Dual Little Strings and their Partition Functions

Dual Little Strings and their Partition Functions Brice Bastian§ , Stefan Hohenegger§ , Amer Iqbal†? , Soo-Jong Rey

§

arXiv:1710.02455v1 [hep-th] 6 Oct 2017



Universit´e de Lyon UMR 5822, CNRS/IN2P3, Institut de Physique Nucl´eaire de Lyon, 4 rue Enrico Fermi, 69622 Villeurbanne Cedex FRANCE † Abdus Salam School of Mathematical Sciences, G.C. University, Lahore PAKISTAN ? Center for Theoretical Physics, Lahore, PAKISTAN School of Physics and Astronomy & Center for Theoretical Physics, Seoul National University, Seoul 08826 KOREA (Dated: October 9, 2017) We study the topological string partition function of a class of toric, double elliptically fibered Calabi-Yau threefolds XN,M at a generic point in the K¨ ahler moduli space. These manifolds engineer little string theories in five dimensions or lower and are dual to stacks of M5-branes probing a transverse orbifold singularity. Using the refined topological vertex formalism, we explicitly calculate a generic building block which allows to compute the topological string partition function of XN,M as a series expansion in different K¨ ahler parameters. Using this result we give further explicit proof for a duality found previously in the literature, which relates XN,M ∼ XN 0 ,M 0 for N M = N 0 M 0 and gcd(N, M ) = gcd(N 0 , M 0 ).

I.

{α}

INTRODUCTION

Recently, the study of little string theories (LSTs) has received renewed interest: first proposed two decades ago [1–4] (see [5, 6] for a review), during the last years, the construction of LSTs from various M-brane constructions and their dual geometric description in F-theory [7] has led to a better understanding of supersymmetric quantum theories in six dimensions. Moreover, in the process new and unexpected dualities have been unravelled. The geometric description of these in terms of F-theory compactification on Calabi-Yau threefolds has been particularly useful in understanding various stringy properties such as LST T-duality [2, 8, 9]. A particular class of LSTs, dual to N M5-branes transverse to a ZM orbifold, preserving eight supercharges can be realized in F-theory using toric, non-compact CalabiYau manifolds denoted by XN,M [10, 11]. The latter have the structure of a double elliptic fibration, in which one elliptic fibration has a singularity of type IN −1 and the other one of IM −1 . The exchange of the two elliptic fibrations is related to the T-duality of LST [7, 11]. Moreover, using for example the refined topological vertex formalism, the topological string partition function ZN,M on XN,M can be computed explicitly [11]. By analysing the web diagrams associated with XN,M , it was shown in [12] that XN,M ∼ XN 0 ,M 0 for M N = M 0 N 0 and gcd(M, N ) = k = gcd(M 0 , N 0 ), i.e. the two Calabi-Yau threefolds lie in the same extended K¨ ahler moduli space and can be related by flop transitions. The partition function of the two dual theories should be the same and so it was expected that the topological string partition function of XN,M and XN 0 ,M 0 should be the same i.e. ZN,M (ω, 1,2 ) = ZN 0 ,M 0 (ω 0 , 1,2 ) 0

(I.1)

for ω and ω the K¨ ahler forms of the two Calabi-Yau threefolds related by flop transitions. Using a general

building block W{β} to compute ZN,M (ω, 1,2 ), we explicitly confirm Eq.(I.1) for gcd(N, M ) = 1, thus verifying the duality proposed in [12] explicitly at the level of the partition function. This paper is organised as follows. In section 2, we discuss the (N, M ) = (3, 2) case in detail and show that it is dual to the (6, 1) case, i.e. related by a combination of flop and symmetry transforms. We also discuss the relation between the general (N, M ) and (k, M N/k) case. In section 3, we calculate the partition function of the (3, 2) case and show that, after a change of K¨ ahler parameters, it is equal to the partition function of the (6, 1) configuration. In section 4, we discuss the open string amplitudes which make up the building block for generic brane configurations. We also discuss the general “twisted” (1, L) case, which is related by flop transitions to the standard (1, L) case. In section 5, we present our conclusions and future directions. Some of the detailed calculations and our notation and conventions are relegated to three appendices. II.

LITTLE STRINGS AND DUAL BRANE WEBS

At low energies, a stack of N coincident M5-branes engineers a six-dimensional N = (2, 0) superconformal field theory of AN −1 -type. Upon replacing the R5 transverse to the M5-branes by an Rtrans × R4 /ZM orbifold geometry, the world-volume theory on the branes becomes an N = (1, 0) SCFT. One may move away from the conformal point by separating the M5-branes along the transverse Rtrans , as massive states appear in the form of M2branes that end on the separated M5-branes. Note that if Rtrans is compactified to a circle S1ρ of circumference ρ so that the M5-branes are points on S1ρ , then the theory arising on the M5-branes (whether coincident or not) gives rise to a LST whose defining scale is ρ.

2

−N −4

...

M

...

−5

... = ... ... ... ...

−3 −2

=

2

1

=

N

3

−5

=

−4

2

=

3

=

... ... ... −

=

−1

M

=

As a next step, we compactify the worldvolume of the M5-branes on a circle S1τ of circumference τ . This Mbrane configuration is dual [13] to a (p, q) 5-brane web in type IIB string theory given by N coincident NS5-branes wrapped on S1τ that intersect with M coincident D5branes wrapped on S1ρ . Turning on deformations which separate the intersections of NS5-branes and D5-branes and stretching the NS5-D5 bound-state branes [14] yields the brane web shown in Fig. 1. Finally, this brane web in type IIB string theory is dual to a toric Calabi-Yau threefold [15], which was first studied in [11] and was denoted by XN,M , and provides a more geometric description of the theory via F-theory compactification [16, 17]. A general classification of toric and non-toric base manifolds in F-theory models was discussed in [18, 19] and, more recently, in [20]. For general properties of topological strings on elliptic Calabi-Yau threefolds, see [21–23].

