PPrepared for submission to JHEP
KIAS-P20045
Bootstrapping ADE M-strings
Zhihao Duan and June Nahmgoong
School of Physics, Korea Institute for Advanced Study, Seoul 02455, Korea
E-mail: [email protected] , [email protected] Abstract:
We study elliptic genera of ADE-type M-strings in 6d (2,0) SCFTs from theirmodularity and explore the relation to topological string partition functions. We find anovel kinematical constraint that elliptic genera should follow, which determines ellipticgenera at low base degrees and helps us to conjecture a vanishing bound for the refinedGopakumar-Vafa invariants of related geometries. Using this, we can bootstrap the ellipticgenera to arbitrary base degree, including D/E-type theories for which explicit formulasare only partially known. We utilize our results to obtain the 6d Cardy formulas and thesuperconformal indices for (2,0) theories. a r X i v : . [ h e p - t h ] O c t ontents (2 , SCFTs and geometries 43 Elliptic genera of ADE M-strings 10 R × T W -algebra 23 Recently we have witnessed a revival of interest in six-dimensional quantum field theories.In particular, 6d N = (2 , superconformal field theories (SCFTs) play an important rolein our understanding of lower dimensional dualities [1–4]. Although still lacking Lagrangiandescriptions, they are well-known to be labeled by simply-laced, i.e., ADE Lie algebras, andallow for self-dual string solitons.Historically, 6d (2,0) SCFTs were first constructed from type IIB superstring com-pactifications [5, 6]. In this paper, we consider a more unifying approach, which involvesF-theory [7–9]. Roughly speaking, F-theory is a non-perturbative completion of type IIBstring theory, by allowing the non-constant axion-dilaton field over the moduli. Its VEVcan be dictated by the complex structure of an elliptic curve, which hints at the class ofCalabi-Yau (CY) manifolds having elliptic fibrations. F-theory compactification on ellip-tically fibered CY threefolds can engineer six-dimensional supersymmetric gauge theory.In [10, 11], it is conjectured that all possible six-dimensional N = (1 , SCFTs can beobtained from this approach. As particular examples, (2 , SCFTs correspond to CYs withtrivial elliptic fibrations. – 1 –n the other hand, CY threefolds are central objects in another area known as topo-logical string theory. Given a CY threefold X , we first consider a sigma model of a two-dimensional worldsheet with fixed metric mapping to X . Its action needs to be topologicallytwisted in order to be topological. Then we couple the system to two-dimensional gravity,i.e., integrate over all possible metrics and topologies and obtain what is known as thetopological string.Furthermore, people discovered that for local CY threefolds, M-theory compactificationengineers 5d gauge theory with N = 1 supersymmetry, as an uplift of the relation betweentype IIA superstring and 4d physics [12]. After a sequence of seminal works [13–16], [17, 18]proposed a refinement of topological string theory, which interprets gauge theory results interms of geometric invariants known as refined Gopakumar-Vafa (GV) invariants. For el-liptically fibered CY threefolds that engineer 5d theories having UV completions to 6d, therelevant gauge-theoretic quantity is precisely the elliptic genus of self-dual strings. There-fore, understanding it becomes an important task for both communities.Currently, a plethora of ways to compute the elliptic genera for various 6d SCFTs hasalready been introduced. Based on their starting point, we roughly divide them into twoclasses and make a very partial list. From the point of view of higher dimensional gaugetheories, one can utilize the brane system if it exists. For example, concerning the E-stringsor M-strings, where one considers M2-branes in a chain of M5-branes with or without M9-brane, the domain wall operator method [19–21] can be used to compute their elliptic genera.Also, if one works out the 2d quiver gauge theories living on the self-dual strings, one canapply localization technique through the Jeffrey-Kirwan residue prescription [22, 23]. In[19, 24–30], this was successfully carried out to tackle E-strings, M-strings, some minimalSCFTs [25] and many more. From the point of view of topological strings, traditionalmethods such as the topological vertex or mirror symmetry are not that straightforward toapply. The non-toric topological vertex [24, 31–33] is developed to compute elliptic generaof E-strings, M-strings, and certain limit of elliptic genera for several other (1,0) minimalSCFTs. B-model technique is explored in [34, 35], but restricted to the first few genusexpansions. Recently, [36–39] found a powerful way to compute the elliptic genera using 6dblow-up equations, which in particular covers all the theories in the non-higgsable clusters[40].Last but not least, since elliptic genus is known to transform as a Jacobi modular form,one could fully explore its modular property. In other words, we write down a modularansatz with a finite number of undetermined coefficients, discussed in detail in section 3,then impose sufficient constraints either from gauge theory or topological string theory tofix the unknowns. This approach is often dubbed modular bootstrap [41–46], and wassuccessfully applied to compute the elliptic genera of E-strings, M-strings, various minimialSCFTs, conformal matters, etc., at low base degrees, sometimes in the unflavored limit.However, even though much progress has been made in understanding the N = (1,0)– 2 –inimal SCFTs, higher rank (2,0) SCFTs have not yet been fully explored in the currentliterature. The A-type theory has a clear brane picture as a stack of M5-branes [19].However, elliptic genera of strings in D/E-type theories were partially studied in the masslesslimit only [27, 47]. In this paper, along the line of the modular bootstrap approach, weinitiate a uniform investigation of the elliptic genera of all ADE-type (2,0) SCFTs withall chemical potentials fully refined. Our approach captures the elliptic genera of massiveM-strings for D- and E-type, which, as far as we know, have not been written down before. We will discuss two different ways to bootstrap the elliptic genera. The first one isbased on the fact that after a circle compactification, 6d (2,0) g -type SCFT becomes 5dmaximally supersymmetric Yang-Mills (SYM) theory with gauge group G . Then, the 6dpartition function has an expansion in terms of instanton numbers. Although the instantonpartition function is unknown if G is exceptional, the perturbative part is well-known. Also,from the definition of the elliptic genus, there is a kinematical constraint that we called "flipsymmetry" in subsection 3.1. Combining the two, we are able to fix many elliptic generaat low base degrees.The second approach is based on the property of refined GV invariants. They dependon the curve classes α ∈ H ( X, Z ) and two spins related to the representation of 5d littlegroup SO (4) , so we label them by N αj − j + . As will be explained in subsection 3.2, thereexists a vanishing bound for those invariants. Namely, for a fixed α , N αj − j + vanishes wheneither j − or j + is sufficiently large. Nevertheless, it is, in general, not easy to derive theprecise bound from the geometry. In subsection 3.2, based on the data obtained from theelliptic genera of low base degrees, we experimentally conjecture a vanishing bound andpropose that it should be able to determine the elliptic genera of arbitrary base degrees inprinciple.This article is organized as follows. In section 2, we give a brief review of ADE-type(2,0) SCFTs, elliptic genera, topological strings and their close relationship. In section 3,we discuss two ways to compute the elliptic genera. More specifically, in subsection 3.1,we first introduce the modular ansatz then show how to combine the 5d perturbative partwith flip symmetry to bootstrap the elliptic genera. In subsection 3.2, we first argue theexistence of vanishing condition and conjecture an empirical bound for GV invariants ofrelated geometries. We also show how to make use of this constraint to bootstrap theelliptic genera recursively. In section 4, we give two applications of our results: the 6dCardy formulas for all (2,0) SCFTs and the (2,0) superconformal index in the W -algebralimit. In section 5, we finish the paper with concluding remarks. We notice that some elliptic genera of D4-type theories were computed in [42, 44]. – 3 – (2 , SCFTs and geometries
In this section, we briefly introduce 6d ADE (2,0) SCFTs and the role of the self-dual stringsfrom the point of view of F-theory. We will further discuss the supersymmetric partitionfunction of the 6d SCFTs on R × T in the Ω -background and the elliptic genus of theM-strings. Finally, we explain the connection of the whole story to topological strings.To start with, we discuss the geometric construction of six-dimensional SCFTs. Thisis achieved through compactifying F-theory on an elliptically fibered CY threefold. Inother words, we assume the CY threefold X enjoys a torus fibration E over a non-compactcomplex surface B , T = E (cid:47) (cid:47) X π (cid:15) (cid:15) B How does geometry see whether a given field theory can reach a CFT fixed point or not?Remember that although lacking Lagrangian descriptions, six-dimensional quantum fieldtheories have stringy solitons. This can be naturally explained in the geometric setup sincea D3-brane wrapped on a two-cycle in B becomes a non-critical BPS string in 6d, whosetension is proportional to the volume of the two-cycle. Such a string is called the ‘self-dualstring’ because it inherits the self-duality condition from the D3-brane in ten dimensions. Ifwe assume that all the compact divisors inside B are contractible, then the self-dual stringbecomes tensionless by shrinking down all such curves, which signals a superconformal fieldtheory.For cases of our interest, i.e., N = (2 , ADE-type SCFTs, the base B in the singularor conformal limit is C / Γ , with Γ the discrete subgroups of SU (2) . The famous Mckaycorrespondence states that there is a one-to-one map between those subgroups and simplylaced, i.e., ADE-type Lie algebras. This can be understood as follows. In the orbifold C / Γ , the origin is singular, which can be resolved by a series of blow-ups. This yieldsexceptional curves intersecting each other according to the negative of the Cartan matricesof ADE Lie algebras. Also, notice that the resolved spaces, also known as ALE spaces,are themselves CY manifolds, so E is trivially fibered over them. However, X must bedeformed to incorporate the M-string mass defined below, making E no longer simply atrivial fibration. This means that the elliptic fiber also carries non-trivial information,which is an important reason why we choose a more general approach. We present theDynkin diagrams of ADE Lie algebras in Figure 1, where the numbers inside the node fixour convention on the ordering.Next, bearing the above geometric picture in mind, let us put our 6d (2,0) SCFTs on R × T and define the BPS index. We will declare one circle of T as a temporal circle,and turn on the Ω -background on R so that the self-dual string is wrapped on T andlocalized at the origin of R . Then, one can define the supersymmetric partition function– 4 – M -1 1 A EB D B DC C xA r : D r : r r E : E : E : Figure 1 : ADE Dynkin diagrams with numbers specifying the order.on R × T as follows, Z R × T ( τ, (cid:15) , , m, v ) = Tr (cid:104) ( − F q P e − n · v e − (cid:15) J − (cid:15) J e − ( (cid:15) + + m ) R − ( (cid:15) + − m ) R (cid:105) . (2.1)Here, J , are the two angular momenta on R which rotate two orthogonal planes in R . R , are the two Cartan charges of SO (5) R-symmetry of (2,0) supersymmetry. Weintroduce the corresponding chemical potentials as (cid:15) , for J , and (cid:15) + ± m for R , , with m a mass deformation henceforth referred as the M-string mass and (cid:15) ± ≡ (cid:15) ± (cid:15) . Also, P is a KK momentum along the spatial circle in T with the conjugate fugacity q ≡ e πiτ .Here, τ = i r r where r is the radius of the temporal circle, and r is the radius of thespatial circle in T . Lastly, n = ( n , n , ..., n r ) denotes the number of the self-dual stringswrapping on each two-cycle in B . For this reason, we also call n as a base degree. Theconjugate chemical potential is given by v = ( v , v , ..., v r ) which are proportional to thevolumes of the two-cycles.Let us take a closer look at the index. Suppose we consider a 6d (2,0) g -type SCFT,then the following expansion for the R × T index (2.1) exists, Z R × T = Z · Z T = Z · (1 + (cid:88) n (cid:54) =0 ∈ Z r ≥ Z n e − n · v ) , (2.2)which is valid when the string tension v is large. The prefactor Z comes from the KK towerof BPS particles decoupled from the self-dual strings. Its form will be given in section 3. Onthe other hand, the expansion coefficient Z n denotes the contribution from the n number– 5 –f self-dual strings on T . Therefore, it is given by the elliptic genus of the 2d (4,4) SCFTon the n self-dual strings [19].The elliptic genus is well known to be a weight zero Jacobi modular form (definitionscan be found in appendix A). Thus what remains to determine is its index. It turns out thatthe index is tightly related to the anomaly polynomial of BPS strings, which we explainbelow.To start with, the 6d N = (2,0) theory has various ’t Hooft anomalies, summarized interms of the eight-form anomaly polynomial. Then based on the anomaly inflow mecha-nism, [29, 48] successfully computed the general four-form anomaly polynomial on the 2dworldsheet of BPS strings. In our case, it gets simplified, I = r g (cid:88) a,b =1 Ω a,b n a n b c ( L ) − c ( R )) + r g (cid:88) a =1 n a (cid:18) − Tr F SU (2) m + c ( I ) (cid:19) , (2.3)with Ω the Cartan matrix of the simply-laced Lie algebras g with rank r g . c ( L ) and c ( R ) the second Chern classes of two SU (2) bundles that split the normal bundle ofstrings. Moreover, we have split the SO (5) R-symmetry into SU (2) m × SU (2) R in (2.1),and F SU (2) m is the background field strength associated to the SU (2) m while c ( I ) is thesecond Chern class of the SU (2) I bundle inherited from the SU (2) R . I is responsible for a non-trivial index. Indeed, if we perform an equivariant integrationin the Ω -background [49] and get (we rescale the parameters to match our convention), i n = I (cid:16) c ( L ) → − ( (cid:15) − πi ) , c ( R ) → − ( (cid:15) + πi ) , Tr F SU (2) m → − ( m πi ) , c ( I ) → − ( (cid:15) + πi ) (cid:17) = − π − (cid:88) a,b Ω a,b n a n b (cid:15) − − (cid:15) ) + (cid:88) a n a ( m − (cid:15) ) , (2.4)then the elliptic genus has the following modular anomaly under an S-transformation [41], Z n ( − τ | (cid:15) , τ , mτ ) = exp (cid:104) πi i n τ (cid:105) · Z n ( τ | (cid:15) , , m ) . (2.5)Comparing (2.5) with the definition of Jacobi forms ((A.10) in the appendix A), we learnthat i n is precisely its index.Let us make a few comments about the existing literature. The 6d (2,0) A-type theoriesare given by the worldvolume theory on r g + 1 M5-branes, and M2-branes ending on M5-branes become the self-dual strings in 6d. With this brane construction, one can engineerthe 2d worldvolume theory on the n self-dual strings, which is given by the A-type quivergauge theory with gauge group U ( n ) × U ( n ) × ... × U ( n r g ) [19]. The elliptic genus ofan arbitrary base degree can be computed from the localization method. However, for theD/E-type SCFTs, we do not have closed-form expressions. In [27, 47], the strings in D/E-– 6 –ype theories were studied with the mass parameter m turned off. However, as far as weknow, the fully refined elliptic genera have been remained to be unknown.In the final part of this section, we talk about the connection to topological stringtheory. In order to do that, we first spell out some necessary ingredients involved.Given a CY threefold X , if we denote the (complexified) Kähler parameter of two cyclesin α ∈ H ( X, Z ) as t α with its exponential Q α = exp( it α ) , the genus g free energy F g hasthe following expansion, F g ( t ) = (cid:88) α ∈ H ( X, Z ) r αg Q α . (2.6) r αg are genus g Gromov-Witten (GW) invariants, which are in general rational numbers.Furthermore, through lifting type IIA string theory to M-theory [50–52], we can do a partialresummation over g s to obtain another type of expansion, F top ( t ) = (cid:88) g ≥ F g g g − s = (cid:88) g ≥ (cid:88) k ≥ (cid:88) α ∈ H ( X, Z ) ( − g − I αg (cid:16) k g s (cid:17) g − Q kα k . (2.7)Now I αg are always