Higher S-dualities and Shephard-Todd groups

Seiberg and Witten have shown that in N=2 SQCD with $N_f=2N_c=4$ the S-duality group PSL(2,Z) acts on the flavor charges, which are weights of Spin(8), by triality. There are other N=2 SCFTs in which SU(2) SYM is coupled to strongly-interacting non-Lagrangian matter: their matter charges are weights of $E_6$, $E_7$ and $E_8$ instead of Spin(8). The S-duality group PSL(2,Z) acts on these weights: what replaces Spin(8) triality for the $E_6,E_7,E_8$ root lattices? In this paper we answer the question. The action on the matter charges of (a finite central extension of) PSL(2,Z) factorizes trough the action of the exceptional Shephard--Todd groups $G_4$ and $G_8$ which should be seen as complex analogs of the usual triality group $\mathfrak{S}_3\simeq \mathrm{Weyl}(A_2)$. Our analysis is based on the identification of S-duality for SU(2) gauge SCFTs with the group of automorphisms of the cluster category of weighted projective lines of tubular type.


Introduction and Summary
We have a complete classification of the 4d N = 2 gauge theories where SU(2) SYM is coupled to vector-less matter (possibly non-Lagrangian) [1]. By vector-less matter we mean N = 2 QFTs whose BPS spectra, in all chambers, consist only of hypermultiplets: the ones having a gaugeable SU(2) symmetry are precisely the Argyres-Douglas (AD) theories of type D p (p ≥ 2) [1,2] quark doublets being the case p = 2. From the classification we learn that the SCFTs in this class are in one-to-one correspondence with the orbifolds of an elliptic curve E i.e. E/Z p where p = 1, 2, 3, 4, 6. (1.1) For p > 2 the curve E should have complex multiplication by the appropriate quadratic field namely 1 Q(ω) for p = 3, 6 and Q(i) for p = 4. This observation plays a crucial role below.
We have five SCFTs in this class of SU(2) gauge theories. The SCFT is a Lagrangian model iff the modulus τ of E is a free parameter, i.e. for p = 1, 2 which correspond, respectively, to N = 2 * and SQCD with N f = 4. p = 3, 4, 6 yield three additional non-Lagrangian (≡ intrinsically strongly coupled) SCFTs. From the viewpoint of [1] it is more natural to state this classification as a one-to-one correspondence between this class of SCFTs and the star graphs (possibly with multiple edges 2 ) which are affine Dynkin diagrams. There are five such affine stars 6 , E which correspond to the five orbifolds (1.1). In the first two models, the Lagrangian ones, g (1) is also the affinization of the flavor symmetry algebra g which is, respectively, su(2) and so (8): thus in the Lagrangian models the flavor charges are weights of g. In the non-Lagrangian theories the matter consists of strongly interacting systems with their own conserved electric and magnetic charges in addition to the flavor ones. Although the Dirac pairing between the internal charges of the matter is no longer trivial, yet it remains true that the matter charges 3 take value in the weight lattice of the corresponding finite-dimensional Lie algebra g which, for the non-Lagrangian models, is E 6 , E 7 , or E 8 (see §. 2.1). These theories, already described in [1,3,4], recently have been constructed also as toroidal compactifications of certain 6d (1, 0) SCFTs [5]. From the internal torus, all five (mass deformed) 4d SCFTs inherit a P SL(2, Z) group of S-dualities. In the p = 2 case Seiberg and Witten [6] have shown that P SL(2, Z) acts on the flavor charges by SO(8) triality; the triality group S 3 being identified with the modular quotient P SL(2, Z)/Γ (2). By the same token, for p = 3, 4, 6 we have a non-trivial action of S-duality, hence of the modular group P SL(2, Z), on the lattice of the matter charges, i.e. on the weight lattices of E 6 , E 7 , E 8 . This action should be thought of as a generalization of the triality action on the weights of SO (8) to the weights of the exceptional Lie groups E 6 , E 7 , E 8 .
At first sight this statement seems rather odd: from Lie algebra theory we do not expect any higher rank analog of SO (8) triality. Yet physics predicts its existence.
The full duality group S is actually an extension of the modular group 1 → W → S → P SL(2, Z) → 1, (1.3) where W is the 'obvious' group of physical symmetries acting on the lattice of conserved charges. For the Lagrangian models W is simply the Weyl group of the flavor symmetry.
In the general case W is a well-understood finite group of symmetries of the matter system (seen as decoupled from the Yang-Mills sector) which fixes the charge to be gauged. W is also the kernel of the action of S on the Yang-Mills electric/magnetic charges, on which only the quotient group P SL(2, Z) acts effectively. The quotient group of S which acts effectively on the matter charges, S matter , is the finite group 1 → W → S matter → P SL(2, Z/p Z) → 1, p = 1, 2, 3, 4, 6, (1.4) whose action on the root lattice of g preserves the Cartan inner product as well as the Dirac skew-symmetric pairing. For p = 1, 2 eqn. (1.4) was obtained by Seiberg and Witten [6]. Its extension to the non-Lagrangian cases looks rather natural, a simple 'analytic continuation in p'.
