Refined geometric transition and qq-characters
YYITP-17-47OU-HET 933
Refined geometric transition and qq -characters Taro Kimura ∗ , Hironori Mori † , and Yuji Sugimoto ‡∗ Department of Physics, Keio University, Kanagawa 223-8521, JapanFields, Gravity & Strings, CTPU, Institute for Basic Science, Daejeon 34047, Korea † Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan ‡ Department of Physics, Graduate School of Science, Osaka University,Toyonaka, Osaka 560-0043, Japan
Abstract
We show the refinement of the prescription for the geometric transition in the refinedtopological string theory and, as its application, discuss a possibility to describe qq -characters from the string theory point of view. Though the suggested way to operatethe refined geometric transition has passed through several checks, it is additionallyfound in this paper that the presence of the preferred direction brings a nontrivial effect.We provide the modified formula involving this point. We then apply our prescription ofthe refined geometric transition to proposing the stringy description of doubly quantizedSeiberg–Witten curves called qq -characters in certain cases. ∗ [email protected] † [email protected] ‡ [email protected] a r X i v : . [ h e p - t h ] O c t ontents qq -characters from refined geometric transition 13 qq -character . . . . . . . . . . . . . . . . . . . . 133.1.1 Y -operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.2 A quiver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.2.1 U(1) theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.2.2 SU( N ) theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.2.3 Higher qq -character . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.3 A quiver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.4 Generic quiver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.4.1 A r quiver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.4.2 DE quiver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.4.3 Beyond ADE quiver . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
A.1 Mathematical preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32A.2 Refined topological vertex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
B Regularity 36
B.1 A quiver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36B.1.1 U(1) gauge theory with single Y -operator . . . . . . . . . . . . . . . . 36B.1.2 U(1) gauge theory with two Y -operators . . . . . . . . . . . . . . . . . 37B.2 A quiver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391 Introduction
We have encountered the great developments of exact methods and a variety of their ap-plications in quantum field theory, for instance, the Seiberg–Witten theory [1, 2] and theNekrasov partition function for instanton counting problem [3, 4] as prominent landmarks,which are part of subjects in this paper. Correspondingly, the string theory and M-theoryrealization of these ingredients have been established and passed through a pile of checks inliteratures. Specifically, (the 5d uplift of) the Nekrasov partition function can be systemat-ically obtained by using the topological vertex [5] that is a powerful ingredient to calculatethe amplitude in the topological string theory [6, 7, 8, 9] for a given Calabi–Yau threefold asthe target space. The free energy of the topological string amplitude is expanded standardlywith respect to the genus and the string coupling constant. The latter is translated into theΩ-backgrounds ( (cid:15) , (cid:15) ) in a special limit (cid:15) + (cid:15) = 0 which is called the unrefined (self-dual)limit. Since the Nekrasov partition function could be actually formulated for a general valueof ( (cid:15) , (cid:15) ), the refined version of the topological vertex to include two parameters given by q = e π i (cid:15) , q = e π i (cid:15) (1.1)has been suggested by [10, 11], which was named the refined topological vertex . Their defini-tion could successfully reproduce the Nekrasov partition function with general Ω-backgroundin many circumstances and bring us to meaningful outcomes from string theory to super-symmetric gauge theories (basically with eight supercharges).It has been shown in [12, 13, 14] that the open string and closed string sector in theusual (i.e. unrefined) topological sting theory is just linked by the geometric transition(open/closed duality). However, underlying physics for the geometric transition in the refinedtopological string theory that we would refer to as the refined geometric transition is not yetwell understood mainly because there is no known world-sheet interpretation of it. Recently,great quantitative support for the refined geometric transition was reported by [15].The prescription for geometric transition in terms of the refined topological vertex hasbeen proposed [16] and basically checked in the context of the AGT correspondence [17, 18],but it is not complete due to the possible choice of the so-called the preferred direction onthe refined topological vertex. The topological vertex is graphically a trivalent vertex, andthe proper point of the refined vertex different from the unrefined one is the existence of thepreferred direction that is a special direction out of three edges of the vertex. This does resultfrom the inclusion of ( q , q ) into the topological vertex. It is labeled by a Young diagramassigned on each edge, and as well, we pick up two of three edges to put ( q , q ) on. Thismeans that the the preferred direction as the last edge has a special role on the computation The convention here is translated into ( q, t ) for the Ω-backgrounds used in [11] as ( q, t − ) → ( q , q ).
2f the refined topological string amplitude. In the first half of the paper, it will be argued thatthe refined geometric transition has to be sensitive to the choice of the preferred direction,and we will provide another prescription to implement the refined geometric transition onthe web diagram constructed by vertices with the preferred direction that differs from theconventional one mentioned above.In order to check the consistency of our prescription, we explore double quantization of theSeiberg–Witten geometry, which is called the qq -character, by utilizing the refined geometrictransition. The qq -character has been recently introduced by Nekrasov in the context of theBPS/CFT correspondence [19, 20, 21]. It is a natural gauge theoretical generalization of the q -character of quantum affine algebra [22], corresponding to the Nekrasov–Shatashvili limit( q , q ) → ( e (cid:126) ,
0) [23, 24], because the qq -character is obtained with generic Ω-backgroundparameter ( q , q ). There are a lot of interesting connections with, for example, quiver gaugetheory construction of W-algebra (quiver W-algebra) [25, 26] , double affine Hecke algebra(DAHA) and Ding–Iohara–Miki (DIM) algebra [28, 29, 30, 31, 32], and so on.The qq -character plays a role as a generating function of the chiral ring operator, and isrealized as a defect operator. For example, it becomes a line operator in 5d gauge theory,which is a codimension-4 defect [33]. In this paper we propose how to realize the qq -characterin refined topological string by the brane insertion, analyzed using the refined geometrictransition. In particular, the codimension-2 defect operator, corresponding to the surfaceoperator in gauge theory, is obtained by inserting a defect brane to the Lagrangian submani-fold of the Calabi–Yau threefold [34, 16, 35, 36, 37, 38]. We show that the Y -operator, whichis a codimension-4 building block of the qq -character, can be constructed by inserting twocodimension-2 defect operators. Although the Y -operator itself has a pole singularity, weobtain the qq -character, having no singularity, as a proper combination of Y -operators. Theregularity of the qq -character is a nontrivial check of our prescription for refined geometrictransition.The remaining part of this paper is organized as follows: In Sec. 2 we propose a newprescription for geometric transition in refined topological string. In order to obtain a propercontribution of the Lagrange submanifold, we have to consider the shift of parameters, whichis not realized as a shift of the K¨ahler parameter, when the defect brane is inserted to theinner brane. In Sec. 3 we apply the prescription of the refined geometric transition to the qq -character, which is a generating function of the chiral ring operator. We examine severalexamples, especially A and A quivers, and obtain a consistent result with quiver gaugetheory. This shows a nontrivial check of our prescription of refined transition. We conclude See also an overview article [27]. The (log of) Y -operator plays essentially the same role as the resolvent in matrix model, which is agenerating function of the gauge invariant single-trace operator. We would upgrade the operation of the geometric transition in the refined topological stringtheory where the partition function can be in principle evaluated by the refined topologicalvertex [10, 11] for a given Calabi–Yau geometry (see Appendix A.2 for our convention). Asthere is a much wide variety of Calabi–Yau geometries, for simplicity and a purpose of theapplication to qq -characters, we restrict our argument to a simple class of the geometriesvisualized by a web diagram in Fig. 1. The thin dotted line connecting the upper and lowerend of the diagram represents a compactified direction in the geometry. Note that this typeis essentially equipped with the structure of the resolved conifold. It is known that thisgeometrical data can be dualized to type IIB string theory with D5-branes, NS5-branes, and(1 , .1 Conventional prescription Since there is no established world–sheet description of the refined topological string theoryso far, one need to fix a guiding principle for the refined geometric transition from anothercontext. One of frameworks to provide such a principle is the AGT correspondence [17] andits 5d uplift [39, 40]. This duality can be encoded into type IIB string theory presented bythe ( p, q )-fivebrane web diagram like Fig. 1. The dictionary between the ( p, q )-web and thegeometry allows us to compute the partition function of the corresponding gauge theory byutilizing the refined topological vertex [10, 11], which in the 4d limit turns out to be consistentwith the correlation function on the 2d conformal field theory (CFT) side in some cases. Soonafter finding the AGT relation, its statement has been extended to include the correspondencebetween a surface operator in the 4d N = 2 SU(2) gauge theory and a degenerate field inthe Liouville CFT [18]. This circumstance can also be realized in the framework of the( p, q )-web. The surface operator is engineered by inserting a D3-brane into the ( p, q )-web,which is further mapped to a Lagrangian brane representing a Lagrangian submanifold inthe corresponding Calabi–Yau. The computation of the topological string partition functionmust be incorporated with contributions from open strings when the target space is a Calabi–Yau with specified Lagrangian submanifolds. Although there is no established formula ofthe refined version of the open topological vertex, this can be evaluated by implementing thegeometric transition. In the 4d limit, the result obtained in this way is actually compatiblewith the correlation function in the presence of a degenerate field in the Liouville CFT.We would sketch concretely the rule of the refined geometric transition that has been leadfrom the AGT story. On the web diagram as shown in Fig. 1, each internal line implies thetopology of C P and is equipped with a K¨ahler modulus. Let Q ( s ) a be a K¨ahler modulus forthe s -th diagonal internal segment from the left in the a -th horizontal (uncompactified) linefrom the top (see Fig. 4 for our convention). The point of calculations along the AGT storywith this web diagram is that the preferred direction is chosen on the vertical (compactified)direction, which is depicted as black dots in Fig. 2 (throughout the paper, the vertical axis isalways the compactified direction and the horizontal one is uncompactified). The geometrictransition can be implemented with the horizontal (uncompactified) line: with appropriatelytuning K¨ahler moduli for diagonal lines attached to the b -th horizontal line, this line isdetached from the vertical lines and moved away. The geometric transition for the webdiagram of our interest is essentially the same as that of the conifold, passing through fromthe resolved conifold to the deformed conifold and vice versa. If one would like to suspend aLagrangian brane on the r -th vertical line in the process shown in Fig. 2, the K¨ahler moduli This is often called a toric brane, however, we do not use this term in the paper since the concerneddiagrams here are non-toric. b + 1 b r r + 1 r ←→ b b + 1 r r + 1 r Figure 2: The geometric transition operated on the unpreferred direction. The horizontalaxis is compactified and the dots indicate the preferred direction.are specialized as Q ( r ) b = q m q n √ q q , Q ( s ) b = 1 √ q q for s (cid:54) = r , (2.1)with m, n ∈ Z . This prescription can nicely produce the AGT relation with the surfaceoperator. Consequently, the refined geometric transition associated with the unpreferred direction is operated by (2.1).We are closing the review with commenting on the integers m, n in (2.1). It has beenargued in [16] that, in the 4d limit, the adjustment such that m, n > z m z n = 0 , (2.2)where ( z , z ) are complex coordinates on two-dimensional planes respecting the rotation bythe Ω-background parameters ( (cid:15) , (cid:15) ), respectively. This discussion seems to work for suchphysical surface operators, however, for the present we do not have a requirement to restrictthe range of m, n to be non-negative from the refined topological string point of view. Thisis why we take m, n to run for all integers. Although the refined geometric transition with m, n < q q = 1) contextare referred to as anti-branes [41]. We would return to this point in Section 4. We turn to giving our new prescription for the refined geometric transition that takes theissues of the preferred direction into account. On computing the refined topological string Note that the combination of q and q depends on ones convention. b + 1 b r r + 1 r ←→ b b + 1 r r + 1 r Figure 3: The geometric transition operated on the preferred direction. The horizontal axisis compactified and the dots indicate the preferred direction.amplitude for the web diagram of our main interest, the preferred direction is chosen alongthe horizontal, i.e., uncompactified direction, marked by dots in Fig. 3. The difference of thepreferred direction from the previous situation requires us to introduce small modification forthe refined geometric transition. In this subsection, we write down the process to implementthe refined geometric transition for the current choice of the preferred direction.A point which we should stress is to put the preferred direction on the uncompactified(horizontal) direction where the geometric transition can be carried out. In addition, forconsistency, it is required that the contributions from the Lagrangian brane is not producedif the web with (
M, N ) lines simply reduces to the one with (
M, N −
1) lines without theLagrangian brane after the geometric transition, where M and N stand for the number ofcompactified (vertical) and uncompactified (horizontal) lines, respectively (see Fig. 4(a)).Let us consider the geometric transition that is executed on the b -th horizontal line withthe Lagrangian brane emerging on the r -th vertical line (Fig. 3). Our proposal for the refinedgeometric transition under the above requirement is comprised of the following three steps:0. As a supposition the preferred direction is taken to be the uncompactified axis (hori-zontal here), and then one computes the refined closed topological string amplitude asdone in [42, 43].1. For s ≥ r , variables w ( s ) ab ( i, j ) and u ( s ) ab ( i, j ) defined in (2.7), which appear in the genericpartition function (2.6), are shifted by using the inversion (A.8) and the differenceequation (A.9) of the theta function, θ (cid:16) e π i w ( s ) ab ( i,j ) (cid:17) = Q / τ e π i w ( s ) ab ( i,j ) × θ (cid:16) Q − τ e − π i w ( s ) ab ( i,j ) (cid:17) ,θ (cid:16) e π i u ( s ) ab ( i,j ) (cid:17) = Q / τ e π i u ( s ) ab ( i,j ) × θ (cid:16) Q − τ e − π i u ( s ) ab ( i,j ) (cid:17) . (2.3)7. Then, tuning the K¨ahler moduli as Q ( s ) b = 1 √ q q ( s < r ) , Q ( r ) b = q m q n √ q q , Q ( s ) b = √ q q ( s > r ) , (2.4)with m, n ∈ Z .3. Finally, shifting variables w ( r ) ab ( i, j ) for all a by hand, w ( r ) ab ( i, j ) → w ( r ) ab ( i, j ) − (cid:15) − (cid:15) , (2.5)while others in (2.7) are kept unchanged.We should make a comment on the shift of step 3 in our prescription. The shift (2.5) hasnothing to do with the K¨ahler parameters: any K¨ahler parameter is not shifted togetherwith this operation, but rather, with viewing w ( r ) ab ( i, j ) as a single variable, it is just to add − (cid:15) − (cid:15) to it. This is purely a technical thing which is originated from the difference of thespecialization (2.4) of the K¨ahler moduli Q ( s ) b for s < r and s > r . The reason why we needthis shift is to satisfy the requirement for consistency that the refined geometric transitionwithout generating a Lagrangian brane reproduces the closed topological string amplitude(see below for numerical details). The step 2 and 3 reflect the dependence of the refinedgeometric transition on the preferred direction. Indeed, it is expected that, even thoughthe closed topological string amplitude should be independent of the preferred direction, theopen one really depends on whether or not the Lagrangian brane is attached to the preferreddirection (see, e.g., [44, 45]). This is basically because the Lagrangian brane can end on the( p, q )-fivebrane with general ( p, q ), therefore the geometric transition should be characterizedby ( p, q ) in addition to ( m, n ). This implies that the position of the preferred direction put onthe ( p, q )-fivebrane leads to the inequivalent result of the open topological string amplitude.Both procedures of the refined geometric transition can reproduce correctly the identicalresult in the unrefined limit q q = 1 as expected. Our prescription seems compatible withthis suggestion.A Lagrangian brane appears on only one vertical line upon a single sequence of the abovegeometric transition. If one desires to generate several Lagrangian branes on different verticallines for a web diagram, it is necessary to consider a bigger web and repeat the procedure(2.3)-(2.5) many times (as demonstrated in Section 3).We will devote the next subsection to showing quantitative clarification how this processworks and produces the refined topological string amplitude incorporating the contribution ofthe Lagrangian brane. In Section 3, it will be discussed that the refined geometric transitioninitiated by our prescription gives possibly how to realize the qq -character from string theory.8 ines M lines N Q (1) f, Q (1) f, Q (1) f,N Q (2) f,N Q (2) f, Q (2) f, Q ( M f, Q ( M f, Q ( M f,N (a) Entire web Q ( s +1) a Q ( s a Q ( s ) a Q ( s a Q ( s ) a Q ( s +1) a Q ( s +1) a +1 Q ( s ) a +1 Q ( s a +1 ˜ Q ( s a ˜ Q ( s a ˜ Q ( s ) a ˜ Q ( s ) a Q ( s a Q ( s ) a Q ( s ) a Q ( s a ˜ Q ( s +1) a ˜ Q ( s +1) a Q ( s a Q ( s a Q ( s ) a Q ( s ) a (b) Internal hexagons Figure 4: The web diagram with M vertical and N horizontal lines. Our prescription given above seems a bit intricate rather than (2.1), and we would explainhere why this works when the uncompactified line assigned with the preferred direction isremoved upon the geometric transition.
We are now concentrating on the compactified web shown as Fig. 4 with general (