integers. They are related to M2-branes in the M-theory. From the 5dperspective, BPS M2-branes wrapped on two cycles α in X give rise to BPS states in thespacetime. They are naturally labeled by two spins j − and j + indicating their representationunder the little group SO (4) , and I αg is related to a Witten-like index over their Hilbertspace H α BPS , Tr H α BPS ( − j + µ j − = (cid:88) j ± ∈ N / ( − j + (2 j + + 1) χ j − ( µ ) N αj − j + = ∞ (cid:88) g =0 I αg ( µ + µ − ) g . (2.8)We introduce the symbol χ j ( x ) to denote the following Laurent polynomial of an irreducible SU (2) highest-weight representation with spin j ∈ N / , χ j ( x ) = x − j + x − j +2 + · · · + x j , (2.9)and N αj − j + is the number of multiplets for BPS states in H α BPS with spin j − and j + . Ingeneral, as we move inside the moduli space of X , N αj − j + may vary, but I αg remains un-changed due to the property of the index. I αg are dubbed (unrefined) Gopakumar-Vafa(GV) invariants.For local, i.e., non-compact CY threefolds, M-theory compactification engineers fivedimensional gauge theory with N = 1 supersymmetry. Putting the five dimensionalspacetime under the Ω -background, we have the Nekrasov partition function Z ( (cid:15) , (cid:15) , t ) [13, 14], where (cid:15) and (cid:15) are two formal parameters associated to the Cartan subalgebra of We rescale the conventional definition of g s by √− to match the convention from gauge theory. Sameas for (cid:15) ± in (2.10). – 7 – o (4) = su (2) × su (2) . Z ( (cid:15) , (cid:15) , t ) can be used to refine the topological string partition functions [16–18]. Itis worth emphasizing that this also holds for elliptically fibered CYs, which engineer fivedimensional theories having six dimensional UV completions. The upshot is that the freeenergy enjoys a refined GV expansion, F ref = log Z ref = log Z ( (cid:15) , (cid:15) , t )= (cid:88) j ± ∈ N (cid:88) k ≥ (cid:88) α ∈ H ( X, Z ) N αj − j + ( − j − + j + ) χ j − ( u k ) χ j + ( v k ) v k + v − k − u k − u − k Q kα k , (2.10)with u = exp( − (cid:15) − ) , v = exp( − (cid:15) + ) and (cid:15) ± = (cid:15) ± (cid:15) . Due to non-compactness, the n αj − j + defined earlier no longer depend on the moduli so are themselves well-defined quantities,known as refined GV invariants [15, 16, 53]. Notice that they are always non-negative sincethey are counting numbers of BPS states.The connection to topological string theory lies in the duality between F-theory andM-theory [54]. It is known that F-theory compactification on X × S is dual to M-theorycompactification on X . As discussed before, in the F-theory picture, BPS strings arise fromD3-branes wrapping on curves Σ i inside B and can further wrap on T in the spacetime.While in the M-theory picture, they are dual to M2-branes wrapping on the same curves in B and the elliptic fiber T in X . The Kähler parameters of the rational curves are identified,while the radius of the extra S in F-theory is inversely proportional to the volume of ellipticfiber in M-theory. Therefore, the fiber of X not only has a complex structure τ but alsoacquires a (complexified) Kähler parameter t , which justifies the wrapping of M2-branes.After turning on the Ω -background, the partition functions of self-dual BPS stringsin F-theory get identified with the refined topological string partition function Z ref of X in M-theory. To make the precise identification, in Z R × T , we are only interested in theelliptic genera of BPS strings Z T (2.2). While corresponding in Z ref we are only summingover two-cycles that contains a non-trivial class in H ( B, Z ) . We abuse our notation andstill call it Z ref , hoping that no confusion will occur, Z ref = 1 + (cid:88) n (cid:54) =0 ∈ Z r ≥ Z n e − n · v . (2.11)We summarize the duality in table 1.Last but not least, the modular anomaly can also be motivated from topological stringtheory. This is not essential to the main result of this paper, and we include it here just forcompleteness. This part can be safely skipped for an uninterested reader.Above, what we already talked about is actually only the type-A topological string, In fact, the two Z n still differ by some overall factor in general [42, 43], which happens to be one for(2,0) theories. – 8 – [ X × S ] M [ X ]1 /R S Vol ( T ) q = exp(2 πiτ ) q = exp(2 πit ) D3-branes on Σ i and T M2-branes on Σ i and T elliptic genus : Z T partition function : Z ref Table 1 : Duality between F-theory and M-theory.and in fact, there exists a type-B topological string theory, based on coupling gravity to B-model on the worldsheet. In this setup, we also have a genus expansion of partition function,which now depends on the holomorphic structure rather than the Kähler structure of theCY manifold. At genus zero, the partition function is holomorphic. However, for the genuslarger than zero, the anti-holomorphic part no longer decouples and gives rise to the so-called holomorphic anomaly [55, 56]. After a careful analysis of anti-holomorphic dependence, theauthors in [55, 56] found a set of recursive equations satisfied by the partition function,which are also known as the BCOV holomorphic anomaly equations.In [57], it is pointed out that the holomorphic anomaly is tightly related to the modularanomaly. For cases of complex structure moduli space being one adimensional, this boilsdown to simply the choice of the quasimodular form E versus almost holomorphic modularform ˆ E ( τ ) (see appendix A) in a suitable parameterization. If we choose the quasimodularform E , the partition function will be holomorphic but no longer modular.For some classes of geometries that exhibit an elliptic fibration, [58] proposed thefollowing form of the anomaly equations, (cid:18) ∂ E + 112 n · ( n + K B )2 (cid:19) Z n = 0 , (2.12)where the modular parameter of E is identified with the complex parameter of the ellipticfiber, n is now regarded as an element in H ( B, Z ) , K B means the canonical divisor of B and the dot denotes intersection inside the base surface. In [42], this was generalized to therefined case with M-string mass turned on. In particular, for geometries related to ADEM-strings, K B is trivial, and they propose (we change it slightly to match our convention), (cid:32) ∂ E + π (cid:34) − π (cid:32) − n (cid:15) − (cid:15) − ) − ( (cid:88) i n i ) (cid:15) + ( (cid:88) i n i ) m (cid:33)(cid:35)(cid:33) Z n = 0 . (2.13)By comparing (2.13) and (A.19), we learn that (2.13) motivates Z n to be a Jacobi form andthe term inside the bracket is precisely its index. Since the curves inside the resolved ALEspaces intersect according to the negative of the corresponding Cartan matrix, we find a– 9 –erfect agreement between topological string theory and gauge theory. In this section, we obtain the elliptic genera of ADE M-strings using the modular boot-strap. In subsection 3.1, we will show that a few consistency conditions can determine theelliptic genera at low base degrees. In subsection 3.2, we study the GV invariants and theirvanishing bounds, which allow us to compute the elliptic genera at an arbitrary base degree.
In this subsection, we discuss some consistency conditions that the elliptic genus shouldfollow, including the ‘flip symmetry,’ which is a purely kinematical constraint for the self-dual strings in (2,0) theories. We will show that the elliptic genera of ADE-type M-stringscan be bootstrapped with those consistency conditions up to some low base degrees. Theextension to an arbitrary base degree will be discussed in subsection 3.2.As we have discussed in section 2, the elliptic genus is a modular form with the modularparameter τ , and the elliptic parameters (cid:15) , and m . Therefore, the elliptic genus of self-dualstrings can be written as follows [41], Z n = η ( τ ) n · N n ( τ, m, (cid:15) , ) D n ( τ, (cid:15) , ) . (3.1)Here η ( τ ) is the Dedekind eta function, and n is the weight of the elliptic genus which is0 in our case. Also, N is a numerator, and D is a denominator of the elliptic genus whichwill be explained in the following paragraphs.First, let us consider the denominator structure of the elliptic genus. In the thermo-dynamic limit (cid:15) , → , the free energy of the 6d R × T index should have the volumedivergence as log Z R × T ∝ (cid:15) (cid:15) . Therefore, the elliptic genus also has a pole structure at (cid:15) , = 0 . We take our denominator to capture all those volume divergences of R of theelliptic genus, and it can be written in the following form [41, 42, 59], D n = r g (cid:89) a =1 n a (cid:89) k =1 θ ( k(cid:15) πi ) η ( τ ) θ ( k(cid:15) πi ) η ( τ ) (3.2)where r g is the rank of the Lie algebra g which is the type of the (2,0) SCFT. Note thatthe denominator itself is a modular form, and it transforms as follows under S-duality, D n ( − τ , (cid:15) , τ ) = τ − (cid:80) a n a exp (cid:104) πiτ r g (cid:88) a =1 n a (cid:88) k =1 k ( (cid:15) + (cid:15) ) (cid:105) · D n ( τ, (cid:15) , ) (3.3)which means that the denominator has the weight w D = − (cid:80) a n a and the index i D = πi ) (cid:80) r g a =1 (cid:80) n a k =1 k ( (cid:15) + (cid:15) ) . – 10 –ow, let us move on to the structure of the numerator N n . Note that it depends on theelliptic parameters m and (cid:15) , , which are also the chemical potentials of the SU (2) globalsymmetries. Therefore, one can capture the dependence of m and (cid:15) ± in the elliptic genusin terms of the SU (2) Weyl-invariant Jacobi forms: φ , ( τ, z ) and φ − , ( τ, z ) . They are themodular forms which are invariant under the SU (2) Weyl transformation of z , and φ k,l ( τ, z ) has an index lz and weight k . See the appendix A for detailed information. Then, one canwrite down the numerator in the following form, N n = (cid:88) p C p E ( τ ) p E ( τ ) p (cid:89) i =1 [ ϕ − , ( τ, z i πi )] p − ,i [ ϕ , ( τ, z i πi )] p ,i . (3.4)Here, E , ( τ ) are the Eisenstein series, and chemical potentials are written as ( z , z , z ) =( (cid:15) + , (cid:15) − , m ) . Note that we express the numerator in terms of the Eisenstein series andthe SU (2) -invariant Jacobi forms. The coefficient is denoted as C p which depends on thevector p = ( p , p , p − , , p , , p − , , p , , p − , , p , ) ∈ ( Z ≥ ) . (3.2) and (3.4) is precisely ourmodular ansatz for Z n .The elliptic genus (3.1) is determined once we can fully fix the coefficient C p of thenumerator (3.4). Fortunately, even without knowing the 2d CFT on the self-dual string,there are some consistency conditions for the elliptic genus that we can use to fix thenumerator. First, one can use the modular property (2.4) of the elliptic genus. Then, theindex and the weight of the numerator is given as follows, w N = − r g (cid:88) a =1 n a i N = 12(2 πi ) (cid:16) (cid:15) (cid:15) r G (cid:88) a,b Ω a,b n a n b + 2( m − (cid:15) ) r g (cid:88) a =1 n a + r g (cid:88) a =1 n a (cid:88) k =1 k ( (cid:15) + (cid:15) ) (cid:17) . (3.5)From (3.5), it is straightforwards to see that the power vector p should satisfy the followingtwo relations, w N = − (cid:88) i =1 p − ,i + 4 p + 6 p , i N = (cid:88) i =1 ( p − ,i + p ,i )( z i πi ) . (3.6)Since all p ’s are non-negative integers, there are only a finite number of solutions for (3.6).All the coefficient C p ’s are zero if p does not satisfy (3.6). The remaining problem is todetermine the finite number of the allowed coefficients.One way to determine those non-zero coefficients is to use the instanton expansion ofthe 5d maximally supersymmetric Yang-Mills theory (MSYM) with gauge group G , whichis the circle compactified 6d (2,0) g -type theory. Until now, we considered the elliptic genus– 11 –xpansion (2.2) of the 6d R × T index, Z R × T ( τ, m, (cid:15) ± , v ) = Z (cid:16) (cid:88) n Z n ( τ, m, (cid:15) ± ) e − n · v (cid:17) (3.7)whose expansion parameter is the string fugacity e − v and the momentum mode Z is givenas follows, Z = PE (cid:104) r g sinh m ± (cid:15) − sinh (cid:15) , q − q (cid:105) . (3.8)Here, PE stands for the plethystic exponential, defined asPE [ f ( x , x , · · · , x n )] := exp (cid:32) ∞ (cid:88) k =1 k f ( x k , x k , · · · , x kn ) (cid:33) (3.9)with x ’s are fugacity-like variables. Instead, one can consider a different form of the expan-sion as follows, Z R × T ( τ, m, (cid:15) ± , v ) = Z pert (cid:16) ∞ (cid:88) k =1 Z k ( m, (cid:15) ± , v ) q k (cid:17) (3.10)where the expansion parameter is the instanton fugacity q = e πiτ . In the above formulation,one views the 6d g -type (2,0) SCFTs on R × T as the 5d MSYM on R × S with gaugegroup G . Here, the KK spectrum of the compactified circle becomes the instantons in 5d[60]. Z pert is the perturbative partition function of the 5d MSYM given as follows [61], Z pert = PE (cid:104) sinh m ± (cid:15) + sinh (cid:15) , (cid:88) α ∈ Ψ + g e − α ( v ) (cid:105) (3.11)where Ψ + g is a set of positive roots of the Lie algebra g . Also, Z k is the k -instanton partitionfunction. It can be computed from ADHM quantum mechanics when G is A- or D-type[62, 63]. However, when G is E-type, there are no known results for the 5d MSYM instantonpartition function. Since we are interested in the elliptic genus of the self-dual strings ingeneric 6d (2,0) SCFTs including the exceptional Lie algebras, we shall not use the instantonpartition function to determine the elliptic genus. Instead, we shall use the perturbativedata only, by imposing the following condition, [ Z Z n ( τ, m, (cid:15) ± )] q = [ Z pert ( m, (cid:15) ± , v )] e − n · v . (3.12)Here [ f ] q denotes the coefficient of q of f , and [ f ] e − n · v denotes the coefficient of e − n · v of f .Lastly, there is another condition, which we will call a ‘flip symmetry,’ that can be usedto fix the elliptic genus. The flip symmetry is defined as the invariance of the elliptic genus– 12 –ase degree Allowed terms Perturbative data Flip symmetry (cid:0) (cid:1)
44 38 6 (cid:0) (cid:1)
103 89 14 (cid:0) (cid:1)
165 140 25 (cid:18)
11 2 1 (cid:19)
221 181 40 (cid:18)
11 1 2 1 (cid:19)
292 231 61 (cid:18)
11 1 2 1 1 (cid:19)
374 286 88 (cid:18)
11 1 2 1 1 1 (cid:19)
468 346 122 (cid:18)
11 1 2 1 1 1 1 (cid:19)
575 411 164
Table 2 : The ‘allowed terms’ column denotes the number of solutions of the modularconstraints (3.5). The ‘perturbative data’ column denotes the number of the terms whosecoefficients can be fixed by the perturbative data using (3.12). The ‘flip symmetry’ columndenotes the number of the terms whose coefficients can be further fixed by the flip symmetryusing (3.13).under the following flip transformation, e − (cid:15) + → − e − (cid:15) + , e − (cid:15) − → − e − (cid:15) − , e − m → − e − m . (3.13)Under the flip transformation, the elliptic genus changes as follows, Z n = Tr [( − F q P e − (cid:15) J − (cid:15) J e − ( (cid:15) + + m ) Q − ( (cid:15) + − m ) Q ] → Tr [ e − πi ( J + Q ) · ( − F q P e − (cid:15) J − (cid:15) J e − ( (cid:15) + + m ) Q − ( (cid:15) + − m ) Q ] . (3.14)Now, recall that J , are the spins of the SO (4) Lorentz symmetry of the tangent R , and Q , are the spins of the unbroken SO (4) R-symmetry of the normal R . Therefore, J and Q are integers for bosons and half-integers for fermions, i.e, e − πiJ = e − πiQ = ( − F . Then, it is trivial to check that e − πi ( J + Q ) = 1 and the elliptic genus is invariant underthe transformation (3.13)The flip symmetry is a purely kinematic constraint in the sense that it does not dependon the microscopic details of the 2d CFT on the self-dual strings. However, it providesstrong constraints when we fix the elliptic genus with the modular bootstrap. Surprisingly,the elliptic genera up to some low base degree can be completely determined from theconditions that we have mentioned so far. More precisely, the elliptic genus of the self-dual strings in 6d (2,0) SCFT can be fully determined up to some base degrees from the This relation also plays an important role in the study of the modified indices [64–66]. – 13 –hree conditions: the modular property (3.6), the 5d perturbative data (3.12), and the flipsymmetry (3.13). In table 2, we tabulated the number of the coefficients of some ellipticgenera whose coefficients can be totally fixed by the aforementioned three conditions. Thefull expression of the elliptic genera can be found in appendix B.Although the flip symmetry can give non-trivial results for the elliptic genera, it doesnot give us the answer up to an arbitrarily high base degree. In the next subsection, basedon the data obtained from the flip symmetry, we will establish a more powerful methodthat enables us to compute the elliptic genus at an arbitrary base degree.
