Gromov-Witten invariants and membrane indices of fivefolds via the topological vertex

GROMOV –WITTEN INV ARIANTS AND MEMBRANE INDICES OF FIVEFOLDS VIA THE TOPOLOGICAL VER TEX Y ANNIK SCHULER A B S T R A C T . W e conjecture the existence of almost integer invariants governing the all-genus equivariant Gromov–W itten theory of Calabi–Y au ﬁvefolds with a torus action. W e prove the conjecture for skeletal, locally anti-diagonal torus actions by establishing a vertex formalism evaluating the Gromov–W itten invari- ants via the topological vertex of Aganagic, Klemm, Mariño and V afa. W e apply the formalism in several examples. C O N T E N T S Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1. The vertex formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2. Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3. Proof of the vertex formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 Appendix A. The plethystic logarithm in localised K-theory . . . . . . . . . . . . . 34 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 I N T R O D U C T I O N 0.1. Gromov–W itten invariants and the membrane index. In recent work of Brini and the author [ BS24 ], a conjecture was put forward relating the equivariant Gromov–W itten invariants of a Calabi–Y au ﬁvefold to the so-called membrane index. Concretely , let 𝑍 be a Calabi–Y au ﬁvefold equipped with the action of a torus T ﬁxing the holomorphic ﬁveform. The conjectur e asserts that the series GW 𝛽 ( 𝑍 , T ) B  𝑔 ≥ 0 ∫ [ 𝑀 𝑔 ( 𝑍 ,𝛽 ) ] virt T 1 of equivariant Gr omov–W itten invariants equals the ˆ 𝐴 -genus of the mathemati- cally yet-to-be-constructed moduli space of M2-branes on 𝑍 . While the Gromov– W itten series is a formal power series in the torus weights 𝜖 𝑖 , the ˆ 𝐴 -genus admits a lift to K-theory under reasonable assumptions on the moduli space of M2-branes. This makes the latter a rational function in e 𝜖 𝑖 . Thus, at a numerical level, the conjecture pr edicts a lift of the Gromov–W itten series to a rational function. 1 2 Y ANNIK SCHULER Conjecture A. Let 𝑍 be a Calabi–Y au ﬁvefold with a Calabi–Y au T -action. There exist rational functions Ω 𝛽 ( 𝑞 𝑖 ) ∈ Z  1 2        𝑞 ± 1 / 2 1 , . . . , 𝑞 ± 1 / 2 dim T , ( 1 1 − Î 𝑖 𝑞 𝑛 𝑖 / 2 𝑖 ) 𝒏 ∈ Z dim T \{ 0 }       labelled by curve classes 𝛽 in 𝑍 such that under the change of variables 𝑞 𝑖 = e 𝜖 𝑖 we have (1) GW 𝛽 ( 𝑍 , T ) =  𝑘 | 𝛽 1 𝑘 Ω 𝛽 / 𝑘 ( 𝑞 𝑘 𝑖 ) . Moreover , Ω 𝛽 has coefﬁcients in Z if the T -action on 𝑍 is skeletal. W e will refer to the invariant Ω 𝛽 as the membrane index of 𝑍 in curve class 𝛽 . This index has earlier been investigated in r elation to K-theoretic Donaldson–Thomas theory by Nekrasov and Okounkov [ NO16 ]. In this paper we prove Conjectur e A for a special class of geometries. Theorem B. (Corollary 1.11 ) Conjectur e A holds if the torus action on 𝑍 is skeletal and locally anti-diagonal and curve classes are supported away fr om anti-diagonal strata. W e call a torus action skeletal if the number of ﬁxed points and one-dimensional orbits is ﬁnite. W e say that such an action is locally anti-diagonal if the weight decomposition of the tangent space at every ﬁxed point of 𝑍 features two torus weights which are opposite to each other (see Deﬁnition 1.1 ). The assumption on curve classes ensures that stable maps do not interact with strata of the one- skeleton of 𝑍 on which the torus acts with opposite weights. 0.2. A vertex formalism for locally anti-diagonal torus actions. Theorem B is proven by a dir ect evaluation of the Gromov–W itten series. Theorem C. (Corollary 1.8 ) The disconnected Gromov–W itten invariants of a Calabi– Y au ﬁvefold 𝑍 with a Calabi–Y au, skeletal and locally anti-diagonal action by a torus T in a curve class 𝛽 supported away from anti-diagonal strata are computed via the topological vertex of Aganagic–Klemm–Mariño–V afa [ AKMV05 ]: (2) GW • 𝛽 ( 𝑍 , T ) =  𝝁 Ö 𝑒 ∈ 𝐸 ( Γ ) 𝐸 ( 𝑒 , 𝝁 ) Ö 𝑣 ∈ 𝑉 ( Γ ) W ( 𝑣 , 𝝁 ) . Here, the sum ranges over tuples of partitions decorating the half-edges of the one-skeleton Γ of 𝑍 . Modulo a change of basis one may think of these partitions as indicating a contact order of relative stable maps mapping to a degeneration of the one-skeleton. Every edge 𝑒 of the one-skeleton is weighted by an explicit monomial 𝐸 ( 𝑒, 𝝁 ) in the exponentiated weights 𝑞 𝑖 = e 𝜖 𝑖 in formula ( 2 ). V ertices GW INV ARIANTS AND MEMBRANE INDICES OF 5-FOLDS VIA TV 3 𝑣 are weighted by the thr ee-leg topological vertex W ( 𝑣, 𝝁 ) of Aganagic–Klemm– Mariño–V afa [ AKMV05 ]. Theorem B then follows fr om Theorem C since both edge and vertex weights are rational functions in 𝑞 1 / 2 𝑖 with constrained poles. The reader might be surprised to encounter the topological vertex in our vertex formula ( 2 ) as it is an object usually appearing in the Gromov–W itten theory of threefolds as opposed to ﬁvefolds. This is explained as follows: By our assump- tion that the torus action is locally anti-diagonal a ﬁvefold vertex weight r educes to a thr eefold one with the T -weight of the attached anti-diagonal stratum taking the role of the genus counting variable. This last comment is probably best illustrated in the situation where 𝑍 is the product of a toric Calabi–Y au threefold 𝑋 with C 2 . W e assume that the threefold is acted on by its two-dimensional Calabi–Y au tor us T ′ and C × is acting anti- diagonally on the afﬁne plane: T = T ′ × C × ⟳ 𝑍 = 𝑋 × C 2 . In this situation the Gromov–W itten invariants of 𝑍 simply coincide with the ones of 𝑋 with the equivariant parameter 𝜖 associated with the weight-one repr esenta- tion of C × taking the role of the genus counting variable: (3) GW 𝛽 ( 𝑍 , T ) =  𝑔 ≥ 0 ( − 𝜖 2 ) 𝑔 − 1 ∫ [ 𝑀 𝑔 ( 𝑋 , 𝛽 ) ] virt T ′ 1 . This fact is a consequence of Mumfor d’s relation for the Chern classes of the Hodge bundle [ Mum83 ]. See [ BS24 , Sec. 2.4] for more details. Consistent with this dimensional reduction, our vertex formula ( 2 ) specialises to the original topo- logical vertex formula [ AKMV05 ] formalised in Gromov–W itten theory by Li, Liu, Liu and Zhou [ LLLZ09 ]. See Section 1.7.2 for details. The key idea in the proof of Theorem C is that one can employ the same dimen- sional r eduction trick ( 3 ) locally at each vertex of the one-skeleton of 𝑍 under the assumption that ther e are two tangent directions with opposite T -weights. The same idea was r ecently pursued by Y u–Zong [ YZ26 ] in the context of the one-leg orbifold vertex. 0.3. Relation to Gopakumar –V afa invariants. Let us also quickly comment on the specialisation of Conjecture A to the product case 𝑍 = 𝑋 × C 2 with the action by T ′ × C × we just discussed. Here, we do not necessarily assume 𝑋 to be toric. W e claim that in this setting the conjecture specialises to a weak version of the Gopakumar–V afa integrality conjecture [ GV98 ] which was recently proven by Ionel–Parker and Doan–Ionel–Parker [ IP18 ; DIW21 ]. Indeed, suppose Conjecture A holds for 𝑋 × C 2 . Then since by the MNOP conjec- ture [ MNOP06a ; Par23 ] the equivariant Gr omov–W itten invariants of 𝑋 are just 4 Y ANNIK SCHULER numbers independent of any T ′ -weights, the functions Ω 𝛽 must satisfy Ω 𝛽 ∈ Z  1 2  h 𝑞 ± 1 / 2 ,   1 − 𝑞 𝑛 / 2  − 1  𝑛 ∈ Z \ { 0 } i where 𝑞 = e 𝜖 . Now if we additionally assume that Ω 𝛽 has integer coefﬁcients, that it only featur es integer powers of 𝑞 , and that Ω 𝛽 can have at worst a double pole at 𝑞 = 1 and no other poles but at zer o and inﬁnity , we can expand Ω 𝛽 = 𝑔 max  𝑔 = 0 𝑛 𝑔,𝛽  𝑞 1 / 2 − 𝑞 − 1 / 2  2 𝑔 − 2 with 𝑛 𝑔,𝛽 ∈ Z . Her e, we also used that the Gr omov–W itten series is invariant under 𝜖 ↦→ − 𝜖 . In combination with ( 3 ), this exactly yields the statement of the Gopakumar–V afa conjecture:  𝑔 ≥ 0 ( − 𝜖 2 ) 𝑔 − 1 ∫ [ 𝑀 𝑔 ( 𝑋 , 𝛽 ) ] virt T ′ 1 =  𝑘 | 𝛽 𝑔 max  𝑔 = 0 𝑛 𝑔,𝛽 / 𝑘 𝑘  𝑞 𝑘 / 2 − 𝑞 − 𝑘 / 2  2 𝑔 − 2 . Hence, we see that, modulo denominators of two and additional constraints on poles and exponents of 𝑞 , Conjecture A r ecovers the Gopakumar –V afa integrality conjecture for products of Calabi–Y au threefolds with the af ﬁne plane when T ﬁxes the holomorphic threeform. When T acts non-trivially on the holomorphic threeform, it is expected that for some Calabi–Y au threefolds the membrane index recovers r eﬁned Gopakumar–V afa invariants [ BS24 , Conj. 7.9]. 0.4. Examples. W e apply our vertex formalism to the following geometries with the action by a torus meeting the assumptions of Theor em C : (i) T ot 𝑋  L ⊕ L ∨  where L is a line bundle on a Calabi–Y au threefold 𝑋 (§ 1.7 ) (ii) T ot P 1  O ( − 2 )  × C 3 (§ 2.1 ) (iii) Products of strip geometries with C 2 (§ 2.2 ) (iv) T ot P 1 × P 1  O ( − 1 , 0 ) ⊕ 2 ⊕ O ( 0 , − 2 )  (§ 2.3 ) (v) T ot P 2  O ( − 1 ) ⊕ 3  (§ 2.4 ) (vi) T ot P 3  O ( − 2 ) ⊕ 2  (§ 2.5 ) Example (i) is a mild generalisation of the product situation 𝑋 × C 2 discussed be- fore. For (ii) and (iii) we are able to prove closed formulae for the Gromov–W itten series. Based on computer experiments we conjectur e a closed form solution for the Gromov–W itten series of (iv) and pr edict structural properties satisﬁed by the membrane indices of (v) and (vi). See Conjectures 2.4 , 2.7 and 2.8 . Out of all geometries studied in this article, example (v) probably best showcases all features of Conjectur e A . Most notably , for special torus actions the membrane indices of this Calabi–Y au ﬁvefold do feature denominators of two. Moreover , in GW INV ARIANTS AND MEMBRANE INDICES OF 5-FOLDS VIA TV 5 accordance with Conjecture A fractions of two seem indeed to be the worst type of denominators to occur . See Remark 2.6 for details. 0.5. Outline of the paper . The vertex formalism together with geometric prelimi- naries is presented in Section 1 . This section is aimed at a reader who is interested in applying the vertex formalism to a concrete example. W e present several such applications in Section 2 while the proof of the vertex formalism is deferred to Sec- tion 3 . Our proof follows the idea of Li, Liu, Liu and Zhou [ LLLZ09 ] of trading the weights in the graph sum resulting fr om torus localisation for relative Gromov– W itten invariants of partial compactiﬁcations of torus orbits in the one-skeleton of the target. This process is referr ed to as capped localisation in [ MOOP11 ]. The proof of Conjecture A in the setting where our vertex formalism applies is found in Section 1.6 . T ogether with the observation that each weight in the vertex formula is a rational function with monic denominators and numerators having integer coefﬁcients, our pr oof crucially uses that these features are preserved when taking the plethystic logarithm. This last fact is proven in Appendix A following ideas of Konishi and Peng [ Kon06 ; Pen07 ]. 