−3 −2

1

−1

FIG. 1. The 5-brane web in type IIB string theory dual to the Calabi-Yau threefold XN,M . The K¨ ahler parameters associated with the horizontal, vertical and diagonal lines in the web are collectively denoted by h, v and m, respectively. More precise labelings will be given for concrete examples below (see e.g. Fig. 2(a)).

The brane web shown in Fig. 1 has 3N M parameters, related to the sizes of various line segments corresponding to the 5-branes. Specifically, we denote the parameters associated with horizontal (green) lines as h = {h1 , . . . , hN M }, with vertical (red) lines as v = {v1 , . . . , vN M } and with diagonal (blue) lines as m = {m1 , . . . , mN M }. Out of these, only N M + 2 are independent (after imposing a number of consistency conditions related to the the periodic identification of the web diagram [24]). In the dual Calabi-Yau threefold XN,M , these (N M +2) parameters are independent K¨ ahler moduli. For a discussion of the geometry of XN,M and the corresponding mirror curves, see [25].

The supersymmetric or BPS-state counting partition function of the above mentioned LST can be computed as the topological string partition function ZN,M (h, v, m, 1 , 2 ) of XN,M . The latter depends on the K¨ahler parameters of XN,M (taking into account that only N M + 2 of them are independent) as well as two regularisation parameters 1,2 introduced to render the partition function well-defined. From the perspective of the field theory that describes the low-energy limit of the LST engineered by XN,M , these parameters can be interpreted geometrically as the Ω-background [26, 27]. The quantization of the K¨ahler parameters in units of unrefined , which correspond to Coulomb branch parameters [28], leads to a description of the partition function in terms of an irreducible representation of an affine group. For details, see [29]. Given the web diagram, ZN,M can be computed efficiently using the (refined) topological vertex formalism: to this end, a preferred direction in the web needs to be chosen, which from the perspective of LSTs determines the parameters that play the role of coupling constants. Concretely, different choices of the preferred direction lead to different (but equivalent) power series expansions of ZN,M in terms of either h, v or m. In this paper, we shall use this fact to prove a duality (that was first proposed in [12]) at the level of the partition function for generic values of the K¨ahler parameters: indeed, it was argued in [12] that XN,M and XN 0 ,M 0 are related to each other through a series of SL(2, Z) symmetry and flop transformations if N M = N 0 M 0 and gcd(N, M ) = k = gcd(N 0 , M 0 ). This suggests that ZN,M (h, v, m, 1 , 2 ) = ZN 0 ,M 0 (h0 , v0 , m0 , 1 , 2 ) , (II.1) where the duality map (h, v, m) 7−→ (h0 , v0 , m0 ) was proposed in [12] (see also appendix C for an example), taking into account the consistency conditions on both sides. The relation (II.1) was tested in [12] (based on a conjecture put forward in [30]) at a particular region in the K¨ahler moduli space of XN,M (with hi = hj , vi = vj and mi = mj for i, j = 1, . . . , N M ) and the Nekrasov-Shatashvili limit 2 → 0 [31, 32]. In the following, by computing ZN,M at a generic point in the K¨ ahler moduli space, we give explicit evidence of (II.1) in general. However, for simplicity, we shall limit ourselves to gcd(N, M ) = 1.

III.

PARTITION FUNCTION OF (3, 2) CASE

As checking (II.1) is rather complicated for generic (N, M ), before presenting the generic case in the following section, we first discuss as a (nontrivial) example the case (N, M ) = (3, 2). Specifically, we shall demonstrate in the following that Z3,2 (ω, 1 , 2 ) = Z6,1 (ω 0 , 1 , 2 ) ,

(III.1)

3 where ω and ω 0 denote the (independent) K¨ ahler parameters of XN,M and XN 0 ,M 0 , respectively. We shall verify (III.1) explicitly at a generic point in the moduli space of K¨ ahler parameters by performing a novel expansion of the left hand side of this relation.

A.

Consistency Conditions and Parametrisation

The web diagram of X3,2 is shown in Fig. 2(a) and the K¨ ahler parameters labelling various rational curves are given by h1,...,6 ,

v1,...,6 ,

m1,...,6 .

(III.2)

v3 = 2M + H1 + H2 + H3 , v4 = 2M + H2 + H3 + H4 , v5 = 2M + H3 + H4 + H5 , v6 = 2M + H4 + H5 + H6 , m1 m2 m3 m4 m5 m6

= 2M = 2M = 2M = 2M = 2M = 2M

= m4 + v1 ,

(III.3)

= m2 + v2 ,

(III.4)

= m6 + v3 ,

(III.5)

= m1 + v4 ,

(III.6)

= m5 + v5 ,

(III.7)

= m3 + v6 .