In this subsection, we first argue the existence of vanishing bound for the GV invariants.Then we present our conjecture on such a bound based on the data at low base degrees.Together with the modular ansatz introduced in the previous subsection, it can recursivelybootstrap all the elliptic genera of ADE-type (2,0) theories in principle. Finally, we addressthe issue of uniqueness for our solutions from modular bootstrap and make a comparisonwith the approach in subsection 3.1.As mentioned in the introduction, a very interesting feature for the refined GV invari-ants N αj − j + is its vanishing property. But to understand that, let us first start from unrefinedGV invariants I αg . We would like to first show that for a fixed curve class α ∈ H ( X, Z ) , I αg all vanish when g is sufficiently large. If the CY manifold is a total space of canonical linebundle over a complex surface, this can be argued for using the adjunction formula [42].However, the geometries we consider in this paper do not belong to that class, so we needa more general argument. Below, we sketch how the vanishing condition can be shown. In the mathematical literature, there is a class of closely related enumerative invariantscalled stable pair or Pandharipande-Thomas (PT) invariants P n,α [68]. The starting pointis to consider a torsion sheaf F having dimension one support inside X , together with anon-trivial holomorphic section s over it. We impose certain stability conditions, such thatthe zeros of sections are only a bunch of points. Physically speaking, the one-dimensionalsupport corresponds to a D2-brane, and the points correspond to D0-branes. Therefore, astable pair ( F, s ) can be roughly thought of as a D0-D2 brane system.The set of all possible stable pairs with a fixed holomorphic Euler characteristic n anda fixed class of support α naturally forms a moduli space P n ( X, α ) . The existence of avirtual fundamental cycle was shown [68], which happens to be zero-dimensional for anyCY threefold. Then the stable pair or PT invariants P n,α just count the number of certainpoints inside P n ( X, α ) . The vanishing of P n,α for a fixed α and small enough n can beeasily explained because a curve with a given homology class can not have arbitrarily smallholomorphic Euler characteristic, rendering the moduli space empty. Another possible way of reasoning is to use the geometric model developed in [67]. – 14 –n the other hand, P n,α can be expressed as combinations of I αg [68]. For instance, foran irreducible class β ∈ H ( X, Z ) , they are related in terms of formal power series in q , (cid:88) n P n,β q n = (cid:88) g ≥ I βg q − g ( q + 1) g − . (3.15)It turns out that to satisfy the vanishing property of P n,β , I βg must also vanish when g issufficiently large. Then we argue inductively and extend the vanishing result for I βg to allcurve classes in H ( X, Z ) . In contrast, the GW invariants do not have such a nice propertydue to the presence of "bubbling phenomenon" [69].As for the refined GV invariants, the geometric model is more complicated so it is harderto argue from the geometry. Instead, we adopt a different strategy, through exploring therelation between the refined GV invariants and the unrefined GV invariants.First, we need to make an important assumption: The refined GV invariants satisfy aso-called checkerboard pattern: fix a given curve class α and tabulate all non-zero invariantswith two axes j + and j − . Then any two occupied blocks are either disconnected orconnected through a diagonal, which makes the table resemble a checkerboard. This patternwas first noticed in [70] and holds true for all non-compact CY threefolds that the authorsknow of. This also applies to the geometries considered in this paper, and we refer readersto appendix C for examples.Assuming the above, we next compare (2.7) and (2.10). For the reader’s convenience,we repeat them below, F top ( t ) = (cid:88) g ≥ (cid:88) k ≥ (cid:88) α ∈ H ( X, Z ) ( − g − I αg (cid:16) k g s (cid:17) g − Q kα k , F ref = (cid:88) j ± ∈ N (cid:88) k ≥ (cid:88) α ∈ H ( X, Z ) N αj − j + ( − j − + j + ) χ j − ( u k ) χ j + ( v k ) v k + v − k − u k − u − k Q kα k . (3.16)The unrefined limit is (cid:15) = − (cid:15) = g s , from which we can deduce the following, (cid:88) j ± ∈ N N αj − j + ( − j − + j + ) (2 j + + 1) χ j − ( e − kg s ) = (cid:88) g I αg ( − g (cid:16) k g s (cid:17) g , (3.17)for given α ∈ H ( X, Z ) and k ∈ Z + .Thanks to the checkerboard pattern, (2 j − + 2 j + ) is always even or odd for a fixedcurve class α . Moreover, χ j − ( e − kg s ) can be decomposed into a polynomial with variable t = − (cid:0) k g s ) (cid:1) , χ j − ( e − kg s ) = a j − t j − + a j − − t j − − + · · · + a . (3.18)It is easy to show that a k > for k even and a k < for k odd always, regardless of j − .– 15 –ecall that N αj − j + is always a non-negative integer, this means for a given genus g , I αg canbe written as a sum of non-zero N αj − j + with j − ≥ g , and each term has the same sign.As a consequence, there must exist vanishing bound for j + since I αg is a finite number,and also for j − , since there is a vanishing bound for g in the unrefined case. As a by-product, this explains the pattern of alternating signs of unrefined GV invariants when weincrease the genus. However, the bound for j ± beyond which refined GV invariants all vanish depends onthe details of geometry. Geometries involved here are elliptic fibrations over ALE spaceswith mass deformations thereof. In practice, there are two issues. The first one is that it isnot clear to us how to construct a compact embedding of X into an ambient toric variety,perhaps except for the A r case. This prevents us from using many powerful techniques suchas mirror symmetry. The other is that how to mass deform the geometry away from thetrivial fibration is not known, except for the A case [38]. For some relevant discussions, see[38, 39]. Last but not least, even if one finds ways to overcome the two issues, it is perhapsequally hard to show a useful vanishing condition based on the geometry.In short, we are far away from being able to derive an optimal vanishing bound fromfirst principles. So instead, we guess an empirical bound for the GV invariants based onthe data at low base degrees obtained from the "flip symmetry" method in subsection 3.1.The bound is not complicated, so it doesn’t saturate for many cases, but we believe that itis efficient enough for practical use.To state our conjecture, we first fix the notation. Specialize the general form (2.10) or(3.16) to our case, F ref = (cid:88) j ± ∈ N (cid:88) k ≥ (cid:88) n ,e,µ N n ,e,µj − j + k ( − j − + j + ) χ j − ( u k ) χ j + ( v k ) v k + v − k − u k − u − k Q k n q ke Q kµm , (3.19)where Q = e − v , q = e πiτ and Q m = e − m are formal exponential of Kähler parametersassociated to exceptional curves in B , elliptic fiber and two-cycle of M-string mass respec-tively. In this representation n and e are non-negative, but µ also takes negative values.It is possible to redefine q to get rid of negative µ , such that the expansion is convergentat large radius limit. However, we choose to use the present form, in order to have thesymmetry µ → − µ in F ref . Then due to N n ,e,µj − j + = N n ,e, − µj − j + , we only need to focus on µ being non-negative. Our conjecture for the vanishing bound is the following. Conjecture.
For elliptically fibered CY threefolds that engineer six-dimensional N = (2 , ADE-type superconformal theories, their refined GV invariants satisfy uniformly the follow- See appendix C for some examples. – 16 – ng vanishing bound: N n ,e,µ ≥ j − j + = 0 for j − > e · max { n i } − µ − e − · H ( µ − e − j + > ( e + 2) · max { n i } − µ − e − · H ( µ − e − (3.20)On the right-hand side, H ( x ) is the Heaviside step function, H ( x ) = , if x < , otherwise (3.21)We explain briefly how to use the above constraint to determine the partition functionor elliptic genus recursively. This starts from the relation between the partition functionand the free energy, Z ref = 1 + (cid:88) n (cid:54) =0 ∈ Z r ≥ Z n Q n = exp ( F ref ) = exp (cid:88) n (cid:54) =0 ∈ Z r ≥ F n Q n , (3.22)where in the first equality we use (2.11), and the second equality holds after we Taylorexpand the exponential in terms of Q . Therefore, for a given n α = { n , · · · , n r } we have, Z n α = F n α + ∞ (cid:88) j =2 j ! (cid:88) n ,..., n j> (cid:80) n i = n α j (cid:89) i =1 F n i . (3.23)Clearly, the second term in the right-hand side only involves F n of base degrees strictlysmaller than n α . To have some feeling of this equation, we work out the expansion of thefirst few degrees for the E6-type, Z { , , , , , } = F { , , , , , } ,Z { , , , , , } = F { , , , , , } + (cid:0) F { , , , , , } · F { , , , , , } (cid:1) ,Z { , , , , , } = F { , , , , , } + (cid:0) F { , , , , , } · F { , , , , , } + F { , , , , , } · F { , , , , , } (cid:1) + (cid:0) F { , , , , , } · F { , , , , , } (cid:1) . (3.24)For our purpose, it is better to invert the above relations, F { , , , , , } = Z { , , , , , } ,F { , , , , , } = Z { , , , , , } − (cid:0) Z { , , , , , } · Z { , , , , , } (cid:1) ,F { , , , , , } = Z { , , , , , } − (cid:0) Z { , , , , , } · Z { , , , , , } + Z { , , , , , } · Z { , , , , , } (cid:1) + (cid:0) Z { , , , , , } · Z { , , , , , } (cid:1) . (3.25)– 17 –n general we can take the logarithm of both sides of (3.22), F n α = Z n α + ∞ (cid:88) j =2 j ! (cid:88) n ,..., n j> (cid:80) n i = n α a ( n α ) n ,..., n j j (cid:89) i =1 Z n i , (3.26)with a ( n α ) n ,..., n j some integer coefficients that can be determined inductively. Clearly, thesecond term in the right-hand side only involves Z n of base degrees strictly smaller than n α . Our strategy goes as follows. We choose n α small enough such that all the partitionfunctions with degree lower than n α are known, e.g., from the "flip symmetry" method insubsection 3.1. Then we impose both the vanishing condition in the left-hand side and themodular ansatz in the right-hand side of (3.26). This turns to give enough constraints tocompletely fix F n α hence Z n α . Increasing the degree one step at each time, we are able todetermine the elliptic genus for any value of n α in principle.There is one important issue that needs careful discussion. Suppose we find one con-sistent solution to the above equation, how can we guarantee that it corresponds to thesolution we want? In other words, can we show that the solution is unique? Note that ifthe solution is unique, then the vanishing bound must be able to fix all the coefficients inthe ansatz. Happily, this point was already discussed in depth in [42]. By invoking the GVexpansion and the fact that all weak Jacobi forms have a non-negative index, the authorsshow the following criterion [42]: Criterion.