0.6. Context & Prospects. 0.6.1. Limitations. Our pr oof of Conjecture A hinges on the fact that we are able to provide closed formulae for every factor in our vertex formula. For this all assumptions made in the statement of Theorem C are crucial: The torus action being skeletal allows us to use capped localisation. This assumption is ther efore responsible for graph sum decomposition of the Gromov–W itten invariant. The assumption that the torus action is locally anti-diagonal r educes vertex and edge terms from ﬁve to three dimensions. T ogether with the fact that the torus action is Calabi–Y au one can identify vertex contributions with the topological vertex weight and evaluate edge terms explicitly . T o drop the assumption of being locally anti-diagonal a better understanding of quintuple Hodge integrals and rubber integrals with four Hodge insertions will be r equired. First steps into this direction will be presented in [ GPS26 ]. See Remark 3.4 for details. Let us also note that the requir ement of being locally anti-diagonal prevents us fr om applying our vertex formalism to interesting examples such as to products 𝑋 × C 2 with a torus action that engineer supersymmetric gauge theories on C 2 with general Ω background, that is, beyond an anti-diagonal torus action on the afﬁne plane (see Remark 2.3 ). Finally , the assumption on curve classes to be supported away from anti-diagonal strata is r equir ed to ensur e that at most thr ee-legged vertices occur . T o drop this assumption, better control over the descendant threefold vertex is requir ed. See Remarks 3.1 and 3.5 for details. 6 Y ANNIK SCHULER 0.6.2. M2-branes. Let us quickly motivate how the speculation that T -equivariant Gromov–W itten invariants of a Calabi–Y au ﬁvefold 𝑍 equate to the ˆ 𝐴 -genus of the moduli space of M2-branes of 𝑍 implies the statement of Conjecture A . In its most optimistic form, the speculation claims the existence of a sufﬁciently well- behaved moduli space M2 𝛽 ( 𝑍 ) labelled by curve classes in 𝑍 . The ˆ 𝐴 -genus of this moduli space Ω 𝛽 = ˆ 𝐴 T  M2 𝛽 ( 𝑍 )  should then equate to the Gromov–W itten series via equation ( 1 ). Now by Hirzebruch–Riemann–Roch, the ˆ 𝐴 -genus lifts to equivariant K-theory . It equates to the Euler characteristic of a square root of the (virtual) canonical bundle Ω 𝛽 = 𝜒 T  M2 𝛽 ( 𝑍 ) , O virt M2 𝛽 ( 𝑍 ) ⊗ 𝐾 1 / 2 virt  . In case 𝐾 1 / 2 virt is an honest line bundle on the T -ﬁxed locus of M2 𝛽 the last equality implies that Ω 𝛽 ∈ 𝐾 T ( pt ) loc  Z h 𝑞 ± 1 1 , . . . , 𝑞 ± 1 dim T ,  ( 1 − Î 𝑖 𝑞 𝑛 𝑖 𝑖 ) − 1  𝒏 ∈ Z dim T \{ 0 } i . There are, however , obstructions towards the existence of such a line bundle: First, it may only be well-deﬁned after passing to a cover of T to allow for frac- tional characters. This is captured in Conjecture A by permitting square roots of T -characters. Second, the square root of 𝐾 virt may only be well-deﬁned as an element in K-theory after inverting two: Ω 𝛽 ∈ 𝐾 T ( pt ) loc ⊗ Z [ 1 2 ] (cf. [ OT23 , Sec. 5.1]). This explains why we permit denominators of two in Conjectur e A . Finally , Nekrasov and Okounkov argue that the (virtual) canonical bundle of M2 𝛽 ( 𝑍 ) relative to the Chow variety should indeed admit a squar e root in the Picard group. Now if the T -action on 𝑍 is skeletal then all ﬁxed points of the Chow variety ar e isolated. As a consequence, in this setting we should ﬁnd that Ω 𝛽 has integer coefﬁcients. 0.6.3. Denominators of two. Let us expand on the occurrence of denominators of two. In Theorem C we indeed verify that for skeletal torus actions (satisfying certain additional conditions) membrane indices have integer coefﬁcients. In Section 2.4 we consider the example of a ﬁvefold whose membrane indices de- velop denominators of two precisely when the torus action turns non-skeletal. It appears worthwhile to investigate whether starting fr om our vertex formula ( 2 ) one can argue combinatorially that factors of two are indeed the worst type of denominators that may appear under restriction of the tor us action. 0.6.4. Pandharipande–Zinger. If the moduli space of stable maps to the Calabi–Y au ﬁvefold 𝑍 is proper for all genera and curve classes, then the denominators of Ω 𝛽 should be even more constrained: For Conjecture A to be compatible with the Gopakumar–V afa integrality conjecture of Pandharipande–Zinger [ PZ10 , Conj. 1] in the non-equivariant limit one must have Ω 𝛽 ∈ 1 8 Z . Whether ther e is any GW INV ARIANTS AND MEMBRANE INDICES OF 5-FOLDS VIA TV 7 geometric r elation between Pandharipande–Zinger and membrane indices is not clear to the author . 0.6.5. Nekrasov–Okounkov. Nekrasov and Okounkov conjecture that the generat- ing series of K-theor etic stable pair invariants of a threefold 𝑋 with an appropri- ate insertion depending on the choice of two line bundles L 4 , L 5 coincides with Laurent expansion of the M2-brane index of 𝑍 = T ot 𝑋 L 4 ⊕ L 5 [ NO16 , Conj. 2.1]. Here, the box-counting variable 𝑞 on the stable pairs side gets identiﬁed with the coordinate of C × acting anti-diagonally on L 4 ⊕ L 5 . This conjecturally implies a maps/sheaves correspondence generalising the MNOP conjectur e without inser - tions [ MNOP06a ; MNOP06b ]. Beyond the situation where the torus action on 𝑋 is Calabi–Y au, it is not immediately clear , however , how the vertex formalisms governing the respective sides of the correspondence can be related. In or der to realise the K-theoretic vertex [ NO16 ; Arb21 ; KOO21 ] governing the sheaf side in Gromov–W itten theory a better understanding of the stable maps vertex beyond anti-diagonal torus actions, i.e. quintuple Hodge integrals, will be crucial. Even the limit of the reﬁned topological vertex [ IKV09 ] is currently out of reach in Gromov–W itten theory since it governs a limit where 𝜖 𝑖 → ±∞ . Accessing this limit would requir e ﬁnding an analytic continuation of the Gromov–W itten series in the torus weights which is currently out of reach. See Remarks 2.1 and 2.2 for comparisons of our vertex formalism with the reﬁned topological vertex in concrete examples. Conversely , the vertex formalism of Theor em C is har d to r ealise in Donaldson– Thomas theory since in general it would requir e the specialisation of the box- counting variable 𝑞 to equivariant variables locally at the vertices. This operation is only well-deﬁned after passing to an analytic continuation in 𝑞 which is cur- rently out of r each beyond the two-leg vertex [ KOO21 ]. 0.6.6. M5-branes. Suppose now that 𝑍 is a toric Calabi-Y au ﬁvefold with a Hamil- tonian action by a torus meeting the requir ements of Theorem C . W e may factor the afﬁne neighbour hood of a torus ﬁxed point of 𝑍 into C 3 × C 2 so that the torus action on both factors is Calabi–Y au. Now suppose that in such an af ﬁne neigh- bourhood we are given a submanifold 𝐿 × C × { 0 } wher e 𝐿 is an Aganagic–V afa Lagrangian submanifold of C 3 intersecting a non-compact stratum of 𝑍 . Then following the arguments of Fang and Liu [ FL13 ] one should be able to pr ove that stable maps fr om Riemann surfaces with boundary mapping to 𝐿 × C × { 0 } are enumerated by a vertex formalism similar to Theorem C . As a consequence the generating series of these open Gromov–W itten invariants should be governed by index type invariants. Such invariants would generalise LMOV invariants [ LM01 ; OV00 ; LMV00 ; MV02 ] and should pr obably admit an interpretation as indices of M2-M5-bound states. W ith the methods of [ Y u24 ] one should be able 8 Y ANNIK SCHULER to prove integrality of these invariants for skeletal and locally anti-diagonal torus actions analogous to Theorem B . 0.6.7. Higher dimensions. By the assumption that each afﬁne neighbourhood of a torus ﬁxed point factors into C 3 × C 2 with the torus acting anti-diagonally on the second factor , the Gr omov–W itten vertex reduces from ﬁve to three dimensions. Since the threefold vertex admits an explicit formula this observation is the key insight that allows us to pr ove Theorem C from which we deduce Theorem B . One can apply the same trick in arbitrarily high odd dimension. Suppose 𝑍 is of dimension 3 + 2 𝑛 with a skeletal and Calabi–Y au action by a torus T so that locally at every torus ﬁxed point 𝑍 looks like C 3 × C 2 × · · · × C 2 with the induced T -action on each factor being Calabi–Y au. Under this assumption the generating series of Gromov–W itten invariants of 𝑍 can again be evaluated via the topological vertex. As a consequence the generating series admits a pr esentation as a rational func- tion in variables of the form exp ( Î 𝑖 𝜖 𝑖 / Î 𝑗 𝜖 ′ 𝑗 ) wher e 𝜖 𝑖 and 𝜖 ′ 𝑗 are torus weights. As in the statement of Conjecture A the poles of these rational functions are highly constrained. It is tempting to wonder whether this feature persists for arbitrary torus actions. The author is, however , unaware of any geometry 𝑍 showcasing such a feature beyond dimension ﬁve. Acknowledgements. The author beneﬁted from discussions with Alessandro Giacchetto, Daniel Holmes, Davesh Maulik and Andrei Okounkov . The author was supported by the DFG W alter Benjamin Fellowship 576663726 and the SNF grant SNF-200020-21936. 