(III.8)

The above conditions indeed leave 8 independent parameters. However, there is a priori an ambiguity in choosing the latter. A suitable parametrisation can be found by comparison with the web diagram of X6,1 in Fig. 3, which has been shown in [12] to be dual to X3,2 by flop and symmetry transforms. The (6, 1) diagram is parametrised in terms of the maximal set of variables (M, V, H1,...,6 ), which satisfies all consistency conditions. Through the explicit duality map found in [12], we can also express the (h, v, m) of the (3, 2) web diagram in terms of (M, V, H1,...,6 ) h1 = −M − H5 − H6 , h3 = −M − H1 − H2 , h5 = −M − H3 − H4 ,

h2 = −M − H1 − H6 h4 = −M − H2 − H3 h6 = −M − H4 − H5

v1 = 2M + H1 + H5 + H6 , v2 = 2M + H1 + H2 + H6 ,

, , , , , .

(III.9)

One can indeed check that (III.9) identically satisfies (III.3) – (III.8). Moreover, as we shall discuss in the following, the parametrisation (III.9) is crucial for showing the identity (III.1). To this end, we first review the topological string partition function of X6,1 in the following.

Of these 18 parameters, however, only 8 are independent: indeed, for each of the six compact divisors of X3,2 (each of which is a P1 × P1 blown up at two points), we have two consistency conditions. These conditions arise because each compact divisor (represented as a hexagon in the web) has six rational curves which are toric divisors of P1 × P1 blown up at two points but only four of these are independent. Explicitly, we therefore obtain the following conditions: • hexagon I : h4 + m4 = m3 + h1 , v3 + m3 • hexagon II : h5 + m2 = m4 + h2 , v4 + m4 • hexagon III : h6 + m6 = m2 + h3 , v5 + m2 • hexagon IV : h1 + m1 = m6 + h4 , v6 + m6 • hexagon V : h2 + m5 = m1 + h5 , v1 + m1 • hexagon VI : h3 + m3 = m5 + h6 , v2 + m5

+ H3 + 2H4 + 2H5 + H6 + V + H1 + 2H2 + 2H3 + H4 + V + 2H1 + H2 + H5 + 2H6 + V + 2H1 + 2H2 + H3 + H6 + V + H1 + H4 + 2H5 + 2H6 + V + H2 + 2H3 + 2H4 + H5 + V

B.

The Partition Function Z6,1

The (6, 1) web is shown in Fig. 3 in which the K¨ ahler parameters are also labelled. Here, the parameters (M, V, H1,...,6 ) are the same that appear in (III.9) (after applying the duality map proposed in [12]). Furthermore, the partition function Z6,1 was explicitly calculated in [10, 24, 33] using the (refined) topological vertex [34, 35]. The latter has a preferred direction which can be chosen in different ways to obtain different series expansions of Z6,1 . For example, choosing the preferred direction to be vertical (i.e. along the (0, 1)-direction in the web diagram in Fig. 3), the topological string partition function takes the form Z6,1 (V, M, Hi , 1,2 ) = W6 (∅)

(III.10) ! X ϑ (Q ; ρ) α α M a a p × (−QV QM )|α1 |+...+|α6 | α1 ,...,α6 a=1 ϑαa αa ( t/q; ρ)   −1 Y ϑ (Q Q ; ρ) ϑ (Q Q ; ρ) αa αb ab M αa αb ab M , p p × ϑ (Q t/q; ρ) ϑ (Q ab αa αb ab q/t; ρ) 1≤a 1.

that changes δ −→ δ + 1, which (by induction) explicitly proves (II.1) for gcd(N, M ) = 1.

1.

Parametrisation

In this section, we consider a twisted web diagram of length L = N M in which the external (diagonal) legs are glued together (albeit with a shift δ), so not all line segments in Fig. 9 are independent one another. Instead, they have to satisfy the following consistency conditions vi + mi = vi+δ+1 + mi+1 , hi+1 + mi+1 = hi+δ+1 + mi ,

(IV.15)

which leave a total of L + 2 independent parameters. Eliminating the mi from (IV.15), we obtain b ai = vi + hi+1 = vi+δ+1 + hi+δ+1 ,

Furthermore, W in (IV.18) follows from the generic expression (IV.10) by identifying the partitions

(IV.16)

βi = αi+L−δ−1 ,

which in terms of the variables (IV.6) corresponds to b ai = bbi+δ+1 , as has already been taken into account in

1 ...αL WααL−δ+1 ai , S, 1 , 2 ) = WL (∅) × ...αL−δ (b

h t  L−1 2 q

QS = e−S ,

i|α1 |+···+|αL |

QρL−δ−1

with ρ = and

Qρ = e2πiρ ,

(IV.21)

(IV.19)

where we recall that the latter are cyclically identified, i.e. αi = αi+L ). Concretely, we find

QL S

Here, we introduced

∀i = 1, . . . , L

i 2π

×

L Y

b i,i−j−δ ; ρ) ϑαi αj (Q p ϑ (Qi,i−j q/t; ρ) i,j=1 αi αj

PL

ai . i=1 b

(IV.20)

Furthermore, we have used the

13 identities (A.5) and (A.6) to combine the Jαβ in (IV.10) into ϑ-functions (cancelling the factor Zˆ in the process). The full partition function (IV.18) also depends on the parameters mi , which are not all independent. Indeed, by rewriting (IV.15) in terms of (independent) parameters S and b ai , we get the following recursive relation for mi mi+1 = mi +

L−1 X

α bi−k −

L−δ−1 X

k=δ+1

α bi−k .

(IV.22)

k=1

This implies that only one of the L many parameters mi can be chosen to be independent. This freedom is parametrised by R(δ) : in Fig. 9, it is shown as the vertical distance between the external legs labelled by the partition α1 and α1t . Similar to the parameter S due to the consistency conditions (IV.15) (and equivalently (IV.22)), this length is the same for any pair of partitions (αi , αit ) R(δ) = mi −

L−δ−1 X

vi−k ,

2.