Since the action of W on the matter charges is obvious, to understand the 'higher versions of triality' it is enough to understand the action of the quotient group P SL(2, Z/p Z) ≃ P SL(2, Z) Γ(p) for p = 2, 3, 4, 6, (1.5) where Γ(p) ⊂ P SL(2, Z) is the principal congruence subgroup of level p [7]. p = 6 is special since 4 P SL(2, Z/6Z) = P SL(2, Z/2Z) × P SL(2, Z/3Z). (1.6) However, it is preferable to study the action of a subgroup G matter ⊂ S matter which is a central extension of P SL(2, Z/p Z) by a finite Abelian group of the form (Z/2Z) k . Studying the central extension G matter , rather than P SL(2, Z/p Z) itself, allows to discuss all five models in an unified way via the theory of reflection groups. Let us explain. Weyl groups should be thought of as reflection groups defined over the rationals Q, and Coxeter groups as real reflection groups. For N f = 4 SQCD (p = 2) the action of P SL(2, Z/2Z) is through SO(8) triality which is a rational reflection group: indeed, in concrete terms, the triality group is the quotient S matter Weyl(SO (8)) ≡ Weyl(F 4 ) Weyl(SO (8)).
(1.7)  In passing from p = 2 to p = 3, 4, 6 what we have to do is to replace reflection groups defined over Q with reflection groups defined over the appropriate complex multiplication fields Q(ω) or Q(i). Roughly speaking, the abstract form of S-duality is the same for p > 2 as for the SQCD p = 2 model (where it is given by SO(8) triality) but structures that in the p = 2 case are defined over the ground field Q get replaced by structures defined over the complex quadratic fields Q(ω) and Q(i).
Reflection groups defined over such quadratic fields are special instances of complex reflection groups. The finite complex reflection groups have been fully classified by Shephard and Todd [9]. From their classification we read the complete list of the "higher triality" groups G matter which act on the matter charges of the p = 3, 4, 6 models, i.e. on the weight lattices Γ w g of E 6 , E 7 and E 8 . The classification also yields the decomposition of the vector space Γ w g ⊗ C in irreducible representations of G matter , and thus completely specifies how P SL(2, Z/p Z), and hence the full modular group P SL(2, Z), acts on the matter charges.
The relevant reflection groups for the three complex multiplication fields Q, Q(ω) and Q(i) are listed in table 1. In the second column we write the group, seen as a concrete reflection group acting on a two-dimensional space, in the Shephard-Todd notation. In the third column we write the standard name of the corresponding abstract group. In the fourth column we describe the same group seen as a quotient of a central extension of the modular group P SL(2, Z); this column specifies how the action of S-duality group S on Γ w g factorizes through a representation of the reflection group G matter . In the fifth column we draw the graph of the reflection group [10]: we stress that, in all three cases, it is the A 2 Dynkin graph but with order p at the nodes. The uniformity of the graph expresses our rough idea that the structure of S-duality is independent of p up to a change of the ground field. In particular, all three groups are realized as a concrete group of reflections by faithful two-dimensional unitary representations which we denote as W , R and F , respectively (see §.4.2.1 for full details). In the last column of the table we recall the affine Dynkin graph which is related to the given reflection group by the McKay correspondence [11]; more precisely, the affine graph shown in the table is the one associated to the finite SU(2) subgroup which has the same image in P SU(2) as the reflection group G matter viewed as a subgroup of U(2) via its defining two-dimensional representation. Note that to the E (1.8) For p = 2 the symmetric group S 3 acts on the weights of D 4 by triality. In this paper we describe the corresponding action of G matter for p = 3, 4, 6. We do so in two ways. First we list the irreducible representations of the matter group in eqn.(1.8) acting on the root vector where: 1 is the trivial representation; W , R and F are the defining 2-dimensional representations of S 3 , G 4 and G 8 , respectively; R, F are their conjugates (W is real); χ is a one-dimensional representation, namely a primitive character of the Abelian quotient 5 In eqn.(1.10) by W we mean the 2-dimensional representation of Z/2Z⋉SL(2, Z/4Z) defined by the degree 2 representation W of S 3 via the canonical mod 2 surjection The action of G matter on Γ Er ⊗ C preserves the lattice Γ Er , the inner product in root space given by the Cartan matrix, and the Dirac electro-magnetic pairing. For concreteness, in appendix A we also give a very explicit realization of the action of S-duality on the charges in terms of integral (r(g) + 2) × (r(g) + 2) matrices acting on the full charge lattice Γ (which includes the matter charges as well as the Yang-Mills electric and magnetic ones).
The rest of the paper is organized as follows. In section 2 we collect the basic tools of the homological approach to S-duality. In section 3 we reconsider N = 2 SQCD with two colors and four flavors as a warm-up; here we recover the Seiberg-Witten result in two ways: first in a rather naive but very concrete approach, and then from a more intrinsic group-theoretical perspective. In section 4 we describe the E cases. In appendix A we write explicit (r(g) + 2) × (r(g) + 2) matrices which represent the action of S-duality on the conserved charges of the theory, and list some of the beautiful identities they satisfy. In appendix B we show how the present approach is related to the one in [1,26] by cluster-tilting.
Remark. For definiteness, here we use the 4d definition of the relevant SCFTs, see refs. [1,3]. Alternatively, one could have adopted the 6d viewpoint of ref. [5], and in particular their mirror Landau-Ginzburg description of the E/Z p orbifolds. By a theorem of Orlov ([19] Theorem 2.5.(iii), see also the discussion in [20]), the category of B-branes for the relevant Landau-Ginzburg models is equivalent to the derived category of coherent sheaves on the corresponding weighted projective lines of tubular type, which is the central object of our analysis, see §.2.2. Then all our considerations apply directly to the Landau-Ginzburg set up. Needless to say, the equivalence of the two categories is an instance of the 2d/4d correspondence advocated in [21].