1. T H E V E R T E X F O R M A L I S M 1.1. Geometric preliminaries. Let 𝑍 be a smooth quasi-projective Calabi–Y au variety with the action by a torus T  ( C × ) 𝑚 . Most of the time we will need to impose the following assumptions on the torus action. Deﬁnition 1.1. W e say that a T -action on 𝑍 is • skeletal if the number of ﬁxed points and one-dimensional orbits is ﬁnite; • Calabi–Y au if the action ﬁxes the holomorphic volume form; • locally anti-diagonal if for each connected component 𝐹 of the ﬁxed locus of 𝑍 there appear at least two non-zero, opposite T -weights in the weight decomposition of the normal bundle of 𝐹 in 𝑍 . For instance, if 𝑍 is toric then the action by its dense torus b T is skeletal. However , the action by this torus is generally not Calabi–Y au. Only once we pass to a suitable codimension one subtorus T ⊂ b T the induced action by this torus will GW INV ARIANTS AND MEMBRANE INDICES OF 5-FOLDS VIA TV 9 be Calabi–Y au. In the context of toric varieties we will often refer to such a torus as the Calabi–Y au torus of 𝑍 . Suppose now we are provided with a skeletal T -action on 𝑍 . W e will associate a decorated graph Γ to ( 𝑍 , T ) which we call the T -diagram of 𝑍 . It recor ds the geometry locally around the one-skeleton of 𝑍 : • vertices 𝑣 ∈ 𝑉 ( Γ ) correspond to T -ﬁxed points 𝑝 𝑣 of 𝑍 ; • edges 𝑒 ∈ 𝐸 ( Γ ) correspond to compact T -preserved lines 𝐶 𝑒 each connect- ing two ﬁxed points; • leafs 𝑙 ∈ 𝐿 ( Γ ) correspond to non-compact T -preserved lines 𝐶 𝑙 containing one ﬁxed point; • half-edges ℎ = ( 𝑣 , 𝑒 ) ∈ 𝐻 ( Γ )  𝐸 ( Γ ) 2 ⊔ 𝐿 ( Γ ) ar e decorated with the T -weight 𝜖 ℎ B 𝑐 1 ( 𝑇 𝑝 𝑣 𝐶 𝑒 ) ∈ H 2 T ( pt , Z ) of the induced torus action on the tangent space of 𝐶 𝑒 at the ﬁxed point 𝑝 𝑣 . W ith this notation at hand we remark that a skeletal T -action on 𝑍 is Calabi–Y au if and only if 0 = Í ℎ ∋ 𝑣 𝜖 ℎ for any (and thus every) ﬁxed point 𝑣 ∈ 𝑉 ( Γ ) . The action is locally anti-diagonal if for every vertex 𝑣 there exist two half-edges ℎ ≠ ℎ ′ adjacent to 𝑣 with 𝜖 ℎ = − 𝜖 ℎ ′ . In case the action of the maximal compact subgroup T R ⊂ T on 𝑍 is Hamiltonian (with respect to some sufﬁciently generic symplectic form on 𝑍 ) the moment map 𝜇 : 𝑍 − → Lie T R  H 2 T ( pt , R )  R 𝑚 provides us with an embedding of Γ into R 𝑚 . As in the following example we will often use this embedding for a better visualisation. Example 1.2. Let us illustrate the setup in a concrete example: 𝑍 = T ot P 1  O ( − 2 )  × C 3 . Let b T  ( C × ) 5 be a dense tor us of 𝑍 . W e assume that it acts on the coordinate lines of C 3 with tangent weights 𝜖 3 , 𝜖 4 , 𝜖 5 respectively . Moreover , we denote by 𝜖 1 the tangent weight at 0 ∈ P 1 ⊂ 𝑍 and ﬁnally by − 𝜖 2 the b T -weight on the holomorphic two-form of T ot P 1 ( O ( − 2 ) ) . The ﬁvefold 𝑍 has two ﬁxed points 0 , ∞ ∈ P 1 ⊂ 𝑍 with tangent weights ( 𝜖 1 , 𝜖 2 − 𝜖 1 , 𝜖 3 , 𝜖 4 , 𝜖 5 ) and ( − 𝜖 1 , 𝜖 2 + 𝜖 1 , 𝜖 3 , 𝜖 4 , 𝜖 5 ) respectively . If we pass to a subtorus T  ( C × ) 4 where the relation Í 5 𝑖 = 2 𝜖 𝑖 = 0 holds then the T -action on 𝑍 is Calabi–Y au. However , this torus action is not locally anti-diagonal. T o get such an action we have to impose additional constraints for which we have essentially two options: (A) one imposes that 𝜖 𝑖 = − 𝜖 𝑗 or (B) 10 Y ANNIK SCHULER ◦ 𝜖 𝑖 ◦ 𝜖 𝑖 𝜖 𝑗 𝜖 𝑗 (A) ◦ 𝜖 𝑖 𝜖 𝑖 𝜖 𝑗 ◦ 𝜖 𝑗 (B) F I G U R E 1 . Illustration of (A) the embedding of the T A -diagram in Lie T A , R  R 3 and (B) the T B -diagram in Lie T B , R  R 2 . The blue, gr een and orange lines indicate the three torus pr eserved coordinate lines of C 3 . The circle highlights the choice of a distinct direction at each vertex as will be introduced in Section 1.3.1 . that 𝜖 𝑖 = 𝜖 1 − 𝜖 2 and 𝜖 𝑗 = − 𝜖 1 − 𝜖 2 for some 𝑖 , 𝑗 ∈ { 3 , 4 , 5 } . W e denote the three- dimensional, r espectively two-dimensional, subtori r ealising these constraints by T A and T B . One checks that the action by these tori is indeed locally anti-diagonal. See Figure 1 for an illustration of the two tor us diagrams. 1.2. Gromov–W itten invariants. From now on let 𝑍 always denote a Calabi–Y au ﬁvefold together with a skeletal, Calabi–Y au and locally anti-diagonal action by a torus T . W e consider the T -equivariant genus- 𝑔 Gromov–W itten invariants of 𝑍 in curve class 𝛽 : GW 𝑔,𝛽 ( 𝑍 , T ) B ∫ [ 𝑀 𝑔 ( 𝑍 ,𝛽 ) ] virt T 1 . In case the moduli space is not pr oper this invariant is deﬁned as a T -equivariant residue assuming that the T -ﬁxed locus is pr oper . The invariants GW 𝑔,𝛽 ( 𝑍 , T ) are rational functions in the torus weights of homoge- nous degree 2 𝑔 − 2 . Hence, by the degree grading GW 𝛽 ( 𝑍 , T ) B  𝑔 ≥ 0 GW 𝑔,𝛽 ( 𝑍 , T ) is a well-deﬁned formal series. Moreover , we denote GW ( 𝑍 , T ) B  𝛽 ≠ 0 𝑄 𝛽 GW 𝛽 ( 𝑍 , T ) where the sum r uns over all ef fective curve classes of 𝑍 and we write 1 +  𝛽 ≠ 0 𝑄 𝛽 GW • 𝛽 ( 𝑍 , T ) B GW • ( 𝑍 , T ) B exp GW ( 𝑍 , T ) for the generating series of disconnected invariants. GW INV ARIANTS AND MEMBRANE INDICES OF 5-FOLDS VIA TV 11 V irtual localisation [ GP99 ] decomposes the Gromov–W itten invariants of 𝑍 into contributions of the individual T -ﬁxed loci of the moduli space: (4) GW • 𝛽 ( 𝑍 , T ) =  𝑔 ≥ 0 ∫ [ 𝑀 • 𝑔 ( 𝑍 ,𝛽 ) ] virt T 1 =  𝛾 ∫ [ 𝐹 𝛾 ] virt 1 𝑒 T ( 𝑁 virt 𝛾 ) . The ﬁxed loci are labelled by decorated graphs 𝛾 whose vertices correspond to components of the domain curve being contracted to a ﬁxed point and edges correspond to rational covers of torus preserved lines fully ramiﬁed over the ﬁxed points. Edges are decorated by the degree of their associated cover . W e denote by 𝑑 𝑒 the degree of the covering of the torus orbit in 𝑍 labelled by 𝑒 ∈ 𝐸 ( Γ ) . This yields an assignment 𝒅 : 𝐸 ( Γ ) → Z ≥ 0 which is subject to condition 𝛽 = Í 𝑒 𝑑 𝑒 [ 𝐶 𝑒 ] . W e call 𝒅 the skeletal degree of 𝛾 . W e write GW • 𝒅 ( 𝑍 , T ) B  𝛾 has skeletal degree 𝒅 ∫ [ 𝐹 𝛾 ] virt 1 𝑒 T ( 𝑁 virt 𝛾 ) . for the partial contribution of the ﬁxed loci with ﬁxed skeletal degree to the overall Gr omov–W itten invariant. W e use a similar notation to denote partial contributions of connected invariants as well. 1.3. Combinatorial preparations. Our vertex formalism will depend on a certain combinatorial choice which we will describe now . 1.3.1. The choice of distinct dir ections and orders. Let Γ be the T -diagram of 𝑍 . For each vertex 𝑣 ∈ 𝑉 ( Γ ) we make the following choices. By our assumption that the torus action is locally anti-diagonal there must be (at least) two half-edges ℎ, ℎ ′ at 𝑣 whose weights are opposite: 𝜖 ℎ = − 𝜖 ℎ ′ . W e will call them the anti-diagonal half-edges at 𝑣 . Out of the two, we choose a distinct direction ℎ 𝑣 ∈ { ℎ, ℎ ′ } and write 𝜖 𝑣 B 𝜖 ℎ 𝑣 . Moreover , we ﬁx a cyclic order 𝜎 𝑣 of the remaining thr ee half-edges adjacent to 𝑣 . Collectively , we will r efer to such a choice ( ℎ 𝑣 , 𝜎 𝑣 ) 𝑣 ∈ 𝑉 ( Γ ) for every vertex as a choice of distinct directions and or ders. Example 1.3. Let us revisit Example 1.2 where we considered 𝑍 = T ot P 1  O ( − 2 )  × C 3 . W e identiﬁed two tori T A and T B whose action on 𝑍 is Calabi–Y au and locally anti-diagonal. The planar embedding of the torus diagrams pr esented in Figure 1 naturally provides us with a choice of cyclic order after ﬁxing an orientation of R 2 . For the torus T A we may choose the same distinct direction 𝜖 𝑣 = 𝜖 𝑖 at both ﬁxed points. For T B we choose 𝜖 𝑖 at 0 and 𝜖 𝑗 at ∞ . As in Figure 1 we will indicate the choice of a distinct direction by a cir cle on the associated stratum. 12 Y ANNIK SCHULER 1.3.2. Framing and mixing parity. Suppose we ﬁxed a choice of distinct directions and orders. Then to every edge 𝑒 = ( ℎ, ℎ ′ ) none of whose half-edges is an anti- diagonal half-edge we assign what we call a framing parity 𝑓 𝑒 ∈ Z / 2 Z and a mixing parity 𝑝 𝑒 ∈ Z / 2 Z : Suppose 𝑒 links the vertices 𝑣 and 𝑣 ′ . Then the weights 𝜖 ℎ , 𝜖 𝜎 𝑣 ( ℎ ) , 𝜖 𝑣 , 𝜖 𝜎 𝑣 ′ ( ℎ ′ ) and 𝜖 𝑣 ′ either satisfy linear relations (5) 𝜖 𝜎 𝑣 ( ℎ ) = ( − 1 ) 𝑝 1 𝜖 𝜎 𝑣 ′ ( ℎ ′ ) + 𝑓 1 𝜖 ℎ and 𝜖 𝑣 = ( − 1 ) 𝑝 2 𝜖 𝑣 ′ + 𝑓 2 𝜖 ℎ or (6) 𝜖 𝜎 𝑣 ( ℎ ) = ( − 1 ) 𝑝 1 𝜖 𝑣 ′ + 𝑓 1 𝜖 ℎ and 𝜖 𝑣 = ( − 1 ) 𝑝 2 𝜖 𝜎 𝑣 ′ ( ℎ ′ ) + 𝑓 2 𝜖 ℎ for some 𝑓 1 , 𝑓 2 ∈ Z and 𝑝 1 , 𝑝 2 ∈ Z / 2 Z . W e deﬁne 𝑓 𝑒 ≡ 𝑓 1 + 𝑓 2 , 𝑝 𝑒 ≡ 𝑝 1 + 𝑝 2 . There is an alternative characterisation of the mixing parity which is often mor e practical: The normal bundle of the line 𝐶 𝑒 splits as 𝑁 𝐶 𝑒 𝑍  O P 1 ( 𝑎 1 ) ⊕ O P 1 ( 𝑎 2 ) ⊕ O P 1 ( 𝑎 3 ) ⊕ O P 1 ( 𝑎 4 ) . Every factor can be associated with one of the half-edges other than ℎ and ℎ ′ at each of the ﬁxed points 𝑣 and 𝑣 ′ . Hence, the splitting induces a bijection 𝛼 :  ℎ 𝑣 , ℎ 𝜎 ( ℎ ) , ℎ ′ 𝑣 , ℎ 𝜎 2 ( ℎ )  ∼ − →  1 , 2 , 3 , 4  ∼ − →  ℎ 𝑣 ′ , ℎ 𝜎 ( ℎ ′ ) , ℎ ′ 𝑣 ′ , ℎ 𝜎 2 ( ℎ ′ )  between the half-edges at 𝑣 and 𝑣 ′ . Then the mixing parity of 𝑒 can be identiﬁed with the cardinality 𝑝 𝑒 ≡   𝛼  { ℎ 𝑣 , ℎ 𝜎 ( ℎ ) }  ∩ { ℎ 𝑣 ′ , ℎ 𝜎 ( ℎ ′ ) }   . Example 1.4. W e continue our running example 𝑍 = T ot P 1  O ( − 2 )  × C 3 . There is one compact edge 𝑒 associated with the zero section P 1 ⊂ 𝑍 . One checks that in both cases (A) and (B) 𝑓 𝑒 is even while 𝑝 𝑒 is odd. 1.4. Diagrammatic rules. In this section we will state the diagrammatic rules that allow the evaluation of the Gromov–W itten invariants GW • 𝒅 ( 𝑍 , T ) after hav- ing ﬁxed a choice of distinct dir ections and or ders. There is, however , a technical condition we need to impose on the skeletal degree 𝒅 . Deﬁnition 1.5. W e say that 𝒅 ∈ Z 𝐸 ( Γ ) ≥ 0 is supported away from anti-diagonal strata if 𝑑 𝑒 = 0 whenever at least one of the half-edges ℎ , ℎ ′ of 𝑒 = ( ℎ, ℎ ′ ) is an anti- diagonal half-edge. Example 1.6. W ith the distinct direction and order we ﬁxed in Example 1.3 for our running example we can infer from Figure 1 that in both cases (A) and (B) any multiple of the zero section is supported away from anti-diagonal strata as all anti-diagonal half-edges are non-compact legs. GW INV ARIANTS AND MEMBRANE INDICES OF 5-FOLDS VIA TV 13 1.4.1. Partition labels. W e ar e now ready to state the r ules of our vertex formalism. W e decorate each half-edge ℎ of the T -diagram Γ with a partition 𝜇 ℎ subject to the following conditions: • 𝜇 ℎ = ∅ whenever ℎ is a leaf or an anti-diagonal half-edge; • for each compact edge 𝑒 = ( ℎ, ℎ ′ ) ∈ 𝐸 ( Γ ) we have 𝜇 ℎ = 𝜇 ℎ ′ if the mixing parity is even and 𝜇 ℎ = 𝜇 t ℎ ′ otherwise; • for all edges 𝑒 = ( ℎ, ℎ ′ ) we have | 𝜇 ℎ | = | 𝜇 ℎ ′ | = 𝑑 𝑒 . W e denote the set of all decorations of half-edges by partitions satisfying the above conditions by P Γ , 𝒑 , 𝒅 . Note that thr ough the mixing parity , the second condition depends on the choice of distinct directions and or ders we ﬁxed. 1.4.2. Edge weights. Given a partition label 𝝁 ∈ P Γ , 𝒑 , 𝒅 we assign a weight to each edge 𝑒 as follows: Denote the half-edges associated to this edge by ℎ = ( 𝑣 , 𝑒 ) and ℎ ′ = ( 𝑣 ′ , 𝑒 ) . Then 𝑒 gets assigned the weight 𝐸 ( 𝑒 , 𝝁 ) B ( − 1 ) ( 𝑓 𝑒 + 𝑝 𝑒 ) | 𝜇 ℎ | exp  𝜅 ( 𝜇 ℎ ) 2 𝜖 𝑣 𝜖 𝜎 ( ℎ ) + ( − 1 ) 𝑝 𝑒 + 1 𝜖 𝑣 ′ 𝜖 𝜎 ( ℎ ′ ) 𝜖 ℎ  where 𝜅 ( 𝜇 ) B Í ℓ ( 𝜇 ) 𝑖 = 1 𝜇 𝑖 ( 𝜇 𝑖 − 2 𝑖 + 1 ) denotes the second Casimir invariant. 