Flop Transformations

Our strategy for proving (II.1) for gcd(N, M ) = 1 is to show that (δ)

(δ+1)

ZL,1 (ω, 1 , 2 ) = ZL,1 (ω 0 , 1 , 2 )

(IV.23)

for the partition function defined in (IV.18): indeed, (δ+1) ZL,1 is associated with a twisted web diagram (with shift δ + 1) that can be related to Fig. 9 through a series of flop and symmetry transformations, that also relate the K¨ahler parameters ω and ω 0 . This duality was first discussed in detail in [12] and is reviewed in appendix C. The parameters (IV.16) and (IV.17) in the transformed diagram are the same as those defined in the original diagram b a0i = vi0 + h0i+1 = −hi + hi + vi + hi+1 = b ai S 0 = hi−(L−δ−1) + b ai−(L−δ−1) +

∀ i = 1, . . . , L = N M .

L−δ−2 X

b ai−r = S .

r=1

k=1

We remark that (R(δ) , S, b a1,...,L ) are L + 2 = N M + 2 independent variables and therefore describe a maximally independent set of K¨ ahler parameters for XN,M .

1 ...αL WααL−δ+1 ...αL−δ = WL (∅) ×

h t  L−1 2 q

Hence, the b ai always parametrise the distance between two adjacent legs and S measures the vertical distance between two identified legs. In order to show that (IV.23) holds, we first rewrite the building block (IV.20) in a way that makes the δ dependence explicit

 =C L z }| { i|α1 |+···+|αL | Y 1−K  QL Q · S ρ

 1

ϑ (Qi,i−j i,j=1 αi αj

=B

=A

{ z

}| 

z

 p q/t; ρ)  }|

{

  Y   Y  Y  Y ϑαi αj (Q−1 ϑαi αj (Qj,i QS Q−1 ϑαi αj (Q−1 ϑαi αj (Qj,i QS )  × ρ ) , i,j QS Qρ ) ·  i,j QS )  i>j i−j>δ

i>j i−j≤δ

i≤j j−i≥K

i≤j j−ij i−j≤δ 0 0

C 0 = Q1−K , ρ

k=0 1 ...αL The difference between WααL−δ+1 ...αL−δ for the original 1 ...αL diagram and WααL−δ ...αL−δ−1 for the diagram obtained by flop and symmetry transformations rests in the three terms A, B and C. Their respective counterparts in the shifted diagram, denoted by A0 , B 0 and C 0 , are given by         Y  Y  Q Q S A0 =  ϑαi αj QQi,j ϑαi αj QSi,jρ  ,  i≤j j−ij i−j>δ 0

(IV.26)

where K 0 = K −1 and δ 0 = δ+1. The difference between A and A0 (respectively, B and B 0 ) lies in the arguments of those ϑ-functions for which j−i = K 0 (resp. i−j = δ+1): they differ by a factor of Qρ . The difference between C and C 0 is also a factor of Qρ .

Finally, we also need to take account of the factors of Qmi that appear in the full partition function (IV.18).

14 In the flopped diagram, these are given by (see (C.4)) ( Qmi Q2S if δ 0 = L 2 Qmi QS (IV.27) Qm0i = QL QL−δ−1 else Q Q r=δ+2

ai−r

r=1

ai−r

where we defined Qai = exp(−b ai ). In order to show that the equality (IV.23) holds, we now show that the differ(δ) (δ+1) ence between ZL,1 (ω, 1 , 2 ) and ZL,1 (ω 0 , 1 , 2 ) can be canceled by merely applying the shift identity (III.19) to the ϑ-functions mentioned above in (IV.26). First, we consider the case when the twist in the external legs is δ 0 = L. Shifting the required ϑ-functions to regain the ϑ-structure of the δ = L − 1 strip gives L Y i=1

ϑαi αi (QS Qρ ) =

L Y

−1 |αi | (Q−2 ϑαi αi (QS ) (IV.28) S Qρ )

i=1

The prefactors in (IV.28) resulting from the shift identity combine with (IV.26) and (IV.27) to reproduce the (δ) expression for ZL,1 (ω, 1 , 2 ), thus proving that (IV.23) holds for δ = L − 1. The computations when δ 0 6= L are more involved, so we will simply sketch them. Below we present the ϑ1 ...αL functions from WααL−δ ...αL−δ−1 that need to be shifted in α1 ...αL order to regain WαL−δ+1 ...αL−δ . We need to distinguish different cases depending on the partition αi in question. For the sake of clarity, we focus only on terms resulting from shifts that come to the power |αi | in each separate case. 1. For i ≤ min(K 0 , δ 0 ), we shift the following ϑfunctions   QS Qρ ϑαi+δ+1 αi (Qi,i+δ+1 QS )ϑαi αi+K 0 Qi,i+K 0   Qi,i+δ+1 QS −2 −1 |αi | 0 ∼ (QS Qi,i+δ+1 Qi,i+K ) ϑαi+δ+1 αi Qρ   QS × ϑαi αi+K 0 (IV.29) Qi,i+K 0 2. For i > max(K 0 , δ 0 ), we shift the following ϑfunctions   QS Qρ ϑαi αi−δ−1 (Qi−δ−1,i QS )ϑαi−K 0 αi Qi−K 0 ,i   Qi−δ−1,i QS −2 −1 |αi | ∼ (QS Qi−δ−1,i Qi−K 0 ,i ) ϑαi αi−δ−1 Qρ   QS × ϑαi−K 0 αi (IV.30) Qi−K 0 ,i 3. For min(K 0 , δ 0 ) < i ≤ max(K 0 , δ 0 ), we need to distinguish between two cases: (a) When K 0 > δ 0 we shift ϑαi+δ+1 αi (Qi,i+δ+1 QS )ϑαi αi−δ−1 (QS Qi−δ−1,i )