Overview
The relation between eqns.(1.1),(1.2) and the mass-deformed SCFTs is as follows 6 . The BPS particles of the QFT are given by the (quantization of) continuous families of stable objects in a certain orbifold category C(E/Z p ) of the derived category of coherent sheaves on the orbifold E/Z p or, equivalently [22], on the orbifold P 1 /Γ p where Γ p ⊂ SU(2) is the finite subgroup associated to the affine Lie algebra g (1) by the McKay correspondence 7 [11] Here C is an infinite cyclic subgroup of Aut(D b coh E/Z p ) to be described later. C(E/Z p ) is the cluster category [17,36] of the orbifold E/Z p whose relevance for the physics of the BPS sector stems from the Kontsevich-Soibelman wall-crossing formula [23], see refs. [21,24]. In the present context, stable means that the lift of the object in D b coh E/Z p is stable in the sense of ref. [25]; the stability condition on D b coh E/Z p is defined by the mass deformation we consider and the chosen point in the Coulomb branch. Since all five QFTs are complete in the sense of [1], we have only zero-and one-dimensional families of stable objects which yield, respectively, hypermultiplets and vector multiplets of N = 2 susy.
Charge lattices. The lattice of conserved QFT charges, Γ, may then be identified with the Grothendieck group of the associated additive categories 8 where Y → Y [1] denotes the shift equivalence in the triangle category D b coh E/Z p . The Dirac pairing in Γ is just the anti-symmetric part of the Euler one The Tits form is the integral quadratic form on Γ The symmetric bilinear form associated to the Tits quadratic form will be written ·, · sym . For X stable one has [26,27] q( where MaxSpin([X]) is the largest possible spin for a BPS particle of charge [X]. Since for a complete N = 2 theory [1,27], the spin of the BPS particles is bounded by 1, the quadratic form q([X]) is positive semi-definite; its radical is a sublattice of rank 2 which may be identified with the lattice of electric/magnetic Yang- 8 The fact that the Grothendieck group is a finite-rank lattice follows from the fact that the Abelian category coh E/Z p admits a tilting object [12][13][14].
Mills SU(2) charges; in particular, the restriction to rad q of the Dirac pairing has the form 9 We define the matter charges lattice as Γ modulo the sublattice of Yang-Mills charges Γ matter ≃ Γ rad q. (2.9) The above categorical identification of the charges is not the conventional one in physics (but it coincides for Lagrangian QFTs). For the three non-Lagrangian models, our SU(2) electric charge is a linear combination of the physical electric charge and matter ones; the present conventions make the group actions more transparent (in facts, they are the obvious generalization of the factor 2 difference of normalizations for the p = 1 and p = 2 models pointed out in §.16 of the original Seiberg-Witten paper [6]). By general theory of integral quadratic forms [28] (reviewed in the present context in [27]) the Tits form q induces an integral quadratic form q on Γ rad q which is positive-definite, hence Z-equivalent to the Tits form q g on the root lattice Γ g of a Lie algebra g of ADE type q ≃ q g hence Γ matter ≃ Γ g where g = A 1 , D 4 , E 6 , E 7 , E 8 respectively. (2.11) In the language of refs. [1,27] the two statements rank rad q = 2 and q ≃ q g are summarized in the fact that the mutation class of quivers with superpotentials (Q, W) which describe the BPS sector of our five theories is given by the elliptic Dynkin graphs g (1,1) of respective type g = A 1 , D 4 , E 6 , E 7 , E 8 (with all triangles oriented). The elliptic (or toroidal [29]) Lie algebra g (1,1) of type g is obtained by affinization of the affine Lie algebra g (1) of the same type; see [27] for more details. As a basis of the matter charges we take the simple roots α a of g. Then the matter charges of the BPS particle corresponding to the stable object X ∈ D b coh E/Z p are f a (X) = [X], α a sym ∈ Z, a = 1, 2, . . . , r(g). (2.12) In particular, for p = 1, 2 we get back that the flavor charges take values in the weight lattice of Sp(1) and SO(8), respectively. The same statement holds, in the present sense, for p = 3, 4, 6.
Serre duality and Coxeter transformation. The Abelian category coh E/Z p satisfies Serre duality in the form 10 where ω is the dualizing sheaf. τ is an auto-equivalence of coh E/Z p , and hence of the derived category D b coh E/Z p ; τ also plays the role of Auslander-Reiten translation [30]. Given that τ is an auto-equivalence, coh E/Z p has no non-zero injectives or projectives and the Abelian category coh E/Z p is hereditary (global dimension 1).
Since ω p ≃ O, one has τ p = Id. The Coxeter transformation Φ : Γ → Γ is defined by (2.14) In particular, for all i.e. the Yang-Mills magnetic/electric charges are the (+1)-eigenvectors of Φ. Likewise, the flavor charges are the (−1)-eigenvectors of Φ. The eigenvectors associated to eigenvalues λ = ±1 correspond to internal electric/magnetic charges of the matter AD systems. Note that Φ p = 1, in fact (for p > 1) where s is the number of branches of the associated star graph and p i are the number of nodes in the i-th branch (counting the vertex node), while p ≡ l.c.m.{p i }.