M, N )lines. On the technique of the refined topological vertex (A.2), the partition function Z M,N for this web diagram has been derived as [43] Z M,N = (cid:88) { µ ( s ) a } s =1 , ··· ,M − a =1 , ··· ,N M − (cid:89) s =1 N (cid:89) a =1 (cid:16) ¯ Q ( s ) f,a (cid:17) | µ ( s ) a | (cid:89) ( i,j ) ∈ µ ( s ) a N (cid:89) b =1 θ ( e π i z ( s ) ab ( i,j ) ) θ ( e π i w ( s ) ab ( i,j ) ) θ ( e π i u ( s ) ab ( i,j ) ) θ ( e π i v ( s ) ab ( i,j ) ) , (2.6)9here e π i z ( s ) ab ( i,j ) = (cid:16) Q ( s +1) ab (cid:17) − q − µ ( s +1) ,tb,j + i − / q µ ( s ) a,i − j +1 / ,e π i w ( s ) ab ( i,j ) = (cid:16) Q ( s ) ba (cid:17) − q µ ( s − ,tb,j − i +1 / q − µ ( s ) a,i + j − / ,e π i u ( s ) ab ( i,j ) = (cid:16) ˆ Q ( s ) ba (cid:17) − q µ ( s ) ,tb,j − i +11 q − µ ( s ) a,i + j ,e π i v ( s ) ab ( i,j ) = (cid:16) ˆ Q ( s ) ab (cid:17) − q − µ ( s ) ,tb,j + i q µ ( s ) a,i − j +12 , (2.7)with t representing the transpose of the Young diagram (Fig. 11(c)). We collect the defini-tions and notations in Appendix A. Note that, for simplicity, we omit a complex parameter Q τ := e π i τ in the theta function as θ ( x ) unless otherwise stated. We denote the K¨ahlermoduli for diagonal, vertical, and horizontal internal segments by Q ( s ) a , ˜ Q ( s ) a , and Q ( s ) f,a , re-spectively, which are visualized in Fig. 4(a) and 4(b). The weights in the partition function,corresponding to instanton counting parameters in the Nekrasov partition function, are givenby ¯ Q ( s ) f,a = ( q q ) N − Q ( s ) f,a N (cid:89) b =1 Q ( s ) b . (2.8)Also, we use the simplified symbols for the products of the K¨ahler moduli in the variables(2.7), defined as Q ( s ) ab = Q ( s ) a N (cid:89) i = b Q ( s ) τ i (mod Q τ ) for a = 1 ,Q ( s ) a a − (cid:89) i =1 Q ( s ) τ i N (cid:89) j = b Q ( s ) τ j (mod Q τ ) for a (cid:54) = 1 , (2.9)for the numerator, and ˆ Q ( s ) ab = a − (cid:89) i = b Q ( s ) τ i for a > b, a = b,Q τ (cid:44) b − (cid:89) i = a Q ( s ) τ i for a < b, (2.10)for the denominator, where Q ( s ) τ i := Q ( s ) i ˜ Q ( s ) i = ˜ Q ( s +1) i Q ( s +1) i +1 . (2.11)The second equality follows from the consistency condition to form internal hexagons (Fig. 4(b)).It has been revealed that this geometry actually realizes an elliptically fibered Calabi–Yau10ith the complex modulus Q τ identified as Q τ = N (cid:89) a =1 Q ( s ) τ a for ∀ s . (2.12)We comment on the M-theory uplift of this picture. It is well known that type IIBstring theory compactified on S is dual to M-theory on the torus T . The web as inFig. 4 is rendered to the M-theory brane configuration where the stacks of M2-branes aresuspended between separated M M5-branes on an asymptotically locally Euclidean (ALE)space equipped with Z N orbifolding. This duality supports the fact that the low energytheory on the present ( p, q )-web are described by the tensor branch of the corresponding 6d N = (1 ,
0) theory. It has been argued in [42, 43] that the partition function (2.6) capturesthe spectra of self-dual strings, called M-strings, wrapping T in the 6d theory, and theYoung diagrams µ ( s ) a label the ground states of M-strings. When we perform the geometric transition on the b -th horizontal line such that the La-grangian brane appears on the r -th vertical line, the main contribution that should be care-fully treated is (cid:89) ( i,j ) ∈ µ ( s ) a θ (cid:16) e π i z ( s ) ab ( i,j ) (cid:17) θ (cid:16) e π i w ( s ) ab ( i,j ) (cid:17) θ (cid:16) e π i u ( s ) ab ( i,j ) (cid:17) θ (cid:16) e π i v ( s ) ab ( i,j ) (cid:17) (2.13)for all a . We would divide the calculation process for this into two parts with s < r and s ≥ r . Remark that we sometimes implicitly use the relation (2.12) to change the variables. For s < r We firstly focus on the sector for s < r where the geometric transition (2.4) can straight-forwardly work. One can easily see that (2.13) does not produce the nontrivial contributionunless µ ( s ) b = ∅ for ∀ s . (2.14)Accordingly, this condition is necessary in order to obtain the appropriate result for the par-tition function obtained via the geometric transition. Then, variables z ( s ) ab ( i, j ) and w ( s ) ab ( i, j )can be rewritten as θ (cid:16) e π i z ( s ) ab ( i,j ) (cid:17) = θ (cid:18)(cid:16) √ q q Q ( s +1) b (cid:17) − e π i v ( s ) ab ( i,j ) (cid:19) , (2.15) θ (cid:16) e π i w ( s ) ab ( i,j ) (cid:17) = θ (cid:18)(cid:16) √ q q Q ( s ) b (cid:17) − e π i u ( s ) ab ( i,j ) (cid:19) , (2.16)11here we used the relation (2.11) for z ( s ) ab ( i, j ). With these expressions, the specialization(2.4) of the K¨ahler moduli results in(2.13) → s < r − , (2.17)and (2.13) → (cid:89) ( i,j ) ∈ µ ( r − a θ (cid:16) q m q n e π i v ( r − ab ( i,j ) (cid:17) θ (cid:16) e π i v ( r − ab ( i,j ) (cid:17) for s = r − . (2.18)Indeed, (2.18) is the half of the contributions of the Lagrangian brane. This is just what wewant because this reduces to 1 when m = n = 0, namely, no Lagrangian brane appear, asrequired. Actually, this expression matches with the result of [46]. Moreover, the weights inthe sum of the Young diagrams change as¯ Q ( s ) f,a → ( q q ) ( N − − Q ( s ) f,a N (cid:89) c =1 c (cid:54) = b Q ( s ) c , (2.19)and this is nothing but the ones in the partition function for the web diagram with ( M, N − s < r . For s ≥ r Let us turn to the sector for s ≥ r . In addition to the first step (2.3), by using (2.11), therelation θ (cid:16) e π i z ( s ) ab ( i,j ) (cid:17) = θ (cid:32) Q ( s +1) b Q τ √ q q (cid:33) − e − π i u ( s ) ab ( i,j ) (2.20)holds under the restriction (2.14). As a result, we have(2.13) → (cid:89) ( i,j ) ∈ µ ( s ) a e π i( w ( s ) ab ( i,j ) − u ( s ) ab ( i,j )) θ (cid:32)(cid:18) Q ( s +1) b √ q q (cid:19) − Q − τ e − π i u ( s ) ab ( i,j ) (cid:33) θ (cid:16) Q − τ e − π i w ( s ) ab ( i,j ) (cid:17) θ (cid:16) Q − τ e − π i u ( s ) ab ( i,j ) (cid:17) θ (cid:16) e π i v ( s ) ab ( i,j ) (cid:17) = (cid:16) √ q q Q ( s ) b (cid:17) −| µ ( s ) a | (cid:89) ( i,j ) ∈ µ ( s ) a θ (cid:32)(cid:18) Q ( s +1) b √ q q (cid:19) − Q − τ e − π i u ( s ) ab ( i,j ) (cid:33) θ (cid:16) Q − τ e − π i w ( s ) ab ( i,j ) (cid:17) θ (cid:16) Q − τ e − π i u ( s ) ab ( i,j ) (cid:17) θ (cid:16) e π i v ( s ) ab ( i,j ) (cid:17) , (2.21)Similarly for (2.19), the overall factor can be absorbed into the associated weight so that¯ Q ( s ) f,a (cid:16) √ q q Q ( s ) b (cid:17) − = ( q q ) ( N − − Q ( s ) f,a N (cid:89) c =1 c (cid:54) = b Q ( s ) c , (2.22)12hich becomes the one for the web diagram with ( M, N −
1) lines. This is the actual reasonwhy our prescription needs the first step (2.3). Then, the parameter tuning (2.4) as thesecond step leads to (2.21) → s > r , (2.23)and (2.21) → (cid:89) ( i,j ) ∈ µ ( r ) a θ (cid:16) Q − τ e − π i( w ( r ) ab ( i,j ) − (cid:15) − (cid:15) ) (cid:17) θ (cid:16) e π i v ( r ) ab ( i,j ) (cid:17) = (cid:89) ( i,j ) ∈ µ ( r ) a θ (cid:18) Q − τ Q ( r ) ba q i +1 / q µ ( r ) a,i − j +3 / (cid:19) θ (cid:18) Q − τ Q ( r ) ba q i +1 / m q µ ( r ) a,i − j +3 / n (cid:19) for s = r , (2.24)where we implemented the shift (2.5) as the third step of our prescription. Note that Q − τ Q ( s ) ba does not contain Q ( s ) a due to (2.12). Namely, the shift (2.5) allows the remaining contribution(2.24) to satisfy the requirement that this becomes trivial when m = n = 0.As the conclusion of this subsection, the refined geometric transition in our scheme cor-rectly produces open string contributions associated to the Lagrangian brane given by (2.18)and (2.24). qq -characters from refined geometric transition In this section, we apply our prescription for the geometric transition to the qq -character,which has been recently proposed in the context of the BPS/CFT correspondence [19, 20, 21].We propose that when we consider the geometric transition so that two Lagrange sub-manifolds emerge, the contributions of two Lagrange submanifolds becomes Y -operator, de-pending on the position of the brane insertion. Let us examine our prescription with someexamples. qq -character Let us briefly remark some background of the qq -character in gauge theory. Nekrasov–Pestun [47] pointed out an interesting connection between the quiver gauge theory and therepresentation theory of the corresponding quiver. Their statement is that the Seiberg–Witten geometry of the Γ-quiver gauge theory in 4d is described by the characters of thefundamental representations of G Γ -group, where G Γ is the finite Lie group associated withthe (ADE) quiver Γ under the identification of the quiver with the Dynkin diagram. In13act, the prescription of Nekrasov–Pestun uses the Weyl reflection to generate the Seiberg–Witten curve, which is generic and applicable to any quiver, even if there is no finite group G Γ corresponding to the quiver Γ. Let us check this process with A quiver, which is thesimplest example. Since G Γ = SU(2) for Γ = A , the fundamental character is given by χ (SU(2)) = y + y − , (3.1)where the first term corresponds to the fundamental weight y , and the second term is gen-erated by the Weyl reflection y → y − . On the other hand, the Seiberg–Witten curve is analgebraic curve given as a zero locus of the algebraic functionΣ = { ( x, y ) | H ( x, y ) = 0 } , (3.2)where ( x, y ) ∈ C × C ∗ for 4d and ( x, y ) ∈ C ∗ × C ∗ for 5d gauge theory. For A quiver gaugetheory with SU( N ) vector multiplet without any matter fields, the function H ( x, y ) turnsout to be H ( x, y ) = y + y − − T N ( x ) , (3.3)where T N ( x ) is a degree N monic polynomial in x , T N ( x ) = x N + · · · . In other words, thecurve is characterized by the polynomial relation y + y − = T N ( x ) . (3.4)Now it is obvious that the LHS agrees with the SU(2) character (3.1). It is possible toderive this polynomial relation from the Γ-quiver gauge theory partition function with theΩ-deformation [3], and taking the Seiberg–Witten limit ( (cid:15) , (cid:15) ) → (0 , y -variable appearing in thealgebraic relation is realized as an expectation value of the Y -operator, which we focus on inthis paper, y ( x ) = (cid:68) Y ( x ) (cid:69) . (3.5)The Y -operator is a generating function of the chiral ring operators, so that it is realized asa codimension-4 defect operator. See [33] for its realization as the line operator in 5d gaugetheory. Furthermore the Y -operator itself has a cut singularity in the complex plane x ∈ C in the Seiberg–Witten limit, and its crossing-cut behavior is indeed described by the Weylreflection. This is the reason why the Weyl reflection generates the Seiberg–Witten curve.It was then shown by Nekrasov–Pestun–Shatashvili [24] that this representation theo-retic structure in gauge theory has a natural q -deformation: The Seiberg–Witten curve in There should be the coupling constant dependence on the LHS, but it is be now absorbed by redefinitionof the y -variable. (cid:15) , (cid:15) ) → ( (cid:126) ,