1.4.3. V ertex weights. Given three partitions 𝜇 1 , 𝜇 2 and 𝜇 3 we introduce the topo- logical vertex function (7) W 𝜇 1 ,𝜇 2 ,𝜇 3 ( 𝑞 ) = 𝑞 𝜅 ( 𝜇 1 ) / 2 𝑠 𝜇 3  𝑞 𝜌   𝜈 𝑠 𝜇 t 1 𝜈  𝑞 𝜌 + 𝜇 3  𝑠 𝜇 2 𝜈  𝑞 𝜌 + 𝜇 t 3  . Here, 𝑠 𝛼 / 𝛽 ( 𝑞 𝜌 + 𝛾 ) denotes the skew Schur function 𝑠 𝛼 𝛽 ( 𝑥 1 , 𝑥 2 , . . . ) evaluated at 𝑥 𝑖 = 𝑞 − 𝑖 + 1 / 2 + 𝛾 𝑖 . A priori, this makes 𝑠 𝛼 / 𝛽 ( 𝑞 𝜌 + 𝛾 ) a formal Laurent series in 𝑞 − 1 / 2 . It can, however , be shown that the series conver ges to a rational function implying that also ( 7 ) is a rational function in 𝑞 1 / 2 . (W e will r ecall this fact in mor e detail in the proof of Theor em 1.10 .) Now given a partition label 𝝁 ∈ P Γ , 𝒑 , 𝒅 we assign a weight to each vertex 𝑣 as follows: Remember that we ﬁxed a cyclic permutation 𝜎 𝑣 = ( ℎ 1 ℎ 2 ℎ 3 ) of three half- edges at 𝑣 and that our choice of distinct direction singled out a distinct torus weight 𝜖 𝑣 . The weight we assign to 𝑣 is W ( 𝑣 , 𝝁 ) B W 𝜇 ℎ 1 ,𝜇 ℎ 2 ,𝜇 ℎ 3 ( e 𝜖 𝑣 ) . This assignment is well deﬁned since the topological vertex function is invariant under cyclic permutations of partitions. 14 Y ANNIK SCHULER 1.5. The vertex formula. W ith these diagrammatic rules at our disposal we are ﬁnally able to state the ﬁrst main result of this paper . Theorem 1.7. Let 𝑍 be a Calabi–Y au ﬁvefold with a skeletal, Calabi–Y au and locally anti-diagonal action by a torus T . Fix a choice of distinct directions and orders. Then for all skeletal degrees 𝒅 supported away from anti-diagonal strata we have (8) GW • 𝒅 ( 𝑍 , T ) =  𝝁 ∈ P Γ , 𝒑 , 𝒅 Ö 𝑒 ∈ 𝐸 ( Γ ) 𝐸 ( 𝑒 , 𝝁 ) Ö 𝑣 ∈ 𝑉 ( Γ ) W ( 𝑣 , 𝝁 ) . W e recover the disconnected Gromov–W itten invariants in a curve class 𝛽 by summing over all skeletal degrees 𝒅 satisfying 𝛽 = Í 𝑒 𝑑 𝑒 [ 𝐶 𝑒 ] : GW • 𝛽 ( 𝑍 , T ) =  𝒅 GW • 𝒅 ( 𝑍 , T ) . T o be able to apply Theor em 1.7 one has to assume that none of the skeletal degrees in the above sum is supported on anti-diagonal strata. In this case we say that 𝛽 is supported away from anti-diagonal strata. Corollary 1.8. (Theor em C ) Let 𝑍 be a Calabi–Y au ﬁvefold with a skeletal, Calabi–Y au and locally anti-diagonal action by a torus T . Fix a choice of distinct directions and orders. Then for all effective curve classes 𝛽 supported away from anti-diagonal strata we have GW • 𝛽 ( 𝑍 , T ) =  𝒅  𝝁 ∈ P Γ , 𝒑 , 𝒅 Ö 𝑒 ∈ 𝐸 ( Γ ) 𝐸 ( 𝑒 , 𝝁 ) Ö 𝑣 ∈ 𝑉 ( Γ ) W ( 𝑣 , 𝝁 ) . □ (9) Remark 1.9. W e r emark that it actually suf ﬁces to impose a slightly weaker con- dition on the curve class 𝛽 : Suppose that for every 𝒅 satisfying 𝛽 = Í 𝑒 𝑑 𝑒 [ 𝐶 𝑒 ] we have GW • 𝒅 ( 𝑍 , T ) = 0 whenever there is an anti-diagonal edge 𝑒 with 𝑑 𝑒 > 0 . Then in this case ( 9 ) still holds true with the ﬁrst sum only ranging over those skeletal degrees that are supported away fr om anti-diagonal strata. In Section 2.5 we will see an application where this extra fr eedom is indeed crucial. The pr oof of Theorem 1.7 is deferred to Section 3 . In the remaining parts of this section we will ﬁrst explain how the vertex formula implies Conjecture A and second we will compare our formula with the original vertex formalism for toric Calabi–Y au thr eefolds. T o see the vertex formula at work in several examples we refer the r eader to Section 2 . 1.6. On Conjecture A . W e will pr ove Conjecture A in the setting where our ver- tex formalism applies in a slightly stronger version than it was stated in Theo- rem B in the intr oduction. Theorem 1.10. Let 𝑍 be a Calabi–Y au ﬁvefold with a skeletal, Calabi–Y au and locally anti-diagonal action by a torus T . Fix a basis 𝐻 2 T ( pt , Z )  Z [ 𝜖 1 , . . . , 𝜖 𝑚 ] and a choice of GW INV ARIANTS AND MEMBRANE INDICES OF 5-FOLDS VIA TV 15 distinct directions and orders. Then ther e exist rational functions Ω 𝒅 ( 𝑞 𝑖 ) ∈ Z  𝑞 ± 1 / 2 1 , . . . , 𝑞 ± 1 / 2 𝑚 , n  1 − Î 𝑖 𝑞 𝑛 𝑖 / 2 𝑖  − 1 o 𝒏 ∈ Z 𝑚 \{ 0 }  labelled by skeletal degrees 𝒅 supported away fr om anti-diagonal strata such that under the change of variables 𝑞 𝑖 = e 𝜖 𝑖 we have GW 𝒅 ( 𝑍 , T ) =  𝑘 | 𝒅 1 𝑘 Ω 𝒅 / 𝑘  𝑞 𝑘 𝑖  . Proof. W e may identify 𝑅 = Z [ 𝑞 ± 1 / 2 1 , . . . , 𝑞 ± 1 / 2 𝑚 , { ( 1 − Î 𝑖 𝑞 𝑛 𝑖 / 2 𝑖 ) − 1 } 𝒏 ] with the ring of virtual repr esentations of a double cover of T localised at the augmentation ideal. Since by Lemma A.1 the plethystic logarithm maps a series with coefﬁcients in 𝑅 to one with coefﬁcients in 𝑅 , it sufﬁces to pr ove that GW • 𝒅 ( 𝑍 , T ) ∈ 𝑅 for all 𝒅 supported away from anti-diagonal strata. This is true if we show that actually every individual factor in our vertex formula ( 8 ) is an element in 𝑅 . Indeed, for edge terms this is a consequence of the fact that 𝜖 ℎ divides 𝜖 𝑣 𝜖 𝜎 ( ℎ ) − ( − 1 ) 𝑝 𝑒 𝜖 𝑣 ′ 𝜖 𝜎 ( ℎ ′ ) for all edges 𝑒 = ( ℎ, ℎ ′ ) by the linear relations ( 5 ) and ( 6 ). This can be seen in a case-by-case analysis. As a consequence, we can write an edge term as 𝐸 ( 𝑒 , 𝝁 ) = ( − 1 ) ( 𝑓 𝑒 + 𝑝 𝑒 ) | 𝜇 ℎ | 𝑞 𝜅 ( 𝜇 ℎ ) / 2 ℎ where 𝑞 ℎ is a T -character . Since 𝜅 ( 𝜇 ℎ ) is integer , we thus deduce that every edge weight lies in 𝑅 . Regarding vertex weights, let us show that W 𝜇 1 ,𝜇 2 ,𝜇 3 ( 𝑞 ) is a rational function in 𝑞 1 / 2 with poles only at zero and roots of unity . Indeed, up to a leading monomial fac- tor , the topological vertex depends on 𝑞 only through the specialised skew Schur functions 𝑠 𝛼 / 𝛽 ( 𝑞 𝜌 + 𝜇 ) . The latter are uniquely determined fr om the specialisation of the power functions 𝑝 𝑘 . The rationality and pole constraint claimed thus follows from the evaluation 𝑝 𝑘  𝑞 𝜌 + 𝜇  = 1 𝑞 𝑘 / 2 − 𝑞 − 𝑘 / 2 + ℓ ( 𝜇 )  𝑖 = 1 𝑞 ( − 𝑖 + 1 / 2 + 𝜇 𝑖 ) 𝑘 − 𝑞 ( − 𝑖 + 1 / 2 ) 𝑘 . Alternatively , one may also interpr et the rationality and the restriction of poles in the vertex weight as a consequence of the r elative Gromov–W itten/Donaldson– Thomas correspondence for toric thr eefolds [ MOOP11 , Thm. 1 & 3]. □ Again, we obtain the analogue of Theorem 1.10 for invariants labelled by curve classes by summing over skeletal degrees. 16 Y ANNIK SCHULER Corollary 1.11. (Theorem B ) Under the assumptions of Theorem 1.10 there exist rational functions Ω 𝛽 ( 𝑞 𝑖 ) ∈ Z  𝑞 ± 1 / 2 1 , . . . , 𝑞 ± 1 / 2 𝑚 , n  1 − Î 𝑖 𝑞 𝑛 𝑖 / 2 𝑖  − 1 o 𝒏 ∈ Z 𝑚 \{ 0 }  labelled by curve classes 𝛽 supported away from anti-diagonal strata such that under the change of variables 𝑞 𝑖 = e 𝜖 𝑖 we have GW 𝛽 ( 𝑍 , T ) =  𝑘 | 𝛽 1 𝑘 Ω 𝛽 / 𝑘  𝑞 𝑘 𝑖  . □ Remark 1.12. The same result holds under the slightly weaker but mor e technical assumption stated in Remark 1.9 . Remark 1.13. The fact that a lift of the Gr omov–W itten series labelled by skeletal degrees to a rational function exists by Theorem 1.10 suggests there should be a reﬁnement of Conjecture A along the following lines. The yet-to-be-constructed moduli space M2 𝛽 ( 𝑍 ) of M2-branes in curve class 𝛽 should admit a morphism to the Chow variety . Also the moduli space of stable maps admits such a morphism by taking the support of a stable map: M2 𝛽 ( 𝑍 ) 𝑀 • ( 𝑍 , 𝛽 ) Chow 𝛽 ( 𝑍 ) Pushing forward an appr opriate K-theory class along the left arrow should yield an element b Ω 𝛽 ∈ 𝐾 T  Chow 𝛽 ( 𝑍 )  loc . It should be related to the push-forward of the virtual fundamental class along the right arrow via the Chern character and the plethysm on the Chow variety (cf. [ NO16 , Sec. 2.3.5]) so that Conjecture A is recover ed by pushing the reﬁned identity forward to a point. Since for skeletal torus actions all ﬁxed points of Chow 𝛽 ( 𝑍 ) are isolated and labelled by the skeletal degr ees, we see that with Theorem 1.10 we actually pr oved such a reﬁned version of Conjecture A . 1.7. The globally anti-diagonal situation. In this section we will specialise our vertex formalism to compute the local contribution of a Calabi–Y au threefold 𝑋 ⊂ 𝑍 which is the ﬁxed locus of a Calabi–Y au C × 𝑞 -action on 𝑍 . 1.7.1. The general case. Let us describe the local setup. Let 𝑋 be a smooth toric Calabi–Y au threefold together with the action by its Calabi–Y au torus T ′  ( C × ) 2 . GW INV ARIANTS AND MEMBRANE INDICES OF 5-FOLDS VIA TV 17 Let L be a T ′ -equivariant line bundle on 𝑋 and let C × 𝑞 act on its ﬁbr es with charac- ter 𝑞 − 1 . W e obtain a Calabi–Y au torus action on the local Calabi–Y au ﬁvefold 𝑍 B T ot 𝑋 L ⊕ L ∨ ⟳ T ′ × C × 𝑞 C T . T o apply our vertex formalism in this situation, we ﬁrst ﬁx a speciﬁc choice of distinct directions and or ders: For each vertex 𝑣 of the T -diagram, that is for each torus ﬁxed point 𝑝 𝑣 ∈ 𝑋 , we choose the distinct direction to be the half-edge associated with the line bundle L : 𝜖 𝑣 = − 𝑐 1 ( L | 𝑝 𝑣 ) . Assuming that the induced action by T ′ R is Hamiltonian, the moment map 𝜇 : 𝑋 → H 2 T ′ ( pt , R ) yields an embedding of the T -diagram into H 2 T ′ ( pt , R )  R 2 . Hence, ﬁxing an orientation of R 2 yields a cyclic or der for the three half-edges associated to 𝑇 𝑝 𝑣 𝑋 at each vertex 𝑣 . W ith this choice of distinct dir ections and orders the mixing parity of each edge is odd. T o determine the edge weights, note that the normal bundle of each torus pr eserved line splits into 𝑁 𝐶 𝑒 𝑋 ⊕ L | 𝐶 𝑒 ⊕ L ∨ | 𝐶 𝑒  O P 1 ( − 1 + 𝑓 𝑒 , 1 ) ⊕ O P 1 ( − 1 − 𝑓 𝑒 , 1 ) ⊕ O P 1 ( 𝑓 𝑒 , 2 ) ⊕ O P 1 ( − 𝑓 𝑒 , 2 ) for some 𝑓 𝑒 , 1 , 𝑓 𝑒 , 2 ∈ Z . W riting ℎ = ( 𝑣 , 𝑒 ) and ℎ ′ = ( 𝑣 ′ , 𝑒 ) for the half-edges of 𝑒 the torus weights at the ﬁxed points ar e r elated by 𝜖 𝜎 𝑣 ( ℎ ) = − 𝜖 𝜎 𝑣 ′ ( ℎ ′ ) + 𝑓 𝑒 , 1 𝜖 ℎ , 𝜖 𝑣 = 𝜖 𝑣 ′ + 𝑓 𝑒 , 2 𝜖 ℎ . Thus, writing 𝑞 𝑖 = e 𝜖 𝑖 our vertex formula ( 8 ) specialises to GW • 𝒅 ( 𝑍 , T ) =  𝝁 ∈ P Γ , 𝒑 , 𝒅 Ö 𝑒 ∈ 𝐸 ( Γ ) ( − 1 ) ( 𝑓 𝑒 , 1 + 𝑓 𝑒 , 2 + 1 ) | 𝜇 ℎ |  𝑞 𝑓 𝑒 , 1 𝑣 𝑞 𝑓 𝑒 , 2 𝜎 𝑣 ( ℎ ) 𝑞 − 𝑓 𝑒 , 1 𝑓 𝑒 , 2 ℎ  𝜅 ( 𝜇 ℎ ) / 2 Ö 𝑣 ∈ 𝑉 ( Γ ) W 𝜇 ℎ 𝑣 1 ,𝜇 ℎ 𝑣 2 ,𝜇 ℎ 𝑣 3 ( 𝑞 𝑣 ) . (10) 1.7.2. The threefold limit. Let us further specialise to the case wher e L  O 𝑋 or in other words to the situation wher e 𝑍 is the product of 𝑋 with the afﬁne plane: 𝑍 = 𝑋 × C 2 . The torus C × 𝑞 acts anti-diagonally on the afﬁne plane with weights ± 𝜖 B ± 𝑐 1 ( 𝑞 ) . As explained in [ BS24 , Sec. 2.4], in this situation the T -equivariant Gromov–W itten invariants of 𝑍 recover the ones of the threefold 𝑋 wher e the weight 𝜖 takes the role of the genus counting variable: (11) GW • 𝒅 ( 𝑍 , T ) =  𝑔 ∈ Z ( − 𝜖 2 ) 𝑔 − 1 GW • 𝒅 ( 𝑋 , T ) . 18 Y ANNIK SCHULER Specialising 𝜖 𝑣 = 𝜖 and 𝑓 𝑒 , 2 = 0 in equation ( 10 ), our vertex formalism thus yields the following formula for these Gromov–W itten invariants:  𝑔 ∈ Z ( − 𝜖 2 ) 𝑔 − 1 GW • 𝒅 ( 𝑋 , T ) =  𝝁 ∈ P Γ , 𝒑 , 𝒅 Ö 𝑒 ∈ 𝐸 ( Γ ) ( − 1 ) ( 𝑓 𝑒 + 1 ) | 𝜇 ℎ | 𝑞 𝜅 ( 𝜇 ℎ ) 𝑓 𝑒 / 2 Ö 𝑣 ∈ 𝑉 ( Γ ) W 𝜇 ℎ 𝑣 1 ,𝜇 ℎ 𝑣 2 ,𝜇 ℎ 𝑣 3 ( 𝑞 ) . This is pr ecisely the topological vertex formula for toric Calabi–Y au threefolds of Aganagic–Klemm–Mariño–V afa [ AKMV05 ] as stated in [ LLLZ09 ]. Note also that in the product case 𝑋 × C 2 one may identify 𝑓 𝑒 with what is usually called the framing factor; this is why in the general case we chose to call its congruence class modulo two the framing parity . 