Auto-equivalences of the derived category. Suppose is an auto-equivalence (of triangulated categories). K induces an automorphism K of the Grothendieck group Γ Choosing a Z-basis in Γ, Φ and K may be seen as rank Γ × rank Γ matrices with integral entries. For all auto-equivalences K we have τ K = Kτ (since the Auslander-Reiten translation is unique). Then Φ and K, as matrices, commute and, in particular, K preserves the radical sublattice rad q ⊂ Γ; this already follows from the fact that all auto-equivalences are isometries of the Euler form Comparing with (2.9), we see that all auto-equivalence K of the derived category D b coh E/Z p induces a reduced additive map which commutes with the reduced Coxeter element Φ and is an isometry of the reduced Tits form q(·) It follows that the image of the automorphism group of the derived category, under the homomorphism ̺ : K → K is a subgroup of the finite group O(Γ g ) of the Zisometries of the positive-definite Tits form q g . More precisely, the image is a subgroup of the centralizer of the reduced Coxeter element Φ where D g is the Dynkin graph of g.
Remark. The triality of so(8) is a group of outer automorphisms; from the point of view of eqn. (2.25) this means that its image is not contained in Weyl(so (8)). For, say, g = E 8 , Aut(D g ) is trivial and the image of Aut(D b coh E/Z p ) is a subgroup of the Weyl group. The reader then may wonder in which sense the S-duality action is a generalization of triality which is an outer action. The point is that the duality action is outer with respect to the natural group of 'inner' automorphisms which is W (see eqn.(2.101)). For SQCD N f = 4 W is the full Weyl group but it is a small subgroup for E 6 , E 7 , E 8 .

Coherent sheaves on weighted projective lines
With the exclusion 11 of N = 2 * , the complete N = 2 gauge theories with gauge group SU(2) are in one-to-one correspondence with the weighted projective lines having non-negative Euler characteristic χ ≥ 0 [1,3]. SU(2) SYM coupled to a set of AD matter systems of types D p i (i = 1, .., s) corresponds to the weighted projective line with weights (p) = (p 1 , p 2 , . . . , p s ). The superconformal theories in this class are precisely the ones associated with the χ = 0 weighted projective lines 12 ; there are four such lines with weights (p) = (2, 2, 2, 2), (3,3,3), (2,4,4), (2,3,6), (2.26) the i-th weight p i being equal to the number of nodes in the i-th branch of the corresponding affine star graph (counting the vertex). The χ = 0 weighted projective lines will be written X p (where p ≡ l.c.m.(p i ) = 2, 3, 4, 6) or simply X. We have [12,22,32] and passing from elliptic orbifolds E/Z p to weighted projective lines X p is just a convenient shift in language.
Weighted projective lines [12-16, 33, 34]. Given a set of positive integral weights 13 p = (p 1 , p 2 , . . . , p s ) we define L(p) to be the Abelian group over the generators c is called the canonical element of L(p), while the dual element is Given the weights p and s distinct points (λ i : µ i ) ∈ P 1 we define a ring graded by L(p) The case of N = 2 * , i.e. p = 1, is rather similar. Indeed, the theory of coherent sheaves on the weighted projective lines of tubular type was constructed by Geigle and Lenzing in [12] using as a model the Atiyah description of coh E [31]. The main technical difference is that for p = 1 the canonical sheaf is trivial, while for p > 1 is a p-torsion sheaf. Then for p = 1 there is no tilting object. 12 Indeed, the coefficient of the β-function of the Yang-Mills coupling, g YM , is −2χ, see [1,3,4]. 13 For definiteness we write the p i 's in a non-decreasing order. Without loss we may assume p i ≥ 2.
where the degree of X i is x i and the degree of u, v is c. The weighted projective line X(p) is defined to be the projective scheme Proj S(p). Its Euler characteristic is The Picard group of X(p) (i.e. the group of its invertible coherent sheaves ≡ line bundles) is isomorphic to the group L(p) i.e. all line bundles are obtained from the structure sheaf O ≡ O(0) by shifting its degree in Any non-zero morphism between line bundles is a monomorphism [13,14]. In particular, for all line bundles L, End L = C. Hence, if (λ : µ) ∈ P 1 is not one of the special s points (λ i : µ i ), we have the exact sequence which defines a coherent sheaf S (λ:µ) concentrated at (λ : µ) ∈ P 1 . It is a simple object in the category coh X(p) (the 'skyscraper'). At the special points (λ i : µ i ) ∈ P 1 the skyscraper is not a simple object but rather it is an indecomposable of length p i . The simple sheaves localized at the i-th special point (λ i : Applying τ to these sequences we get In conclusion we have 14 [13,14] coh where H 0 is the full Abelian subcategory of finite length objects (which is a uniserial category) and H + is the subcategory of bundles. Any non-zero morphism from a line bundle L to a bundle E is a monomorphism. For all bundles E we have a filtration [13,14] Then we have an additive function rank : K 0 (coh X(p)) → Z, the rank, which is τ -invariant, zero on H 0 and positive on H + . rank E is the length ℓ of the filtration (2.39); line bundles have rank 1. In physical terms [3] H 0 is the 'light category' which encodes the zero Yang-Mills coupling limit g YM → 0; hence H 0 is well understood in terms of 'perturbative' physics [3].
We define the additive function degree, deg : Physically, rank is the Yang-Mills magnetic charge while deg is (a linear combination of) the Yang-Mills electric charge (and matter charges) normalized so that the W boson has charge +1. For the four weighted projective lines X p with χ(p) = 0, eqn.(2.26), the Riemann-Roch theorem reduces to the equality [12][13][14] 1 p (2.41) Explicit formulae in the canonical basis. To write explicit expressions, it is convenient to choose a set of homological generators of coh X(p); their classes then give a Z-basis of the Grothendieck group K 0 (coh X(p)). It is convenient to choose the generators to be the direct summands of a tilting object of coh X(p) [12][13][14]. We choose the canonical 15 such tilting object whose endomorphism algebra is the Ringel canonical algebra Λ(p) of type (p) [12][13][14].