0) [23] is promoted to the q -character,which was originally introduced in the context of the quantum affine algebra [22] with em-phasis on its connection with the quantum integrable system. See also [48, 49, 50] for furtherdevelopments. This means that the polynomial relation holds in the NS limit just by replac-ing the character with q -character. In this case, the Seiberg–Witten curve is not an algebraiccurve any longer, but lifted to a quantum curve, which is a difference equation. For example,for A quiver theory, it is given by y ( x ) + 1 y ( q − x ) = T N ( x ; (cid:15) ) . (3.6)The RHS is again a degree N monic polynomial in x , but the coefficients may depend onthe equivariant parameter (cid:15) . In particular, this difference equation, also called the quan-tum (deformed) Seiberg–Witten curve [51, 52, 53], is equivalent to (precisely speaking, thedegenerate version of) the TQ-relation of the G Γ XXX/XXZ/XYZ spin chain for 4d/5d/6dgauge theory. Then, as a corollary, the SUSY vacuum (twisted F-term) condition of the 4dgauge theory in the NS limit is equivalent to the Bethe ansatz equation of the G Γ XXX spinchain.Recently it has been shown in the context of BPS/CFT correspondence [19, 20, 21] thata similar polynomial relation actually holds even with generic Ω-background parameters( (cid:15) , (cid:15) ) by replacing the q -character for the NS limit with a further generalized character,called the qq -character. For A quiver, in the 5d notation, it is given by y ( x ) + 1 y ( q − x ) = T N ( x ; (cid:15) , (cid:15) ) . (3.7)The qq -character has a gauge theoretical definition due to the invariance under the deformedWeyl reflection, which is called the iWeyl reflection, reflecting the non-perturbative aspects ofthe instanton moduli space. This qq -character relation is interpreted as a (non-perturbativeversion of) Ward identity or Schwinger–Dyson equation since it gives a relation betweencorrelation functions in quiver gauge theory.The y -function, which is the gauge theory average of the Y -operator, has pole singu-larities. But such singularities are canceled in the combination of y ( x ) and y ( q − x ) − . Ingeneral, the iWeyl reflection shows how to cancel the pole singularity of the y -function. Y -operator Before discussing the topological string setup, let us mention more about the Y -operator tofix our notation. For generic quiver theory, we define Y -operator associated with each gauge We use the 5d notation ( q , q ) = ( e (cid:15) , e (cid:15) ) and define q = q q = e (cid:15) + (cid:15) . The unrefined limit is givenby q = q − , namely q = 1. Precisely speaking, y ( q − x ) − means (cid:68) Y ( q − x ) − (cid:69) here. Y i for i ∈ Γ where Γ is a set of nodes in the quiver Γ. Then, in the 5d (K-theoretic)notation, the contribution of the Y -operator for the configuration µ is [47, 24] Y i,µ ( x ) = (cid:89) x (cid:48) ∈X i − x (cid:48) /x − q x (cid:48) /x (3.8)where we put SU( N i ) gauge group for the i -th node, and define X i = { x i,α,k } α =1 ,...,N i ,k =1 ,..., ∞ , x i,α,k = q µ i,α,k q k − Q i,α , Q i,α = e a i,α . (3.9)The parameter Q i,α is the multiplicative (K-theoretic) Coulomb moduli. The Y -operator hasseveral expressions Y i,µ ( x ) = N i (cid:89) α =1 (cid:89) ( j,k ) ∈ ∂ + µ i,α (cid:32) − q j − q k − Q i,α x (cid:33) (cid:89) ( j,k ) ∈ ∂ − µ i,α (cid:32) − q j q k Q i,α x (cid:33) − = N i (cid:89) α =1 (cid:18) − Q i,α x (cid:19) (cid:89) ( j,k ) ∈ µ i,α (1 − q j q k − Q i,α /x )(1 − q j − q k Q i,α /x )(1 − q j q k Q i,α /x )(1 − q j − q k − Q i,α /x ) (3.10)where ∂ ± µ is the outer/inner boundary of the partition µ , where we can add/remove a box,and q j − q k − Q i,α is the q -content of the box ( j, k ) ∈ µ i,α . From this expression it is easy tosee the asymptotic behavior of the Y -operator, which does not depend on the configuration µ , Y i,µ ( x ) −→ x → ∞ )( − N i Q i x − N i ( x →
0) (3.11)where we define the Coulomb moduli product Q i = N i (cid:89) α =1 Q i,α . (3.12)We remark that Q i = 1 for SU( N i ) theory, but keep it for latter convenience.In addition, from the expression (3.8) we obtain Y i,µ ( x ) = exp (cid:32) ∞ (cid:88) n =1 − x − n n O i,n (cid:33) , O i,n = (1 − q n ) (cid:88) x ∈X i x n . (3.13)Here O i,n is the contribution of the chiral ring operator for the configuration µ , which is givenby the single trace operator with respect to the complex adjoint scalar field O i,n = Tr Φ ni in4d, and the loop/surface operator wrapping the compactified S / T in 5d/6d. Actually, forthe gauge theory on R × T , the variable x takes a value in x ∈ ˇ T where ˇ T is a dual torusof T [47]. Thus the Y -operator is interpreted as a codimension-4 defect operator, whichplays a role as the generating function of the chiral ring operator.16et us introduce the elliptic Y -operator corresponding to 6d gauge theory, which is ob-tained by replacing the factors in (3.8) with the elliptic functions, Y i,µ ( x ) = (cid:89) x (cid:48) ∈X i θ ( x (cid:48) /x ) θ ( q x (cid:48) /x ) . (3.14)This is reduced to the operator in 5d gauge theory (3.8) in the limit Im τ → ∞ . We alsohave a similar combinatorial expression to (3.10) in the elliptic theory, Y i,µ ( x ) = N i (cid:89) α =1 θ ( Q i,α /x ) (cid:89) ( j,k ) ∈ µ i,α θ ( q j q k − Q i,α /x ) θ ( q j − q k Q i,α /x ) θ ( q j q k Q i,α /x ) θ ( q j − q k − Q i,α /x ) . (3.15)We will use this expression in the following sections. A quiver Let us consider the Y -operator in A quiver gauge theory. The Y -operator is a codimension-4defect operator, and we here try to find its realization using the lower codimension surfacedefects. Here we give the prescription:1. Consider the geometric transition so that the brane and the anti-brane emerge, andtune the distance between these branes.2. We shift the K¨ahler parameters Q ( s ) i → Q ( s ) i √ q q in order to make agreement withthe Nekrasov partition function.3. Finally, identifying the K¨ahler parameter which corresponds to the position of thebranes as the x -variable, the summation of all possible configurations of the braneand anti-brane in the Calabi–Yau is regular (invariant under the iWeyl reflection) forarbitrary x with a suitable µ -independent normalization factor.Let us demonstrate this prescription in several examples. For simplicity let us first consider the Abelian gauge theory. Comparing the Y -operator(3.15) with the contribution of the defect insertion shown in (2.24), it turns out to be a half of the Y -operator. Thus we can construct the Y -operator by merging two surface operatorswith respect to the q -brane and anti- q -brane, corresponding to the geometric transitionshown in Fig. 5. Now the dashed lines on the right and on the left denote the q -brane andanti- q -brane, respectively. We remark that the coupling constant is given by q − for the anti- q -brane instead of q , since the sign of the string coupling is opposite to the ordinary one [41], The convention of the theta function used here (Dirac) is different from that used in Ref. [26] (Dolbeault). Z M =2 ,N =3 defined in (2.6). For thispartition function, by setting Q (1)2 = Q (1)3 = ( q q ) − ,Q (2)2 = q ( q q ) − , Q (2)3 = q − ( q q ) − , Q (2) τ = q q − , (3.16)the partition function reduces Z , = (cid:88) µ ( ¯ Q f, ) | µ | (cid:89) ( i,j ) ∈ µ θ ( Q (2) − q µ i − j q i − ) θ ( Q (1) − q − µ i + j − q − i ) θ ( q − µ i + j q µ tj − i +11 ) θ ( q − µ i + j − q µ tj − i ) × (cid:89) ( i,j ) ∈ µ θ (( q − q Q (2)1 Q (2) τ ) − q j − q i − ) θ (( Q (2)1 Q (2) τ ) − q j − q i − ) θ (( q − Q (2)1 Q (2) τ ) − q j − q i − ) θ (( q Q (2)1 Q (2) τ ) − q j − q i − ) , (3.17)where we shift Q (1 , → √ q q Q (1 , . The products in the first line are the contributionsof the M-strings without the Lagrange submanifolds. Thus the products in the second linecorrespond to the contribution of the Lagrange submanifolds. The latter contributions areconsistent with the Y -operator defined in (3.15) for U(1) theory under the identification Q x := Q x = ( q Q (2)1 Q (2) τ ) − , (3.18)where Q is the multiplicative Coulomb moduli of U(1) theory. Thus the partition function Z , gives rise to the average of the Y -operator Z , −→ (cid:68) Y ( x ) (cid:69) . (3.19)18his average is defined with respect to the partition function Z , , which is the 6d U(1) N f = 2 Nekrasov function (cid:68) O ( x ) (cid:69) = (cid:88) µ O µ ( x ) Z U(1) µ (3.20a) Z U(1) µ = ( ¯ Q f, ) | µ | (cid:89) ( i,j ) ∈ µ θ ( Q (2) − q µ i − j q i − ) θ ( Q (1) − q − µ i + j − q − i ) θ ( q − µ i + j q µ tj − i +11 ) θ ( q − µ i + j − q µ tj − i ) (3.20b)where the parameters ¯ Q f, , and Q ( s )1 correspond to the gauge coupling and the (anti)fundamentalmass, respectively. We remark that we have to multiply the factor θ ( Q x ) to obtain a preciseagreement with the definition of Y -operator [26] because the µ -independent factor cannot befixed in the current formalism.We can also consider the following geometric transition, corresponding to the partitionfunction Z , as well. This configuration corresponds to the parametrization given byFigure 6: The geometric transition. Q (1)2 = q ( q q ) − , Q (1)3 = q − ( q q ) − , Q (1) τ = q q − ,Q (2)2 = Q (2)3 = ( q q ) , (3.21)and define Q x = Q x = ( q Q (1) τ ) − . (3.22)In this case the contribution of the Lagrange submanifolds reads (cid:89) ( i,j ) ∈ µ θ ( q i q j Q x ) θ ( q i − q j − Q x ) θ ( q i +12 q j Q x ) θ ( q i q j +11 Q x ) . (3.23)However this naive expression does not work. We have to shift the argument in the numeratoras discussed in Sec. 2.3, to obtain a consistent result, θ ( q i q j Q x ) θ ( q i − q j − Q x ) −→ θ ( q i +12 q j +11 Q x ) θ ( q i q j Q x ) . (3.24)19nder the identification Q x = Q /x , this configuration gives rise to the Y -operator inverseby multiplying a factor θ ( qQ x ) − ,1 Y µ ( q − x ) = θ ( qQ x ) − (cid:89) ( i,j ) ∈ µ θ ( q i q j ( qQ x )) θ ( q i − q j − ( qQ x )) θ ( q i q j − ( qQ x )) θ ( q i − q j ( qQ x )) . (3.25)Thus the partition function Z , under the parametrization (3.21) leads to the average of the Y -operator inverse Z , −→ (cid:28) Y ( q − x ) (cid:29) . (3.26)Although the Y -operator and its inverse themselves have pole singularities, we can con-struct a regular function using these two operators, as discussed in Sec. 3.1. In this case, thefundamental qq -character of A quiver, which has no singularity, is given by the