2. E X A M P L E S 2.1. T ot P 1 ( O (− 2 ) ) × C 3 (continued). Let us apply the vertex formalism to our running example 𝑍 = T ot P 1 ( O ( − 2 ) ) × C 3 (Examples 1.2 – 1.4 and 1.6 ). First, we con- sider case (A) which is a special instance of the situation described in Section 1.7.2 : W e have 𝑍 = 𝑋 × C 2 and T A acts on the coordinate lines of the afﬁne plane with opposite weights. Hence, the formalism reduces to the usual topological vertex method which yields (12) GW • ( 𝑍 , T A ) =  𝜇 𝑄 | 𝜇 | 𝑞 𝜅 ( 𝜇 ) / 2 𝑖 𝑠 𝜇  𝑞 𝜌 𝑖  𝑠 𝜇 t  𝑞 𝜌 𝑖  = Exp 𝑄  𝑞 1 / 2 𝑖 − 𝑞 − 1 / 2 𝑖  2 where we write 𝑞 𝑖 = e 𝜖 𝑖 and Exp denotes the plethystic exponential Exp  𝑓 ( 𝑞, 𝑄 )  B exp  𝑘 > 0 1 𝑘 𝑓  𝑞 𝑘 , 𝑄 𝑘  ! . Situation (B) is more interesting. Here, the mixing parity of the unique compact edge is odd too but now the vertices carry differ ent choices for the distinct direc- tion. If we apply Theorem 1.7 we get the formula (13) GW • ( 𝑍 , T B ) =  𝜇 𝑄 | 𝜇 | 𝑞 − 𝜅 ( 𝜇 ) / 2 𝑖 𝑠 𝜇  𝑞 − 𝜌 𝑖  𝑠 𝜇 t  𝑞 𝜌 𝑗  = Exp 𝑄  𝑞 1 / 2 𝑖 − 𝑞 − 1 / 2 𝑖   𝑞 1 / 2 𝑗 − 𝑞 − 1 / 2 𝑗  . Formula ( 12 ) and the above are both specialisations of the following conjectural formula for the four dimensional Calabi–Y au tor us T acting on 𝑍 [ BS24 , Conj. 3.4]: (14) GW • ( 𝑍 , T ) = Exp  𝑞 1 / 2 2 − 𝑞 − 1 / 2 2  𝑄  𝑞 1 / 2 3 − 𝑞 − 1 / 2 3   𝑞 1 / 2 4 − 𝑞 − 1 / 2 4   𝑞 1 / 2 5 − 𝑞 − 1 / 2 5  . Remark 2.1. As already explained in [ BS24 , Sec. 7.2.5], the fact that neither of the formulae for torus actions (A) and (B) agree with any quantity computed via the reﬁned topological vertex [ IKV09 ] is due to the non-compactness of the moduli space of stable maps to the threefold T ot P 1 ( O ( − 2 ) ) × C . The reﬁned topological GW INV ARIANTS AND MEMBRANE INDICES OF 5-FOLDS VIA TV 19 vertex evaluates the 𝜖 3 → ±∞ limit of formula ( 14 ). In the following section we will analyse a larger class of toric thr eefolds exhibiting the same phenomenon. 2.2. Strip geometries. Generalising the last example, our formalism applies to a wider class of so-called strip geometries. By this we mean a product 𝑍 = 𝑋 × C 2 where 𝑋 is the toric variety whose fan is the cone over a triangulated strip placed at height one. The torus diagram of 𝑋 takes the following shape: (15) Here, we presented its embedding in R 2 provided by the moment map of the two-dimensional Calabi–Y au torus of 𝑋 . Now let T ′  ( C × ) 4 be a Calabi–Y au torus acting on 𝑍 . Note that by construction all T ′ -weights attached to upwar ds pointing legs coincide. W e denote their weight by 𝜖 2 . Similarly , let us write 𝜖 3 for the T ′ -weight associated to downwards pointing legs. Finally , denote the tangent weights at the origin of C 2 by 𝜖 4 and 𝜖 5 . W e remark that in T ′ -equivariant cohomology 𝜖 2 , . . . , 𝜖 5 are linearly independent. Now consider the two-dimensional subtorus T ⊂ T ′ on which the relations 𝜖 2 = − 𝜖 4 and 𝜖 3 = − 𝜖 5 hold. W ith these constraints the induced T -action on 𝑍 is locally anti-diagonal and we can apply Theorem 1.7 . For this we ﬁx a distinct direction at each vertex by choosing 𝜖 𝑣 = 𝜖 4 whenever a vertex 𝑣 carries an upwards pointing leg and 𝜖 𝑣 = − 𝜖 5 otherwise. A choice of distinct direction at each vertex is ﬁxed by choosing an orientation of the plane R 2 for the embedding of the diagram ( 15 ). W ith this choice the mixing parity of an edge is even if and only if it connects two vertices where one has an upwards and the other one a downwards pointing leg attached. T o state the vertex formula we label the vertices in ( 15 ) fr om left to right by 𝑣 1 , . . . , 𝑣 𝑁 . The index set decomposes into 𝐼 u ⊔ 𝐼 d = { 1 , . . . , 𝑁 } labelling vertices with an upwards respectively downwar ds pointing leg. W e decorate the edge 𝑒 20 Y ANNIK SCHULER connecting 𝑣 𝑖 with 𝑣 𝑖 + 1 with a partition 𝜇 𝑖 and write 𝑄 𝑖 = 𝑄 𝐶 𝑒 . W ith this notation Theorem 1.7 yields the formula GW • ( 𝑍 , T ) =  𝜇 1 ,. . . ,𝜇 𝑁 + 1 𝑁 − 1 Ö 𝑖 = 1 𝑄 | 𝜇 𝑖 | 𝑖 𝑞 − 𝜅 ( 𝜇 𝑖 ) / 2 𝑖 , 𝑖 + 1 Ö 𝑖 ∈ 𝐼 u W 𝜇 𝑖 ,𝜇 t 𝑖 − 1 , ∅  𝑞 4  Ö 𝑖 ∈ 𝐼 d W 𝜇 t 𝑖 − 1 ,𝜇 𝑖 , ∅  𝑞 − 1 5  where 𝑞 𝑖 , 𝑖 + 1 =                𝑞 4 𝑖 , 𝑖 + 1 ∈ 𝐼 u 𝑞 5 𝑖 , 𝑖 + 1 ∈ 𝐼 d 𝑞 4 𝑞 5 𝑖 ∈ 𝐼 u and 𝑖 + 1 ∈ 𝐼 d 1 otherwise. If we plug in the topological vertex formula ( 7 ) and use the identities 𝜅  𝜇 t  = − 𝜅  𝜇  , 𝑞 − 𝜅 ( 𝜇 ) / 2 𝑠 𝜇  𝑞 𝜌  = 𝑠 𝜇 t  𝑞 − 𝜌  we get GW • ( 𝑍 , T ) =  𝜇 1 ,. . . ,𝜇 𝑁 − 1 𝜈 2 ,. . . , 𝜈 𝑁 − 1 𝑁 − 1 Ö 𝑖 = 1 𝑄 | 𝜇 𝑖 | 𝑖 𝑠 𝜇 1  𝑞 − 𝜌 𝑣 1  ©   « Ö 1 < 𝑖 < 𝑁 𝑖 ∈ 𝐼 u 𝑠 𝜇 t 𝑖 − 1 𝜈 𝑖  𝑞 𝜌 4  𝑠 𝜇 t 𝑖 𝜈 𝑖  𝑞 𝜌 4  ª ® ® ¬ ©    « Ö 1 < 𝑖 < 𝑁 𝑖 ∈ 𝐼 d 𝑠 𝜇 𝑖 − 1 𝜈 𝑖  𝑞 − 𝜌 5  𝑠 𝜇 𝑖 𝜈 𝑖  𝑞 − 𝜌 5  ª ® ® ® ¬ 𝑠 𝜇 t 𝑁 − 1  𝑞 𝜌 𝑣 𝑁  where we write 𝑞 𝑣 = 𝑞 4 for a vertex 𝑣 carrying an upwards pointing leg and 𝑞 𝑣 = 𝑞 5 otherwise. Following the approach of [ IK06b ], one may evaluate the sum over partitions 𝜇 𝑖 , 𝜈 𝑖 using the homogeneity of Schur functions 𝑄 | 𝜇 | −| 𝜈 | 𝑠 𝜇 / 𝜈 ( 𝑥 ) = 𝑠 𝜇 / 𝜈 ( 𝑄 𝑥 ) and repeatedly applying the following specialisation of the skew Cauchy identities [ Mac95 , Sec. I.5]:  𝜇 𝑠 𝜇 𝜈 1  𝑄 𝑞 𝜌  𝑠 𝜇 𝜈 2  𝑡 𝜌  = Exp  𝑄 ( 𝑞 1 / 2 − 𝑞 − 1 / 2 ) ( 𝑡 1 / 2 − 𝑡 − 1 / 2 )  ·  𝜇 𝑠 𝜈 2 𝜇  𝑄 𝑞 𝜌  𝑠 𝜈 1 𝜇  𝑡 𝜌  ,  𝜇 𝑠 𝜇 t 𝜈 1  𝑄 𝑞 𝜌  𝑠 𝜇 𝜈 2  𝑡 𝜌  = Exp  − 𝑄 ( 𝑞 1 / 2 − 𝑞 − 1 / 2 ) ( 𝑡 1 / 2 − 𝑡 − 1 / 2 )  ·  𝜇 𝑠 𝜈 t 2 𝜇  𝑄 𝑞 𝜌  𝑠 𝜈 t 1 𝜇 t  𝑡 𝜌  . The resulting formula is (16) GW • ( 𝑍 , T ) = Exp  1 ≤ 𝑚 ≤ 𝑛 ≤ 𝑁 Î 𝑛 𝑘 = 𝑚 𝑄 𝑘  𝑞 1 / 2 𝑣 𝑚 − 𝑞 − 1 / 2 𝑣 𝑚   𝑞 1 / 2 𝑣 𝑛 − 𝑞 − 1 / 2 𝑣 𝑛  ! . Conjectural formulas for strip geometries beyond locally anti-diagonal torus ac- tion like ( 14 ) will be presented in [ HS26 ]. Remark 2.2. In general, the above expression agr ees with formulae produced via the reﬁned topological vertex only when the moduli space of stable maps to 𝑋 is GW INV ARIANTS AND MEMBRANE INDICES OF 5-FOLDS VIA TV 21 proper in all genera and curve classes. This is for instance the case for the resolved conifold. For non-proper the moduli spaces the quantities generally disagr ee. Remark 2.3. A toric Calabi–Y au threefold 𝑋 engineering supersymmetric SU ( 𝑁 ) gauge theory on C 2 may be obtained by gluing two strip geometries with all legs pointing upwar ds r esp. downwar ds along the vertical non-compact directions. So from the above discussion the reader might be tempted to hope that reﬁned invariants of such an 𝑋 (that is invariants on the so-called general Ω background 𝜖 4 , 𝜖 5 ) may be computed via our vertex formalism. This is, however , unfortunately impossible because for the gluing to be compatible with the torus action one has to impose the constraint 𝜖 2 = − 𝜖 3 which in turn forces 𝜖 4 = − 𝜖 5 . This means the vertex formalism pr esented in this note cannot compute the Gromov–W itten invariants of 𝑋 × C 2 beyond the self-dual limit which is already well-studied in the literature [ IK06a ]. 2.3. The GW dual of rank-two DT theory on the resolved conifold. Let us consider our ﬁrst example of a ﬁvefold featuring a compact four cycle: 𝑍 = T ot P 1 × P 1  O ( − 1 , 0 ) ⊕ O ( − 1 , 0 ) ⊕ O ( 0 , − 2 )  − → P 1 × P 1 This ﬁvefold is the product of the r esolved conifold and the resolution of the 𝐴 1 surface singularity . W e assume that the dense torus b T  ( C × ) 5 acts with tangent weights 𝜖 1 , 𝜖 2 at the ﬁxed point ( 0 , 0 ) ∈ P 1 × P 1 and on the ﬁbre of each line bundle over ( 0 , 0 ) with tangent weight 𝜖 3 , 𝜖 4 and 𝜖 5 respectively . W e denote by T the two-dimensional subtorus of b T for which 𝜖 3 = − 𝜖 2 , 𝜖 4 = 𝜖 2 , 𝜖 5 = − 𝜖 1 − 𝜖 2 . Figure 2 illustrates the resulting T -diagram. Observe that this torus action is both Calabi–Y au and locally anti-diagonal. T o apply Theorem 1.7 we pick distinct directions at each vertex as indicated in Figure 2 . W e order the remaining half-edges clockwise for the bottom and anti- clockwise for the top vertices. The resulting mixing paritys may be inferr ed from Figure 2 from how we decorate half-edges by partitions. In this situation Corollary 1.8 pr ovides us with the following formula: GW • ( 𝑍 , T ) =  𝜇 1 ,𝜇 2 𝜈 1 , 𝜈 2 𝑄 | 𝜇 1 | +| 𝜇 2 | 1 𝑄 | 𝜈 1 | +| 𝜈 2 | 2 ( − 1 ) | 𝜈 1 | W 𝜈 1 ,𝜇 1 , ∅  𝑞 − 1 2  W 𝜇 1 , 𝜈 2 , ∅  𝑞 1 𝑞 2  W 𝜇 2 , 𝜈 2 , ∅  𝑞 1 𝑞 − 1 2  W 𝜈 t 1 ,𝜇 2 , ∅  𝑞 2  . Based on computer experiments we expect that the above sum over partitions can be carried out explicitly to yield the following formula. 22 Y ANNIK SCHULER ◦ ◦ ◦ ◦ 𝜇 1 𝜇 1 𝜈 2 𝜈 2 𝜇 2 𝜇 2 𝜈 1 𝜈 t 1 𝑄 1 𝑄 2 𝜖 1 𝜖 2 F I G U R E 2 . The T -diagram of T ot O ( − 1 , 0 ) ⊕ O ( − 1 , 0 ) ⊕ O ( 0 , − 2 ) with a choice of distinct direction at each vertex and half-edges decorated by partitions. The half-edges associated with a line bundle ar e coloured blue, green and orange respectively . All vertical lines on the left should be parallel. The tilt of the coloured half-edges is solely for display purposes. Conjecture 2.4. We have GW • ( 𝑍 , T ) = Exp  𝑞 1 / 2 1 + 𝑞 3 / 2 1  𝑄 1  1 − 𝑞 1 𝑞 2   1 − 𝑞 1 𝑞 − 1 2  ! · Exp  1 − 𝑞 1  2 𝑞 2 𝑄 2  1 − 𝑞 2  2  1 − 𝑞 1 𝑞 2   1 − 𝑞 1 𝑞 − 1 2  ! . Remark 2.5. The above observed factorisation into a contribution coming from the r esolved conifold and another coming fr om the resolution of the 𝐴 1 singu- larity without the pr esence of cr oss-terms conjecturally occurs in mor e a general situation: Such a factorisation should happen for all products of the form 𝑋 × A 𝑟 where 𝑋 is a Calabi–Y au thr eefold and A 𝑟 is the resolution of the 𝐴 𝑟 surface singularity . When the torus action on A 𝑟 is Calabi–Y au the absence of cross-terms is a consequence of the vanishing of the virtual fundamental class due to the nowhere vanishing holomorphic two-form on the surface. Numerical evidence beyond Calabi–Y au torus actions will be pr esented in [ HS26 ]. 