The canonical generating set consists of the following n ≡ i ( By definition of tilting object, Ext 1 vanishes between any pair of sheaves in eqn.(2.42), while the only non-zero Hom spaces are We write (φ 1 , . . . , φ n ) for the elements of the basis of K 0 (coh X(p)) given by the Grothendieck classes of the n line bundles in eqn. (2.42 listed in the (i, j) lexicographic order. The YM magnetic and electric charges of the generating sheaves (2.42) are Specializing to the χ(p) = 0 case. For the χ(p) = 0 weighted projective lines, eqn.(2.26), the electric charges of the sheaves (2.42) are just the Coxeter labels on the i-th branch of the associated affine star Dynkin graph g (1) numbered in increasing order from the most peripheral node to the vertex of the star (which has label p). Explicitly, The Euler form between the elements of the basis E ab = φ a , φ b E is given by the unipotent (upper triangular) block matrix where 1 p stands for the column (p − 1)-vector with 1's in all entries, and T p is the (p − 1) × (p − 1) triangular matrix with 1's along the main diagonal and everywhere above it.
In the canonical basis the Coxeter element is represented by the matrix Φ ab such that (2.49) The radical of the Tits form q is generated by the two vectors 50) In particular, we note that the last two entries of the radical vector E −1 (pM − Q) are y 1,1 y 3,4 y 3,3 y 3,2 y 3,1 y ⋆ y 2,1 y 2,2 Figure 1: The assignments of integral variables (y) to the nodes of the E 8 graph. For D 4 , E 6 , E 7 just restrict to the corresponding Dynkin subgraph.
(· · · , 1, 0) t . It follows that in this basis we may identify the matter charge lattice Γ matter ≡ Γ/rad q with the sublattice Γ matter ⊂ Γ of vectors of the form Since E is upper triangular and unimodular, it maps a basis of the sublattice Γ matter into a Z-equivalent basis. Then we write the vectors (2.52) in the form LetĖ (resp.Ë) be the principal submatrix of E obtained by omitting the last (resp. the last two) row(s) and column(s). Essentially by definition, the Cartan matrix of the finite-type Lie algebra g. (2.55) Then the restricted Tits form q on Γ matter ≃ Γ matter is where q g (y) = 1 2 y t C g y is the Tits form of the finite-dimensional Lie algebra g. The integers (y) = (y ⋆ , y i,j ) are attached to the vertices of the Dynkin graph g as in figure 1. In particular, q and q g are Z-equivalent; the isometry between the two Tits forms is just multiplication by Remark. Here we defined the matter charges as equivalence classes in Γ/rad q. Of course, the physical matter charges are specific representatives of these classes. In §.3.1 we shall use the physical definition. However, the action of the S-duality group is independent of the choice of representatives, and often a different choice simplifies the computations.
Structure of the derived category D b coh X(p). The Abelian category H = coh X(p) is hereditary. The derived category D b H of a hereditary Abelian category H coincides with its repetitive category rep H [13,14].
Definition. Let A be an Abelian category. Its repetitive category rep A is The slope µ(E) of a coherent sheaf E is the ratio of its degree and rank 16 The χ(p) = 0 case. We restrict ourselves to the four weighted projective lines with χ = 0, i.e. to X p with p = 2, 3, 4, 6. In this case all indecomposable coherent sheaf is semistable [12,13]. Let H (q) be the full (hereditary) Abelian subcategory of semi-stable objects of slope q ∈ Q ∪ ∞. For all q one has H (q) ≃ H 0 the full subcategory of finite-length objects. Then and hence where the notation implies that Hom H (H (q) , H (q ′ ) ) = 0 for q > q ′ . H 0 (and then H (q) for all q) is a P 1 -family of stable tubes all of which are homogenous but the ones over the three (or four) special points (λ i : µ i ) which have periods p i > 1.
In conclusion, all indecomposable object X of the derived category D b coh X p belongs to a H (q) [n] for some q ∈ Q and n ∈ Z; all these subcategories H (q) [n] are equivalent to the 'perturbative' category H 0 and hence physically well understood [3].
2.3 Telescopic functors and B 3 braid group action on D b coh X p If our N = 2 theory has S-duality, the duality should be, in particular, a property of its BPS sector. Hence the duality should act by automorphisms of the relevant derived category or, more precisely, of its orbit category (2.1). For the five SU(2) SCFTs we expect the S-duality group S to contain a P SL(2, Z) from the internal torus of its 6d construction. Before entering in the technical details, let us see why this fact is rather natural in view of the peculiar form of the derived category as described in eqn.(2.63). We may identify Q ≡ Q ∪ ∞ with the projective line over the field Q. The group P SL(2, Z) naturally acts on P 1 (Q). Then the structure in the rhs of (2.63) suggests the existence of an action of P SL(2, Z) on D b coh X p which sends an object X of slope q ∈ P 1 (Q) into an object of slope q ′ = γ · q ∈ P 1 (Q) for γ ∈ P SL(2, Z). Since the slope is essentially the ratio of the Yang-Mills electric and magnetic charges, such a natural P SL(2, Z) action will have the physical interpretation of the electro-magnetic S-duality.
In order to implement this idea, we first need to normalize correctly the Yang-Mills charges so that they are integral, while the degree is quantized in units of 1/p. So, for all object X of the derived category, we set To prove that the electro-magnetic duality group SL(2, Z) is part of Aut(D b coh X p ) we need to construct two functors T and L, which are auto-equivalences of the triangulated category D b coh X p , and have the property that so that the subgroup of Aut(D b coh X p ) generated by the two functors T, L will induce an SL(2, Z) action on the Yang-Mills charges YM(·). Such auto-equivalences T , L do exist: they are called telescopic functors [15,16,33].