average ofthe T -operator defined χ ( A ; q , q ) = (cid:68) T ( x ) (cid:69) := (cid:68) Y ( x ) (cid:69) + q P ( x ) (cid:68) Y ( q − x ) − (cid:69) (3.27)with the gauge coupling q = ¯ Q f, and the matter factor P ( x ) = θ ( Q (1)1 Q − x ) θ ( Q (2) − Q − q − x ) . (3.28)The average is taken with respect to the 6d U(1) Nekrasov function (3.20) as before. Thisshows that the T -operator average is given by the qq -character discussed in Sec. 3.1, and itsregularity is proven using the iWeyl reflection Y ( x ) −→ q P ( x ) Y ( q − x ) . (3.29)We provide a proof of the regularity of this qq -character in Appendix B. We remark that,comparing with (3.7), we have additional factors q and P ( x ) in this case. The former onecan be absorbed by redefinition of the Y -operator Y → q Y , and the latter is due to the(anti)fundamental matters, which is necessary for gauge/modular anomaly cancellation in6d gauge theory.The Y -operator and its inverse Y − correspond to the brane insertion to the right andleft NS5-branes, respectively, as shown in Figs. 5 and 6. These are all the possibilitiesfor the brane insertion because there are only two NS5-branes for A quiver theory wherethe right and left branes are connected by a suspended D5-brane. On the other hand, asmentioned in Sec. 3.1, the qq -character is generated by the iWeyl reflection (3.29) convertingthe Y -operator to its inverse, Y ( x ) → Y ( q − x ) − . The iWeyl reflection is a consequenceof creation/annihilation of instantons [19], which is a fluctuation on the suspended brane.Since the fluctuation affects the branes on the both sides, the brane insertion on the right istransferred to the left through the iWeyl reflection.20 .2.2 SU( N ) theory One can easily generalize this result to the non-Abelian case. Let us consider the followinggeometric transition corresponding to SU( N ) theory with the insertion (Fig. 7). In this case N+2 lines
N lines
N lines case 1 case 2
Figure 7: The geometric transition.we have two possible brane insertion to the right and left NS5-branes, which is actually thesame as U(1) theory discussed in Sec. 3.2.1. For the case 1, where the defect brane is insertedto the right NS5-brane, we obtain the Y -operator Y (cid:126)µ ( x ) = N (cid:89) a =1 θ ( Q a /x ) (cid:89) ( j,k ) ∈ µ a θ ( q j q k − Q a /x ) θ ( q j − q k Q a /x ) θ ( q j q k Q a /x ) θ ( q j − q k − Q a /x ) (3.30)under the parametrization Q (1) N +1 = Q (1) N +2 = ( q q ) − ,Q (2) N +1 = q ( q q ) − , Q (2) N +2 = q − ( q q ) − , Q (2) τ N +2 = q q − , (3.31a) ( Q (2)1 ) − ˜ Q (2) − N +2 =: Q /x ( a = 1)( Q (2) a (cid:81) a − i =1 Q (2) τ i ) − ˜ Q (2) − N +2 =: Q a /x ( a = 2 , . . . , N ) (3.31b)where we define N -tuple partition (cid:126)µ = ( µ , µ , . . . , µ N ), and the µ -independent factor N (cid:89) a =1 θ ( Q a /x ) is multiplied by hand. Thus the partition function Z ,N +2 tuned with theparameters (3.31) gives rise to the average of the Y -operator Z ,N +2 (3.31) −→ (cid:68) Y ( x ) (cid:69) . (3.32)The operator average is now taken with respect to 6d SU( N ) N f = 2 N Nekrasov function (cid:68) O ( x ) (cid:69) = (cid:88) (cid:126)µ O (cid:126)µ ( x ) Z SU( N ) (cid:126)µ (3.33a) Z SU( N ) (cid:126)µ = Q | (cid:126)µ | f N (cid:89) a =1 (cid:89) ( i,j ) ∈ µ a N (cid:89) b =1 θ ( Q (2) − ab q µ a,i − j q i − ) θ ( Q (1) − ba q − µ a,i + j − q − i ) θ ( ˆ Q (1) − ba q − µ a,i + j q µ tb,j − i +11 ) θ ( ˆ Q (1) − ab q µ a,i − j +12 q − µ tb,j + i )(3.33b)21here we define the total instanton number | (cid:126)µ | = N (cid:88) a =1 | µ a | . Imposing the condition Q (1) τ i = Q (2) τ i Q (2) − i Q (2) i +1 , the Coulomb moduli parameter in this SU( N ) Nekrasov function is relatedto that defined in (3.31b) as ˆ Q (1) ab = Q b /Q a ( a > b ) Q τ Q b /Q a ( a < b ) . (3.34)Similarly we obtain the Y -operator inverse Y − from the case 2 with the defect braneinserted to the left. The Y -operator and its inverse have pole singularities as before, but wecan use essentially the same combination as (3.27) to obtain a regular function, which is the qq -character χ ( A ; q , q ) = (cid:68) T ( x ) (cid:69) = (cid:68) Y ( x ) (cid:69) + q P ( x ) (cid:68) Y ( q − x ) − (cid:69) (3.35)where the coupling constant and the (anti)fundamental contribution are now given by q = Q f , and P ( x ) := N (cid:89) a =1 θ ( Q (1) a Q − a x ) θ ( Q (2) − a Q − a q − x ) . (3.36)One can show the regularity of the qq -character (the T -operator average) in a similar wayto U(1) theory, using the iWeyl reflection (3.29). We remark that the expression of the qq -character for SU( N ) theory (3.35) coincides with that for U(1) theory (3.7) apart from thematter factor P ( x ). The qq -character provides a universal relation, which does not dependon the gauge group rank, but does only on the quiver structure. qq -character The Seiberg–Witten curve and its quantizations for Γ-quiver theory are described using thefundamental ( q - and qq -)characters of G Γ -group. In addition, we can consider the higher-representation qq -character, which plays a role to determine the OPE of the generatingcurrents of quiver W-algebras [25]. In this case, we have to consider several Y -operators atthe same time, and construct a regular function which is invariant under the iWeyl reflection.Let us demonstrate how to treat multiple Y -operators in U(1) theory for simplicity.We start with the web diagram shown in Fig. 8. In this case we tune the followingparameters to obtain two Y -operators, Q (1)2 , Q (1)3 , Q (2)2 , Q (2)3 , (3.37a) Q (1)4 , Q (1)5 , Q (2)4 , Q (2)5 . (3.37b)22 ase 1 case 2 case 3 case 4 Figure 8: In this geometric transition we obtain the T -operator which consists of two Y -operators for A quiver. We set the K¨ahler parameters in the blue and red parts.The parameters (3.37a) and (3.37b) correspond to the blue brane and the red brane in Fig. 8,respectively. We show how to set the parameter in order to realize the brane configurationin each case:Case 1 : Y ( x ) Y ( x ) Q (1)2 = ( q q ) − , Q (1)3 = ( q q ) − , Q (1)4 = ( q q ) − , Q (1)5 = ( q q ) − ,Q (2)2 = q ( q q ) − , Q (2)3 = q − ( q q ) − , Q (2)4 = q ( q q ) − , Q (2)5 = q − ( q q ) − (3.38a)Case 2 : Y ( x ) / Y ( q − x ) Q (1)2 = ( q q ) − , Q (1)3 = ( q q ) − , Q (1)4 = q ( q q ) − , Q (1)5 = q − ( q q ) − ,Q (2)2 = q ( q q ) − , Q (2)3 = q − ( q q ) − , Q (2)4 = ( q q ) , Q (2)5 = ( q q ) (3.38b)Case 3 : Y ( x ) / Y ( q − x ) Q (1)2 = q ( q q ) − , Q (1)3 = q − ( q q ) − , Q (1)4 = ( q q ) − , Q (1)5 = ( q q ) − ,Q (2)2 = ( q q ) , Q (2)3 = ( q q ) , Q (2)4 = q ( q q ) − , Q (2)5 = q − ( q q ) − (3.38c)Case 4 : (cid:0) Y ( q − x ) Y ( q − x ) (cid:1) − Q (1)2 = q ( q q ) − , Q (1)3 = q − ( q q ) − , Q (1)4 = q ( q q ) − , Q (1)5 = q − ( q q ) − ,Q (2)2 = ( q q ) , Q (2)3 = ( q q ) , Q (2)4 = ( q q ) , Q (2)5 = ( q q ) (3.38d)23n the cases 2, 3, 4, we have to perform the q q -shift as before, where we define Q x := Q x = ( q Q (2)1 5 (cid:89) i =3 Q (2) τ i ) − = ( q (cid:89) i =3 Q (1) τ i ) , (3.39a) Q x := Q x = ( q Q (2)1 Q (2) τ ) − = ( q Q (1) τ ) (3.39b)Then the partition function Z , gives rise to the two-point function of the Y -operator, bymultiplying the µ -independent factor, Z , −→ (cid:68) Y ( x ) Y ( x ) (cid:69) (case 1) (cid:28) Y ( x ) Y ( q − x ) (cid:29) (case 2) (cid:28) Y ( x ) Y ( q − x ) (cid:29) (case 3) (cid:68)(cid:0) Y ( q − x ) Y ( q − x ) (cid:1) − (cid:69) (case 4) (3.40)where the average is taken with respect to the U(1) Nekrasov function (3.20). Then theaverage of the T -operator defined χ ( A ; q , q ) = (cid:68) T [2] ( x , x ) (cid:69) := (cid:68) Y ( x ) Y ( x ) (cid:69) + q P ( x ) S (cid:18) x x (cid:19) (cid:28) Y ( x ) Y ( q − x ) (cid:29) + q P ( x ) S (cid:18) x x (cid:19) (cid:28) Y ( x ) Y ( q − x ) (cid:29) + q P ( x ) P ( x ) Y ( q − x ) Y ( q − x ) (3.41)yields the qq -character of the degree-2 symmetric representation for A quiver, and its reg-ularity is again shown using the iWeyl reflection (3.29). Now the S -factor is defined [26] S ( x ) = θ ( q x ) θ ( q x ) θ ( qx ) θ ( x ) (3.42)and the matter factor P ( x ) is the same as (3.28). This qq -character is regular even in thecollision limit x → x , involving a derivative term, which is a specific feature to the qq -character [19]. In this limit, the cycle between the blue and red ones shrinks in Fig. 8. Weshow the proof of the regularity in Appendix B. We remark that we put the µ -independentfactors S ( x ) and P ( x ) to define the T -operator because it’s a matter of the normalization ofthe partition function.In general, the n -point function of the Y -operator for SU( N ) theory is obtained from thepartition function Z ,N +2 n with 2 n possible brane insertions, Z ,N +2 n −→ (cid:68) Y ( x ) · · · Y ( x n ) (cid:69) , (cid:28) Y ( x ) · · · Y ( x n ) Y ( q − x ) (cid:29) , (cid:28) Y ( x ) · · · Y ( x n ) Y ( q − x ) Y ( q − x ) (cid:29) , . . . (3.43)24e can construct the qq -character of the degree- n representation R n = (cid:3) · · · (cid:3) (cid:124) (cid:123)(cid:122) (cid:125) n for A quiverby summing up all the possible n -point functions of the Y -operator [19, 25, 26], with asuitable S -factor inserted, χ R n ( A ; q , q ) = (cid:68) T [ n ] ( x , . . . , x n ) (cid:69) := (cid:68) Y ( x ) · · · Y ( x n ) (cid:69) + · · · . (3.44) A quiver Next we consider the A quiver gauge theory to examine the qq -character using the refinedgeometric transition. As mentioned in Sec. 3.1, the Seiberg–Witten curve and its quantizationare associated with the fundamental representation character of G Γ -group for Γ-quiver gaugetheory. Thus in this case it is deeply related to the representation theory of SU(3) group.Since the qq -character generated by the iWeyl reflection does not depend on the gauge grouprank, let us focus on the Abelian A quiver theory, U(1) × U(1), for simplicity. We havethree possible ways to insert the defect brane as shown in Fig. 9. case1 case2 case3
Figure 9: In this geometric transition we obtain the T -operator for A quiver. Case 1
