = d = 1 and coefficient in (3.7) is nonzero,so there is no relation. • Case d + d = 1 . We have two terms: with d = 1 , d = 0 and d = 0 , d = 1. Both coefficientsin (3.7) equal 1. We obtain relation (cid:88) n ∈ Z Z ( uq n ; q , q q − | q z ) Z ( uq n ; q q − , q | q z ) = (cid:88) n ∈ Z Z ( uq n ; q , q q − | q − z ) Z ( uq n ; q q − , q | q − z ) , (3.18)this is conjectured relation (4.24) from [BGM17].10 Case d + d = 0 . We have three terms: with d = 0 , d = 0, d = 1 , d = − d = − , d =1. Coefficcients in (3.7) are equal to 1, 1 − ( q q ) − / z / and 1 − ( q q ) / z / respectively. Weobtain relations (cid:88) n ∈ Z Z ( uq n ; q , q q − | q z ) Z ( uq n ; q q − , q | q z )= (1 − ( q q ) / z / ) (cid:88) n ∈ Z Z ( uq n ; q , q q − | z ) Z ( uq n ; q q − , q | z ) (3.19)and (cid:88) n ∈ Z Z ( uq n ; q , q q − | q − z ) Z ( uq n ; q q − , q | q − z )= (1 − ( q q ) − / z / ) (cid:88) n ∈ Z Z ( uq n ; q , q q − | z ) Z ( uq n ; q q − , q | z ) (3.20)which is equivalent to previous relation (3.19) under substitution q , q (cid:55)→ q − , q − according toRemark 2.1. Relation (3.19) is conjectured relation (B.5) from [BS16q] and (4.25) from [BGM17]One can see easily that cases d + d = − , − According to Prop. 3.1 (ii) we start from itemizing different vanishing convolutions. After that wewill look for appropriate pairs of convolutions and write corresponding − • Case d + d = 2 . We have only one term with d = d = 1 and coefficient in (3.7) is − ( q q z ) / (1 + (cid:15) ), so we have vanishing convolution only for (cid:15) = − • Case d + d = 1 . We have two terms: with d = 1 , d = 0 and d = 0 , d = 1. Coefficients in(3.7) equal − ( q z ) / and − (cid:15) ( q q ) / ( q z ) / respectively, so we have two vanishing convolutionswith (cid:15) = ± • Case d + d = 0 . We have three terms: with d = 0 , d = 0, d = 1 , d = − d = − , d =1. Coefficcients are equal to 0, ( q q − z ) / ( (cid:15)q − −
1) and (cid:15)q ( q q − z ) / ( (cid:15)q − −
1) respectively,so we have vanishing convolutions consisting from first term or from next two terms, both with (cid:15) = ± • Case d + d = − . We have two terms: with d = − , d = 0 and d = 0 , d = −
1. Coeffi-cients in (3.7) equal ( q − z ) / and (cid:15) ( q q ) − / ( q − z ) / respectively, so we have two vanishingconvolutions with (cid:15) = ± • Case d + d = − . We have only one term with d = d = − q − q − z ) / (1 + (cid:15) ), so we have vanishing convolution only for (cid:15) = − − d + d = ± (cid:15) = ± (cid:88) n ∈ Z +1 / (cid:15) n Z ( uq n ; q , q q − | q z ) Z ( uq n ; q q − , q | q z )= (cid:15) ( q q ) / (cid:88) n ∈ Z +1 / (cid:15) n Z ( uq n ; q , q q − | q − z ) Z ( uq n ; q q − , q | q − z ) . (3.21)This is just relation (4.23) from [BGM17], it could be seen by taking sum and difference of relationswith two different (cid:15) . 11e also have − d + d = 0and d + d = ± (cid:15) = − (cid:88) n ∈ Z +1 / ( − n Z ( uq n ; q , q q − | z ) Z ( uq n ; q q − , q | z ) = 0 . (3.22)This is just relation (4.22) from [BGM17].This exhausts the relations, which follow from the above vanishing convolutions. Obtaining − Z [ l ] ( u ; q , q | z ) by our approach seems to be much more subtle andcumbersome. Let us illustrate this approach on relation (cid:88) n ∈ Z Z [1] ( uq n ; q , q q − | q z ) Z [1] ( uq n ; q q − , q | q z )= (1 − ( q q ) / z / ) (cid:88) n ∈ Z Z [1] ( uq n ; q , q q − | q z ) Z [1] ( uq n ; q q − , q | q z ) , (3.23)which for q q = 1 was proposed in [BGM18] (in terms of tau functions for k = 1 , N = 2 see formula(3.7) in loc. cit.).We should modify our convolution (3.3) for l = 1 in the following way. Split our − z (up to common factor) z − log2 u q q ( + ∞ (cid:88) k =0 c k ( u ) z k + (cid:15) + ∞ (cid:88) k =0 c k +1 ( u ) z k +1 / ) = (cid:98) Z [1]0 ( z ) + (cid:98) Z [1]1 ( z ) (3.24)Then make a modified convolution (braces denote fractional part of the number) (cid:88) m ∈ Z ,p =0 , Z [1] ( u ( q q ) m ; q q , q − | ( q q ) l (2 { ( m + p ) / }− z ) (cid:98) Z [1] n ( uq m , q , q ; q l (2 { ( m + p ) / }− z ) , (3.25)namely, convolution shift become dependent on relative parity of m and p As before, using Nakajima-Yoshioka blowup relations (2.21) twice we obtain β − β + β − ( q q , q q − | q q z ) β ( q q , q | z ) = β β + β β − ( q q ) / z / ( β β + q − / β − β ) , (3.26)where we wrote dependence on variables only where it is necessary. This expression turns out to bean identity.Finally, above convolution, written as sum of z -powers, is (cid:88) m ∈ Z (cid:15) m z m + ∞ (cid:88) p =0 b p ( uq m q m )(( q q ) l (2 { ( m + p ) / }− z ) p + ∞ (cid:88) k =0 c k ( uq m )( q l (2 { ( m + p ) / }− z ) k/ = 0 , (3.27)and, as before, c n = 0 , n ≥ Using our approach in opposite direction, we could obtain Nakajima-Yoshioka blowup relation (2.21)for l = 0, j = 0, d = 2. We could always write such relation with unknown function β , , which apriori12s a power series in z and dependens on u . Let’s calculate it. Let us take convolution (3.3) of (3.19),taking d = 0 , d = 2 and d = d = 1 for the corresponding summands of the relation. Because (3.19)is already proved, reducing convolution to linear combination of Nekrasov partition functions as wasdone in Subsection 3.1, we obtain β β ( u ; q q , q | z ) + β β = (1 − ( q q z ) / )( β β + β ( q q , q q − | q q z ) β ( q q , q | z )) = 1 − q q z, (4.1)where we wrote dependence on variables only where it is necessary. So, thanks to β = 0, we find β β ( q , q | z ) = 1 − q q z (4.2)and, according to (2.22) β − ( q , q | z ) = 1 − q − q − z. (4.3)In the same manner we can find β , [1]0 ( q , q | z ) for l = 1 (in fact, it differs from above β ). Analogouscalculation with (3.23) and convolution (3.25), where other shift l (2 { ( m + p ) / } − (cid:55)→ l (2 { ( m + p ) / } −
1) is taken, we obtain β β ( u ; q q , q | z )+ β β = β β + β ( q q , q q − | q q z ) β ( q q , q | z ) − ( q q ) / z / ( β β ( u ; q q , q | z )+ q − β β ) , (4.4)so we find that β , [1]0 ( q , q | z ) = 1, which differ from (4.2). (2.10) and (2.9) This Subsection starts from proving equality (2.10), using Nakajima-Yoshioka blowup equations (2.21)for l = 2 and l = 0 in sector j = 0.As already mentioned in [BS18], in terms of topological strings this relation means a relationbetween the geometry of local F = P × P and local Hirzebruch surface F . Particularly, the relationbetween Gopakumar-Vafa invariants of these manifolds is given in e.g. [IKP02, eq. (94)]. Case q q = 1 of (2.10) appear in [BGM18], it is proved together with cases q q = 1 , q q = 1 in [BS18,Prop. 4.3.]. Before this papers we have not found equality (2.10) in the literature, but it is maybeknown. For full Nekrasov functions we have the same equality, because Z cl and Z − loop do not dependon l Proposition 4.1.