T is simply the functor which shifts the L(p) degree of the sheaf by x 3 [14, 16, 33] where we ordered the weights so that p 3 ≡ p is the largest one. One can see the map X −→ T (X) as the completion of a canonical map to a triangle of D b coh X p [15,16,33] p−1 j=0 Explicitly, the action on the generating set O, S i,j is given by Thus T preserves the rank, while increases the degree by 1/p times the rank; therefore as required. The definition of the second functor L is similar; one introduces the triangle and proves that X −→ L(X) is an auto-equivalence of the derived category, see [15,16]. The action of L on the Grothendieck group is then where we used the Riemann-Roch theorem (2.41). Thus we get the desired property YM(L(X)) = L YM(X). (2.77) The explicit action on the generating set O, S i,j (j = 0, 1, . . . , p i − 1) is which, in the particular case p i = p, reduces to The B 3 braid group relation. It is easy to see that [15,16] LT L = T LT. (2.81) As an illustration (and to establish a few useful equalities), we check that the two sides of the equality act in the same way on the structure sheaf O and on the simple sheaves S 3,j (assuming p 3 = p). By proposition 5.3.4 of [16] this suffices to conclude that the two automorphisms of D b coh X are at least isomorphic. Since τ commutes with T, L,  where Σ stands for the shift functor, Σ(X) = X [1]. Hence (T L) 3 induces an automorphism (T L) 3 of the Grothendieck lattice Γ of order 2, 2, 4, 2 for p = 2, 3, 4, 6, respectively. Consider the automorphism τ 3 (T L) 3 Σ −1 : (i) it fixes the structure sheaf and the simples S 3,j , (ii) it preserves the slope µ(X) = deg(X)/rank(X), and (iii) its action on Γ has order dividing 2, 2, 4, 2, respectively. The group of slope preserving automorphisms of D b coh X which fix O is precisely Aut(X) i.e. (essentially) the permutations of equal weight special points in P 1 . Thus τ 3 (T L) 3 Σ −1 is a permutation of the special points which respects their weights and leaves the third one fixed (by convention p 3 = p). Hence τ 3 (T L) 3 Σ −1 = 1 for p = 4, 6. For p = 2, τ 3 (T L) 3 Σ −1 is either the identity or an order 2 permutation of three objects (x 1 , x 2 , x 4 ) which treats the three objects on the same footing; hence τ 3 (T L) 3 is either the identity or the permutation π 12 of the first two special points. From their explicit action on the Grothendieck group, see §.A.1, we conclude for the second possibility. Hence where π 12 is the automorphism which interchanges the first two special points i.e., in terms of the canonical generating set (2.42), We note that (T L) 3 acts as −1 on the Yang-Mills charges and as +1 on the flavor charges.

2.4
The groups Aut(D b coh X) and S ≡ Aut C(X) The subgroup B 3 generated by L, T is not the full automorphism group Aut(D b coh X).
We have a surjection of the automorphism group on B 3 whose kernel is the group of the automorphisms of the Abelian category coh X which fix degree and rank. This is the group Pic(X) 0 ⋉ Aut(X), where Pic(X) 0 is the group of degree zero line bundles and Aut(X) is the group of geometric automorphisms of X, essentially 18 the group of permutations of the special points having the same weight p i . Thus, for a weighted projective line X having zero Euler characteristic, χ(X) = 0, we have [15,16] In particular, the derived auto-equivalence the situation is slightly subtler [15,16]. We shall ignore this aspect.
The cluster category of the weighted projective line X is defined to be the orbit category of D b coh X with respect to the cyclic subgroup generated by τ −1 Σ . Its quotient group acting effectively on the Yang-Mills charges (modulo the Weyl action of the gauge group) is P SL(2, Z). The physical interpretation of the kernel subgroup will be discussed in section 3.

Some useful formulae
Note that both our generating functors L, T ∈ Aut(D b coh X) are of the form X −→ L Y (X) where, for a fixed object Y , the functor L Y is defined by the triangle A basic theorem (see theorem 5.1.3 of [16]) states that, for a weighted projective line with χ(X) = 0, a functor X −→ L Y (X) is an auto-equivalence of the derived category iff (i) p is length of the τ -orbit of the object Y , and (ii) the object Y is quasi-simple. These two conditions imply, in particular, 19 104) 19 Here δ from which we get We shall see in §.4.2 that this last equation guarantees that the induced action in the Grothendieck group is an isometry of the Euler form i.e.
which is (obviously) a necessary condition in order L Y to be an auto-equivalence. In the X 2 case τ 2 = Id, so Φ 2 = 1. The Yang-Mills charges correspond to the (+1)eigenvectors of Φ and the flavor charges to the (−1)-eigenvectors. To make everything very transparent, in the next subsection we illustrate how flavor Spin(8) triality arise in concrete terms i.e. writing down explicit expressions in the basis of Γ matter which is standard in physics. Then in §.3.2 we present a more elegant abstract viewpoint. The reader may prefer to skip §.3.1.