If the index of either (cid:15) + or (cid:15) − for Z n α is smaller than − , then the solution to (3.26) must be unique. Let us apply it to our situation. It so happens that the index of (cid:15) − takes the form, i (cid:15) − = − n · Ω · n T , (3.27)where Ω is again the Cartan matrix. If we embed the simple roots into the Euclidean space,then Ω will map to the standard Euclidean inner-product and − i (cid:15) − the norm-square ofa given root. Since they are all even integral lattices, the smallest non-zero possible valuefor length-square is two, and the next one is four. Thus based on the criterion 3.2, onlynon-zero vectors with the shortest length may cause issues. For simply-laced Lie algebras,they are nothing but the positive roots. Although this could be treated generally, let usjust do a case-by-case analysis here. For a , a , a , d , d , e , e and e , there are 1, 3, 6,12, 20, 36, 63 and 120 in total respectively, and it is straightforward to enumerate themon a computer. Then we need to determine the index of i (cid:15) + for those vectors. It turns outthat in each case, for most of them, the index i (cid:15) + is, fortunately, smaller than − , and allexceptional vectors are from low enough base degrees such that their free energies can be– 18 –nambiguously determined by the "flip symmetry" method in subsection 3.1. To summarize, combining information from both gauge theories and topological strings,we are able to show that our bootstrapping procedure gives the correct answer for the ellipticgenera, provided that the conjecture 3.20 is valid.The maximal degrees that we have computed for each theory are listed in table 3 -8 for A , , , D , , and E , , -type theories. As a consistency check, one can compare ourbootstrapped elliptic genera with the known results. For A-type theories, its localizationmethod was studied in [19], where the formula is given. We checked that the bootstrapgives exactly the same results for all the A-type elliptic genera we computed in table 3. ForD/E-type theories, its elliptic genera is currently unknown unless m is turned off [27, 47].Instead, one can use the instanton partition function of 5d D-type MSYM to test our ellipticgenus. We checked that the bootstrap gives exactly the same results with the instantonpartition function up to O ( q ) for all the D-type elliptic genera we computed in table 4and 5. Especially for the D case, the elliptic genera of some low base degrees are alreadycomputed in [42, 44]. We also confirmed that our results are the same.Lastly, for the E-type cases, there have been no known fully refined results before thispaper. Therefore, we would like to emphasize that the results given in table 6, 7, and 8 arethe first non-perturbative computations for the fully refined elliptic genera of the 6d (2,0)E-type SCFTs.As a final remark, let’s make a comparison between the "flip symmetry" method insubsection 3.1 and the "vanishing bound" method in this subsection. On the one hand,they both rely on the modular ansatz for the partition function to start with. On theother hand, although the flip symmetry is only a kinematical constraint, it can be imposeddirectly on the partition function itself. Together with the perturbative part, it usuallyfixes most, if not all, of the coefficients in the modular ansatz at low base degrees. The restcan possibly be determined if we further demand, e.g., the GV form for the free energy,instanton partition functions expansion if available. On the other hand, the vanishingbound for GV invariants is more powerful, albeit conjectural. To make use of it, we need tosubtract all the previous partition functions of smaller base degrees, which often becomesquite time-consuming in practice. This explains why we could not go very far beyond thecases already covered by the "flip symmetry" method. Also, its efficiency depends cruciallyon the precision of the vanishing bound. This section discusses the two applications of the elliptic genera that we obtained from themodular bootstrap. We will mainly focus on the generalization of the results previouslyknown only for A or D-type to all ADE theories. First, we shall compute the 6d (2,0) For example, they are not larger than the case where the degrees wrapping exceptional cycles are allequal to one. – 19 –ardy formulas [71] on R × T to all ADE-type. Second, we shall compute the (2,0)superconformal index and show that it becomes the W -algebra character in the unrefinedlimit for all ADE-type. R × T In this subsection, we compute the Cardy formulas of general 6d (2,0) SCFTs on R × T from the elliptic genera. [71] initiated the Cardy limit study of the 6d SCFTs on R × T for the (2,0) A-type theory and the higher rank E-string theory. In [71], the authorscomputed the 6d free energy by evaluating the elliptic genus summation with the continuumapproximation in the Cardy limit. We will mainly follow the same method pioneered in[71], but extend their studies to (2,0) D/E-type SCFTs.The Cardy limit of R × T is defined as the large momenta limit, i.e., the limit where J , and P are large in (2.1). In terms of the canonical ensemble, this limit can be achievedby setting the conjugate chemical potentials to be small as follows, | (cid:15) , | (cid:28) , | β | (cid:28) (4.1)where β ≡ − πiτ is a conjugate chemical potential to P . Now, recall that (cid:15) , are the IRregulators on R by adjusting the effective volume as vol ( R ) ∼ (cid:15) (cid:15) . Also, β is inverselyproportional to the radius of the spatial circle in T . Therefore, the Cardy limit (4.1)corresponds to the thermodynamic limit where the spatial volume diverges. As a result,the leading free energy in the Cardy limit should scale as log Z R × T ∼ O ( (cid:15) (cid:15) β ) which isthe spatial volume factor of the 6d background.In general, one may consider the complexified chemical potentials for (cid:15) , and β . Suchnon-trivial phases of the chemical potentials are important to observe the deconfining be-havior of the superconformal indices [72], including 6d SCFTs on S × S [66]. However,on R × T , one can observe the deconfining free energy even at the real chemical potentialsetting. It can be achieved by setting the Ω -background parameters to be close to theself-dual point (cid:15) = − (cid:15) by giving them different signs. Therefore, for simplicity, we shallconsider the following chemical potential setting in the rest of this subsection, (cid:15) > , (cid:15) < , β > , Re [ m ] = 0 . (4.2)Note that the flavor chemical potential m is purely imaginary with Im [ m ] of order O (1) .Also, we will set the tensor VEVs v to be sufficiently close to the origin of the tensor branch v = 0 . More precisely, we will assume that v (cid:28) β − .We expect that the leading volume divergence log Z R × T ∼ O ( (cid:15) (cid:15) β ) does not dependon the order of taking the Cardy limit (4.1). Hence, let us take the small β limit first.In this limit, the asymptotic form of the elliptic genus can be easily computed from themodular property. Using (2.4) and (2.5), the S-dual transformation of the elliptic genus– 20 –an be written as follows, Z n ( τ | (cid:15) , , m ) = exp (cid:104) β (cid:16) (cid:15) (cid:15) (cid:88) a,b Ω a,b n a n b + ( m − (cid:15) ) (cid:88) a n a (cid:17)(cid:105) · Z n ( − τ | (cid:15) , τ , mτ ) . (4.3)Now, we consider the asymptotics of the dual elliptic genus Z n ( − τ | (cid:15) , τ , mτ ) in the Cardylimit. In the right hand side of (4.3), the dual instanton fugacity q D = e − π /β is muchsmaller than . Therefore, one might guess that the dual instanton corrections in Z n ( − τ | (cid:15) , τ , mτ ) can be always ignored in the Cardy limit. However, one should be careful due to the presenceof other fugacities such as e ± m/τ which can give additional growth factor that can overcomethe dual instanton suppression. After carefully investigating the suppression/growth factorsfrom each fugacities, we found that asymptotics of the dual elliptic genus Z n ( − τ | (cid:15) , τ , mτ ) isgiven as follows, Z n ( − τ | (cid:15) , τ , mτ ) = exp (cid:104) − πim (2 p + 1) − π ( p + p ) + O ( (cid:15) , ) β (cid:88) a n a + o ( β − ) (cid:105) (4.4)which we checked for the results given in the appendix B. Here, p is an integer such that πp < Im [ m ] < π ( p + 1) . When p = 0 , the dual elliptic genus is dominated by thedual perturbative part at O (( q D ) ) . When p (cid:54) = 0 , the dominant contribution in the dualelliptic genus comes from the ( p + p ) (cid:80) n a number of dual instantons. As we shall showlater, considering the effect of dual instantons at generic p is important to recover the πi periodicity of m .For simplicity, let us first consider the p = 0 chamber where < Im [ m ] < π . Byplugging (4.4) into (4.3), one obtains the following form, log Z n = 1 β (cid:16) (cid:15) (cid:15) (cid:88) a,b Ω a,b n a n b + m ( m − πi ) (cid:88) a n a + O ( (cid:15) , ) (cid:17) + o ( β − ) . (4.5)Now, let us evaluate the 6d R × T index by summing over the elliptic genera as follows, Z R × T = Z (cid:88) n Z n e − n · v = Z (cid:88) n exp (cid:104) β (cid:16) (cid:15) (cid:15) (cid:88) a,b Ω a,b n a n b + m ( m − πi ) (cid:88) a n a + O ( (cid:15) , ) (cid:17) + o ( β − ) (cid:105) . (4.6)Note that the effect of the string fugacity e − n · v is subleading since we assumed that v (cid:28) β − . Also, the summation is convergent since det ( (cid:15) (cid:15) β Ω) < . Now, let us further takethe remaining Cardy limit, i.e., (cid:15) , → limit. We take a continuum approximation of the– 21 –tring number by defining following continuous variables, x a ≡ − (cid:15) (cid:15) n a . (4.7)Although n a is a discrete variable with ∆ n a = 1 , x a can be viewed as an almost continuousvariable since ∆ x a = − (cid:15) (cid:15) (cid:28) in the Cardy limit. Then, the elliptic genus summation(4.6) can be approximated to the integral as follows, Z R × T = Z (cid:90) ∞ [ r g (cid:89) a =1 dx a ] exp (cid:104) (cid:15) (cid:15) β (cid:16) (cid:88) a,b Ω a,b x a x b + m (2 πi − m ) (cid:88) a x a (cid:17) + o ( 1 (cid:15) (cid:15) β ) (cid:105) . (4.8)The free energy log Z R × T can be evaluated from the saddle point approximation of theabove integral. The peak of the Gaussian is located at x a = m ( m − πi ) (cid:88) b (Ω − ) a,b + O ( (cid:15) , ) . (4.9)Here, notice that the vector (cid:80) b (Ω − ) a,b e b = ρ is the Weyl vector of the Lie algebra g , i.e. ρ = (cid:80) α ∈ Ψ + g α . For the simply-laced Lie algebras, the Weyl vector is given as follows, ρ = r (cid:88) a =1 a ( r + 1 − a ) e a , g = A r = r − (cid:88) a =1 a (2 r − − a ) e a + r ( r − e r − + e r ) , g = D r = (16 , , , , , , g = E = (34 , , , , , , , g = E = (92 , , , , , , , , g = E . (4.10)In the Cardy limit, the string number n has a non-zero VEV on the tensor branch, givenby (cid:104) n a (cid:105) = − x a (cid:15) (cid:15) [71]. Therefore, the Weyl vector of g determines the distribution of theself-dual strings in 6d (2,0) g -type SCFT in the Cardy limit.The contribution of the elliptic genera is localized near the peak (4.9), and the leadingfree energy can be obtained from the value of the integrand at the peak. As a result, weobtain the following free energy, log Z R × T (cid:39) log Z − (cid:80) a,b (Ω − ) a,b m (2 πi − m ) (cid:15) (cid:15) β (4.11)where we ignored the subleading corrections in the Cardy limit. From the expression of Z given in (3.8), one can compute that log Z = − r g m (2 πi − m ) (cid:15) (cid:15) β [73]. Also, the group– 22 –heoretic constant (cid:80) a,b (Ω) − a,b is equal to h ∨ g d g where h ∨ g is a dual Coxeter number and d ∨ g is a dimension of the Lie algebra g . Therefore, we obtain the Cardy formula of 6d (2,0) g -type theory on R × T as follows, log Z R × T (cid:39) − h ∨ g d g + r g m (2 πi − m ) (cid:15) (cid:15) β . (4.12)For the simply-laced Lie algebra, the overall constant h ∨ g d g + r g is given as follows, h ∨ g d g + r g = ( r + 1) − , g = A r = 4 r − r + 3 r, g = D r = 942 , , , g = E , , (4.13)Note that in our chemical potential setting (4.2) and < Im [ m ] < π , we can see that log Z (cid:29) , which signals the macroscopic number of deconfining degrees of freedom. Theabove result is a straightforward generalization of the (2,0) A-type formula in [71] to generalADE-type theories. The overall factor of the free energy h ∨ g d g + r g was also observed inthe 6d Cardy formula on S × S [66], which explains the black hole entropy in the dualgravity.Lastly, let us consider the free energy in the general chamber of m given by πp < Im [ m ] < π ( p + 1) . In this chamber, one should use a general form (4.4) with p (cid:54) = 0 .Inserting (4.4) to (4.3) yields, log Z n = 1 β (cid:16) (cid:15) (cid:15) (cid:88) a,b Ω a,b n a n b + ( m − πip )( m − πi ( p + 1)) (cid:88) a n a + O ( (cid:15) , ) (cid:17) + o ( β − ) . (4.14)After following the similar calculations explained so far, we obtain the following 6d freeenergy, log Z R × T (cid:39) − h ∨ G d G + r G
24 ( m − πip ) (2 πi ( p + 1) − m ) (cid:15) (cid:15) β (4.15)One can see that the above expression is periodic under m ∼ m + 2 πi . Therefore, as weemphasized, the instanton correction in the dual elliptic genus (4.4) is essential to recoverthe periodicity of m . W -algebra In this subsection, we compute the superconformal indices of the 6d (2,0) SCFTs from theelliptic genus method. We will focus on the unrefined limit of the chemical potentials wherethe superconformal indices are expected to be reduced to the W -algebra character [74]. Weshow that our results agree with the prediction, including E-type cases.– 23 –et us briefly review the superconformal index of 6d (2,0) SCFT. See [75] for a detailedreview. The bosonic part of the 6d (2,0) superconformal algebra is given by SO (6 , conformal symmetry and SO (5) R-symmetry. Let us denote E and J , , as the charges for SO (6 , and Q , as the charges for SO (5) , which are all normalized to be ± for spinors.We take a defining supercharge of the index to give the BPS bound E ≥ Q + 2 Q + J + J + J . Then, the superconformal index is defined as follows, I = Tr (cid:104) ( − F e − ω J − ω J − ω J e − ∆ Q − ∆ Q (cid:105) (4.16)where the trace is taken over the Hilbert space of the radially quantized 6d SCFT on S × R time . The chemical potentials are constrained by ∆ + ∆ − ω − ω − ω = 0 topreserve the supersymmetry. We will use the notation ∆ R = ∆ + ∆ and ∆ L = ∆ − ∆ in this subsection.The 6d superconformal index was studied from localization of 5d MSYM on S [76–82].The path integral on S is localized at the three fixed points, and the contribution fromeach fixed point is given by the R × T index on the tensor branch. The full S × S indexcan be obtained from the three copies of R × T indices as follows [75], I = e − E − S bkgd W g (cid:112) | Ω g | (cid:90) [ r (cid:89) a =1 dv a ] e − S ˆ Z [1] ˆ Z [2] ˆ Z [3] . (4.17)Let us explain the various factors in (4.17). First, ˆ Z [ k ] is the Weyl-symmetrized R × T index. It is basically same with the R × T index in (3.10) discussed in this paper so far,but multiplied with the Weyl-symmetric factor as follows, ˆ Z = Z Weyl · Z, Z