We consider the defect brane inserted to the right-most NS5-brane. In this case, the calcu-lation is essentially the same as that for A quiver shown in Fig. 5. We apply the followingconfiguration Q (1)2 = Q (1)3 = ( q q ) − , Q (2)2 = Q (2)3 = ( q q ) − ,Q (3)2 = q ( q q ) − , Q (3)3 = q − ( q q ) − , Q (3) τ = q q − , (3.45)25ith the Coulomb moduli parameter Q , x = ( q Q (3)1 Q (3) τ ) − . (3.46)Comparing with the Y -operator definition (3.15), the contribution of the defect brane leadsto Y ,µ ( x ) by multiplying the factor θ ( Q , /x ). Thus the partition function Z , gives riseto the average of Y ( x ) under the parametrization (3.45): Z , −→ (cid:68) Y ( x ) (cid:69) (3.47)where the operator average is taken with respect to 6d U(1) × U(1) Nekrasov function (cid:68) O ( x ) (cid:69) = (cid:88) µ ,µ O µ ,µ ( x ) Z U(1) × U(1) µ ,µ (3.48a) Z U(1) × U(1) µ ,µ = ¯ Q | µ | f, ¯ Q | µ | f, (cid:89) ( i,j ) ∈ µ θ ( Q (3) − q µ ,i − j q i − ) θ ( Q (2) − q − µ ,i + j − q µ t ,j − i ) θ ( q − µ ,i + j q µ t ,j − i +11 ) θ ( q − µ ,i + j − q µ t ,j − i ) × (cid:89) ( i,j ) ∈ µ θ ( Q (2) − q µ ,i − j q − µ t ,j + i − ) θ ( Q (1) − q − µ ,i + j − q − i ) θ ( q − µ ,i + j q µ t ,j − i +11 ) θ ( q − µ ,i + j − q µ t ,j − i ) , (3.48b)where we define the gauge couplings ¯ Q f, , and the Young diagrams µ , as follows,¯ Q f, = ¯ Q (2) f, , ¯ Q f, = ¯ Q (1) f, , (3.49) µ = µ (2)1 , µ = µ (1)1 . (3.50) Case 2
In this case, the defect brane is inserted to the middle brane. This configuration correspondsto the following parametrization Q (1)2 = Q (1)3 = ( q q ) − , Q (3)2 = Q (3)3 = ( q q ) ,Q (2)2 = q ( q q ) − , Q (2)3 = q − ( q q ) − , Q (3) τ = q q − , (3.51)and two Coulomb moduli parameters defined Q , x = ( q Q (2) τ ) − , Q , x = ( q Q (2)1 Q (2) τ ) − . (3.52)We remark that the difference between Q , and Q , is given by the factor Q (2)1 =: Q m ,which is interpreted as the bifundamental mass parameter, because such a bifundamentalmass can be absorbed by the shift of U(1) Coulomb moduli [24]. In this paper we do notexplicitly write the bifundamental mass parameter.In this case, the contribution of the Lagrange submanifolds reads (cid:89) ( i,j ) ∈ µ θ ( q i q j − Q , /x ) θ ( q i − q j Q , /x ) θ ( q i q j Q , /x ) θ ( q i − q j − Q , /x ) (cid:89) ( i,j ) ∈ µ θ ( q i − q j − Q , /x ) θ ( q i q j Q , /x ) θ ( q i +12 q j Q , /x ) θ ( q i q j +11 Q , /x ) . (3.53)26n order to obtain a consistent result, we have to shift the parameters of the numerator inthe second factor, as discussed in Sec. 2.3, θ ( q i − q j − Q , /x ) θ ( q i q j Q , /x ) −→ θ ( q i q j Q , /x ) θ ( q i +12 q j +11 Q , /x ) . (3.54)Multiplying the µ -independent factors, θ ( Q , /x ) and θ ( qQ , /x ) − , the µ - and µ -contributionsare written as Y ( x ) and Y − ( q − x ), respectively. Thus the partition function Z , becomesthe average of the Y -operator ratio, by tuning the parameters as (3.51), Z , −→ (cid:28) Y ( x ) Y ( q − x ) (cid:29) . (3.55)The average is again taken with respect to the U(1) × U(1) Nekrasov function (3.48).
Case 3
The remaining situation is that the defect brane is inserted to the left-most brane. In thiscase, the calculation is essentially the same as Fig. 6 for A quiver theory. Applying theparametrization Q (2)2 = Q (2)3 = ( q q ) , Q (3)2 = Q (3)3 = ( q q ) ,Q (1)2 = q ( q q ) − , Q (1)3 = q − ( q q ) − , Q (3) τ = q q − ,Q , x = ( q Q (2)1 Q (1) τ ) − (3.56)with a suitable q q -shift of the arguments to be consistent with the geometric transition,the partition function Z , yields Z , −→ (cid:28) Y ( q − x ) (cid:29) . (3.57) qq -characters Now we can construct the qq -character using all the possible brane insertions. The qq -character of the fundamental representation for A quiver theory, denoted by , is given bythe T -operator average, χ ( A ; q , q ) = (cid:68) T ( x ) (cid:69) := (cid:68) Y ( x ) (cid:69) + q P ( x ) (cid:28) Y ( x ) Y ( q − x ) (cid:29) + q q P ( x ) P ( x ) (cid:28) Y ( q − x ) (cid:29) (3.58)where the coupling constants are given by q = ¯ Q f, and q = ¯ Q f, , and the matter factorsare defined P ( x ) = θ ( q − Q (3) − Q − , x ) , P ( x ) = θ ( Q (1) − Q (2) − Q , /x ) . (3.59)27lthough each factor in (3.58) has pole singularities as before, the qq -character itself is aregular entire function in x , as shown in Appendix B. The local pole cancellation is performedby the iWeyl reflection Y ( x ) −→ q P ( x ) Y ( x ) Y ( q − x ) , Y ( x ) −→ q P ( x ) Y ( q − x ) Y ( q − x ) . (3.60)For A quiver, we have another representation, which is the anti-fundamental represen-tation denoted by ¯ . The corresponding qq -character is generated by applying the iWeylreflection (3.60) to the highest weight Y ( x ), χ ¯ ( A ; q , q ) = (cid:68) T ( x ) (cid:69) := (cid:68) Y ( x ) (cid:69) + q P ( x ) (cid:28) Y ( q − x ) Y ( q − x ) (cid:29) + q q P ( q − x ) P ( x ) (cid:28) Y ( q − x ) (cid:29) . (3.61)We remark that the operator Y ( x ) itself cannot be constructed by a single insertion of thedefect brane, but is realized as a composite operator: Y ( x ) = Y ( q − x ) × Y ( x ) Y ( q − x ) . (3.62)In other words, the operator Y ( x ) is obtained by two insertions of the defect branes to theright-most and the middle branes (see the case 1 in Fig. 10). Similarly the remaining terms case1 case2 case3 Figure 10: The geometric transition which emerge the two defect branes. The summationof them corresponds to the qq -character of ¯ .28n (3.61) are obtained as Y ( q − x ) Y ( q − x ) = Y ( q − x ) × Y ( q − x ) (case 2) (3.63)1 Y ( q − x ) = Y ( q − x ) Y ( q − x ) × Y ( q − x ) (case 3) (3.64)Thus the qq -character of ¯ for A quiver is given by summing all the possible configurationswith two defect branes shown in Fig. 10. The argument discussed above is extended to generic (simply-laced) quiver gauge theory. A r quiver For A r quiver, there exist r weights, associated with the gauge nodes, and the fundamentalrepresentation is obtained from each (highest) weight, which is the antisymmetric repre-sentation of SU( r + 1). The qq -character of the degree n antisymmetric representation R (cid:48) n ( n = 1 , . . . , r ) is given by [19] χ R (cid:48) n ( A r ; q , q ) = (cid:68) T n ( x ) (cid:69) := (cid:32) n (cid:89) k =1 Q k (cid:33) − P ( q − n x ) (cid:88) ≤ i < ···
For
ADE quiver, all the fundamental representations are finite dimensional, and thus the( qq -)character is given by a finite (elementary symmetric) polynomial of { Λ i } , which is a ratioof the Y -operator (3.66). In general, we can consider the quiver, which does not correspondto the finite ADE -type Dynkin diagram, namely affine and hyperbolic quivers. Although, insuch a case, the fundamental representations become infinite dimensional, we can discuss the qq -character generated by the iWeyl reflection. For example, the affine quiver ˆ A r is realizedusing the infinitely-long linear quiver A ∞ by imposing periodicity. Thus there are infinitelymany possibilities for the brane insertion. This is a geometric interpretation of the infinitesum in the affine qq -character. For the simplest case ˆ A corresponding to 4d N = 2 ∗ (5d N = 1 ∗ ) theory, the qq -character is described as a summation over the partition [19, 25]. In this paper, we have proposed the prescription of the geometric transition in the refinedtopological string enforced along the preferred direction. In order to obtain a proper contri-bution of the brane insertion, in addition to the specialization of the K¨ahler moduli, we haveto shift the variable by hand to satisfy consistency, which becomes trivial in the unrefinedlimit. We then have applied this prescription to the codimension-4 defect operator, calledthe Y -operator as its stringy realization. The pole singularity of the Y -operator is cancelledout in a proper combination of the Y -operators, which is given by the qq -character. We haveexamined the pole cancellation in the qq -character as a nontrivial check of our prescription30f the refined geometric transition.Let us finally provide several open questions which we would like to resolve. As com-mented, the refined large N duality between the resolved and deformed conifold has beenclarified in terms of the refined Chern–Simons theory [15]. Nevertheless, the correspondingbrane configuration is not clear from their argument, and as the first issue, we would pursuethat our geometric transition may give a actual brane picture compatible with their result.Second, it may be possible that our prescription in Section 2.2 is generalized so as to in-corporate the labels ( p, q ) of the fivebrane charges, as mentioned there. The third thing isconcerned with the exact definition of the refined version of the open topological vertex for-malism. As far as we know, it is not yet established, and thus, the direct computation of theopen string amplitude respecting the Lagrangian brane on the inner brane is still a nontrivialproblem. In the unrefined case, the Schur function is suitable to capture the holonomy ofD-branes corresponding to the insertion of the Lagrangian brane. It is expected from theresults of [15] that the Schur function would be replaced with the Macdonald function inthe refined case as done for the refined topological vertex in [10]. Combining the expressionobtained via the refined geometric transition, we hope that the successful direct approachwould be reported in the near future.We also hold some technical and qualitative issues on the Y -operator. In the topologicalstring approach, there is an ambiguity of the normalization. Actually the Y -operator andthe qq -character have factors independent of the partition µ , and we need to add such afactor by hand to obtain a proper result. It would be interesting to clarify a systematic wayto discuss the µ -independent factor in the framework of refined topological string.The brane configuration of the Y -operator proposed in this paper is due to the comparisonwith the gauge theory definition. The current construction of the codimension-4 Y -operatoruses the codimension-2 surface defects with the q -brane and anti- q -brane. Such a relationbetween defect operators with different codimensions is not yet obvious. One possible inter-pretation is the tachyon condensation, which could be related to the (refined) supergroupChern–Simons theory [41]. For example, it is interesting to compare the Y-operator con-tribution with the partition function of the refined U(1 |
1) Chern–Simons theory [58]. Moredetailed analysis is necessary for understanding its geometric meaning in refined theory.