−1 −1 |αi | ∼ (Q−2 ϑαi+δ+1 αi S Qρ Qi−δ−1,i Qi,i+δ+1 )   QS Qi−δ−1,i × ϑαi αi−δ−1 Qρ



Qi,i+δ+1 QS Qρ



(IV.31)

(b) For K 0 < δ 0 , we shift     QS Qρ QS Qρ ϑαi−K 0 αi ϑαi αi+K 0 Qi,i+K 0 Qi−K 0 ,i ∼

−1 |αi | (Q−2 S Qρ Qi−K 0 ,i Qi,i+K 0 )

 × ϑαi−K 0 αi

QS

 ϑαi αi+K 0

QS



Qi,i+K 0



Qi−K 0 ,i

(IV.32)

In each case, the factors resulting from shifting the ϑfunctions combine with (IV.26) and (IV.27) to repro(δ) duce the expression for ZL,1 (ω, 1 , 2 ), thus showing that (IV.23) holds for a generic off-set δ with ω and ω 0 related trough the duality transformations (C.4). As the relation (IV.23) can be applied iteratively to the point (δ=L) δ = L (for which ZL,1 = ZL,1 ), this further implies (for gcd(N, M ) = 1) ZN,M (h, v, m, 1,2 ) = ZN M,1 (h0 , v0 , m0 , 1,2 ) , (IV.33) where the K¨ahler parameters (h, v, m) and (h0 , v0 , m0 ) are related by the duality map implied by (C.4). This proves (II.1) for gcd(N, M ) = 1. As was already argued in [12], this behavior is expected to hold also for gcd(N, M ) > 1, however, extending the above computations to these cases is technically more involved and will be left for forthcoming study.

V.

CONCLUSIONS

In this paper, we studied the topological string partition functions of double elliptically fibered Calabi-Yau threefolds XN,M that give rise to a class of LSTs with 8 supercharges via F-theory compactification. We have shown by an explicit example that X3,2 and X6,1 , which are related by flop and symmetry transforms, have the same topological string partition function, hence providing an explicit proof of the duality proposed in [12] between XN,M and X N M ,k (k = gcd(N, M )). Ink deed, while the case discussed here were characterised by gcd(N, M ) = 1 (in order to simplify the computations), we expect the duality to straightforwardly extend also to gcd(N, M ) > 1 on general grounds [39–42]. A logically natural question is whether the diagonal expansion of ZN,M , performed in this paper, can also be interpreted as an instanton expansion of a new gauge theory engineered from the XN,M web. As we will discuss in [43], this turns out indeed the case. The parameters mi can be expressed in terms of coupling constants of a quiver gauge theory related by U-dualities to the

15 usual quiver theory on the compactified M5-branes dual to XN,M . The dual gauge theories coming from the horizontal and vertical description of the XN,M brane web, discussed in [11], together with this new dual gauge theory associated with the same web gives us a ”triality” of quiver gauge theories. The Calabi-Yau threefolds XN,M are resolution of ZN × ZM orbifold of X1,1 . Therefore, another very interesting and natural question is whether similar duality related by flop transitions can be obtained for different types of orbifolds where ZN × ZM is replaced by Γ1 × Γ2 , with Γ1 , Γ2 some other discrete subgroups of SU (2).

Y

×

  t 1 1 ϑ x−1 q µj −i+ 2 tνi −j+ 2 ; ρ ,

(A.3)

(i,j)∈ν

where (with x = e2πiz and Qρ = e2πiρ ) 1

1

ϑ(x; ρ) = (x 2 − x− 2 )

∞ Y

(1 − x Qkρ )(1 − x−1 Qkρ )

k=1

=

−1 i Qρ 8 θ1 (ρ; z) Q∞ k k=1 (1 − Qρ )

.

(A.4)

Pairs of Jµν -functions can be combined into ϑµν functions in Eq. (A.3) by utilizing the following identities Jµν (x; q, t)Jνµ (Qρ x−1 ; q, t)

ACKNOWLEDGEMENT

=x We are grateful to Dongsu Bak, Kang-Sin Choi, Taro Kimura and Washington Taylor for useful discussions. A.I. would like to acknowledge the “2017 Simons Summer Workshop on Mathematics and Physics” for hospitality during this work. A.I. was supported in part by the Higher Education Commission grant HEC-20-2518.

|µ|+|ν| 2

q

||ν t ||2 −||µt ||2 4

t

||µ||2 −||ν||2 4

ϑµν (x; ρ) ,

(A.5)

as well as (−1)|µ| t|

||µ||2 2

Jµµ (Qρ = ϑµµ (

q

1 q

||µt ||2 2

Z˜µ (q, t)Z˜µt (t, q) 1 pq = pq t ϑµµ ( t ; ρ) q ; q, t)Jµµ (Qρ t ; q, t) q

t q ; ρ)

.