Weyl = PE (cid:104)
12 sinh m ± (cid:15) + sinh (cid:15) , (cid:88) α ∈ Ψ + g ( e α ( v ) − e − α ( v ) ) (cid:105) . (4.18)The reason that we introduce the Weyl symmetric factor is that the elliptic genus ex-pansion Z = Z (cid:80) n Z n e − n · v is not symmetric under the Weyl reflection v → − v . Bymultiplying the prefactor Z Weyl , one can make the full index ˆ Z = Z Weyl Z symmetric un-der the Weyl reflection. Also, the contribution at the k th fixed point on S is given by ˆ Z [ k ] = ˆ Z R × T ( τ [ k ] , v [ k ] , (cid:15) , [ k ] , (cid:15) , [ k ] , m [ k ] ) , and the chemical potentials are given as follows[75], τ [ k ] = 2 πiω k , v [ k ] ,a = 2 πωω k v a ,(cid:15) , [ k ] = 2 πi ( ω k +1 ω k − , (cid:15) , [ k ] = 2 πi ( ω k − ω k − , m [ k ] = − πi ( ∆ L ω k + 1) . (4.19)where ω = ω + ω + ω . Second, S is the classical action of the 5d MSYM on the squashed– 24 – , and it is given as follows, S = 2 π ω ω ω ω r (cid:88) a,b =1 (Ω − ) a,b v a v b . (4.20)Third, S bkgd is the background action which gives a divergent behavior when the chemicalpotentials are small ∆ , ∼ ω , , (cid:28) . Its form is given as follows [78, 83], S bkgd = r g π ω + ω + ω − ω ω − ω ω − ω ω − ∆ L ω ω ω . (4.21)Also, E is the Casimir energy which gives a divergent behavior when the chemical potentialsare large ∆ , ∼ ω , , (cid:29) . Its form is given as follows [84], E = h ∨ G d G
384 (∆ R − ∆ L ) ω ω ω + r G ∆ R + ∆ L r G
384 (∆ R − ∆ L )(2 ω − ∆ R − ∆ L )(2 ω − ∆ R − ∆ L )(2 ω − ∆ R − ∆ L ) ω ω ω . (4.22)Lastly, | Ω g | is the determinant of the Cartan matrix, and W g is the dimension of the Weylgroup of g . For the simply-laced Lie algebra, they are given as follows, g = A r , | Ω g | = r + 1 , W g =( r + 1)! g = D r , | Ω g | = 4 , W g =2 r − r ! g = E , | Ω g | = 3 , W g =51 , g = E , | Ω g | = 2 , W g =2 , , g = E , | Ω g | = 1 , W g =696 , , . (4.23)Although the 6d superconformal indices have been studied extensively in various places,the proper prescription to evaluate the integral (4.17) is currently unknown when the chem-ical potentials are fully refined. The main reason is that the chemical potentials in theintegrand of (4.17) is ‘S-dualized’ in the sense that τ [ k ] ∼ ω k . Therefore, the integral ex-pression (4.17) does not have a proper fugacity expansion structure with respect to e − ω , , and e − ∆ , .Therefore, we will study the index (4.16) with unrefined chemical potentials. Even-tually, we would take a single chemical potential to be independent by setting ∆ = 2 ω , ∆ = ω , and ω , , = ω . However, directly substituting these unrefined values makes theintegrand of (4.17) singular due to the vanishing Ω -background. Therefore, we shall firstconsider the following unrefinement, ∆ L = ω + ω − ω , (4.24)– 25 –nd then take ω , , → ω limit later. In the above setting, the chemical potentials of each R × T indices in the integrand satisfy a simple relation. According to (4.19), we obtain e − m [1] = e + (cid:15) − , [1] , e − m [2] = e − (cid:15) − , [2] , e − m [3] = e + (cid:15) + , [3] (4.25)Here, e − m = e ± (cid:15) + and e − m = e ± (cid:15) − are special points that the elliptic genera become muchsimplified. When e − m = e ± (cid:15) + , all elliptic genera we obtain become unless the base degreeis empty. When e − m = e ± (cid:15) − , all elliptic genera we obtain become +1 , − , or . For all thedata we obtained, we check that the following two properties hold, e − m = e ± (cid:15) + : (cid:88) n Z n e − n · v = 1 e − m = e ± (cid:15) − : (cid:88) n Z n e − n · v = (cid:89) α ∈ Ψ + g (1 − e − α · v ) . (4.26)According to (4.26), the maximal base degree for the non-vanishing elliptic genus when e − m = e ± (cid:15) − is expected to be n = 2 ρ where ρ is a Weyl vector (4.10). Unfortunately, wecould not reach that bound due to the limited computing power. However, our data inappendix B still provide non-trivial evidence for (4.26) inclulding E-type cases.Similarly, the pure momentum contribution Z in (3.8) and the Weyl prefactor in (4.18)become e − m = e ± (cid:15) + : Z = q r g η ( τ ) − r g Z Weyl = 1 ,e − m = e ± (cid:15) − : Z = 1 Z Weyl = (cid:89) α ∈ Ψ + G (cid:16) − e α ( v ) − e − α ( v ) (cid:17) . (4.27)Then, one can rewrite the integral expression (4.17) as follows, I = e − h ∨ g d g ω ω ω ω ω [ e − ω η ( 2 πiω )] − r g × W g (cid:112) | Ω g | (cid:90) [ r g (cid:89) a =1 dv a ] e − S (cid:89) α ∈ Ψ + g (cid:16) πωα ( v ) ω (cid:17)(cid:16) πωα ( v ) ω (cid:17) . (4.28)Now, the integrand in (4.28) becomes Gaussian, and the evaluation is straightforward.After taking the further unrefinement ω , , = ω and using the S-duality of the Dedekindeta function η ( πiω ) = (cid:112) ω π η ( − ω πi ) , we obtain that I = PE (cid:104) r g e − ω − e − ω (cid:105) (cid:89) α ∈ Ψ + g (1 − e − ωα · ρ ) (4.29)where ρ = (cid:80) α ∈ Ψ + g α is a Weyl vector. The above unrefined index can be also written in– 26 –he following form, I = PE (cid:104) (cid:80) n ∈ D g e − nω − e − ω (cid:105) . (4.30)Here, D g is a set of dimensions of the Casimir operators of the Lie algebra g , and it is givenas follows, D A N = { , , ..., N } , D D N = { , , , ..., N − , N } D E = { , , , , , } , D E = { , , , , , , } , D E = { , , , , , , , } . (4.31)The unrefined limit corresponds to the ‘chiral algebra limit’ of the 6d theory, and thesuperconformal index (4.30) is the same with the W -algebra character of type g [74]. Theunrefined superconformal index (4.30) was previously computed for A/D-type (2,0) theorieswith the instanton partition function method. However, it had been still a conjecture thatthe index should become the W -algebra character for E-type also. In this paper, we partiallycheck that the conjecture is true by computing the self-dual strings’ elliptic genera in (2,0)E-type theories. In this paper, we studied the elliptic genera of the self-dual strings in general 6d (2,0)SCFTs. We found that the elliptic genus is invariant under the flip symmetry, which is anovel kinematical constraint that we found in this paper. Using the 5d perturbative dataas an input, we obtained elliptic genera at low base degrees using the modular propertyand the flip symmetry. Then, we could further conjecture the vanishing bound of the GVinvariants that enables us to bootstrap the elliptic genera up to arbitrary base degrees.We did various consistency checks with our results from the bootstrap. For the ellipticgenera of (2,0) A-type theories, our result is exactly the same as the localization computationin [19]. The general formulas for the elliptic genera in D/E-type theories are currentlyunknown. However, we check that our results agree with the instanton partition functionof 5d maximal SYM with the D-type gauge group [62, 63]. For E-type theories, our resultsgive the first non-perturbative computation beyond the 5d perturbative data.As straightforward applications, we utilize our data to compute the 6d Cardy formulaon R × T [71] and the unrefined (2,0) superconformal index, which is previously computedfor A, or D-type theories only. There seem to be many other important topics that ourresults can be used. Here, we finish our paper by listing some of them.First, it would be interesting to construct the instanton partition functions of 5d E-typemaximal SYM from the elliptic genera of the 6d (2,0) E-type SCFT. In general, one canstudy the instanton partition function from the ADHM construction [85] of the instanton– 27 –uantum mechanics if the gauge group is classical. Unlike classical groups, computing theinstantons of the exceptional gauge group has been a difficult problem. Although instantonpartition functions in many 5d exceptional gauge theories were computed [30, 41, 45, 86],there has been no known result for the maximal SYM case. However, the string fugacityexpansion of the instanton partition function can be computed from our data in appendix B.Therefore, if one can construct an appropriate ansatz for the instanton partition functionsof E-type maximal SYM, the coefficients can be fitted by comparing with the elliptic genera.Second, one can also consider the extension to 6d little string theories (LSTs), whichare non-local QFTs given by the UV-completion of 6d (2,0) SCFTs, or (1,1) SYMs. Theyalso admit affine ADE-type classification, and the elliptic genera were computed in [87]for A-type LSTs. Especially in [44], the authors bootstrapped the elliptic genera of typeLSTs using the T-duality between (2,0) and (1,1) LSTs. However, the T-duality bootstrap isincomplete for D/E-type LSTs. It would be interesting if our flip-symmetry or the vanishingcondition for the GV invariants can give noan-trivial results for the D/E-type LSTs also.Third, our result could provide hints to other methods of computing the elliptic generaof (2,0) SCFTs. In particular, it would be nice to see if one could write down 6d blow-upequations that the elliptic genera are supposed to satisfy, along the line of [36–39].Lastly, let us make a few comments on the modular bootstrap of the elliptic genera in6d (1,0) SCFTs. In this paper, we could bootstrap all the elliptic genera in (2,0) SCFTs onlyfrom the 5d perturbative data and the modular anomaly. However, a similar bootstrap pro-cedure is known to be impossible for general (1,0) SCFTs with gauge groups, unless we knowthe precise vanishing condition [43]. Still, one can utilize the flip symmetry to bootstrapthe elliptic genera in (1,0) theory if it preserves SU (2) flavor symmetry originating fromthe (2,0) SO (5) R-symmetry. If the flip symmetry and other kinematical constraints canfix the elliptic genera of (1,0) SCFTs at low base degrees, it would be helpful to conjecturethe precise vanishing bound from them to initiate the bootstrap program.
Acknowledgements
We are grateful to Kihong Lee, Kimyeong Lee, Amir-Kian Kashani-Poor, Joonho Kim, SeokKim, and Xin Wang for helpful discussions. We would like to thank Amir-Kian Kashani-Poor, Seok Kim, and Kimyeong Lee for their reading of the draft and providing inspiringcomments. In particular, we thank Amir-Kian Kashani-Poor for kindly improving Englishwriting of the draft. JN is supported by KIAS Individual Grant PG076401. ZD is supportedby KIAS Individual Grant PG076901. – 28 –
Modular Forms
In this appendix, we collect some standard facts about the theory of modular forms [88–90].
Elliptic Modular FormsDefinition 1.
Suppose k is an integer. A function f : H → C is called a modular form ofweight k for full modular group SL (2 , Z ) if f is holomorphic on H ∪ {∞} and satisfies thefollowing equation f ( aτ + bcτ + d ) = ( cτ + d ) k f ( τ ) , for any (cid:32) a bc d (cid:33) ∈ SL (2 , Z ) (A.1)In particular, if we choose the matrix T = (cid:32) (cid:33) , we find that f ( τ + 1) = f ( τ ) . Thusif we introduce q = exp(2 πiτ ) , f can be expanded as a power series at τ = i ∞ , f ( τ ) = (cid:88) n ≥ a n q n . (A.2)The condition n ≥ is guaranteed by the holomorphicity at the infinity. If furthermore a n = 0 , then f is called a cusp form.A natural construction of modular forms involves summing over all two dimensionallattice points, which are reshuffled under the SL (2 , Z ) action. Indeed, we can define nor-malized Eisenstein series, E k ( τ ) = 1 ζ ( k ) (cid:88) ( m,n ) ∈ Z ( m,n ) (cid:54) =(0 , m + nτ ) k , (A.3)and it is easy to show that E k is a modular form of weight k when k is an integer largerthan two. When k is odd, it trivially vanishes so we only need to focus on k being even.The overall constant is chosen such that when power expanded in terms of q , the leadingconstant of E k is one. In fact, we have, E k = 1 − kB k ∞ (cid:88) n =1 σ k − ( n ) q n , (A.4)where B k is the k th Bernoulli number and σ i ( n ) denotes the sum of the i th powers of thepositive divisors of n .It is a classical result that the ring of holomorphic (elliptic) modular forms M ∗ = ⊕ k ≥ M k ( SL (2 , Z )) is freely generated by E and E [88, 89]. This is needed to ensure that we can rearrange the order of summation. – 29 –unctions that are not exactly modular are also important. In the main text, we alsoneed the Dedekind η function, η ( τ ) = q ∞ (cid:89) i =1 (1 − q i ) . (A.5)Its th power is a cusp modular form of weight twelve. It can be expressed in terms of thetwo generators, η ( τ ) = 11728 ( E ( τ ) − E ( τ ) ) = q − q + · · · . (A.6)Another important function that is nearly modular is the second Eisenstein series E . UnderSL (2 , Z ) , it transforms as E ( aτ + bcτ + d ) = ( cτ + d ) E ( τ ) − iπ c ( cτ + d ) . (A.7) E is a famous example of quasimodular forms. In fact, we can enlarge our ring M ∗ to bethe ring of quasi-modular forms ˜ M ∗ , by relaxing the condition (A.1) to have lower powersin ( cτ + d ) on the right-hand side. ˜ M ∗ is also finitely generated. Actually, it can be shown[88] that ˜ M ∗ is generated from M ∗ just by adding one extra generator E .If one wants to maintain modularity, we can add an extra piece to E and define ˆ E ( τ ) = E ( τ ) − π Im( τ ) . (A.8) ˆ E ( τ ) can be shown to be modular with weight two, but clearly at the expense of losingholomorphicity. It is an example of almost holomorphic modular forms. Jacobi Modular FormsDefinition 2.