Acknowledgments
We would like to thank Shamil Shakirov, Masato Taki, Satoshi Yamaguchi, and YegorZenkevich for giving helpful comments. The work of T. K. was supported in part byKeio Gijuku Academic Development Funds, JSPS Grant-in-Aid for Scientific Research (No.JP17K18090), the MEXT-Supported Program for the Strategic Research Foundation at Pri-vate Universities “Topological Science” (No. S1511006), and JSPS Grant-in-Aid for Scien-31ific Research on Innovative Areas “Topological Materials Science” (No. JP15H05855). Thework of H. M. and Y. S. was supported in part by the JSPS Research Fellowship for YoungScientists.
A Definitions and notations
A.1 Mathematical preliminariesYoung diagram
To define the Young diagram, we take the decreasing sequence of nonnegative integers thatis regularly used for the instanton counting problem. Let ( i, j ) be positions of boxes in thediagram (shown in Fig. 11(a)), then we denote as µ a Young diagram of the following set of l -tuple diagrams (Fig. 11(b)): µ = { µ i ∈ Z ≥ | µ ≥ µ ≥ · · · ≥ µ l } , µ t = (cid:8) µ tj ∈ Z ≥ | µ tj = { i | µ i ≥ j } (cid:9) , (A.1)where the transpose of µ is indicated by the superscript t (Fig. 11(c)). For a given Youngdiagram µ , we use the following simplified symbols: | µ | = l (cid:88) i =1 µ i , || µ || = l (cid:88) i =1 µ i , (cid:89) ( i,j ) ∈ µ f ( i, j ) = l (cid:89) i =1 µ i (cid:89) j =1 f ( i, j ) . (A.2)The first one in (A.2) is the total number of boxes of µ . The partitions { µ i } and { µ tj } concretely characterize the instanton partition function, which can be removed by using µ i (cid:88) j =1 ( µ i − j ) = µ i (cid:88) j =1 ( j −
1) for fixed i, µ tj (cid:88) i =1 (cid:0) µ tj − i (cid:1) = µ tj (cid:88) i =1 ( i −
1) for fixed j. (A.3)In the paper, these are implicitly applied as expressing the Y -operator in a convenient fashionfrom the general form obtained via the refined geometric transition in Section 2.2. Theta function
The topological string amplitude for the compactified web diagram of our interest is nicelyexpressed in terms of the theta function, θ ( z | τ ) = − ie π i τ e π i z ∞ (cid:89) k =1 (cid:16) − e π i kτ (cid:17) (cid:16) − e π i kτ e πiz (cid:17) (cid:16) − e π i( k − τ e − π i z (cid:17) , (A.4)32 i, j ) i j (a) Positions of boxes µ µ µ l (b) l -tuple diagram µ tl µ t µ t (c) Transpose Figure 11: The Young diagram and its parameters.where a variable is z ∈ C , and τ ∈ C is a constant with Im( τ ) >
0. Equivalently, the thetafunction is frequently used in the multiplicative form, θ ( x ; q ) = − i q x ( q, qx, x − ; q ) ∞ , (A.5)where x := e π i z , q := e π i τ , and the q -Pochhammer symbol ( q -shifted factorial) is definedby ( x ; q ) n = n = 0 , n − (cid:89) k =0 (1 − xq k ) for n ≥ , − n (cid:89) k =1 (1 − xq − k ) − for n ≤ − . (A.6)In addition, ( x ; q ) ∞ := lim n →∞ ( x ; q ) n with | q | < x , x , · · · , x r ; q ) n := ( x ; q ) n ( x ; q ) n · · · ( x r ; q ) n . (A.7)Note that (A.4) and (A.5) are nothing but the Jacobi’s triple product identity. This thetafunction actually has the simple inversion property and satisfies the q -difference equation, θ ( x − ; q ) = − θ ( x ; q ) , (A.8) θ ( xq n ; q ) = ( − x ) − n q − n θ ( x ; q ) for n ∈ Z . (A.9)We further give another type of the theta function defined by θ ( x ; q ) = 1( q ; q ) ∞ (cid:88) n ∈ Z ( − n q n ( n − x n = ( x, qx − ; q ) ∞ . (A.10)33his theta function is simply translated into θ ( x ; q ) via the Jacobi’s triple product identity, θ ( x ; q ) = i q x − ( q ; q ) ∞ θ ( x ; q ) . (A.11)We can immediately verify that this theta function actually satisfies the q -difference equa-tions, θ ( x − ; q ) = − x − θ ( x ; q ) = θ ( xq ; q ) , (A.12) θ ( xq n ; q ) = ( − x ) − n q − n ( n − θ ( x ; q ) , (A.13) θ ( xq n ; q ; p ) m = ( − x ) − nm q − nm ( n − p − nm ( m − θ ( x ; q ; p ) m , (A.14)where we define θ ( x ; q ; p ) m := m − (cid:89) s =0 θ ( xp s ; q ) . (A.15)We remark that the q → q → θ ( x ; q ) = 1 − x. (A.16)It will be turned out that this limiting formula is actually the operation of the dimensionalreduction from 6d to 5d at the level of the partition function. Elliptic gamma function
The elliptic gamma function is defined byΓ e ( x ) := Γ( x ; p, q ) = (cid:89) n,m ≥ − x − p n +1 q m +1 − xp n q m , (A.17)with | p | , | q | <
1, and x ∈ C ∗ . For specific values of x , the elliptic gamma function getsimplified asΓ e ( p ) = ( q ; q ) ∞ ( p ; p ) ∞ , Γ e ( q ) = ( p ; p ) ∞ ( q ; q ) ∞ , Γ e ( −
1) = 12( − p ; p ) ∞ ( − q ; q ) ∞ . (A.18)The certain combinations of elliptic gamma function are related to the theta function definedabove as follows: Γ e ( x )Γ e ( x − ) = 1 θ ( x ; p ) θ ( x − ; q ) = 1 θ ( x ; q ) θ ( x − ; p ) (A.19)because p and q are encoded symmetrically into the elliptic gamma function, in addition, wefind the difference equations involving the theta function,Γ e ( xp ) = θ ( x ; q )Γ e ( x ) , Γ e ( xq ) = θ ( x ; p )Γ e ( x ) , (A.20)Γ e ( xp n ) = θ ( x ; q ; p ) n Γ e ( x ) , Γ e ( xq m ) = θ ( x ; p ; q ) m Γ e ( x ) , (A.21)Γ e ( xp n q m ) = ( − x ) − mn p − nm ( n − q − nm ( m − θ ( x ; q ; p ) n θ ( x ; p ; q ) m Γ e ( x ) . (A.22)34or n, m ∈ Z . Note that the first line represents the finite difference equations of the firstorder [59] that can lead to the second line, in other words, the last relation can be derived inthe recursive manner from the first one. Furthermore, there are the limiting relations [59],lim p → Γ e ( x ) = 1( x ; q ) ∞ , (A.23)lim x → (1 − x )Γ e ( x ) = 1( p ; p ) ∞ ( q ; q ) ∞ . (A.24)Moreover, we have the reflection identity,Γ e (cid:16) ( pq ) a x b (cid:17) Γ e (cid:16) ( pq ) − a x − b (cid:17) = 1 . (A.25)The usage of the elliptic gamma function is underlying a nontrivial property linking itsspecific ratio to the theta function involving Young diagrams [60] (see also [61]), (cid:89) ( i,j ) ∈ µ θ ( Qp µ i − j t ν tj − i +1 ; q ) (cid:89) ( i,j ) ∈ ν θ ( Qp − ν i + j − t − µ tj + i ; q ) = (cid:89) i,j ≥ Γ e ( Qt j − i +1 ; p, q )Γ e ( Qp µ i − ν j t j − i ; p, q )Γ e ( Qt j − i ; p, q )Γ e ( Qp µ i − ν j t j − i +1 ; p, q ) . (A.26)Note that it has been reported in [4] that there exists a similar formula involving the gammafunction for the Nekrasov function for the 4d theory. Further, the 5d Nekrasov function issimilarly written in terms of the q -gamma function. A.2 Refined topological vertex
In this paper, we rely on the Iqbal–Koz¸caz–Vafa formalism [11] for the refined topologicalvertex C λµν ( t, q ) given by C λµν ( t, q ) = t − || µt || q || µ || || ν || ˜ Z ν ( t, q ) (cid:88) η (cid:16) qt (cid:17) | η | + | λ |−| µ | s λ t /η ( t − ρ q − ν ) s µ/η ( t − ν t q − ρ ) , (A.27)where s λ/µ ( x ) is the skew Schur function and˜ Z ν ( t, q ) = (cid:89) ( i,j ) ∈ ν − q ν i − j t ν tj − i +1 , ρ = (cid:26) − , − , − , · · · (cid:27) . (A.28)The function ˜ Z ν ( t, q ) is essentially the Macdonald function P ν ( x ; q, t ) [62] (cid:101) Z ν ( t, q ) = t − || ν T || P ν ( t − ρ ; q, t ) . (A.29)We do not go further details of the refined topological vertex and trace back the calculationof the partition function (2.6) that has been accomplished in [43]. Note that the parameters( q, t − ) are replaced in the main context of the paper with ( q , q ), respectively. We wouldlike to comment on the fact that this partition function is absolutely reproduced by usingthe Awata–Kanno formalism for C λµν ( t, q ) [10, 63].35 Regularity
In this appendix we show the regularity of the qq -character in the case of A , A quiver withthe single Y -operator, and A quiver with two Y -operators. The strategy is as follows:1. We write the partition function and the Y -operator to the infinite product form.2. We calculate the ratio Z U(1) µ / Z U(1) µ +1 and the product Y µ Y µ +1 , where µ + 1 denotes theYoung diagram that we add the one box to some row µ I , namely µ I → µ I + 1.3. Then, we find that the ratio of the partition functions relates to the product of the Y -operators.We will demonstrate these steps. Note that we consider the regularity for the variable Q x instead of the x -variable while we focus on U(1) theory. B.1 A quiver B.1.1