Flavor charges and flavor weigths
The four linear independent flavor charges are Γ flavor is thus identified with the root lattice of Spin (8). An object X of D b coh X 2 then carries a weight (w 1 (X), w 2 (X), w 3 (X), w 4 (X)) of the flavor Spin (8). Explicitly, (3.7) (Note that the Euler form is symmetric if one of its arguments is a flavor charge). Using (3.7) we easily compute the flavor weights of (i) the exceptional simples (3.1), (ii) the degree zero line bundles (which correspond to monopoles with magnetic charge +1 and zero electric charge), and (iii) the degree 1 line bundles (dyons with unit electric and magnetic charges), see table 2. We see that the quarks have the weights of the vector representation v of Spin(8), the monopoles of the spinorial representation s, and the dyons of the spinorial representation c. Of course, this is the physically correct result [6].  sheaf (w 1 , w 2 , w 3 , w 4 ) sheaf (w 1 , w 2 , w 3 , w 4 ) (1, 0, 0, −1)

Spin(8) weights of degree zero line bundles
sheaf (w 1 , w 2 , w 3 , w 4 ) sheaf (w 1 , w 2 , w 3 , w 4 ) Spin(8) weights of degree 1 line bundles in other words by the automorphism α 3 ←→ α 4 of the Dynkin graph (3.6). This is precisely the non-trivial element of the Spin(8) triality group S 3 which maps the weights of v into themselves, consistently with the fact that the T -duality makes sense at weak coupling where the quark fields keep their identity. The functor L acts as which are the weights of the spinorial representation c. Thus L maps the Spin(8) vector representation v into the spinorial representation c. However, the action of L on the weights is not simply the automorphism α 1 ←→ α 4 of the Dynkin graph (3.6) but rather the composition of this graph automorphism with a Spin(8) Weyl transformation. In order to see this, we need to look at the full automorphism group Aut(D b coh X 2 ).
From eqn.(2.96) we know that, besides the braid group generated by by T , L we have the automorphism group Pic(X 2 ) 0 ⋉ Aut(X 2 ). Now Pic(X 2 ) 0 = (Z/2Z) 4 (Z/2Z) diag , (3.14) while, neglecting the subtlety already mentioned 22 , we may effectively take Aut(X 2 ) ≃ S 4 , the permutation of the four special points in P 1 . Then (8)). (3.15) Let Π be the autoequivalence of the derived category associated with the permutation π = (14)(23) of the four special points in P 1 . We define a new auto-equivalence of D b coh X 2 L = ΠL.
(3.16) 22 The points is as follows: a permutation of the four special points does not give back the same X 2 but rather a new one with a different cross-ratio of the four special points. Since this cross-ratio does not enter anywhere in the BPS sector, as far as we are interested to BPS physics, we may identify all X 2 and consider S 4 to be a symmetry of the physics. For a generic configuration of the four points the actual Aut(X 2 ) is the Klein group. Hence the operation Π defined in the text is an actual automorphism of the derived category.
L has the same action on the Yang-Mills charges as L. It follows from eqn. (3.13) that the Spin(8) weights of L(S i,1 ) are equal to those of O( x π(i) ); comparing with table 2 we see that the effect of L on the flavor weights is simply w 1 ( L(X)), w 2 ( L(X)), w 3 ( L(X)), w 4 ( L(X)) = w 4 (X), w 2 (X), w 3 (X), w 1 (X) , (3.17) i.e. L induces the graph automorphism α 1 ↔ α 4 which interchanges the Spin (8) spinorial representations v ↔ c. However, L is less convenient than L since LT L = T LT .

Relation with the F 4 root system
Let us look to the action of B 3 on the flavor charges from a more conceptual standpoint. We start from the formulae proven in §.2.5 which we specialize to N f = 4 SQCD. In this case both O and S 3,0 have τ -period p = 2. For p = 2 the action of L Y on the Grothendieck group, eqn.(2.106), reduces to eigenvectors of Φ i.e. they are a Yang Mills and a matter charge, respectively. Hence, the induced action on the matter lattice is Γ matter ≡ Γ/rad q is simply where ·, · sym is the inner product on Γ matter ≃ Γ D 4 given by the Cartan matrix. The action of an automorphism is an isometry of the positive definite symmetric pairing; this requires which is automatic in view of eqn.(2.105). From eqns.(2.96)(3.15) we conclude that Aut(D b coh X 2 ) acts 23 on the flavor root lattice Γ so (8) as the group of reflections generated by its elements of square-length 2 and 4. This group is isomorphic to Weyl(F 4 ) [35]. One has The triality group is simply Weyl(F 4 )/Weyl(SO (8)). 23 Neglecting the subtlety in footnote 22.

The cluster category
Around eqn.(2.1) we claimed that the triangulated category which describes the physics of the BPS sector is not D b coh X itself but rather an orbit category of the form D b coh X/C where C is a certain subgroup of Aut(D b coh X). The subgroup C is determined by nonperturbative physical considerations, in particular by the study of the quantum monodromy M(q) and its fractional powers [21,24]. In the case of a Lagrangian field theory, however, the group C should also have an elementary interpretation in terms of conventional weakcoupling physics. Then we use N f = 4 SU(2) SQCD as a convenient example to shed light on the physical meaning of the cluster category. We claim that passing from D b coh X 2 to C(X 2 ) in this example corresponds to implementing the Gauss' law on the physical states. Indeed, the center Z(B 3 ) of the braid group B 3 is the infinite cyclic group generated by (T L) 3 = (LT ) 3 . For p = 2 one has the equality and hence (T L) 3 acts on the Yang-Mills charges YM(X) as multiplication by −1, while fixing all flavor charges. Thus (T L) 3 acts on Γ as the Weyl group of the gauge group SU (2), which is a gauge transformation belonging to the connected component of the identity which leaves invariant all physical states satisfying the Gauss' law. Hence (T L) 3 ≡ τ −1 Σ should act trivially on the physical states, i.e. two objects in the same τ −1 Σ-orbit should be considered the same 'physical' object. Then the natural triangulated category which is associated to the physical BPS states is which is precisely the definition of the cluster category for the coherent sheaves on the weighted projective line of type (2, 2, 2, 2) (equivalently of the canonical algebra of the same type) [17,36]. It is convenient to see the orbit category C 2 ≡ D b (coh X 2 )/ τ −1 Σ as a (triangulated) category with the same objects as the triangulated category D b (coh X 2 ) and morphism spaces (3.24) see appendix B for further details. Then the isoclasses of C 2 objects are the orbits of isoclasses of objects of the derived category, and the physical BPS states -satisfying Gauss' law -are in one-to-one correspondence with the isoclasses of stable C 2 objects. The physical meaning of equation (3.24) is that defining the "physical" morphism space to be the direct sum of morphisms in the original category from X to gY for all g, makes the morphism to account correctly for all the possible "relative gauge orientations" of the two objects when they form a BPS bound state.