2.4. T ot P 2  O (− 1 ) ⊕ 3  . In this section we consider the ﬁvefold 𝑍 = T ot P 2  O ( − 1 ) ⊕ O ( − 1 ) ⊕ O ( − 1 )  − → P 2 GW INV ARIANTS AND MEMBRANE INDICES OF 5-FOLDS VIA TV 23 ◦ ◦ ◦ 𝜖 1 − 𝜖 0 𝜖 2 − 𝜖 0 𝜇 1 𝜇 1 𝜇 2 𝜇 2 𝜇 3 𝜇 3 F I G U R E 3 . The T -diagram of T ot O P 2 ( − 1 ) ⊕ 3 with a choice of a dis- tinct direction at each vertex and half-edges decorated by partitions. with r espect to a speciﬁc torus action. The easiest way to describe it is by pr esent- ing 𝑍 as a quotient 𝑍 = C 6 \ 𝑉 ( 𝑥 0 𝑥 1 𝑥 2 )  C × where the tor us C × acts via  𝑡 , ( 𝑥 0 , 𝑥 1 , 𝑥 2 , 𝑦 0 , 𝑦 1 , 𝑦 2 )  ↦− → ( 𝑡 𝑥 0 , 𝑡 𝑥 1 , 𝑡 𝑥 2 , 𝑡 − 1 𝑦 0 , 𝑡 − 1 𝑦 1 , 𝑡 − 1 𝑦 2 ) on afﬁne space. The natural torus action of ( C × ) 6 on C 6 , whose tangent weights we denote by 𝜖 0 , 𝜖 1 , 𝜖 2 , 𝛼 0 , 𝛼 1 , 𝛼 2 , descends to an action on the quotient 𝑍 . At the ﬁxed point [ 1 : 0 : 0 ] ∈ P 2 ⊂ 𝑍 the tangent weights read 𝜖 1 − 𝜖 0 , 𝜖 2 − 𝜖 0 , 𝛼 0 + 𝜖 0 , 𝛼 1 + 𝜖 0 , 𝛼 2 + 𝜖 0 and similar for the two other ﬁxed points. W e denote by T  ( C × ) 3 the subtorus of ( C × ) 6 where the following r elations hold: 𝛼 0 = − 𝜖 0 + 𝜖 1 − 𝜖 2 , 𝛼 1 = − 𝜖 0 − 𝜖 1 + 𝜖 2 , 𝛼 2 = 𝜖 0 − 𝜖 1 − 𝜖 2 . One checks that the action of T on 𝑍 is Calabi–Y au and locally anti-diagonal as illustrated in Figure 3 . W e choose the distinct dir ections as indicated in the ﬁgure and orient all half-edges at the vertices clockwise. Applying Corollary 1.8 to this setup yields the formula GW • ( 𝑍 , T ) =  𝜇 1 ,𝜇 2 ,𝜇 3 ( − 𝑄 ) | 𝜇 1 | +| 𝜇 2 | +| 𝜇 3 |  𝑞 − 1 1 𝑞 2  𝜅 ( 𝜇 1 ) / 2  𝑞 − 1 2 𝑞 0  𝜅 ( 𝜇 2 ) / 2  𝑞 − 1 0 𝑞 1  𝜅 ( 𝜇 3 ) / 2 × W 𝜇 1 ,𝜇 2 , ∅  𝑞 2 𝑞 − 1 0  W 𝜇 2 ,𝜇 3 , ∅  𝑞 0 𝑞 − 1 1  W 𝜇 3 ,𝜇 1 , ∅  𝑞 1 𝑞 − 1 2  . (17) 24 Y ANNIK SCHULER The author is not aware of any trick that allows one to carry out the above sum explicitly . However , one may still use the formula to determine Ω 𝑑 [ 𝐻 ] in low degree 𝑑 . As formula ( 17 ) inherited the full 𝔖 3 W eyl symmetry of P 2 we may expand the membrane indices in terms of elementary symmetric polynomials: 𝑒 1 = 𝑞 1 / 2 0 + 𝑞 1 / 2 1 + 𝑞 1 / 2 2 , 𝑒 2 = ( 𝑞 0 𝑞 1 ) 1 / 2 + ( 𝑞 0 𝑞 2 ) 1 / 2 + ( 𝑞 1 𝑞 2 ) 1 / 2 , 𝑒 3 = ( 𝑞 0 𝑞 1 𝑞 2 ) 1 / 2 . The r esulting expr essions become particularly nice when normalised by the sym- metrised 𝑞 -number [ 𝑛 ] 𝑞 B 𝑞 𝑛 / 2 − 𝑞 − 𝑛 / 2 𝑞 1 / 2 − 𝑞 − 1 / 2 . Moreover , let us write 𝑞 𝑖 B 𝑞 1 / 2 𝑖 𝑞 − 1 / 2 𝑖 + 1 . W ith this notation the membrane indices extracted from ( 17 ) in low degr ee read: Ω 1 [ 𝐻 ] = − 2 Ö 𝑖 = 0 1 [ 2 ] 𝑞 𝑖 Ω 2 [ 𝐻 ] = 2 Ö 𝑖 = 0 1 [ 2 ] 𝑞 𝑖 Ω 3 [ 𝐻 ] = 2 Ö 𝑖 = 0 1 [ 2 ] 𝑞 𝑖 [ 4 ] 𝑞 𝑖 ×  − 𝑒 4 1 𝑒 4 2 𝑒 − 4 3 + 3 𝑒 2 1 𝑒 5 2 𝑒 − 4 3 − 𝑒 6 2 𝑒 − 4 3 + 3 𝑒 5 1 𝑒 2 2 𝑒 − 3 3 − 8 𝑒 3 1 𝑒 3 2 𝑒 − 3 3 − 𝑒 6 1 𝑒 − 2 3 + 11 𝑒 2 1 𝑒 2 2 𝑒 − 2 3 − 3 𝑒 3 2 𝑒 − 2 3 − 3 𝑒 3 1 𝑒 − 1 3  Ω 4 [ 𝐻 ] = 2 Ö 𝑖 = 0 [ 3 ] 𝑞 𝑖 [ 2 ] 𝑞 𝑖 [ 4 ] 𝑞 𝑖 [ 6 ] 𝑞 𝑖 ×  𝑒 9 1 𝑒 9 2 𝑒 − 9 3 − 8 𝑒 7 1 𝑒 10 2 𝑒 − 9 3 + 21 𝑒 5 1 𝑒 11 2 𝑒 − 9 3 − 19 𝑒 3 1 𝑒 12 2 𝑒 − 9 3 + 3 𝑒 1 𝑒 13 2 𝑒 − 9 3 − 8 𝑒 10 1 𝑒 7 2 𝑒 − 8 3 + 63 𝑒 8 1 𝑒 8 2 𝑒 − 8 3 − 155 𝑒 6 1 𝑒 9 2 𝑒 − 8 3 + 109 𝑒 4 1 𝑒 10 2 𝑒 − 8 3 + 15 𝑒 2 1 𝑒 11 2 𝑒 − 8 3 − 2 𝑒 12 2 𝑒 − 8 3 + 21 𝑒 11 1 𝑒 5 2 𝑒 − 7 3 − 155 𝑒 9 1 𝑒 6 2 𝑒 − 7 3 + 295 𝑒 7 1 𝑒 7 2 𝑒 − 7 3 + 56 𝑒 5 1 𝑒 8 2 𝑒 − 7 3 − 332 𝑒 3 1 𝑒 9 2 𝑒 − 7 3 + 8 𝑒 1 𝑒 10 2 𝑒 − 7 3 − 19 𝑒 12 1 𝑒 3 2 𝑒 − 6 3 + 109 𝑒 10 1 𝑒 4 2 𝑒 − 6 3 + 56 𝑒 8 1 𝑒 5 2 𝑒 − 6 3 − 958 𝑒 6 1 𝑒 6 2 𝑒 − 6 3 + 816 𝑒 4 1 𝑒 7 2 𝑒 − 6 3 + 294 𝑒 2 1 𝑒 8 2 𝑒 − 6 3 − 3 𝑒 9 2 𝑒 − 6 3 + 3 𝑒 13 1 𝑒 2 𝑒 − 5 3 + 15 𝑒 11 1 𝑒 2 2 𝑒 − 5 3 − 332 𝑒 9 1 𝑒 3 2 𝑒 − 5 3 + 816 𝑒 7 1 𝑒 4 2 𝑒 − 5 3 + 492 𝑒 5 1 𝑒 5 2 𝑒 − 5 3 − 1356 𝑒 3 1 𝑒 6 2 𝑒 − 5 3 − 129 𝑒 1 𝑒 7 2 𝑒 − 5 3 − 2 𝑒 12 1 𝑒 − 4 3 + 8 𝑒 10 1 𝑒 2 𝑒 − 4 3 + 294 𝑒 8 1 𝑒 2 2 𝑒 − 4 3 − 1356 𝑒 6 1 𝑒 3 2 𝑒 − 4 3 + 554 𝑒 4 1 𝑒 4 2 𝑒 − 4 3 + 950 𝑒 2 1 𝑒 5 2 𝑒 − 4 3 + 35 𝑒 6 2 𝑒 − 4 3 − 3 𝑒 9 1 𝑒 − 3 3 − 129 𝑒 7 1 𝑒 2 𝑒 − 3 3 + 950 𝑒 5 1 𝑒 2 2 𝑒 − 3 3 − 690 𝑒 3 1 𝑒 3 2 𝑒 − 3 3 − 374 𝑒 1 𝑒 4 2 𝑒 − 3 3 + 35 𝑒 6 1 𝑒 − 2 3 − 374 𝑒 4 1 𝑒 2 𝑒 − 2 3 + 305 𝑒 2 1 𝑒 2 2 𝑒 − 2 3 + 70 𝑒 3 2 𝑒 − 2 3 + 70 𝑒 3 1 𝑒 − 1 3 − 60 𝑒 1 𝑒 2 𝑒 − 1 3  GW INV ARIANTS AND MEMBRANE INDICES OF 5-FOLDS VIA TV 25 Remark 2.6. Let us emphasise the following remarkable features of the above formulae. (i) First, observe that all formulae are in agreement with Conjecture A : All expressions ar e elements in localised equivariant K-theory with integer coefﬁcients. (ii) However , in the limit 𝑞 𝑖 → 1 coefﬁcients in the above formulae feature negative powers of 2 . This is in accor dance with Conjectur e A since this is precisely the limit in which the tor us action becomes non-skeletal. (iii) It should also be stressed that powers of 2 are indeed the worst denomi- nators that appear . The cancellations ensuring this are surprisingly ﬁne- tuned and based on numerical data we conjecture the following general behaviour in higher degree. Conjecture 2.7. For any 𝑑 > 1 we have 2 Ö 𝑖 = 0 𝑑 − 1 Ö 𝑛 = 1 [ 2 𝑛 ] 𝑞 𝑖 [ od ( 𝑛 ) ] 𝑞 𝑖 ! · Ω 𝑑 [ 𝐻 ] ∈ Z [ 𝑒 ± 1 1 , 𝑒 2 , 𝑒 3 ] where od ( 𝑛 ) denotes the odd part of an integer 𝑛 . W e checked this conjectur e numerically up to degree ten. Conjectural formulae for low degree Ω 𝛽 on the full four-dimensional Calabi–Y au torus of 𝑍 will be presented in [ HS26 ]. 2.5. T ot P 3  O (− 2 ) ⊕ 2  . In this section we discuss 𝑍 = T ot P 3  O ( − 2 ) ⊕ O ( − 2 )  − → P 3 . As in the last section we present this variety as a quotient 𝑍 = C 6 \ 𝑉 ( 𝑥 0 𝑥 1 𝑥 2 𝑥 3 )  C × where the tor us acts on af ﬁne space via  𝑡 , ( 𝑥 0 , 𝑥 1 , 𝑥 2 , 𝑥 3 , 𝑦 0 , 𝑦 1 )  ↦− → ( 𝑡 𝑥 0 , 𝑡 𝑥 1 , 𝑡 𝑥 2 , 𝑡 𝑥 3 , 𝑡 − 2 𝑦 0 , 𝑡 − 2 𝑦 1 ) . The action of ( C × ) 6 on C 6 , whose tangent weights we denote by 𝜖 0 , 𝜖 1 , 𝜖 2 , 𝜖 3 , 𝛼 0 , 𝛼 1 , descends to the quotient 𝑍 . W e will analyse the equivariant Gr omov–W itten theory of 𝑍 with respect to a four -dimensional tor us T ⊂ ( C × ) 6 which is subject to the constraints 𝛼 0 = − 𝜖 1 and 𝛼 1 = 𝜖 0 − 𝜖 2 − 𝜖 3 . One can check that this torus action is Calabi–Y au and locally anti-diagonal. Indeed, for instance at the ﬁxed point [ 1 : 0 : 0 : 0 ] ∈ P 3 ⊂ 𝑍 we ﬁnd the tangent weights 𝜖 1 − 𝜖 0 , 𝜖 2 − 𝜖 0 , 𝜖 3 − 𝜖 0 , − 𝜖 1 + 𝜖 0 , 2 𝜖 0 − 𝜖 2 − 𝜖 3 . The T -diagram of 𝑍 which is displayed in Figure 4 indicates the tangent weights at the remaining ﬁxed points. 26 Y ANNIK SCHULER 𝜇 1 𝜇 t 1 𝜇 2 𝜇 t 2 𝜇 3 𝜇 t 3 𝜇 4 𝜇 t 4 ◦ ◦ ◦ ◦ F I G U R E 4 . The T -diagram of T ot O P 3 ( − 2 ) ⊕ 2 with a choice of a dis- tinct direction at each vertex and half-edges decorated by partitions. Now note that despite the fact that the T -action on 𝑍 is skeletal, Calabi–Y au and locally anti-diagonal we cannot readily apply Coroll ary 1.8 to compute the Gromov–W itten invariants of 𝑍 — at least not in the form it is stated. The problem is that the class of a line can be supported on both of the two anti-diagonal edges of the T -diagram which are highlighted red in Figure 4 . However , as explained in Remark 1.9 the conclusion of Corollary 1.8 still holds if we can show that GW • 𝒅 ( 𝑍 , T ) = 0 whenever 𝒅 has non-trivial support on one of the two red edges. So let 𝐹 𝛾 be a component of T -ﬁxed locus of 𝑀 ( 𝑍 , 𝛽 ) parametrising stable maps with non-zero support on one of the red edges. It sufﬁces to show that [ 𝐹 𝛾 ] virt = 0 . Indeed, the restriction of the vector bundle 𝑍 → P 3 to each of the red edges is isomorphic to O P 1 ( − 2 ) ⊕ O P 1 ( − 2 ) . Mor eover , T acts on one of the line bundles in a way that the holomorphic two-form of T ot P 1 ( O ( − 2 ) ) is ﬁxed. This nowhere vanishing T - invariant holomorphic two-form thus yields a trivial factor in the obstruction bundle of 𝐹 𝛾 . This implies the vanishing of the virtual class as desired. Hence, we may apply the conclusion of Corollary 1.8 to our case at hand. Note that by the vanishing we have just shown we can decorate the red edges with trivial partitions as indicated in Figure 4 . W e ﬁx a cyclic order at each vertex 𝑣 by GW INV ARIANTS AND MEMBRANE INDICES OF 5-FOLDS VIA TV 27 demanding that 𝜎 𝑣 maps the half-edge decorated with 𝜇 𝑖 to 𝜇 𝑖 + 1 . T ogether with the choice of distinct directions indicated in Figur e 4 this implies that all edges have negative mixing parity . Let us write 𝑄 B 𝑄 [ 𝐿 ] where [ 𝐿 ] is the clas s of a line in P 3 . W e obtain the following formula for the Gromov–W itten invariants of 𝑍 : GW • ( 𝑍 , T ) =  𝜇 1 ,𝜇 2 , 𝜇 3 ,𝜇 4 𝑄 Í 4 𝑖 = 1 | 𝜇 𝑖 |  𝑞 − 2 0 𝑞 2 1 𝑞 − 1 2 𝑞 3  𝜅 ( 𝜇 1 ) / 2  𝑞 0 𝑞 − 1 1 𝑞 − 2 2 𝑞 2 3  𝜅 ( 𝜇 2 ) / 2  𝑞 2 0 𝑞 − 2 1 𝑞 2 𝑞 − 1 3  𝜅 ( 𝜇 3 ) / 2  𝑞 − 1 0 𝑞 1 𝑞 2 2 𝑞 − 2 3  𝜅 ( 𝜇 4 ) / 2 × W 𝜇 t 4 ,𝜇 1 , ∅  𝑞 0 𝑞 − 1 1  W 𝜇 t 1 ,𝜇 2 , ∅  𝑞 2 𝑞 − 1 3  W 𝜇 t 2 ,𝜇 3 , ∅  𝑞 − 1 0 𝑞 1  W 𝜇 t 3 ,𝜇 4 , ∅  𝑞 − 1 2 𝑞 3  . (18) As in the last example we ar e not able to carry out the above sums explicitly but we may still employ the formula to extract membrane indices in low degr ee. Experimentally we observe the following. Conjecture 2.8. Ω 𝑑 [ 𝐿 ] is a Laurent polynomial in 𝑞 0 , 𝑞 1 , 𝑞 2 and 𝑞 3 . Moreover , observe that formula ( 18 ) only depends on the characters 𝑞 0 𝑞 − 1 1 and 𝑞 2 𝑞 − 1 3 and is invariant under 𝑞 0 𝑞 − 1 1 → 𝑞 − 1 0 𝑞 1 and 𝑞 2 𝑞 − 1 3 → 𝑞 − 1 2 𝑞 3 . The latter sym- metry is inherited from the Z / 2 Z × Z / 2 Z symmetry of Figure 4 . Hence, assuming the absence of poles other than zero or inﬁnity , the membrane indices can be expanded as Ω 𝑑 [ 𝐿 ] C  𝑘 1 , 𝑘 2 ≥ 1 𝑁 𝑑 ; 𝑘 1 , 𝑘 2 [ 𝑘 1 ] 𝑞 0 𝑞 − 1 1 [ 𝑘 2 ] 𝑞 2 𝑞 − 1 3 with 𝑁 𝑑 ; 𝑘 1 , 𝑘 2 ∈ Z . In degree 𝑑 ≤ 3 the only non-vanishing invariants are: 𝑁 2;1 , 1 = 2 , 𝑁 3;4 , 4 = − 2 . All non-zer o invariants for 4 ≤ 𝑑 ≤ 6 are listed in Tables 1 – 3 . W e checked that Conjecture 2.8 holds up to degree ten numerically . Conjectural formulae for low- degree membrane indices on the full four dimensional Calabi–Y au torus of 𝑍 will be presented in [ HS26 ]. 1 3 5 7 9 1 -1 2 2 3 2 5 2 2 2 7 9 2 2 2 T A B L E 1 . 𝑁 4; 𝑘 1 , 𝑘 2 2 4 6 8 10 12 14 16 2 -2 -2 -2 4 2 -2 -2 -2 -2 -2 6 -2 -2 -4 -4 -2 -2 -2 8 -2 -2 -4 -4 -2 -2 -2 10 -2 -2 -2 -2 -2 -2 12 -2 -2 -2 -2 -2 -2 14 16 -2 -2 -2 -2 -2 -2 T A B L E 2 . 𝑁 5; 𝑘 1 , 𝑘 2 28 Y ANNIK SCHULER 1 3 5 7 9 11 13 15 17 19 21 23 25 1 7 -3 7 1 10 4 12 4 8 4 4 4 3 -3 -3 -1 3 2 2 2 2 5 7 -1 13 5 14 6 14 6 8 4 4 4 7 1 3 5 9 10 6 10 6 4 2 2 2 9 10 2 14 10 18 8 20 8 12 6 6 6 11 4 2 6 6 8 4 8 4 4 2 2 2 13 12 2 14 10 20 8 20 8 12 6 6 6 15 4 2 6 6 8 4 8 4 4 2 2 2 17 8 8 4 12 4 12 4 8 4 4 4 19 4 4 2 6 2 6 2 4 2 2 2 21 4 4 2 6 2 6 2 4 2 2 2 23 25 4 4 2 6 2 6 2 4 2 2 2 T A B L E 3 . 𝑁 6; 𝑘 1 , 𝑘 2 3. P R O O F O F T H E V E R T E X F O R M A L I S M In this section we will pr ove Theor em 1.7 . Throughout this section let 𝑍 be a Calabi–Y au ﬁvefold with a skeletal, Calabi–Y au and locally anti-diagonal action by a torus T . Moreover , we ﬁx a choice of distinct directions and orders. W e will deduce the vertex formula for Gr omov–W itten invariants of 𝑍 from capped locali- sation. This method repackages the formula for Gromov–W itten invariants result- ing from torus localisation in a more practical form. This idea was pioneered by Li–Liu–Liu–Zhou in [ LLLZ09 ] (see also [ LLZ03 ; LLZ07 ]) and was key in Maulik– Oblomkov–Okunkov–Pandharipande’s proof of the Gromov–W itten/Donaldson– Thomas correspondence for toric thr eefolds [ MOOP11 ]. 