U(1) gauge theory with single Y -operator To begin with, let us consider the simplest case. By using the formula in Appendix A, wewrite the partition function and the Y -operator to the infinite product form as follows, Z U(1) µ = ( − | µ | ( Q (1)1 Q (2) − ) | µ | q | µ | q || µ || q (cid:80) ( i,j ) ∈ µ i × (cid:89) i,j ≥ Γ e ( Q (2) − q − q j − i )Γ e ( Q (2) − q µ j − q j − i − )Γ e ( Q (1)1 q j − i +11 )Γ e ( Q (1)1 q µ j q j − i )Γ e ( Q (2) − q − q j − i − )Γ e ( Q (2) − q µ j − q j − i )Γ e ( Q (1)1 q j − i )Γ e ( Q (1)1 q µ j q j − i +11 ) × Γ e ( q − q j − i − )Γ e ( q − µ i + µ j − q j − i )Γ e ( q − q j − i )Γ e ( q − µ i + µ j +12 q j − i − ) , (B.1) Y µ ( x ) = − i e i πτ Q x (cid:89) i ≥ θ ( Q x q µ i q i − ) θ ( Q x q µ i q i ) , (B.2)where we denote the elliptic gamma function Γ e ( x ; q − , Q τ ) =: Γ e ( x ) for simplicity, and Q x = Q /x . Note that the µ -independent factors are interpreted as the one-loop contribution, andthe remaining ones are the full partition function. By using the reflection of the thetafunction θ ( x ) = − θ ( x − ), the Y -operator can also be written as Y µ ( x ) = i e i πτ Q − x (cid:89) i ≥ θ ( Q − x q − µ i q − i +11 ) θ ( Q − x q − µ i q − i ) . (B.3)36his coincides with the definition in [26], up to a trivial factor. Let us consider the ratio Z U(1) µ / Z U(1) µ +1 and the product Y µ ( q − x ) Y µ +1 ( x ). After some calculations, we have Z U(1) µ Z U(1) µ +1 = − q − µ I − q − I ( Q (1)1 Q (2) − ) − θ ( Q − m q µ I q I − ) θ ( Q m q µ I +12 q I ) (cid:89) i ≥ ,i (cid:54) = I θ ( q − µ I + µ i − q i − I − ) θ ( q − µ i + µ I q I − i ) θ ( q − µ I + µ i − q i − I ) θ ( q − µ i + µ I q I − i − ) , (B.4) Y µ ( q − x ) Y µ +1 ( x ) = q − q − e i πτ (cid:89) i ≥ θ ( Q − x q − µ i − q − i ) θ ( Q x q µ (cid:48) i q i − ) θ ( Q − x q − µ i − q − i − ) θ ( Q x q µ (cid:48) i q i ) , (B.5)where µ + 1 =: µ (cid:48) denotes the Young diagram that we add the one box to some row µ I ,namely µ I → µ I + 1, as we defined in the beginning of this section. Then by using therelation P ( x ) = Q − x ( Q (1)1 Q (2) − ) q − e i πτ θ ( Q (1)1 Q − x ) θ ( Q (2) − q − Q − x ) , (B.6)we find Z U(1) µ Z U(1) µ +1 = − q − Y µ ( q − x ) Y µ +1 ( x ) P ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Q x = q − µI − q − I , (B.7)which implies Res Q x = q − µI − q − I (cid:20) Y µ +1 ( x ) Z U(1) µ +1 + q P ( x ) Y µ ( q − x ) Z U(1) µ (cid:21) = 0 (B.8)This means that the Y -operators Y µ ( x ) and Y µ ( q − x ) − have the poles, but the summationis regular since these poles cancelled with each other. Therefore we obtain the T -operatoraverage for U(1) theory (3.27), which is regular for arbitrary Q x , by the summation over thepartition µ . B.1.2 U(1) gauge theory with two Y -operators In this subsection we show the regularity for the U(1) theory with the two Y -operators. Thecalculation is almost done in the previous subsection. In this case we have to rewrite thefactor S ( x ) in terms of the Y -operator. This factor can be written as S (cid:18) x x (cid:19) = S (cid:18) q − x x (cid:19) = θ ( q − Q x Q − x ) θ ( q − Q x Q − x ) θ ( q − Q x Q − x ) θ ( Q x Q − x ) . (B.9)We remark Q x = Q /x and Q x = Q /x . Also we show the ration of the Y -operator, Y µ +1 ( x ) Y µ ( x ) = θ ( Q x q µ I +12 q I − ) θ ( Q x q µ I q I ) θ ( Q x q µ I +12 q I ) θ ( Q x q µ I q I − ) , (B.10) Y µ +1 ( q − x ) Y µ ( q − x ) = θ ( Q − x q − µ I − q − I ) θ ( Q − x q − µ I − q − I − ) θ ( Q − x q − µ I − q − I − ) θ ( Q − x q − µ I − q − I ) . (B.11)37hese two expressions are related each other, Y µ +1 ( x ) Y µ ( x ) = S (cid:18) x x (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) Q x = q − µI − q − I , Y µ +1 ( q − x ) Y µ ( q − x ) = S (cid:18) x x (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) Q x = q − µI − q − I . (B.12)One can obtain the similar equations for Q x . Then, according to the discussion in theappendix B.1.1, we can show the regularity for the arbitrary Q x and Q x .However, when we take the collision limit Q x = Q x , the S -factor might have the pole.In order to consider this matter, let us consider the following case, Q x = Q x , Q x = wQ x (B.13)and take the limit w →
1. Then, by using the following formula θ ( x ; p ) = ( x ; p ) ∞ ( px − ; p ) ∞ x → −→ (1 − x )( p ; p ) ∞ , (B.14)and O ( wx ) = O ( e log x +log w )= O ( e log x ) + log w ∂∂ log x O ( e log x ) + O ((log w ) )= O ( x ) − (1 − w ) ∂∂ log x O ( x ) + O ((1 − w ) ) , (B.15) (cid:16) log w = log(1 − (1 − w )) = − (1 − w ) + O ((1 − w ) ) (cid:17) we have P ( wx ) S ( w − ) Y µ ( x ) Y µ ( qwx ) + P ( x ) S ( w ) Y µ ( wx ) Y µ ( qx )= S ( w − ) Y µ ( x ) × (cid:18) P ( x ) − (1 − w ) ∂ log x P ( x ) + O ((1 − w ) ) (cid:19)(cid:18) Y µ ( qx ) + (1 − w ) ∂ log x Y µ ( qx ) Y µ ( qx ) + O ((1 − w ) ) (cid:19) + P ( x ) S ( w ) 1 Y µ ( qx ) (cid:18) Y µ ( x ) − (1 − w ) ∂ log x Y µ ( x ) + O ((1 − w ) ) (cid:19) w → −−−→ P ( x ) Y µ ( x ) Y µ ( qx ) (cid:18) c ( q , q ) − θ ( q ) θ ( q ) θ ( q )( Q τ , Q τ ) ∞ ∂ log x log (cid:20) Y µ ( x ) Y µ ( qx ) P ( x ) (cid:21)(cid:19) , (B.16)where c ( q , q ) = lim w → ( S ( w ) + S ( w − ))= lim w → (cid:34) w − θ ( w ) ∂ w (cid:20) θ ( q − w ) θ ( q − w ) θ ( q − w ) (cid:21) + w − − θ ( w − ) ∂ w − (cid:20) θ ( q − w − ) θ ( q − w − ) θ ( q − w − ) (cid:21)(cid:35) . (B.17)One can show that this coefficient c ( q , q ) is regular. Therefore, even if Q x = Q x , theexpectation value of the T -operator is regular.38 .2 A quiver Let us consider the regularity for the T -operator average in A quiver theory. Again by usingsome formulas in Appendix A, we obtain Z U(1) × U(1) µ ,µ = q || µ || + | µ | q (cid:80) ( i,j ) ∈ µ i q − || µ || + | µ | q | µ |− (cid:80) ( i,j ) ∈ µ i × (cid:89) i,j ≥ Γ e ( Q (2)1 q j − i +11 )Γ e ( Q (2)1 q − µ ,i + µ ,j q j − i )Γ e ( Q (2)1 q j − i )Γ e ( Q (2)1 q − µ ,i + µ ,j q j − i +11 ) × (cid:89) i,j ≥ Γ e ( q − q j − i − )Γ e ( q − µ ,i + µ ,j +12 q j − i )Γ e ( q − q j − i )Γ e ( q − µ ,i + µ ,j − q j − i − ) Γ e ( q − q j − i − )Γ e ( q − µ ,i + µ ,j − q j − i )Γ e ( q − q j − i )Γ e ( q − µ ,i + µ ,j − q j − i − ) × (cid:89) i,j ≥ Γ e ( Q (1) − q − q j − i )Γ e ( Q (1) − q µ ,j − q j − i − )Γ e ( Q (3) − q − q j − i )Γ e ( Q (3) − q − µ ,i − q j − i − )Γ e ( Q (1) − q − q j − i − )Γ e ( Q (1) − q µ ,j − q j − i )Γ e ( Q (3) − q − q j − i − )Γ e ( Q (3) − q − µ ,i − q j − i ) . (B.18)Then, we have Z U(1) × U(1) µ ,µ Z U(1) × U(1) µ +1 ,µ = q − q − µ ,I − q − I θ ( Q (3) − q µ ,I q I − ) (cid:89) i ≥ θ ( Q (2)1 q µ ,I − µ ,i +12 q I − i ) θ ( Q (2)1 q µ ,I − µ ,i +12 q I − i +11 ) × (cid:89) i ≥ ,i (cid:54) = I θ ( q − µ ,I + µ ,i − q i − I − ) θ ( q − µ ,i + µ ,I q I − i ) θ ( q − µ ,I + µ ,i − q i − I ) θ ( q − µ ,i + µ ,I q I − i − ) , (B.19) Z U(1) × U(1) µ ,µ Z U(1) × U(1) µ ,µ +1 = q − q µ ,I q I − θ ( Q (1) − q − µ ,I − q − I ) (cid:89) i ≥ θ ( Q (2)1 q − µ ,I + µ ,i q i − I +11 ) θ ( Q (2)1 q − µ ,I + µ ,i q i − I ) × (cid:89) i ≥ ,i (cid:54) = I θ ( q − µ ,I + µ ,i − q i − I − ) θ ( q − µ ,i + µ ,I q I − i ) θ ( q − µ ,I + µ ,i − q i − I ) θ ( q − µ ,i + µ ,I q I − i − ) . (B.20)The product of Y -operators is given by (B.5). Then, we find that Z U(1) × U(1) µ ,µ Z U(1) × U(1) µ +1 ,µ = − q − Y µ ( q − x ) Y µ +1 ( x ) P ( x ) Y µ ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Q x = q − µ ,I − q − I , (B.21) Z U(1) × U(1) µ ,µ Z U(1) × U(1) µ ,µ +1 = − q − Y µ ( x ) Y µ +1 ( q − x ) P ( x ) Y µ ( q − x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Q x = Q (2) − q − µ ,I − q − I . (B.22)Note that the variable x is given by (3.52). Therefore the average (cid:68) T ( x ) (cid:69) is regular for thearbitrary x . References [1] N. Seiberg and E. Witten, “Electric-magnetic duality, monopole condensation, andconfinement in N = 2 supersymmetric Yang–Mills theory,” Nucl. Phys.
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