(A.6)

Appendix A: Notation and Useful Identities

In this appendix, we first introduce some of our notation and provide some useful computational identities. We start by introducing some notation concerning integer partitions. Given an integer partition λ, we denote its transpose by λt . We define |λ| =

l(λ) X

2

||λ|| =

λi ,

l(λ) X

l(λt )

λ2i

t 2

||λ || =

,

(λti )2

i=1

i=1

i=1

X

(A.1) where l(λ) is the length of the partition (i.e. the number of non-zero terms). We also introduce the following functions that are indexed by two integer partitions µ and ν: Jµν (x; t, q) =

∞ Y

Jµν (Qk−1 x; t, q) , ρ

(A.2)

k=1

where

(i,j)∈µ

Y 

t

1

1

1 − x t−µj +i− 2 q −νi +j− 2



(i,j)∈ν

We further define   Y t 1 1 ϑµν (x; ρ) = ϑ x−1 q −νj +i− 2 t−µi +j− 2 ; ρ (i,j)∈µ

X

sηt /µ (x)sη/ν (y) =

η

X η

sη/µ (x)sη/ν (y) =

∞ Y

(1 + xi yj )

X

sν t /τ (x)sµt /τ t (y)

τ

i,j=1 ∞ Y

(1 − xi yj )−1

X

sν t /τ (x)sµ/τ (y)

τ

i,j=1

(A.7) Secondly, we also used the following identities between products over integer partitions and infinite products [36] t ∞ Y Y t 1 − Qq −µj +i−1 t−νi +j = (1 − Qq −µj +i−1 t−νi +j ) i−1 j 1 − Qq t i,j=1 (i,j)∈ν Y t × (1 − Qq νj −i tµi −j+1 )

 Y  t 1 1 Jµν (x; t, q) = 1 − x tνj −i+ 2 q µi −j+ 2 ×

In performing the calculations in Section III and Section IV, we utilized a number of computational identities. Firstly, we found the following two identities helpful for performing sums of skew Schur functions (see [36] (page 93))

.

(i,j)∈µ t ∞ Y Y t 1 − Qq −µj +i−1 t−µi +j = (1 − Qq −µj +i−1 t−µi +j ) i−1 tj 1 − Qq i,j=1

(i,j)∈µ

t

× (1 − Qq µj −i tµi −j+1 ) (A.8)

16 Finally, we also recall the following identity [44], X X µtj = νjt . (i,j)∈ν

nal expansion of Z3,2 , which is introduced in Eq. (III.15). Using the definition of the refined topological vertex (A.9)

(i,j)∈µ

Appendix B: Diagonal Partition Function Z3,2

In this appendix, we present details of the calculation of the building block W for the computation of the diago-

Cλµν (t, q) =

||µ||2 ||µt ||2 − 2 2 q t

||ν||2 q 2

Z˜ν (t, q)

X  q  |η|+|λ|−|µ| 2 η

where sµ/ν are skew Schur functions and Z˜ν (t, q) =

Y 

t

1 − tνj −i+1 q νi −j

−1

,

(B.2)

(i,j)∈ν

t

t

sλt /η (t−ρ q −ν ) sµ/η (q −ρ t−ν )

(B.1)

In what follows, we denote the whole set of variables as bold expressions, i.e. x = {xi }i=1,...,6 and, for simplicity, we shall not consider the prefactor Zˆ in (III.16): indeed, the summation over the skew Schur function in (III.16) can be written in the form

the expression (III.15) can be written in the form (III.16).

G(x, y, w, z) =

6 X Y

(−Qhi )|µi | (−Qvi )|νi | sµi /˜ηi+3 (xi+3 ) sµti /ηi+5 (yi+5 )sνit /ηi (wi )sνi /˜ηi+3 (zi+3 ) .

(B.3)

{µ}{ν} i=1 {η}{˜ η}

We can perform the sum over skew Schur functions using a method similar to those in [10]: repeatedly using the

G(x, y, w, z) = P ×

6 X Y

identities (A.7) for summing skew Schur functions, we can write:

˜ i yi+5 ) (−Qhi )|µi | (−Qvi )|νi | sµi−1 /˜ηi+2 (Qhi Qvi−1 xi+3 )sµti+1 /ηi (Q

{µ}{ν} i=1 {η}{˜ η}

˜ t × sνi−1 ηi+2 (Qhi+1 Qvi zi+3 ) , /ηi−1 (Qi wi )sνi+1 /˜

(B.4)

˜ i = Qh Qv and where we introduced the notation Q i i P =

6 Y ∞ Y

˜ i−1 xi+3,r yi+4,s )(1 − Qv Q ˜ i+1 wi+1,r zi+3,s ) (1 − Qhi Q (1 − Qhi xi+3,r yi+5,s )(1 − Qvi wi,r zi+3,s ) i . ˜ i yi+5,r wi,s ) (1 − Qh Q ˜ i−1 Qv xi+3,r zi+1,s )(1 − Q ˜iQ ˜ i+1 yi+5,r wi+1,s ) (1 − Qhi Qvi−1 xi+3,r zi+2,s )(1 − Q i i−2 i=1 r,s=1

Thus, in (B.4), we find an expression similar to the expression (B.3) except for the difference that the partitions

have been shifted, e.g. the Schur functions for i = 1 have been replaced in the following fashion

˜ 1 y6 )sν t /η (Q ˜ 1 w1 )sν /˜η (Qh Qv z4 ) . sµ1 /˜η4 (x4 )sµt1 /η6 (y6 )sν1t /η1 (w1 )sν1 /˜η4 (z4 ) → sµ6 /˜η3 (Qh1 Qv6 x4 )sµt2 /η1 (Q 2 1 2 5 6 6