3.1. T orus localisation. Recall from Section 1.2 that we refer to an assignment of a non-negative integer to each edge of the T -diagram of 𝑍 as a skeletal degree. Given such an assignment 𝒅 : 𝐸 ( Γ ) → Z ≥ 0 we denoted by (19) GW • 𝒅 ( 𝑍 , T ) B  𝛾 has skeletal support 𝒅 ∫ [ 𝐹 𝛾 ] virt 1 𝑒 T ( 𝑁 virt 𝛾 ) the contribution of all T -ﬁxed loci parametrising stable maps whose cover of the torus orbit associated with an edge 𝑒 has degree 𝑑 𝑒 . 3.2. Capped localisation. Each term on the right-hand side of ( 19 ) can be written as a product of weights labelled by vertices and edges of the T -diagram. V ertices are weighted by Hodge integrals and edges by some closed-form combinatorial GW INV ARIANTS AND MEMBRANE INDICES OF 5-FOLDS VIA TV 29 factors. The idea of capped localisation is to repackage this decomposition in a way that vertex weights become relative Gromov–W itten invariants of a partial compactiﬁcation of C 5 and edge weights become relative invariants of a vector bundle over a rational curve relative to two ﬁbres. The result is a formula express- ing the Gromov–W itten invariant with skeletal degree 𝒅 as a weighted sum over so-called capped markings which r ecord the relative conditions at the interface between partially compactiﬁed vertices and edges. T o be precise, by a capped marking 𝝂 we mean the assignment of a partition 𝜈 ℎ to every half-edge ℎ of the T -diagram of 𝑍 satisfying • 𝜈 ℎ = ∅ whenever ℎ is a leaf; • | 𝜈 ℎ | = | 𝜈 ℎ ′ | = 𝑑 𝑒 when 𝑒 = ( ℎ, ℎ ′ ) . W e denote by P Γ , 𝒅 the set of all such capped markings. W e r emark that since we assume 𝒅 is supported away from anti-diagonal strata we have 𝜈 ℎ = ∅ whenever ℎ is an anti-diagonal half-edge. This assumption will become crucial later in our proof (see Remarks 3.1 and 3.5 ). In the following we will describe the partial compactiﬁcations of vertices and edges and their associated weights needed in order to state the capped localisa- tion formula. W e do so by lifting the constructions described in [ LLLZ09 , Sec. 7] and [ MOOP11 , Sec. 2.3] from the thr eefold to our ﬁvefold setting. 3.2.1. Capped vertices. W e begin by discussing the local model for partial compact- iﬁcations of vertices in Γ . For this denote by 𝑈 the result of blowing up P 1 × P 1 × P 1 along the lines ∞ × P 1 × 0 , 0 × ∞ × P 1 , P 1 × 0 × ∞ and deleting the pre-image of the thr ee ( C × ) 3 -preserved lines through ( ∞ , ∞ , ∞) under the blow-up morphism. The construction provides us with a partial com- pactiﬁcation of C 3 ∋ ( 0 , 0 , 0 ) where the 𝑖 th coordinate axis is compactiﬁed to a P 1 with normal bundle 𝑁 P 1 𝑈 = O P 1 ⊕ O P 1 ( − 1 ) . W e denote by 𝐷 𝑖 ⊂ 𝑈 the divisor at ∞ ∈ P 1 . Note that 𝐷 1 , 𝐷 2 and 𝐷 3 are pairwise disjoint and that whenever the action of a torus T ′ ⊂ ( C × ) 3 on C 3 is Calabi–Y au, then its action on 𝐷 𝑖  C 2 will be so as well. Now let 𝑣 be a vertex of the T -diagram Γ of 𝑍 . Locally at the associated ﬁxed point 𝑝 𝑣 the ﬁvefold 𝑍 looks like C 5  𝑇 𝑝 𝑣 𝑍 with the induced torus action. The choice of a distinct direction 𝜖 𝑣 and cyclic order 𝜎 𝑣 = ( ℎ 1 ℎ 2 ℎ 3 ) singles out a decomposition (20) 𝑇 𝑝 𝑣 𝑍  C 5 = C 3 × C 2 where T acts on the coordinate lines of C 2 with opposite weights 𝜖 𝑣 and − 𝜖 𝑣 and the action on C 3 is Calabi–Y au. W e choose the following partial compactiﬁcation at 𝑝 𝑣 : ( e 𝑈 | e 𝐷 ℎ 1 + e 𝐷 ℎ 2 + e 𝐷 ℎ 3 ) B ( 𝑈 × C 2 | 𝐷 1 × C 2 + 𝐷 2 × C 2 + 𝐷 3 × C 2 ) . 30 Y ANNIK SCHULER The T -action on C 3 × C 2 lifts to 𝑈 × C 2 . Its tangent weights at the origin of 𝐷 𝑖 × C 2  C 4 are 𝜖 𝜎 ( ℎ 𝑖 ) , − 𝜖 𝜎 ( ℎ 𝑖 ) , 𝜖 𝑣 , − 𝜖 𝑣 . Now suppose we are given a capped marking 𝝂 . Then only the partitions 𝜈 ℎ 1 , 𝜈 ℎ 2 and 𝜈 ℎ 3 of the half-edges adjacent to 𝑣 may be non-trivial. W e weight the vertex 𝑣 by the relative Gr omov–W itten invariants of ( e 𝑈 | e 𝐷 ℎ 1 + e 𝐷 ℎ 2 + e 𝐷 ℎ 3 ) : e 𝐶 ( 𝑣, 𝝂 ) B  𝑔 ∈ Z ∫ [ 𝑀 • 𝑔 ( e 𝑈 | e 𝐷 ℎ 1 + e 𝐷 ℎ 2 + e 𝐷 ℎ 3 , 𝜈 ℎ 1 , 𝜈 ℎ 2 , 𝜈 ℎ 3 ) ] virt T 1 . Remark 3.1. W e want to stress that due to our assumption that 𝒅 is supported away from anti-diagonal strata we do not need to compactify vertices in the two anti-diagonal directions as the associated half-edges are decorated with empty partitions. This fact is going to be crucial for evaluating the vertex weights later in Corollary 3.3 . 3.2.2. Capped edges. Let 𝑒 be an edge with none of its half-edges ℎ and ℎ ′ being an anti-diagonal half-edge. At the vertices linked by 𝑒 we have alr eady chosen a partial compactiﬁcation whose interface at ℎ and ℎ ′ is the divisor e 𝐷 ℎ  C 4 and e 𝐷 ℎ ′ respectively . Now let T act on P 1 with tangent weight − 𝜖 ℎ at 0 and − 𝜖 ℎ ′ = 𝜖 ℎ at ∞ . There is a T -equivariant rank-four vector bundle 𝑉 on P 1 whose ﬁbres at 0 and ∞ are e 𝐷 ℎ and e 𝐷 ℎ ′ compatible with the T -actions. The vector bundle 𝑉 will serve as the partial compactiﬁcation for the edge 𝑒 . For a capped marking 𝝂 we weight 𝑒 by the relative Gr omov–W itten invariant e 𝐸 ( 𝑒 , 𝝂 ) B  𝑔 ∈ Z ∫ [ 𝑀 • 𝑔 ( 𝑉 | e 𝐷 ℎ + e 𝐷 ℎ ′ , 𝜈 ℎ , 𝜈 ℎ ′ ) ] virt T 1 . 3.2.3. The formula. Capped localisation then yields the following formula: (21) GW • 𝒅 ( 𝑍 , T ) =  𝝂 ∈ P Γ , 𝒅 Ö 𝑒 ∈ 𝐸 ( Γ ) e 𝐸 ( 𝑒 , 𝝂 ) Ö 𝑣 ∈ 𝑉 ( Γ ) e 𝐶 ( 𝑣, 𝝂 ) Ö ℎ ∈ 𝐻 ( Γ ) 𝔷 ( 𝜈 ℎ )  𝜖 𝜎 𝑣 ( ℎ ) 𝜖 𝑣  2 ℓ ( 𝜈 ℎ ) . The last product features the gluing terms associated with each interface e 𝐷 ℎ where ℓ ( 𝜈 ) denotes the length of a partition 𝜈 and 𝔷 ( 𝜈 ) = | Aut ( 𝜈 ) | Î 𝑖 𝜈 𝑖 . As discussed in [ MOOP11 , Sec. 2.4] and explained in full detail in [ LLLZ09 , Sec. 7] formula ( 21 ) may be proven by inserting appropriate rubber integrals of e 𝐷 ℎ × P 1 and combinatorial factors at the half-edges. 3.3. Simpliﬁcation of the capped localisation formula. W e will deduce our ver- tex formula ( 8 ) from ( 21 ) by explicitly evaluating the vertex and edge weights and reor ganising the resulting expression. GW INV ARIANTS AND MEMBRANE INDICES OF 5-FOLDS VIA TV 31 3.3.1. V ertex weights. W e recall the topological vertex formula for the relative Gromov–W itten invariants of the partial compactiﬁcation ( 𝑈 | 𝐷 1 + 𝐷 2 + 𝐷 3 ) of C 3 described in Section 3.2.1 . Let T be a torus acting on C 3 . Theorem 3.2. Let 𝜈 1 , 𝜈 2 , 𝜈 3 be partitions. If the T -action on C 3 is Calabi–Y au we have  𝑔 𝑢 2 𝑔 − 2 + Í 𝑖 ℓ ( 𝜈 𝑖 ) ∫ [ 𝑀 • 𝑔 ( 𝑈 | 𝐷 1 + 𝐷 2 + 𝐷 3 , 𝜈 1 , 𝜈 2 , 𝜈 3 ) ] virt T 1 = 3 Ö 𝑖 = 1 ( − 1 ) | 𝜈 𝑖 | ( i 𝜖 𝑖 + 1 ) − ℓ ( 𝜈 𝑖 ) !  𝜆 1 ,𝜆 2 ,𝜆 3 W 𝜆 1 ,𝜆 2 ,𝜆 3 ( e i 𝑢 ) 3 Ö 𝑖 = 1 𝜒 𝜆 𝑖 ( 𝜈 𝑖 ) 𝔷 ( 𝜈 𝑖 ) where 𝜒 𝜆 ( 𝜈 ) is the character of the irreducible repr esentation labelled by a partition 𝜆 evaluated at the conjugacy class 𝜈 . This formula was ﬁrst pr oven in the case where two partitions ar e empty [ LLZ03 ; OP04 ] and later for one partition being empty [ LLZ07 ]. The general formula is a consequence of the relative Gromov–W itten/Donaldson–Thomas correspon- dence for toric threefolds [ MOOP11 ] together with the formula for the three-leg Donaldson–Thomas vertex [ OR V06 , Eq. (3.23)]. By our assumption that the T -action on 𝑍 is locally anti-diagonal the above three- fold formula will allow us to evaluate our ﬁvefold vertex weights. Indeed, the partial compactiﬁcation at a vertex 𝑣 is the product e 𝑈 = 𝑈 × C 2 . A direct compari- son of the perfect obstruction theories shows that the relative genus- 𝑔 invariants of e 𝑈 differ fr om those of 𝑈 by an insertion of Λ 𝑔 ( 𝜖 𝑣 ) Λ 𝑔 ( − 𝜖 𝑣 ) where Λ 𝑔 ( 𝜖 ) =  𝑘 ≥ 0 𝜆 𝑘 ( − 1 ) 𝑘 𝜖 𝑔 − 𝑘 − 1 and 𝜆 𝑘 is the 𝑘 th Chern class of the Hodge bundle. Hence, we get the following corollary as an immediate consequence of Theorem 3.2 and Mumford’s relation [ Mum83 , Sec. 5] (22) Λ 𝑔 ( 𝜖 ) Λ 𝑔 ( − 𝜖 ) = ( − 1 ) 𝑔 − 1 𝜖 2 𝑔 − 2 . Corollary 3.3. Let 𝝂 be a capped marking. Then e 𝐶 ( 𝑣, 𝝂 ) = 3 Ö 𝑖 = 1 ( − 1 ) | 𝜈 𝑖 | ( 𝜖 𝜎 ( ℎ 𝑖 ) 𝜖 𝑣 ) − ℓ ( 𝜈 ℎ 𝑖 ) !  𝜆 1 ,𝜆 2 ,𝜆 3 W 𝜆 1 ,𝜆 2 ,𝜆 3 ( e 𝑢𝜖 𝑣 ) 3 Ö 𝑖 = 1 𝜒 𝜆 𝑖 ( 𝜈 𝑖 ) 𝔷 ( 𝜈 𝑖 ) . □ Remark 3.4. W e stress that in or der to deduce the cor ollary it was essential to assume that the torus action on 𝑍 is locally anti-diagonal. Otherwise we could not have used Mumford’s relation to r educe the ﬁvefold invariant to a thr eefold one. In order to establish a vertex formalism which applies beyond the locally anti-diagonal situation a better understanding of quintuple Hodge integrals will be crucial. See [ GPS26 ] for a ﬁrst step in this direction. 32 Y ANNIK SCHULER Remark 3.5. T o be able to evaluate the vertex terms it is also crucial to assume that the skeletal degree is supported away from anti-diagonal strata. It is unclear to the author whether there is an appr opriate substitute for the partial compactiﬁca- tion e 𝑈 we chose at each vertex which lets one dr op this assumption. On the level of bar e vertices (that is Hodge integrals) dropping the r equir ement that 𝑑 𝑒 = 0 for edges ending at anti-diagonal half-edges would requir e closed formulae for the 3-legged threefold vertex with descendants. 3.3.2. Edge weights. Let 𝑒 = ( ℎ, ℎ ′ ) be an edge with none of its half-edges being an anti-diagonal half-edge. W e will evaluate the edge weight e 𝐸 ( 𝑒 , 𝝂 ) via virtual localisation. Recall from Section 3.2.2 that we weight 𝑒 by the relative Gromov– W itten invariant of a rank-four vector bundle 𝑉 on P 1 relative to the ﬁbr es at 0 and ∞ which we identiﬁed with the divisors e 𝐷 ℎ and e 𝐷 ℎ ′ respectively . Further , recall that the tangent T -weights at the origin of these ﬁbres ar e ( 𝜖 𝜎 ( ℎ ) , − 𝜖 𝜎 ( ℎ ) , 𝜖 𝑣 , − 𝜖 𝑣 ) , ( 𝜖 𝜎 ( ℎ ′ ) , − 𝜖 𝜎 ( ℎ ′ ) , 𝜖 𝑣 ′ , − 𝜖 ′ 𝑣 ) . Evaluating the r elative Gr omov–W itten invariants of ( 𝑉 | e 𝐷 ℎ + e 𝐷 ℎ ′ ) via virtual localisation [ GP99 ; GV05 ] yields a decomposition of the edge weight into (23) e 𝐸 ( 𝑒 , 𝝂 ) =  𝜆 ⊢ 𝑑 𝑒 𝐴 ( 𝜈 ℎ , 𝜆 ) · 𝐵 ( 𝜆 ) · 𝐴 ′ ( 𝜆, 𝜈 ℎ ′ ) · ( Î 𝑖 𝜆 𝑖 ) 𝔷 ( 𝜆 ) ( 𝜖 𝜎 ( ℎ ) 𝜖 𝑣 𝜖 𝜎 ( ℎ ′ ) 𝜖 𝑣 ′ ) 2 ℓ ( 𝜆 ) where the middle factor 𝐵 ( 𝜆 ) = 1 Î 𝑖 𝜆 𝑖 ℓ ( 𝜆 ) Ö 𝑖 = 1  𝜆 𝑖 𝜖 ℎ  4 Γ  𝜆 𝑖 𝜖 𝜎 𝑣 ′ ( ℎ ′ ) 𝜖 ℎ  Γ  − 𝜆 𝑖 𝜖 𝜎 𝑣 ′ ( ℎ ′ ) 𝜖 ℎ  Γ  𝜆 𝑖 𝜖 𝑣 ′ 𝜖 ℎ  Γ  − 𝜆 𝑖 𝜖 𝑣 ′ 𝜖 ℎ  Γ  𝜆 𝑖 𝜖 𝜎 𝑣 ( ℎ ) 𝜖 ℎ + 1  Γ  − 𝜆 𝑖 𝜖 𝜎 𝑣 ( ℎ ) 𝜖 ℎ + 1  Γ  𝜆 𝑖 𝜖 𝑣 𝜖 ℎ + 1  Γ  − 𝜆 𝑖 𝜖 𝑣 𝜖 ℎ + 1  comes from components of a torus ﬁxed stable map which are degree 𝜆 𝑖 covers of the zero section P 1 ⊂ 𝑉 fully ramiﬁed over 0 and ∞ . The 𝐴 -factors are rubber integrals arising from domain components mapping into bubbles at 0 and ∞ . More pr ecisely , we have 𝐴 ( 𝜈 ℎ , 𝜆 ) =  𝑔 ∫ [ 𝑀 • 𝑔 ( P 1 , 𝜈 ℎ ,𝜆 ) ∼ ] virt Λ 𝑔 ( 𝜖 𝜎 ( ℎ ) ) Λ 𝑔 ( − 𝜖 𝜎 ( ℎ ) ) Λ 𝑔 ( 𝜖 𝑣 ) Λ 𝑔 ( − 𝜖 𝑣 ) 𝜖 ℎ − 𝜓 ∞ with a similar formula for 𝐴 ′ . The last factor arises from gluing the domains responsible for the 𝐴 , 𝐵 and 𝐴 ′ terms along the nodes over 0 and ∞ . W e chose to present the middle term in a rather non-standar d way to facilitate the evaluation of this factor . Lemma 3.6. We have 𝐵 ( 𝜆 ) = ( − 1 ) 𝑝 𝑒 ℓ ( 𝜆 ) + 𝑓 𝑒 | 𝜆 | ( 𝜖 𝜎 𝑣 ( ℎ ) 𝜖 𝑣 𝜖 𝜎 𝑣 ′ ( ℎ ′ ) 𝜖 𝑣 ′ ) − ℓ ( 𝜆 ) 1 Î 𝑖 𝜆 𝑖 . GW INV ARIANTS AND MEMBRANE INDICES OF 5-FOLDS VIA TV 33 Proof. By the reﬂection formula of the Gamma function Γ ( 𝑧 ) Γ ( 1 − 𝑧 ) = 𝜋 sin 𝜋 𝑧 we can write 𝐵 ( 𝜆 ) as 𝐵 ( 𝜆 ) = ( 𝜖 𝜎 𝑣 ( ℎ ) 𝜖 𝑣 𝜖 𝜎 𝑣 ′ ( ℎ ′ ) 𝜖 𝑣 ′ ) − ℓ ( 𝜆 ) 1 Î 𝑖 𝜆 𝑖 ℓ ( 𝜆 ) Ö 𝑖 = 1 sin 𝜋 𝜆 𝑖 𝜖 𝜎 𝑣 ( ℎ ) 𝜖 ℎ sin 𝜋 𝜆 𝑖 𝜖 𝑣 𝜖 ℎ sin 𝜋 𝜆 𝑖 𝜖 𝜎 𝑣 ′ ( ℎ ′ ) 𝜖 ℎ sin 𝜋 𝜆 𝑖 𝜖 𝑣 ′ 𝜖 ℎ . Now due to the linear relations ( 5 ) or ( 6 ) satisﬁed by the weights 𝜖 ℎ , 𝜖 𝜎 𝑣 ( ℎ ) , 𝜖 𝑣 , 𝜖 𝜎 𝑣 ′ ( ℎ ′ ) and 𝜖 𝑣 ′ the last pr oduct simpliﬁes to a factor ± 1 . One can check in a case-by-case analysis that by our deﬁnition of the framing and mixing parity we have sin 𝜋 𝜆 𝑖 𝜖 𝜎 𝑣 ( ℎ ) 𝜖 ℎ sin 𝜋 𝜆 𝑖 𝜖 𝑣 𝜖 ℎ sin 𝜋 𝜆 𝑖 𝜖 𝜎 𝑣 ′ ( ℎ ′ ) 𝜖 ℎ sin 𝜋 𝜆 𝑖 𝜖 𝑣 ′ 𝜖 ℎ = ( − 1 ) 𝑝 𝑒 + 𝑓 𝑒 𝜆 𝑖 . □ The 𝐴 -terms ar e evaluated as follows. Lemma 3.7. We have 𝐴 ( 𝜈 ℎ , 𝜆 ) = ( 𝜖 𝜎 𝑣 ( ℎ ) 𝜖 𝑣 ) − ℓ ( 𝜈 ℎ ) − ℓ ( 𝜆 )  𝜇 exp  𝜅 ( 𝜇 ) 2 𝜖 𝜎 𝑣 ( ℎ ) 𝜖 𝑣 𝜖 ℎ  𝜒 𝜇 ( 𝜈 ℎ ) 𝔷 ( 𝜈 ℎ ) 𝜒 𝜇 ( 𝜆 ) 𝔷 ( 𝜆 ) Proof. The claim follows from Mumford’s relation ( 22 ) together with the rubber integral formula  𝑔 𝑢 2 𝑔 − 2 + ℓ ( 𝜈 1 ) + ℓ ( 𝜈 2 ) ∫ [ 𝑀 • 𝑔 ( P 1 , 𝜈 1 , 𝜈 2 ) ∼ ] virt 1 𝜖 − 𝜓 ∞ =  𝜇 exp  𝜅 ( 𝜇 ) 2 𝑢 𝜖  𝜒 𝜇 ( 𝜈 1 ) 𝔷 ( 𝜈 1 ) 𝜒 𝜇 ( 𝜈 2 ) 𝔷 ( 𝜈 2 ) . This equation is proven in [ LLZ07 , Pr op. 5.4 & Eq. (17)] by relating the rubber integral to double Hurwitz numbers. □ If we insert the formulae from Lemma 3.6 and 3.7 into equation ( 23 ) and employ the orthogonality relations  𝜆 𝜒 𝜇 1 ( 𝜆 ) 𝜒 𝜇 2 ( 𝜆 ) 𝔷 ( 𝜆 ) = 𝛿 𝜇 1 ,𝜇 2 , (24)  𝜆 ( − 1 ) ℓ ( 𝜆 ) 𝜒 𝜇 1 ( 𝜆 ) 𝜒 𝜇 2 ( 𝜆 ) 𝔷 ( 𝜆 ) = 𝛿 𝜇 1 ,𝜇 t 2 ( − 1 ) | 𝜇 1 | . we arrive at the following expression for the edge weights. 34 Y ANNIK SCHULER Corollary 3.8. Let 𝝂 be a capped marking. Then e 𝐸 ( 𝑒 , 𝝂 ) = ( − 1 ) ( 𝑓 𝑒 + 𝑝 𝑒 ) | 𝜈 ℎ | ( 𝜖 𝜎 𝑣 ( ℎ ) 𝜖 𝑣 ) − ℓ ( 𝜈 ℎ ) ( 𝜖 𝜎 𝑣 ′ ( ℎ ′ ) 𝜖 𝑣 ′ ) − ℓ ( 𝜈 ℎ ′ ) ×  𝜇 ℎ ,𝜇 ℎ ′ exp  𝜅 ( 𝜇 ℎ ) 2 𝜖 𝜎 𝑣 ( ℎ ) 𝜖 𝑣 + ( − 1 ) 𝑝 𝑒 + 1 𝜖 𝜎 𝑣 ′ ( ℎ ′ ) 𝜖 𝑣 ′ 𝜖 ℎ  𝜒 𝜇 ℎ ( 𝜈 ℎ ) 𝔷 ( 𝜈 ℎ ) 𝜒 𝜇 ℎ ′ ( 𝜈 ℎ ′ ) 𝔷 ( 𝜈 ℎ ′ ) where the sum is over tuples of partitions ( 𝜇 ℎ , 𝜇 ℎ ′ ) satisfying 𝜇 ℎ = 𝜇 ℎ ′ if the mixing parity of 𝑒 is even and 𝜇 ℎ = 𝜇 t ℎ ′ if it is odd. □ Remark 3.9. Both in Lemma 3.6 and 3.7 the assumptions that the T -action is Calabi–Y au and locally anti-diagonal are used crucially: First, the assumptions lead to the collapse of the 𝐵 -term and second, they enable us to use Mumford’s relation which is key to evaluate the 𝐴 -terms. T o go beyond locally anti-diagonal torus actions new ideas will be necessary . 3.3.3. Putting everything together. If we plug our formulae for the vertex and edge weights (Corollary 3.3 and 3.8 ) into the capped localisation formula ( 21 ) we obtain GW • 𝒅 ( 𝑍 , T ) =  𝝂 ∈ P Γ , 𝒅  𝝀  𝝁 ∈ P Γ , 𝒑 , 𝒅 Ö 𝑒 = ( ℎ,ℎ ′ ) ( − 1 ) ( 𝑓 𝑒 + 𝑝 𝑒 ) | 𝜇 ℎ | exp  𝜅 ( 𝜇 ℎ ) 2 𝜖 𝜎 𝑣 ( ℎ ) 𝜖 𝑣 + ( − 1 ) 𝑝 𝑒 + 1 𝜖 𝜎 𝑣 ′ ( ℎ ′ ) 𝜖 𝑣 ′ 𝜖 ℎ  × Ö 𝑣 W 𝜆 ℎ 1 ,𝜆 ℎ 2 ,𝜆 ℎ 3 ( e 𝑢𝜖 𝑣 ) Ö ℎ 𝜒 𝜆 ℎ ( 𝜈 ℎ ) 𝜒 𝜇 ℎ ( 𝜈 ℎ ) 𝔷 ( 𝜈 ℎ ) where the second sum runs over tuples of partitions 𝝀 = ( 𝜆 ℎ 1 , 𝜆 ℎ 2 , 𝜆 ℎ 3 ) 𝑣 ∈ 𝑉 ( Γ ) assign- ing a partition to each of the thr ee half-edges permuted by the cyclic order 𝜎 𝑣 = ( ℎ 1 ℎ 2 ℎ 3 ) at a vertex 𝑣 . W e can carry out the sum over 𝝂 using the orthogonality relation ( 24 ). The result of this manipulation is precisely the vertex formula stated in Theorem 1.7 . A P P E N D I X A. T H E P L E T H Y S T I C L O G A R I T H M I N L O C A L I S E D K - T H E O R Y A.1. Integrality. Let T be a torus. W e consider the repr esentation ring of T with coefﬁcients in Z localised at the augmentation ideal, i.e. the ideal of zero dimen- sional virtual repr esentations: 𝑅 B Rep ( T ) loc = 𝐾 T ( pt ) loc . If we ﬁx an isomorphism T  ( C × ) 𝑚 we get the following presentation: 𝑅  Z " 𝑞 ± 1 1 , . . . , 𝑞 ± 1 𝑚 ,  1 Î 𝑚 𝑖 = 1 ( 1 − 𝑞 𝑛 𝑖 𝑖 )  𝒏 ∈ Z 𝑚 \{ 0 } # . GW INV ARIANTS AND MEMBRANE INDICES OF 5-FOLDS VIA TV 35 The ring 𝑅 is a lambda ring. The 𝑘 th Adams operation Ψ 𝑘 acts on one-dimensional repr esentations 𝑞 as Ψ 𝑘 ( 𝑞 ) = 𝑞 𝑘 . W e get a lambda ring structur e on 𝑅 ⟦ 𝑄 1 , . . . , 𝑄 ℓ ⟧ by declaring that it acts on monomials as Ψ 𝑘 ( 𝑟 𝑄 𝒅 ) B Ψ 𝑘 ( 𝑟 ) 𝑄 𝑘 𝒅 . Let 𝐼 ⊂ 𝑅 be the ideal generated by 𝑄 1 , . . . , 𝑄 ℓ . W e deﬁne the plethystic logarithm as Log : 1 + 𝐼 · 𝑅 ⟦ 𝑄 1 , . . . , 𝑄 ℓ ⟧ 𝐼 · 𝑅 ⟦ 𝑄 1 , . . . , 𝑄 ℓ ⟧ ⊗ Q , 𝐺  𝑘 > 0 𝜇 ( 𝑘 ) 𝑘 Ψ 𝑘 ( log 𝐺 ) . A priori, by the above deﬁnition we should expect the image Log ( 𝐺 ) of a power series with coef ﬁcients in 𝑅 to feature coefﬁcients in 𝑅 ⊗ Q — one source of denom- inators being the factor 1 / 𝑘 and the other being the logarithm. In contrast to this expectation, one can show that the plethystic logarithm preserves integrality . Lemma A.1. The image of the plethystic logarithm lies in 𝐼 · 𝑅 ⟦ 𝑄 1 , . . . , 𝑄 ℓ ⟧ . Remark A.2. The analogous statement for Rep ( T ) , that is without localising at the augmentation ideal, holds true since the plethystic exponential acts on a rep- resentation 𝑉 times 𝑄 𝒅 as Exp ( 𝑉 𝑄 𝒅 ) = Í 𝑛 ≥ 0 Sym 𝑛 ( 𝑉 ) 𝑄 𝑛 𝒅 meaning that it preserves integrality . Thus, the same must be true for the plethystic logarithm since the relation Exp ( Log 𝐺 ) = 𝐺 allows to solve for Log 𝐺 recursively . Hence, the insight of Lemma A.1 is that the feature of pr eserving integrality persists after localising at the augmentation ideal. A.2. The proof of Lemma A.1 . T o prove Lemma A.1 we follow Konishi [ Kon06 , Sec. 5] generalising ideas of Peng [ Pen07 ]. Let 𝐺 ∈ 1 + 𝐼 · 𝑅 ⟦ 𝑄 1 , . . . , 𝑄 ℓ ⟧ . W e write 𝐺 𝒅 for the coefﬁcients of this power series and Ω 𝒅 for the coefﬁcients of its plethystic logarithm: Log 1 +  𝒅 ≠ 0 𝑄 𝒅 𝐺 𝒅 ! B Log ( 𝐺 ) C  𝒅 ≠ 0 𝑄 𝒅 Ω 𝒅 . W e need to show that Ω 𝒅 ∈ 𝑅 ⊂ 𝑅 ⊗ Q . T o do so we ﬁrst ﬁnd a formula for these coefﬁcients by expanding the left-hand side of the above equation. Following [ Kon06 , Sec. 5.1], we write 𝐷 ( 𝒅 ) B  𝜹 ∈ Z ℓ ≥ 0 \ { 0 }   𝛿 𝑖 ≤ 𝑑 𝑖 for all 𝑖  and call an element 𝒏 ∈ Z 𝐷 ( 𝒅 ) ≥ 0 a multiplicity of 𝒅 if  𝜹 ∈ 𝐷 ( 𝒅 ) 𝑛 𝜹 𝜹 = 𝒅 . 36 Y ANNIK SCHULER W ith this notation we have 1 (25) Ω 𝒅 = −  𝑘 | gcd ( 𝒅 )  𝒏 1 𝑘 | 𝒏 |  𝑘 ′ | 𝑘 𝜇  𝑘 𝑘 ′  ( 𝑘 ′ | 𝒏 | ) ! Î 𝜹 ∈ 𝐷 ( 𝒅 ) ( 𝑘 ′ 𝑛 𝜹 ) ! ©  « ( − 1 ) | 𝒏 | Ö 𝜹 ∈ 𝐷 ( 𝒅 ) Ψ 𝑘 / 𝑘 ′  𝐺 𝑛 𝜹 𝜹  ª ® ¬ 𝑘 ′ . where 𝜇 is the Möbius function and the second sum runs over all multiplicities 𝒏 of 𝒅 / 𝑘 satisfying gcd ( 𝒏 ) = 1 . T o pr ove Lemma A.1 it ther efor e suf ﬁces to show the following. Proposition A.3. For 𝐺 ∈ 𝑅 , 𝒏 ∈ Z ℓ > 0 with gcd ( 𝒏 ) = 1 and 𝑘 > 0 we have 1 𝑘 | 𝒏 |  𝑘 ′ | 𝑘 𝜇  𝑘 𝑘 ′  ( 𝑘 ′ | 𝒏 | ) ! Î 𝑖 ( 𝑘 ′ 𝑛 𝑖 ) !  Ψ 𝑘 / 𝑘 ′ 𝐺  𝑘 ′ ∈ 𝑅 . W e requir e the following two lemmas. Lemma A.4. [ Kon06 , Lem. A.2] Let 𝒏 ∈ Z ℓ > 0 with gcd ( 𝒏 ) = 1 . (i) For 𝑘 ∈ Z > 0 we have ( 𝑘 | 𝒏 | ) ! Î 𝑖 ( 𝑘 𝑛 𝑖 ) ! ≡ 0 mod | 𝒏 | . (ii) For 𝑝 a prime, 𝑎 ∈ Z > 0 and 𝑘 ∈ Z > 0 not divisible by 𝑝 we have ( 𝑝 𝑎 𝑘 | 𝒏 | ) ! Î 𝑖 ( 𝑝 𝑎 𝑘 𝑛 𝑖 ) ! ≡ ( 𝑝 𝑎 − 1 𝑘 | 𝒏 | ) ! Î 𝑖 ( 𝑝 𝑎 − 1 𝑘 𝑛 𝑖 ) ! mod 𝑝 𝑎 | 𝒏 | . □ Lemma A.5. Let 𝑝 be a prime and 𝑎 ∈ Z > 0 be an integer . Then for all 𝐺 ∈ 𝑅 we have 𝐺 𝑝 𝑎 −  Ψ 𝑝 𝐺  𝑝 𝑎 − 1 ≡ 0 mod 𝑝 𝑎 . Proof. W e ﬁrst prove the relation assuming 𝐺 ∈ Rep ( T ) ⊂ 𝑅 . W e will prove the claim by induction on 𝑎 . For 𝑎 = 1 the claim holds by the standard argument proving additivity of the Frobenius morphism. Now suppose the claim holds for 𝑎 − 1 , that is there exists a 𝑔 ∈ 𝑅 such that 𝐺 𝑝 𝑎 − 1 −  Ψ 𝑝 𝐺  𝑝 𝑎 − 2 = 𝑝 𝑎 − 1 𝑔 . W e ﬁnd that 𝐺 𝑝 𝑎 −  Ψ 𝑝 𝐺  𝑝 𝑎 − 1 =   Ψ 𝑝 𝐺  𝑝 𝑎 − 2 + 𝑝 𝑎 − 1 𝑔  𝑝 −   Ψ 𝑝 𝐺  𝑝 𝑎 − 2  𝑝 =  𝑘 > 0  𝑝 𝑘  𝑝 ( 𝑎 − 1 ) 𝑘 𝑔 𝑘  Ψ 𝑝 𝐺  𝑝 𝑎 − 2 ( 𝑝 − 𝑘 ) ≡ 0 mod 𝑝 𝑎 1 W e remark that there is a small typo in [ Kon06 ] in the formula in Lemma 5.2 and the one for 𝐺 Γ ® 𝑑 ( 𝑞 ) stated before the lemma: The denominator 𝑘 ′ | 𝑛 | should not be present. GW INV ARIANTS AND MEMBRANE INDICES OF 5-FOLDS VIA TV 37 since 𝑝 divides  𝑝 𝑘  for 𝑘 > 0 . This means the claim is proven for 𝐺 ∈ Rep ( T ) . W e now extend to general 𝐺 ∈ 𝑅 by r eduction to the last case. For this we express 𝐺 = 𝐺 1 / 𝐺 2 as a ratio of elements 𝐺 1 ∈ Rep ( T ) and 𝐺 2 = Î 𝑗 ( 1 − 𝜒 𝑗 ) where each 𝜒 𝑗 is a monomial in Rep ( T ) . W e can write 𝐺 𝑝 𝑎 −  Ψ 𝑝 𝐺  𝑝 𝑎 − 1 = 1 𝐺 𝑝 𝑎 2 ( Ψ 𝑝 𝐺 2 ) 𝑝 𝑎 − 1 h 𝐺 𝑝 𝑎 1  ( Ψ 𝑝 𝐺 2 ) 𝑝 𝑎 − 1 − 𝐺 𝑝 𝑎 2  + 𝐺 𝑝 𝑎 2  𝐺 𝑝 𝑎 1 − ( Ψ 𝑝 𝐺 1 ) 𝑝 𝑎 − 1  i . Since 𝐺 1 , 𝐺 2 ∈ Rep ( T ) we know that the numerator on the right-hand side is divisible by 𝑝 𝑎 and so the lemma follows. □ Remark A.6. The proof of Lemma A.5 is the only place in our proof of Lemma A.1 that uses any features special to 𝑅 = Rep ( T ) loc . All other steps in our proof hold for an arbitrary lambda ring. Thus, the conclusion of Lemma A.1 actually holds for any lambda ring 𝑅 satisfying Lemma A.5 . Now we ar e equipped to prove Proposition A.3 from which Lemma A.1 follows by equation ( 25 ). Proof of Pr oposition A.3 . Our proof closely follows Konishi [ Kon06 , Sec. A.2]. First, note that for 𝑘 = 1 the claim of the proposition holds trivially . Thus, assume 𝑘 > 1 and write 𝑘 = Î 𝑠 𝑗 = 1 𝑝 𝑗 𝑎 𝑗 for its prime decomposition. It sufﬁces to show that (26)  𝑘 ′ | 𝑘 𝜇  𝑘 𝑘 ′  ( 𝑘 ′ | 𝒏 | ) ! Î 𝑖 ( 𝑘 ′ 𝑛 𝑖 ) !  Ψ 𝑘 / 𝑘 ′ 𝐺  𝑘 ′ is an element of 𝑅 which is divisible by 𝑝 𝑗 𝑎 𝑗 | 𝒏 | for all 𝑗 ∈ { 1 , . . . , 𝑠 } . W ithout loss of generality we may prove the claim for 𝑗 = 1 . W riting 𝑙 B 𝑘 / 𝑝 1 𝑎 1 observe that since for every divisor 𝑘 ′ of 𝑘 we have 𝜇  𝑘 𝑘 ′  =            𝜇  𝑙 𝑙 ′  𝑘 ′ = 𝑝 𝑎 1 1 𝑙 ′ − 𝜇  𝑙 𝑙 ′  𝑘 ′ = 𝑝 𝑎 1 − 1 1 𝑙 ′ 0 otherwise 38 Y ANNIK SCHULER expression ( 26 ) equates to  𝑙 ′ | 𝑙 𝜇  𝑙 𝑙 ′  ( 𝑝 𝑎 1 1 𝑙 ′ | 𝒏 | ) ! Î 𝑖 ( 𝑝 𝑎 1 1 𝑙 ′ 𝑛 𝑖 ) !  Ψ 𝑙 / 𝑙 ′ 𝐺  𝑝 𝑎 1 1 𝑙 ′ − ( 𝑝 𝑎 1 − 1 1 𝑙 ′ | 𝒏 | ) ! Î 𝑖 ( 𝑝 𝑎 1 − 1 1 𝑙 ′ 𝑛 𝑖 ) !  Ψ 𝑝 1 𝑙 / 𝑙 ′ 𝐺  𝑝 𝑎 1 − 1 1 𝑙 ′ ! =  𝑙 ′ | 𝑙 𝜇  𝑙 𝑙 ′  Ψ 𝑙 / 𝑙 ′ ( 𝐺 ) 𝑝 𝑎 1 1 𝑙 ′ ( 𝑝 𝑎 1 1 𝑙 ′ | 𝒏 | ) ! Î 𝑖 ( 𝑝 𝑎 1 1 𝑙 ′ 𝑛 𝑖 ) ! − ( 𝑝 𝑎 1 − 1 1 𝑙 ′ | 𝒏 | ) ! Î 𝑖 ( 𝑝 𝑎 1 − 1 1 𝑙 ′ 𝑛 𝑖 ) ! ! ( ★ ) +  𝑙 ′ | 𝑙 𝜇  𝑙 𝑙 ′  ( 𝑝 𝑎 1 − 1 1 𝑙 ′ | 𝒏 | ) ! Î 𝑖 ( 𝑝 𝑎 1 − 1 1 𝑙 ′ 𝑛 𝑖 ) ! ( ∗)  Ψ 𝑙 / 𝑙 ′ ( 𝐺 𝑙 ′ ) 𝑝 𝑎 1 1 − Ψ 𝑝 1 𝑙 / 𝑙 ′ ( 𝐺 𝑙 ′ ) 𝑝 𝑎 1 − 1 1  ( †) . 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Gromov-Witten invariants and membrane indices of fivefolds via the topological vertex

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