A Jacobi modular form of weight k and index m is a function φ : H × C → C that depends on a modular parameter τ ∈ H and an elliptic parameter z ∈ C . It transformsunder the action of SL (2 , Z ) on H × C as τ (cid:55)→ τ γ = aτ + bcτ + d , z (cid:55)→ z γ = zcτ + d with (cid:32) a bc d (cid:33) ∈ SL(2 , Z ) , (A.9) as φ k,m ( τ γ , z γ ) = ( cτ + d ) k e πimcz cτ + d φ k,m ( τ, z ) , (A.10) φ k,m ( τ, z + λτ + µ ) = e − πim ( λ τ +2 λz ) φ k,m ( τ, z ) ∀ λ, µ ∈ Z , (A.11)(A.10) is known as the modular transformation which is a generalization of modular– 30 –ransform, while the second one (A.11) is known as the elliptic transform.From the definition, if we choose T = (cid:32) (cid:33) and λ = 0 , µ = 1 in equations (A.10) and(A.11) respectively, we see that the Jacobi form is invariant under the shift τ → τ + 1 and z → z + 1 , hence it enjoys a double Fourier expansion, φ ( τ, z ) = (cid:88) n,r c ( n, r ) q n y r , for q = e πiτ , y = e πiz . (A.12)It can be shown that c ( n, r ) only depends on r and an SL (2 , Z ) invariant combination nm − r . In other words, c ( n, r ) = C (4 nm − r , r ) . We can further define three classesof Jacobi modular forms: holomorphic Jacobi forms J h ∗ , ∗ satisfy the constraint c ( n, r ) = 0 unless nm ≥ r , cusp forms J c ∗ , ∗ satisfy c ( n, r ) = 0 unless nm > r and weak Jacobiforms J w ∗ , ∗ satisfy c ( n, r ) = 0 unless n ≥ . Clearly we have, J c ∗ , ∗ ⊂ J h ∗ , ∗ ⊂ J w ∗ , ∗ . (A.13)An important theorem in [90] shows that weak Jacobi forms J w ∗ , ∗ of integer index isfreely generated over the ring of elliptic modular forms by two generators ϕ − , ( τ, z ) and ϕ , ( τ, z ) . ϕ − , and ϕ , can be defined in terms of the Jacobi theta functions. For a and b ∈ { , / } , we have Θ (cid:104) ab (cid:105) ( τ, z ) = (cid:88) n ∈ Z e πi ( n + a ) τ +2 πiz ( n + a )+2 πibn . (A.14)The four theta functions in our convention are θ = i Θ (cid:34) (cid:35) , θ = Θ (cid:20) (cid:21) , θ = Θ (cid:20) (cid:21) , θ = Θ (cid:20) (cid:21) . (A.15)Based on θ i ( τ, z ) , we can give explicit forms of our two generators, ϕ − , ( τ, z ) = θ ( τ, z ) η ( τ ) ,ϕ , ( τ, z ) = 4 (cid:16) θ ( τ, z ) θ ( τ, + θ ( τ, z ) θ ( τ, + θ ( τ, z ) θ ( τ, (cid:17) . (A.16)We also give the first few terms in their q -expansion for convenience, φ − , ( τ, z ) = − ( y − /y ) + 2( y − y + 6 − /y + 1 /y ) q + · · · ,φ , ( τ, z ) = ( y + 10 + 1 /y ) + (10 y − y + 108 − /y + 10 /y ) q + . . . . (A.17)Finally, we point out an interesting relation between Jacobi forms φ ( τ, z ) and quasi-– 31 –odular forms E . Looking back to the modular transformation (A.10), the exponentialfactor on the right-hand side looks a bit annoying, but actually one can introduce an auto-morphic correction to cancel it. Recall the modular transformation of E (A.7), it is easyto verify that the combination Φ( τ, z ) = e mπ E z φ ( τ, z ) transforms without that factor.In other words, if we consider a series expansion of Φ( τ, z ) in z , Φ( τ, z ) = (cid:88) n ≥ g n ( τ ) z n , (A.18)Each g n ( τ ) now behaves nicely as genuine elliptic modular forms. From this we can extracta differential equation for φ ( τ, z ) , ∂∂E Φ( τ, z ) = 0 ⇒ (cid:18) ∂∂E + mπ z (cid:19) φ ( τ, z ) = 0 . (A.19) B Elliptic genera
In this section, we summarize the elliptic genera of the ADE M-strings of a few base degrees.We consider base degrees with n i (cid:54) = 0 for all ≤ i ≤ r g so that the full chain does notdegenerate into the shorter pieces of the M-string chain. We used a shorthand notationsuch that ϕ − n, ( (cid:15) + πi ) = R n , ϕ − n, ( (cid:15) − πi ) = L n , ϕ − n, ( m πi ) = f n . (B.1)Also, we further use the identity that (cid:16) ϕ − , ( x πi ) ϕ , ( y πi ) − ϕ − , ( y πi ) ϕ , ( x πi ) (cid:17) = θ ( x ± y ) (B.2)where we defined a function θ ( x ) as follows, θ ( x ) ≡ θ ( τ, x πi ) η ( τ ) (B.3)and use a notation that θ ( x ± y ) = θ ( x + y ) θ ( x − y ) . Finally, in the case when the numeratorof a higher rank elliptic genus is factorized into that of a lower rank elliptic genus, we write itas ( higher rank base degree ) = factor × ( lower rank base degree ) in the tables. We find thatattaching a single M-string to another single M-string yields θ ( m ± (cid:15) − ) factor universally.– 32 – N (cid:0) (cid:1) θ ( m ± (cid:15) + ) (cid:0) (cid:1) θ ( m ± (cid:15) + ) θ ( m ± (cid:15) − )(27 R E − R R E − R R E E − R R E − R E − R R E + R ) (cid:0) (cid:1) ∗ θ ( m ± (cid:15) + ) θ ( m ± (cid:15) − ) (cid:16) f L R +4 E f L R +3 E f L R +6 E f L L R +4 E f L L R − f L L R + f L R R +15 E f L R R − E E f L R R +28 E f L L R R − E f L L R R +12 E f L L R R − E f L L R R − E f L L R R − E f L R R − E E f L R R +621 E f L R R − E f L R R − E f L L R R − E E f L L R R +108 E f L L R R − E f L L R R +174 E f L L R R − E f L L R R − E f L R R − E f L R R − E f L R R − E f L R R − E E f L R R − E E f L L R R − E f L L R R − E f L L R R +288 E f L L R R − E E f L L R R +1212 E f L L R R − E f L L R R − E f L R R +1668 E f L R R − E E f L R R − E f L R R +15936 E E f L R R − E f L L R R − E f L L R R − E E f L L R R − E E f L L R R +10656 E f L L R R − E f L L R R +3480 E f L L R R +5856 E E f L L R R − E f L R R +10368 E E f L R R +15228 E f L R R − E E f L R R +19968 E f L R R − E E f L R R − E E f L L R R − E f L L R R − E E f L L R R − E f L L R R +19008 E f L L R R +10800 E E f L L R R +1824 E E f L L R R +12600 E f L L R R +21120 E f L L R R − E E f L R R +11196 E f L R R +18624 E f L R R − E f L R R +89856 E E f L R R +293220 E f L R R − E E f L R R +35640 E f L L R R +11904 E E f L L R R − E f L L R R +182592 E E f L L R R +7056 E E f L L R R − E f L L R R +204480 E E f L L R R − E f L L R R +384 E f L L R R +73584 E E f L L R R +18540 E f L R R − E f L R R +40560 E E f L R R − E f L R R +258816 E E f L R R − E E f L R R +866592 E E f L R R +62976 E f L L R R +328896 E E f L L R R +484056 E f L L R R − E E f L L R R +315360 E f L L R R − E E f L L R R +304128 E f L L R R − E E f L L R R − E E f L L R R +61560 E f L L R R − E E f L L R R +18720 E E f L R R − E f L R R +18432 E E f L R R +920064 E E f L R R − E E f L R R − E f L R R − E f L R R +2505024 E E f L R R − E f L L R R +1506816 E E f L L R R +562176 E E f L L R R +65880 E E f L L R R − E f L L R R +607392 E E f L L R R +69984 E f L L R R − E E f L L R R − E f L L R R − E E f L L R R − E f L L R R − E E f L L R R − E f L R R +162048 E E f L R R +80384 E f L R R − E E f L R R +215298 E f L R R +677888 E f L R R +575424 E E f L R R +5830656 E E f L R R − E E f L R R +2337792 E E f L L R R +191592 E E f L L R R − E f L L R R +413696 E f L L R R +2593152 E E f L L R R − E f L L R R +1523520 E E f L L R R − E E f L L R R +514296 E E f L L R R +34304 E f L L R R − E E f L L R R − E f L L R R − E E f L L R R +52736 E f L R R +107424 E E f L R R − E f L R R − E E f L R R − E E f L R R +3408480 E E f L R R +2267190 E f L R R +10186752 E E f L R R − E E f L R R +2522340 E f L L R R +1716224 E f L L R R − E E f L L R R +2260992 E E f L L R R − E E f L L R R +2867712 E E f L L R R − E E f L L R R +2377512 E f L L R R − E f L L R R − E E f L L R R +728676 E f L L R R − E E f L L R R − E E f L L R R − E E f L L R R +433026 E f L R R − E E f L R R − E E f L R R +52488 E E f L R R +984150 E f L R R − E E f L R R +5686848 E E f L R R +5505024 E f L R R − E E f L R R +6104160 E E f L R R − E E f L L R R +8098704 E E f L L R R +3513780 E f L L R R − E E f L L R R − E E f L L R R − E f L L R R +940032 E f L L R R +876096 E E f L L R R − E E f L L R R +898128 E E f L L R R − E E f L L R R +1812456 E E f L L R R +834948 E f L L R R − E f L L R R − E E f L L R R − E E f L R R +329184 E E f L R R +76788 E f L R R − E f L R R − E E f L R R − E f L R R +5446656 E E f L R R − E E f L R R − E f L R R − E E f L R R +24649920 E E f L R R − E f L L R R − E E f L L R R +12358656 E E f L L R R − E f L L R R − E E f L L R R +10361520 E E f L L R R +184320 E E f L L R R +1391904 E E f L L R R − E f L L R R − E E f L L R R +7988544 E E f L L R R − E f L L R R − E f L L R R +2571264 E E f L L R R − E E f L L R R +85536 E E f L L R R − E f L R R +36864 E f L R R +1022976 E E f L R R +224256 E E f L R R − E E f L R R − E f L R R +14659584 E E f L R R − E E f L R R − E E f L R R +25284096 E E f L R R − E E f L R R − E f L L R R +11759616 E E f L L R R − E E f L L R R +5196312 E f L L R R − E E f L L R R − E E f L L R R +3697488 E f L L R R +2912256 E E f L L R R − E E f L L R R − E f L L R R +17187840 E E f L L R R − E E f L L R R +5379072 E E f L L R R − E E f L L R R − E f L L R R − E E f L L R R − E E f L L R R +1281024 E E f L R R − E E f L R R − E f L R R +442368 E E f L R R − E E f L R R +15138816 E E f L R R +1534464 E E f L R R − E E f L R R +7479540 E f L R R − E f L R R +5935104 E E f L R R − E E f L R R − E f L L R R +17104896 E E f L L R R − E E f L L R R +8552448 E E f L L R R − E E f L L R R +8211456 E E f L L R R +2211840 E f L L R R − E E f L L R R +6636816 E E f L L R R +5143824 E f L L R R +21012480 E E f L L R R − E E f L L R R +250776 E f L L R R +5480448 E E f L L R R − E E f L L R R − E f L L R R − E E f L L R R − E E f L L R R +836892 E f L R R +688128 E E f L R R − E E f L R R +49152 E f L R R − E E f L R R +377136 E E f L R R − E f L R R +5111808 E f L R R +16035840 E E f L R R − E E f L R R +14155776 E E f L R R − E E f L R R +19105632 E E f L R R +7176192 E E f L L R R +4271616 E E f L L R R − E E f L L R R − E f L L R R +3145728 E f L L R R − E E f L L R R +14432256 E E f L L R R − E f L L R R − E E f L L R R +17122752 E E f L L R R +14745600 E E f L L R R − E E f L L R R +10450944 E E f L L R R +1425408 E f L L R R − E E f L L R R +3277584 E E f L L R R +1277208 E f L L R R − E E f L L R R +3421440 E E f L L R R +196608 E f L R R − E E f L R R +1625184 E E f L R R +478953 E f L R R − E E f L R R +539136 E E f L R R +11698176 E E f L R R − E E f L R R − E E f L R R +2066715 E f L R R +12582912 E E f L R R − E E f L R R +8538048 E E f L R R +984150 E f L L R R +786432 E f L L R R +7004160 E E f L L R R − E E f L L R R − E E f L L R R +46697472 E E f L L R R − E E f L L R R − E E f L L R R +11805696 E E f L L R R +1562976 E E f L L R R − E f L L R R +3538944 E f L L R R − E E f L L R R +11492928 E E f L L R R +931662 E f L L R R − E E f L L R R +5308416 E E f L L R R − E E f L L R R − E E f L L R R +5015520 E E f L L R R − E f L R R − E E f L R R +1555200 E E f L R R − E E f L R R − E E f L R R +699840 E E f L R R +885735 E f L R R +3145728 E E f L R R +608256 E E f L R R − E E f L R R +3145728 E f L R R − E E f L R R − E E f L R R +1732104 E E f L R R +7077888 E E f L L R R − E E f L L R R +6482268 E E f L L R R − E f L L R R − E E f L L R R +32624640 E E f L L R R − E E f L L R R +2598156 E f L L R R − E f L L R R − E E f L L R R +1073088 E E f L L R R − E E f L L R R +8584704 E E f L L R R − E E f L L R R − E E f L L R R +898560 E E f L L R R +2449440 E E f L L R R +669222 E f L L R R − E f L L R R − E E f L L R R +3639168 E E f L L