17 By repeating this procedure multiple times, we again obtain the quantity G defined in (B.3), up to a prefactor P1 (The precise form of P1 turns out not to be important) and a shift of all arguments, as ¯ Qy, ¯ Qw, ¯ Qz) ¯ . G(x, y, w, z) = P1 × G(Qx,

(B.5)

further, the relation (B.5) leads to the following recursion relation for all integers n ¯ n x, Q ¯ n y, Q ¯ n w, Q ¯ n z) . (B.6) G(x, y, w, z) = Pn × G(Q ¯ n = 0, we have the relation Using the fact that limn→∞ Q

¯=Q ˜1Q ˜2Q ˜3Q ˜4Q ˜5Q ˜ 6 . Repeating where we abbreviated Q

¯ n x, Q ¯ n y, Q ¯ n w, Q ¯ n z) lim G(Q

n→∞

= lim

n→∞

=

6 X Y ¯ 2 )n wi )sν /˜η (zi+3 ) ¯ 2 )n xi+3 )sµt /η (yi+5 )sν t /η ((Q (−Qhi )|µi | (−Qvi )|νi | sµi /˜ηi+3 ((Q i i+3 i i+5 i i {µ}{ν} i {η}{˜ η}

6 X Y t (−Qhi )|˜ηi+3 | (−Qvi )|ηi | sη˜i+3 ηi+3 (zi+3 ) , /ηi+5 (yi+5 )sηit /˜

(B.7)

{η}{˜ η} i

as the only non-vanishing terms in this limit correspond to µi = η˜i+3 and νit = ηi . The skew Schur functions are non-zero when ηi+2 ⊂ η˜it and η˜i+3 ⊂ ηit , such that we have η3 ⊂ η˜4 ⊂

η˜1t η1t

, η4 ⊂ , η˜5 ⊂

η˜2t η2t

, η5 ⊂ , η˜6 ⊂

η˜3t η3t

, η6 ⊂ , η˜1 ⊂

η˜4t η4t

, η1 ⊂ , η˜2 ⊂

η˜5t η5t

, η2 ⊂ , η˜3 ⊂

η˜6t η6t

η˜4t = η˜5t = η˜6t , such that (B.7) is reduced to ¯ n x, Q ¯ n y, Q ¯ n w, Q ¯ n z) = lim G(Q

n→∞

η3 ⊂ η˜1t ⊂ η4 ⊂ η˜2t ⊂ η5 ⊂ η˜3t ⊂ η6 ⊂ η˜4t ⊂ η1 ⊂ η˜5t ⊂ η2 ⊂ η˜6t ⊂ η3 , which implies that the summation in (B.7) only receives contributions for

¯ |η| = Q

η

.

By considering the transpose of the conditions in the second line conditions, we get

X

∞ Y

¯ k )−1 . (1 − Q

k=1

We thus obtain G(x, y, w, z) = P∞ ·

∞ Y

¯ k )−1 , (1 − Q

(B.8)

k=1

where the multiplicative prefactor can be worked out to be

η1 = η2 = η3 = η4 = η5 = η6 = η˜1t = η˜2t = η˜3t =

P∞ =

6 Y ∞ Y ∞ Y

¯ k−1 xi+3,r yi+5,s )(1 − Qv Q ¯ k−1 wi,r zi+3,s ) (1 − Qhi Q i ¯ k−1 xi+3,r zi+2,s )(1 − Q ˜iQ ¯ k−1 yi+5,r wi,s ) (1 − Qhi Qvi−1 Q i=1 r,s=1 k=1

˜ i−1 Q ¯ k−1 xi+3,r yi+4,s )(1 − Qv Q ˜ i+1 Q ¯ k−1 wi+1,r zi+3,s ) (1 − Qhi Q i k−1 ˜ i−1 Qv Q ¯ ˜iQ ˜ i+1 Q ¯ k−1 yi+5,r wi+1,s ) (1 − Qhi Q xi+3,r zi+1,s )(1 − Q i−2 ˜ i−1 Q ˜ i−2 Q ¯ k−1 xi+3,r yi+3,s )(1 − Qv Q ˜ i+1 Q ˜ i+2 Q ¯ k−1 wi+2,r zi+3,s ) (1 − Qhi Q i × ˜ i−1 Q ˜ i−2 Qv Q ¯ k−1 xi+3,r zi,s )(1 − Q ˜iQ ˜ i+1 Q ˜ i+2 Q ¯ k−1 yi+5,r wi+2,s ) (1 − Qh Q ×

i

i−3

˜ i−1 Q ˜ i−2 Q ˜ i−3 Q ¯ k−1 xi+3,r yi+2,s )(1 − Qv Q ˜ i+1 Q ˜ i+2 Q ˜ i+3 Q ¯ k−1 wi+3,r zi+3,s ) (1 − Qhi Q i × ˜ i−1 Q ˜ i−2 Q ˜ i−3 Qv Q ¯ k−1 xi+3,r zi+5,s )(1 − Q ˜iQ ˜ i+1 Q ˜ i+2 Q ˜ i+3 Q ¯ k+5 yi+5,r wi+3,s ) (1 − Qhi Q i−4 ˜ i−1 Q ˜ i+4 Q ˜ i−3 Q ˜ i+2 Q ¯ k−1 xi+3,r yi+1,s )(1 − Qv Q ˜ i+1 Q ˜ i+2 Q ˜ i+3 Q ˜ i+4 Q ¯ k−1 wi+4,r zi+3,s ) (1 − Qhi Q i × ˜ i−1 Q ˜ i+4 Q ˜ i−3 Q ˜ i−4 Qv Q ¯ k−1 xi+3,r zi−2,s )(1 − Q ˜iQ ˜ i+1 Q ˜ i+2 Q ˜ i+3 Q ˜ i+4 Q ¯ k−1 yi+5,r wi+4,s ) (1 − Qh Q i