Nekrasov function Z [2] is equal to Z [0] up to double q -Pochhammer symbol Z [2] ( u ; q , q | z ) = ( z ; q , q ) ∞ Z [0] ( u ; q , q | z ) , (4.5)We follow proof of [BS18, Prop. 4.3.], sligthly modifying it for arbitrary q , q .Consider Nakajima-Yoshioka blowup relations (2.21) for l = 2, j = 0, d = − , , Z [2] ( u ; q , q | z ) from them (cid:88) n ∈ Z Z [2] ( uq n ; q , q q − | q − z ) Z [2] ( uq n ; q q − , q | q − z ) == (cid:88) n ∈ Z Z [2] ( uq n ; q , q q − | q − z ) Z [2] ( uq n ; q q − , q | q − z ) == (cid:88) n ∈ Z Z [2] ( uq n ; q , q q − | z ) Z [2] ( uq n ; q q − , q | z ) . (4.6)These equations are equations on Z [2] inst , or bilinear equations on coefficients c (1) k , c (2) k , k ∈ Z ≥ ofthe corresponding power series Z [2] inst ( u ; q , q q − | z ) = (cid:80) + ∞ k =0 c (1) k ( u ; q , q ) z k and Z [2] inst ( u ; q q − , q | z ) = (cid:80) + ∞ k =0 c (2) k ( u ; q , q ) z k . Namely, relations (4.6) split into the relations corresponding to powers z k , k ∈ Z ≥ (up to the power Λ − log2 u log q q from Z cl ). 13 emma 4.1. ([BS18, Lemma 4.1.]) Relations (4.6) recursively determine coefficients c (1) k , c (2) k , k ∈ N starting from normalization c (1)0 = c (2)0 = 1 .Proof. Let us take the coefficient of the power z k in the relation (4.6), then coefficients c (1) k , c (2) k appearonly for n = 0. Other coefficients in these relations have lower index, so they are known due to theinduction supposition. Therefore we obtain system of two linear equations on two unknown variables c (1) k , c (2) k . The fundamental matrix of this system is as follows (cid:18) q − k − q − k − q − k − q − k − (cid:19) , (4.7)its determinant equals ( q − k − q − k − q − k − q − k ) which is non-zero iff none of q , q , q /q is a rootof unity.In our sector | q | ≶ , | q | ≷
1, these special cases are not realized.
Proof of the Proposition 4.1.
Consider Nakajima-Yoshioka blowup relations (2.21) for l = 0, j =0, d = − , − , Z [2] ( u ; q , q | z ) (1 − q − q − z ) − (cid:88) n ∈ Z Z [0] ( uq n ; q , q q − | q − z ) Z [0] ( uq n ; q q − , q | q − z ) == (cid:88) n ∈ Z Z [0] ( uq n ; q , q q − | q − z ) Z [0] ( uq n ; q q − , q | q − z ) == (cid:88) n ∈ Z Z [0] ( uq n ; q , q q − | z ) Z [0] ( uq n ; q q − , q | z ) . (4.8)Let us replace Z [0] with (cid:101) Z [2] , formally defined by (4.5). From calculation with q -Pochhammers( q − z ; q , q q − ) ∞ ( q − z ; q q − , q ) ∞ ( z ; q , q q − ) ∞ ( z ; q q − , q ) ∞ = 1 , ( q − z ; q , q q − ) ∞ ( q − z ; q q − , q ) ∞ ( z ; q , q q − ) ∞ ( z ; q q − , q ) ∞ = 11 − q − q − z , (4.9)we obtain that (cid:101) Z [2] satisfies (4.6). Therefore, according to Lemma 4.1 (cid:101) Z [2] = Z [2] (for general q , q ),which is desired relation (4.5).Relation (2.9) is obtained, using the same idea, but even simpler. Proposition 4.2.
Nekrasov instanton partition function Z [1] inst is invariant under q , q (cid:55)→ q − , q − Z [1] inst ( u ; q , q | z ) = Z [1] inst ( u ; q − , q − | z ) . (4.10) Proof.
Using (2.8), we find out that we should prove Z [ − inst ( u ; q , q | z ) = Z [1] inst ( u ; q , q | z ) (4.11)For l = 1 take Nakajima-Yoshioka blowup relations d = 0 , , l = − d = − , ,
1. Excluding Z [ ± ( u ; q , q | z ) from these relations we obtain two bilinear relations on Z [ l ] ( u ; q , q q − | z ) and Z [ l ] ( u ; q q − , q | z ) both for l = − l = 1. And these pairs of relations areidentical. Here analogue of fundamental matrix (4.7) is matrix (cid:32) q k/ − q − k/ q k/ − q − k/ q k/ − q − k/ q k/ − q − k/ (cid:33) , (4.12)which determinant is equal to ( q q ) − k/ ( q − k − q − k − q − k − q − k ) which is non-zero iff none of q , q , q /q is a root of unity. This completes the proof.14 Further questions • Relation (3.19) was already written in terms of quantum tau functions, i.e., tau functions of form(1.1), where [ σ, log s ] = (cid:126) (see (4.15) in [BGM17]) It will be interesting to rewrite our approachin terms of certain quantum tau functions. • Using this approach, we possibly could find − − C / Z p , p > • In terms of quantum tau functions this approach probably will be easy to generalize for SU ( N ), N >
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