R R − E E f L R R +898560 E E f L R R − E E f L R R − E f L R R − E f L R R − E E f L R R +419904 E E f L R R +1048576 E f L R R − E E f L R R +944784 E E f L R R − E f L R R +786432 E E f L R R − E E f L R R +1049760 E E f L R R +826686 E f L L R R +4718592 E E f L L R R − E E f L L R R +3639168 E E f L L R R − E f L L R R +6488064 E E f L L R R +1741824 E E f L L R R − E E f L L R R +294912 E E f L L R R − E E f L L R R +3647916 E E f L L R R − E f L L R R − E E f L L R R +1603584 E E f L L R R +979776 E E f L L R R +196830 E f L L R R − E f L L R R − E E f L L R R +1213056 E E f L L R R − E E f L L R R +1244160 E E f L L R R +516132 E E f L L R R − E f L R − E f L R +184320 E E f L R − E E f L R − E E f L R +124416 E E f L R − E E f L R − E f L R +786432 E E f L R − E E f L R +1049760 E E f L R +1048576 E f L L R − E E f L L R +248832 E E f L L R +472392 E E f L L R +354294 E f L L R − E E f L L R +3317760 E E f L L R − E E f L L R +786432 E E f L L R − E E f L L R +1259712 E E f L L R − E E f L L R +746496 E E f L L R − E E f L L R − E E f L L R +82944 E E f L L R +113724 E E f L L R − E f L L R − E E f L L R +559872 E E f L L R (cid:17) (cid:0) (cid:1) θ ( m ± (cid:15) + ) · θ ( m ± (cid:15) − ) (cid:0) (cid:1) θ ( m ± (cid:15) + ) (cid:16) − f L R R E − f L R E + 81 f L L R E + 18 f L R R E + 36 f L R R E +18 f L L R R E − f L L R R E − f L R E − f L L R E + 24 f L R R E E + 24 f L R R E E − f L L R R E E − f L L R E E + 3 f L R E + 3 f L L R E + 18 f L L R R E − f L L R R E − f L R R E − f L R R E + 64 f L R R E + 32 f L R E − f L L R E + 4 f L R E + 20 f L R R E + 12 f L L R R E + 36 f L L R R E − f L L R R E − f L R R E − f L L R R E − f L R E + f L R − f L L R + 2 f L R R (cid:17) (cid:0) (cid:1) θ ( m ± (cid:15) + ) · θ ( m ± (cid:15) − ) (cid:0) (cid:1) θ ( m ± (cid:15) − ) × (cid:0) (cid:1)(cid:0) (cid:1) θ ( m ± (cid:15) + ) θ ( m ± (cid:15) − ) (cid:16) f L L R E − f L R E − f L R R E − f L L R E − f L L R E − f L L R R E + 36 f L L R R E + 36 f L L R R E + 9 f L R E + 27 f L R R E − f L L R E E +24 f L R R E E + 24 f L R R E E − f L R E − f L R R E − f L L R R E − f L L R R E + 6 f L L R E +18 f L L R R E + 12 f L L R E − f L L R E + 32 f L R E + 96 f L R R E − f L R E − f L L R E − f L L R R E +16 f L L R E + 48 f L L R R E + 8 f L R E + f L R + 3 f L R R − f L L R (cid:17) Table 3 : Elliptic genera of A , , . The result that cannot be obtained from the flip sym-metry is marked with the asterisk. – 33 – N (cid:18)
11 1 1 (cid:19) θ ( m ± (cid:15) + ) · θ ( m ± (cid:15) − ) (cid:18)
12 1 1 (cid:19) θ ( m ± (cid:15) − ) × (cid:0) (cid:1)(cid:18)
11 2 1 (cid:19) θ ( m ± (cid:15) + ) θ ( m ± (cid:15) − ) (cid:16) f R L + f R L − E f R L − E f R R L − f f R R L + 3 f f R R L − f f L R L − E f L R L +12 E f f L R L − E f L R R L + 18 E f f L R R L − E f L R R L + 9 E f L R L + 6 f f L R L − E f L R L − E E f L R L + 54 E f f L R L − E f f L R L − E f L R R L + 108 E f f L R R L + 27 E f f L R R L − E f L R R L + 9 E f f L R R L − f L R L + 36 E f L R L + 18 E f f L R L − E f L R L + 54 E f L R L − E f L R L + 72 E E f f L R L − E f f L R L + 6 E f L R R L − E E f L R R L + 162 E f f L R R L +12 E f f L R R L − E f L R R L − E f f L R R L − E f f L R R L + 45 E f L R L + 12 E f f L R L − E f f L R L − E f L R L − E f f L R L + 192 E f f L R L − E E f f L R L +36 E f L R R L + 72 E E f f L R R L − E f f L R R L + 18 E f L R R L − E f f L R R L − E f f L R R L +6 E f L R L + 24 E E f L R L − E f f L R L + 162 E f f L R L − E f f L R L + 54 E f L R R L + 36 E f L R R L − E E f f L R R L − E f f L R R L + 4 E f L R − E f L R + 32 E f L R − E f L R + 32 E f L R +24 E E f L R R − E f f L R R + 96 E f f L R R + 18 E f L R R + 81 E f f L R R − E f f L R R (cid:17) (cid:18)
12 2 1 (cid:19) ∗ θ ( m ± (cid:15) + ) θ ( m ± (cid:15) − ) (cid:16) f R L − E f R L +48 E f R L +36 E E f f R L +9 E f f R L − E f R R L +108 E E f R R L +36 E f f R R L +12 E f f R R L +63 E f R R L − E f f R R L +2 f R R L − E f R R L − E f f R R L − E f R R L +3 f f R R L +6 f f R R L − f f L R L − E f L R L +432 E E f L R L +162 E f f L R L +144 E f f L R L +72 E E f f L R L − E f L R R L +486 E f L R R L +576 E f L R R L +396 E E f f L R R L +108 E f f L R R L +828 E E f L R R L − E f f L R R L +36 E f f L R R L +162 E f L R R L − E f f L R R L +18 E f f L R R L − f L R R L − E f L R R L +18 E f f L R R L − E f L R R L − f f L R R L +18 E f L R L +18 f f L R L − E E f L R L +972 E f L R L +576 E E f L R L +864 E E f f L R L +144 E f f L R L − E f L R R L +3888 E E f L R R L +486 E f f L R R L +288 E f f L R R L +72 E E f f L R R L +72 E f L R R L +1458 E f L R R L +1872 E f L R R L − E E f f L R R L − E f f L R R L +36 E f L R R L +1080 E E f L R R L − E f f L R R L +216 E f f L R R L +504 E f f L R R L +180 E f f L R R L +18 f L R R L − E f L R R L +126 E f f L R R L − f L R L +72 E f L R L +162 E f L R L − E f L R L +1536 E f L R L +1296 E E f L R L − E f f L R L +1152 E E f f L R L − E E f f L R L +72 E E f L R R L +1458 E f L R R L +5184 E E f L R R L − E E f f L R R L − E f f L R R L − E f f L R R L +216 E f L R R L +5040 E E f L R R L − E f f L R R L − E f f L R R L − E E f f L R R L − E f L R R L +756 E f L R R L +384 E f L R R L − E E f f L R R L − E f f L R R L − E f L R R L − E E f L R R L +450 E f f L R R L − E f f L R R L − E f L R R L +72 E f f L R R L − E f f L R R L +108 E f L R L − E f f L R L − E f f L R L +108 E E f L R L − E f L R L +8640 E E f L R L +1152 E f f L R L − E E f f L R L +486 E f f L R L − E E f f L R L +162 E f L R R L +576 E f L R R L +8064 E f L R R L − E E f L R R L − E f f L R R L − E E f f L R R L − E E f f L R R L +864 E E f L R R L +1440 E E f L R R L − E E f f L R R L − E f f L R R L − E f f L R R L − E E f L R R L − E f f L R R L − E f f L R R L +144 E E f f L R R L − E f L R R L − E f L R R L − E f L R R L − E E f f L R R L +270 E f f L R R L +54 E f L R R L − E E f L R R L − E f f L R R L − E f f L R R L +72 E f L R L +108 E E f L R L − E f f L R L +36 E f f L R L − E f L R L +1440 E E f L R L +9216 E E f L R L − E E f L R L +4374 E f f L R L − E E f f L R L − E f f L R L +1944 E E f f L R L +1836 E E f L R R L − E f L R R L +5184 E E f L R R L − E f f L R R L +2268 E E f f L R R L +3402 E f f L R R L − E E f f L R R L +972 E f L R R L +1008 E f L R R L − E f L R R L +2268 E E f L R R L − E f f L R R L − E E f f L R R L +3780 E E f f L R R L +432 E E f L R R L − E E f L R R L − E E f f L R R L +5832 E f f L R R L +1728 E f f L R R L − E E f L R R L +1782 E f f L R R L +1008 E f f L R R L +4248 E E f f L R R L +252 E f L R R L − E f L R R L − E f L R R L +252 E E f f L R R L +486 E f f L R R L +48 E f L R L − E f L R L +240 E f L R L +54 E f f L R L +2304 E f L R L − E E f L R L +8748 E f L R L +12288 E f L R L − E E f L R L − E E f f L R L +9720 E E f f L R L +576 E E f f L R L +1782 E f L R R L − E E f L R R L +5832 E E f L R R L +18954 E f f L R R L − E E f f L R R L +2304 E f f L R R L +1728 E E f f L R R L +1080 E E f L R R L +10206 E f L R R L − E E f L R R L − E f f L R R L − E E f f L R R L − E f f L R R L +14976 E E f f L R R L − E f L R R L − E f L R R L − E f L R R L +216 E E f L R R L +4212 E f f L R R L +4032 E E f f L R R L +12888 E E f f L R R L − E E f L R R L − E f L R R L +1152 E E f L R R L +7488 E E f f L R R L +1836 E f f L R R L +4464 E f f L R R L − E f L R R L − E E f L R R L +1026 E f f L R R L +288 E f f L R R L +936 E E f f L R R L − E f L R L − E f f L R L − E E f f L R L − E f L R L +1728 E E f L R L − E f f L R L − E f f L R L +31104 E E f f L R L +4608 E E f f L R L − E E f f L R L − E f L R R L +2592 E E f L R R L − E f L R R L − E f L R R L +31104 E E f L R R L − E E f f L R R L +1944 E E f f L R R L − E f f L R R L +13824 E E f f L R R L − E E f L R R L − E E f L R R L +1944 E E f L R R L − E f f L R R L +19008 E E f f L R R L +16128 E f f L R R L − E E f f L R R L − E E f L R R L − E f L R R L +3456 E E f L R R L +6912 E f f L R R L +1944 E E f f L R R L +2916 E f f L R R L +2880 E E f f L R R L − E f L R R L − E f L R R L +2304 E f L R R L − E E f L R R L +486 E f f L R R L +4608 E E f f L R R L +1080 E E f f L R R L − E E f L R R L +486 E f L R R L − E E f L R R L +432 E E f f L R R L − E f f L R R L − E f f L R R L − E E f L R L +243 E f L R L − E E f L R L − E E f f L R L − E f f L R L − E f f L R L +6561 E f f L R L +9216 E f f L R L − E E f f L R L +2916 E f L R R L − E E f L R R L +13122 E f f L R R L +18432 E f f L R R L − E E f f L R R L +4608 E E f f L R R L − E E f f L R R L − E f L R R L +6561 E f L R R L +9216 E f L R R L − E E f L R R L +11520 E E f f L R R L − E E f f L R R L − E f f L R R L +1728 E E f f L R R L − E f L R R L − E E f L R R L +2304 E E f L R R L − E E f L R R L − E f f L R R L +3456 E E f f L R R L − E f f L R R L +5184 E E f f L R R L − E E f L R R L +1152 E f f L R R L − E E f f L R R L +1215 E f f L R R L − E E f f L R R L − E f L R R L − E f L R R L − E f L R R L +972 E E f L R R L − E f f L R R L +288 E E f f L R R L − E E f f L R R L − E f L R − E f L R +108 E E f L R − E f f L R − E E f f L R − E f L R − E f L R +1728 E E f L R − E E f L R R +1296 E E f L R R − E f f L R R − E f f L R R +10368 E E f f L R R +972 E f L R R − E E f L R R − E f f L R R − E f f L R R +5184 E E f f L R R − E E f f L R R +3888 E E f f L R R − E f L R R − E E f L R R − E f L R R − E f L R R +3456 E E f L R R − E E f f L R R +1944 E E f f L R R +4374 E f f L R R − E E f f L R R − E f L R R − E E f L R R − E E f L R R +648 E E f L R R − E E f f L R R − E f f L R R +216 E E f f L R R − E E f L R R +486 E f L R R − E E f L R R +384 E f f L R R − E E f f L R R − E f f L R R − E E f f L R R (cid:17) Table 4 : Elliptic genera of D . The result that cannot be obtained from the flip symmetryis marked with the asterisk. – 34 – N (cid:18)
11 1 1 1 (cid:19) θ ( m ± (cid:15) + ) · θ ( m ± (cid:15) − ) (cid:18)
11 1 1 2 (cid:19)(cid:18)
12 1 1 1 (cid:19) θ ( m ± (cid:15) − ) × (cid:0) (cid:1)(cid:18)
11 2 1 1 (cid:19) θ ( m ± (cid:15) − ) × (cid:0) (cid:1)(cid:18)
11 1 2 1 (cid:19) θ ( m ± (cid:15) − ) × (cid:18)
11 2 1 (cid:19)(cid:18)