i−5

˜ i−1 Q ˜ i−2 Q ˜ i−3 Q ˜ i−4 Q ˜ i−5 Q ¯ k−1 xi+3,r yi,s )(1 − Qv Q ˜ i+1 Q ˜ i+2 Q ˜ i+3 Q ˜ i+4 Q ˜ i+5 Q ¯ k−1 wi+5,r zi+3,s ) (1 − Qhi Q i × . k k ¯ xi+3,r zi+3,s )(1 − Q ¯ yi+5,r wi+5,s ) (1 − Q

(B.9)

18

Finally, using the consistency conditions (III.3)-(III.8), this expression can be simplified to Eq.(III.18).

Appendix C: Flop Transforms for Twisted Diagrams

In this appendix, we recapitulate a series of flop and symmetry transforms of a twisted web diagram of length L with generic shift δ (as shown in Fig. 9 and equivalently in Fig. 10, along with a labelling of all relevant parameters and integer partitions) that relate it to a twisted web diagram of the same length but with shift δ + 1. The duality between the two twisted web diagrams was first discussed in [12]. It can be applied iteratively to obtain a web diagram with shift δ = 0, which was used in [12] to argue for a duality between XN,M and XN 0 ,M 0 for N M = N 0 M 0 and gcd(N, M ) = k = gcd(N 0 , M 0 ). Here, we are primarily interested in the case gcd(N, M ) = 1. We start out by performing an SL(2, Z) transformation of the twisted web diagram in Fig. 9, whose result is shown in Fig. 10. This does not change the K¨ahler parameters, as indicated in Fig. 10.

v100 = v1 + h1 + h2 ,

00 vL = vL + h1 + hL ,

h00i = h0i ,

m00i = m0i ,

∀i ∈ / {1, L} .

(C.3)

We can summarise this in the form of the duality map vi −→ −hi , hi −→ vi−1 + hi + hi−1 , mi −→ mi + hi + hi+δ+1 .

m01 = m1 + hδ+2 , v10 = v1 + h2 , 0 m2 = m2 + h2 + hδ+3 , v20 = v2 + h2 + h3 , ... ... 0 0 mL−1 = mL−1 + hL−1 + hδ , vL−1 = vL−1 + hL−1 + hL , 0 0 mL = mL + hL + hδ+1 , vL = vL + hL , (C.1) which reflect how the various intervals are connected to the ones that are being flopped. As the final step, performing a flop transformation of h1 , we get the web shown in Fig. 12, which is a twisted web diagram with shift δ + 1. Concerning the parameters that are associated with the individual line segments in the web, we have

m001 = m1 + h1 + hδ+2 ,

and vi00 = vi0 ,

As the next step, we perform a flop transformation on the intervals {h2 , . . . , hL }. After suitable cutting and re-gluing, the twisted web can be presented in the form shown in Fig. 11. Notice that not only hi −→ −hi for i = 2, . . . , L, but also the remaining parameters have changed according to

m00L−δ = mL−δ + h1 + hL−δ ,

(C.2)

Moreover, the basis parameters b a1,...,L and S remain the same as in Fig. 10 (i.e. they are invariant under the duality map), while the parameters R(δ) and R(δ+1) are related by R(δ+1) − R(δ) = S .

(C.5)

(C.4)

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19 b a1 1

b a2

m1 h1

a

2 v1

··· b aδ

m2 h2

δ

b aδ + 1

mδ h3

v2 L−δ+1

···

hδ

δ+1

···

b aL−1

mδ+1 L−1

hδ+1

vδ L−δ+2

b aL

mL−1

vδ+1

hδ+2

L

···

hL−1

L mL

vL−1 1

hL

−R(δ)

vL

h1

a

L−δ−1

L−δ

S

···

···

FIG. 10. Twisted web diagram with shift δ. This diagram is obtained from Fig. 9 through an SL(2, Z) transformation. 1 m01

h1

2

v10

m02

−h2

3

v20

m03

δ+1

a −h3

v30

L−δ+1

···

vδ0

m0δ+1

δ+2

0 vδ+1

−hδ+1

m0δ+2

L−1

L−δ+2 −hδ+2

0 vδ+2

L

···

0 vL−2

m0L−1 L

−hL−1

0 vL−1

m0L a

1 −hL

0 vL

L−δ−2

L−δ−1

L−δ

FIG. 11. Web diagram obtained by flopping h2 , . . . , hL in Fig. 10.

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20 b a1 1

a

00 vL

b a2

m001

2

−h1 v100

··· b aδ

m002

−h2

δ+1

v200

L−δ

···

vδ00

b aδ + 1

m00δ+1

δ+2

00 vδ+1

−hδ+1

···

m00δ+2

b aL−1 L−1

L−δ+1 −hδ+2

00 vδ+2

L

···

L

−hL−1 1

b aL

m00L−1

00 vL−2

00 vL−1

−R(δ+1)

m00L

−hL

00 vL

a

L−δ−2

S

···

L−δ−1

···

FIG. 12. Twisted web diagram with shift δ + 1 obtained from Fig. 11 through flop of the curve h1 .

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