11 2 2 1 (cid:19) ∗ θ ( m ± (cid:15) + ) θ ( m ± (cid:15) − ) (cid:16) f R L − E f R L +16 E f R L − E f f R L +48 E E f R R L +36 E f f R R L − E f f R R L +3 f R R L +36 E f R R L − E f f R R L +36 E f f R R L − E f R R L − E f f R R L − f f R R L − E f R R L +6 f f R R L − f f L R L − E f L R L +144 E E f L R L − E f f L R L − E f f L R L − E f L R R L +216 E f L R R L +288 E f L R R L − E E f f L R R L +162 E f f L R R L − E f f L R R L +576 E E f L R R L − E f f L R R L +216 E f f L R R L − E f f L R R L − f L R R L +198 E f L R R L − E f f L R R L +72 E f f L R R L − E f L R R L +12 E f f L R R L +18 f f L R R L − E f L R R L − f f L R R L +21 E f L R L +24 f f L R L +45 E f L R L +324 E f L R L +192 E E f L R L − E E f f L R L +96 E f f L R L +108 E E f f L R L +36 E f L R R L +2016 E E f L R R L − E f f L R R L − E f f L R R L − E E f f L R R L − E f f L R R L − E f L R R L +1296 E f L R R L +1584 E f L R R L − E E f f L R R L − E f f L R R L +1572 E E f L R R L − E f f L R R L +120 E f f L R R L − E f f L R R L +63 E f L R R L +432 E f f L R R L − E f f L R R L − E f L R R L +162 E f f L R R L +92 E f L R L − E f f L R L − f f L R L +324 E E f L R L +512 E f L R L +432 E E f L R L − E f f L R L − E E f f L R L +864 E E f f L R L +972 E f f L R L +224 E f f L R L +180 E f L R R L +1296 E f L R R L +2688 E E f L R R L − E E f f L R R L − E f f L R R L +864 E f f L R R L +672 E E f f L R R L − E f L R R L +6480 E E f L R R L − E f f L R R L − E f f L R R L − E E f f L R R L − E f f L R R L +108 E f L R R L +1620 E f L R R L +2080 E f L R R L − E E f f L R R L +64 E f f L R R L − E E f L R R L +504 E f f L R R L − E f f L R R L − E f f L R R L − E f L R R L +288 E f f L R R L − E f f L R R L +4 f L R L +162 E f L R L − E f f L R L − E f f L R L +54 E f L R L +592 E f L R L − E f L R L +2880 E E f L R L − E f f L R L − E E f f L R L +1944 E f f L R L +1152 E E f f L R L +3204 E E f f L R L − E E f L R R L +4608 E f L R R L − E E f L R R L − E f f L R R L − E E f f L R R L +756 E E f f L R R L +1188 E f f L R R L +576 E f f L R R L − E f L R R L +1620 E f L R R L +5760 E E f L R R L − E E f f L R R L +1296 E f f L R R L +864 E f f L R R L +216 E E f f L R R L +236 E f L R R L +2700 E E f L R R L − E f f L R R L +320 E f f L R R L +4032 E E f f L R R L +1188 E f f L R R L − E f L R R L − E f L R R L − E f L R R L − E E f f L R R L − E f f L R R L − E f f L R R L − E E f L R R L − E f f L R R L − E f f L R R L +288 E f f L R R L +300 E E f L R L − E f f L R L − E f f L R L +96 E f f L R L +396 E E f L R L +3072 E E f L R L − E E f L R L +7776 E f f L R L − E E f f L R L +3072 E f f L R L +2592 E E f f L R L − E f f L R L +5280 E E f f L R L − E f L R R L − E f L R R L − E f L R R L +8064 E E f L R R L − E f f L R R L +1728 E E f f L R R L +7776 E f f L R R L − E E f f L R R L − E E f L R R L +4608 E f L R R L − E E f L R R L − E f f L R R L − E E f f L R R L +13176 E E f f L R R L − E f f L R R L − E f f L R R L +144 E f L R R L +1296 E f L R R L − E E f L R R L +288 E E f f L R R L +9072 E f f L R R L +5568 E f f L R R L +1344 E E f f L R R L − E f L R R L − E E f L R R L − E f f L R R L − E f f L R R L +2520 E E f f L R R L − E f f L R R L − E f L R R L − E f L R R L − E f L R R L − E E f f L R R L +864 E f f L R R L +576 E f f L R R L − E f L R L +54 E f L R L +592 E f L R L +564 E E f f L R L +216 E f f L R L +64 E f f L R L +1458 E f L R L − E E f L R L +2916 E f L R L +4096 E f L R L − E E f L R L − E E f f L R L +10368 E E f f L R L − E f f L R L +17280 E E f f L R L +6016 E f f L R L − E E f f L R L − E E f L R R L +3072 E E f L R R L − E E f L R R L +23328 E f f L R R L − E E f f L R R L +3456 E f f L R R L +4212 E E f f L R R L − E f f L R R L − E E f f L R R L − E f L R R L − E f L R R L − E f L R R L +2880 E E f L R R L +5760 E f f L R R L − E E f f L R R L − E f f L R R L +22464 E E f f L R R L − E E f f L R R L +924 E E f L R R L − E f L R R L +2700 E E f L R R L +10692 E f f L R R L − E E f f L R R L +17712 E E f f L R R L − E f f L R R L +320 E f f L R R L +180 E f L R R L − E f L R R L − E E f L R R L +1512 E E f f L R R L +1296 E f f L R R L +864 E f f L R R L − E E f f L R R L − E f L R R L − E E f L R R L +1188 E f f L R R L +576 E f f L R R L +1188 E E f f L R R L − E f f L R R L − E f L R L +900 E E f L R L +972 E f f L R L +224 E f f L R L +288 E E f f L R L − E f f L R L − E f L R L +2268 E E f L R L − E f f L R L − E f f L R L +27648 E E f f L R L +18432 E E f f L R L − E E f f L R L − E f f L R L +576 E E f f L R L − E f L R R L +96 E E f L R R L − E E f f L R R L +10368 E E f f L R R L − E f f L R R L +12096 E E f f L R R L − E f f L R R L +864 E E f f L R R L +36 E E f L R R L − E f f L R R L +24768 E E f f L R R L +20736 E f f L R R L − E E f f L R R L +4536 E f f L R R L − E E f f L R R L +1620 E f L R R L +2080 E f L R R L +8748 E f L R R L − E E f L R R L − E f f L R R L +10368 E E f f L R R L +5760 E E f f L R R L − E E f f L R R L +1680 E E f L R R L +1152 E f L R R L − E E f L R R L − E f f L R R L +4416 E E f f L R R L − E E f f L R R L − E f f L R R L − E f f L R R L +144 E f L R R L +1620 E f L R R L − E E f L R R L +3168 E E f f L R R L − E f f L R R L +864 E f f L R R L − E E f f L R R L +36 E f L R L +405 E f L R L +288 E E f L R L +972 E E f f L R L +96 E f f L R L − E E f f L R L +1701 E f L R L − E E f L R L +17496 E f f L R L +24576 E f f L R L − E E f f L R L − E E f f L R L +4536 E E f f L R L +3456 E f L R R L − E E f L R R L − E E f f L R R L +1944 E E f f L R R L +13122 E f f L R R L − E E f f L R R L +567 E f L R R L +2592 E E f L R R L +20736 E E f f L R R L − E E f f L R R L − E f f L R R L +3456 E E f f L R R L − E f f L R R L +9288 E E f f L R R L +5508 E E f L R R L − E E f L R R L +5832 E E f L R R L − E f f L R R L +4032 E E f f L R R L − E f f L R R L − E f f L R R L − E E f f L R R L +1296 E f L R R L +1584 E f L R R L − E f L R R L +288 E E f L R R L +4608 E f f L R R L − E E f f L R R L − E E f f L R R L − E E f f L R R L +576 E E f L R R L − E f L R R L +2484 E E f L R R L − E f f L R R L +2304 E E f f L R R L − E E f f L R R L − E f f L R R L − E f f L R R L +48 E E f L R L +384 E f L R L − E E f L R L − E f f L R L +864 E E f f L R L − E f f L R L − E f f L R L − E f f L R L +17280 E E f f L R L − E f L R R L +4032 E E f L R R L − E f f L R R L − E f f L R R L +31104 E E f f L R R L − E E f f L R R L +3888 E E f f L R R L +1920 E f L R R L +756 E E f L R R L +13122 E f f L R R L +18432 E f f L R R L − E E f f L R R L − E E f f L R R L +3888 E E f f L R R L +972 E f f L R R L − E E f f L R R L +1782 E f L R R L +1824 E E f L R R L − E f L R R L − E f L R R L +10368 E E f L R R L − E E f f L R R L +1296 E E f f L R R L +5832 E f f L R R L − E E f f L R R L +5632 E f f L R R L − E E f f L R R L +2016 E E f L R R L +1536 E E f L R R L − E E f L R R L − E f f L R R L +576 E E f f L R R L − E f f L R R L +3240 E E f f L R R L − E f f L R R L +1824 E E f f L R R L +216 E f L R R L +288 E f L R R L − E f L R R L +1152 E E f L R R L +1536 E f f L R R L − E E f f L R R L +1458 E f f L R R L − E E f f L R R L − E E f f L R R L +16 E f L R − E f L R +288 E E f L R +384 E f f L R − E E f f L R +729 E f L R +1024 E f L R − E E f L R +1536 E E f L R R − E E f L R R +4374 E f f L R R +6144 E f f L R R − E E f f L R R − E f L R R +1440 E E f L R R +3840 E E f f L R R − E E f f L R R +128 E f L R R +756 E E f L R R − E f f L R R − E f f L R R +10368 E E f f L R R − E E f f L R R +1944 E E f f L R R − E f f L R R +4032 E E f f L R R +324 E f L R R +192 E E f L R R +2187 E f L R R +3072 E f L R R − E E f L R R − E E f f L R R +648 E E f f L R R +2916 E f f L R R − E E f f L R R +1152 E f f L R R − E E f f L R R +144 E E f L R R +768 E E f L R R − E E f L R R − E f f L R R +576 E E f f L R R − E f f L R R +972 E E f f L R R (cid:17) Table 5 : Elliptic genera of D . The result that cannot be obtained from the flip symmetryis marked with the asterisk. – 35 – N (cid:18)
11 1 1 1 1 (cid:19) θ ( m ± (cid:15) + ) · θ ( m ± (cid:15) − ) (cid:18)
21 1 1 1 1 (cid:19)(cid:18)
12 1 1 1 1 (cid:19) θ ( m ± (cid:15) − ) × (cid:0) (cid:1)(cid:18)
11 2 1 1 1 (cid:19) θ ( m ± (cid:15) − ) × (cid:0) (cid:1)(cid:18)
11 1 2 1 1 (cid:19) θ ( m ± (cid:15) − ) × (cid:18)
11 2 1 (cid:19)
Table 6 : Elliptic genera of E n N (cid:18)
11 1 1 1 1 1 (cid:19) θ ( m ± (cid:15) + ) · θ ( m ± (cid:15) − ) (cid:18)
21 1 1 1 1 1 (cid:19)(cid:18)
12 1 1 1 1 1 (cid:19)(cid:18)
11 1 1 1 1 2 (cid:19) θ ( m ± (cid:15) − ) × (cid:0) (cid:1)(cid:18)
11 2 1 1 1 1 (cid:19)(cid:18)
11 1 1 2 1 1 (cid:19)(cid:18)
11 1 1 1 2 1 (cid:19) θ ( m ± (cid:15) − ) × (cid:0) (cid:1)(cid:18)
11 1 2 1 1 1 (cid:19) θ ( m ± (cid:15) − ) × (cid:18)
11 2 1 (cid:19)
Table 7 : Elliptic genera of E n N (cid:18)
11 1 1 1 1 1 1 (cid:19) θ ( m ± (cid:15) + ) · θ ( m ± (cid:15) − ) (cid:18)
21 1 1 1 1 1 1 (cid:19)(cid:18)
12 1 1 1 1 1 1 (cid:19)(cid:18)
11 1 1 1 1 1 2 (cid:19) θ ( m ± (cid:15) − ) × (cid:0) (cid:1)(cid:18)
11 2 1 1 1 1 1 (cid:19)(cid:18)
11 1 1 2 1 1 1 (cid:19)(cid:18)
11 1 1 1 2 1 1 (cid:19)(cid:18)
11 1 1 1 1 2 1 (cid:19) θ ( m ± (cid:15) − ) × (cid:0) (cid:1)(cid:18)
11 1 2 1 1 1 1 (cid:19) θ ( m ± (cid:15) − ) × (cid:18)
11 2 1 (cid:19)
Table 8 : Elliptic genera of E – 36 – GV invariants
Here, we list some GV invariants of geometries that engineer E-type (2,0) SCFTs, extractedfrom the elliptic genera. { (1 , , , , , , e, } e = g = 0 -158 -14198 -630198 -182946321 34 8323 605826 245438382 -1506 -224344 -138302463 84 40334 42819224 -3640 -8011725 140 92994 { (1 , , , , , , e, } e = g = 0 51 6090 311415 98760001 -8 -3098 -274390 -124592062 477 91798 65401873 -24 -14764 -18687404 1190 3198755 -40 -33634 { (1 , , , , , , e, } e = g = 0 -38 -5266 -2969581 5 2498 2528812 -370 -823903 18 130964 -10465 34 Table 9 : Unrefined GV invariants from (2,0) E SCFT at base degree (1,1,2,1,1,1), fiberdegree 1 to 4 and mass degree 2, 3 and 7. { (1 , , , , , , , } j + = j − = { (1 , , , , , , , } j + = j − = Table 10 : Refined GV invariants from (2,0) E SCFT at base degree (1,1,2,1,1,1), fiberdegree 1, 2 with mass degree 1 and 2 respectively.– 37 – (1 , , , , , , , e, } e = g = 0 323 27832 1319194 424222261 -86 -17200 -1289146 -568794662 3244 482499 318605663 -168 -86166 -97154424 7490 17685865 -280 -197964 { (1 , , , , , , , e, } e = g = 0 -10 -2216 -166600 -70983681 1 868 124584 79227362 -94 -33822 -35800003 3 4175 8511584 -240 -1164665 5 9321 { (1 , , , , , , , e, } e = g = 0 1 438 45561 23465311 -118 -28154 -23170342 7 5970 9025303 -528 -1789304 17 194475 -1122 Table 11 : Unrefined GV invariants from (2,0) E SCFT at base degree (1,1,2,1,1,1,1), fiberdegree 1 to 4 and mass degree 1, 4 and 5. { (1 , , , , , , , , } j + = j − = { (1 , , , , , , , , } j + = j − = Table 12 : Refined GV invariants from (2,0) E SCFT at base degree (1,1,2,1,1,1,1), fiberdegree 1 and 2 with mass degree 1 and 4 respectively.– 38 – (1 , , , , , , , , e, } e = g = 0 1100 57254 1851595 440393851 -378 -37522 -1771932 -553250962 21 7462 611859 272490803 -406 -88228 -65176344 4256 7453395 -31626 { (1 , , , , , , , , e, } e = g = 0 -224 -18256 -731712 -190821441 42 9604 617120 228390162 -1344 -179592 -101003843 35 20146 20978584 -630 -1970365 6146 { (1 , , , , , , , , e, } e = g = 0 21 3948 216538 69957581 -1470 -153062 -71927242 105 34650 27436853 -2590 -4661444 35 324805 -602 Table 13 : Unrefined GV invariants from (2,0) E SCFT at base degree (1,1,1,1,1,1,1,1),fiber degree 2 to 5 and mass degree 3, 4 and 5. { (1 , , , , , , , , , } j + = j − = { (1 , , , , , , , , , } j + = j − = { (1 , , , , , , , , , } j + = j − = Table 14 : Refined GV invariants from (2,0) E SCFT at base degree (1,1,1,1,1,1,1,1), fiberdegree 0, 1 and 2 with mass degree 0, 1 and 2 respectively.– 39 – eferences [1] D. Gaiotto,
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