.1 Generalities
We have seen in §.2.1 that the matter charges of the five g (1,1) r models (g = A 1 , D 4 , E 6 , E 7 , E 8 ) take values in the weight lattice Γ w g of g. By this we mean that they are valued on a lattice endowed with the Cartan (integral) symmetric form ·, · sym which is preserved by all symmetries and dualities of the quantum field theory. On the root lattice Γ g there is, in addition, an integral skew-symmetric form ·, · Dirac which is also preserved by all dualities. Indeed, the reduced Coxeter element Φ centralizes the action of all dualities on Γ g , cfr. eqn. (2.25). However there are two major differences between the Lagrangian models A Indeed, in section 3 we saw that, in the two Lagrangian models, the class of coherent sheaves X with a fixed Yang-Mills charge YM(X) ≡ (p deg X, rank X) t have the correct g weights to form a complete representation of g. This is certainly not true for p = 3, 4, 6. For instance, the number of zero-degree line bundles is equal to the order of the restricted Picard groups and no non-trivial representation of E 6 , E 7 and E 8 have dimension 9, 8, and 6. Alternatively we may note that Weyl(E r ) is not a triangle auto-equivalence for E (1,1) r models. The subgroups of Aut(C(X p )) which fix the rank and the degree are listed in the second column of the following table: They certainly do not contain the Weyl group of any Lie group but those listed in the third column and their subgroups. The lattice of flavor charges has rank 0, 3, 2 for p = 3, 4, 6. For p = 4 Γ fl is generated by the three classes The symmetric form on Γ fl is not related in any obvious way to a Cartan matrix, which is consistent with the fact that (contrary to the p = 2 case studied in sect. 3) physically we do not expect any non-Abelian enhancement of the flavor symmetry for the p = 4 model. Aut(X 4 ) just permutes the two identical matter AD subsystems of type D 4 , and acts on the flavor charges as The two generators of Pic(X 4 ) 0 act as 24 O( x 1 − 2x 2 ) : (α 1 , α 2 , α 3 ) → (α 3 , α 1 + α 2 + α 3 , α 1 ).

Emergence of Shephard-Todd complex reflection groups
From §.2.5 we know that the action of the two functors L and T on the Grothendieck group Γ is given by where we used (4.13) where {φ a } n a=1 is the canonical basis of Γ (cfr. §.2.2) and [X] a ∈ Z, we have 25 (4.14) T and L commute with Φ and, for p = 3, also with the permutation π 12 in eqn.(2.95). We decompose V in eigenspaces of the semisimple linear map Φ, are where v = (1, 0, · · · , 0) t , w = (0, · · · , 0, −1, 1) t .  model. The minimal polynomial for the n×n matrices T , L which give the action of the telescopic functors T , L on the Grothendieck group Γ is then Indeed, more generally, the following identities are true for all analytic function f (z) eqn. (4.24) being the special case f (z) = z p . In addition, the following identities hold (see appendix A) preserves the positive-definite Hermitian product induced by the Euler form as well as the lattice Γ matter ⊂ V matter . Hence the quotient of B 3 which acts effectively on V matter is a finite group G matter . This group is generated by the restriction of the matrices T and L to the (n−2)-dimensional space V matter . By abuse of notation, we denote these 'matter' restrictions by the same symbols T , L. Then the finite group G matter is generated by the two elements T , L subjected at least to the obvious relations Given the underling real structure on the complex vector space V matter induced by the lattice Γ matter we know that the G matter -modules V s and V p−s are conjugate The case V s is one-dimensional. If dim V s = 1 the action of both generators T s , L s is just multiplication by e −2πis/p , and the quotient group of G matter which acts effectively on V s is the cyclic group Z p gcd(s,p) Z. unitary matrices with all but one eigenvalues equal 1 [9], cfr. eqn. (4.29). A group generated by complex reflections is called a complex reflection group. We conclude that the quotient group G s of G matter which acts effectively on V s is a finite complex reflection group.

Finite complex reflection groups
Finite complex reflection groups have been classified by Shephard and Todd [9]: there are three infinite families and 34 exceptional groups denoted G 4 , · · · , G 37 . Real reflection groups (Coxeter groups) and rational reflection groups (Weyl groups) are encoded in Dynkin graphs; complex reflection groups are also encoded in certain graphs of a more general kind [10]. We look for Shephard-Todd groups generated by two elements T , L satisfying relations which imply (4.28) with p = 3, 4, 6. A part for the Weyl group of SU(2) (which corresponds to the already discussed D  G 4 is a subgroup of U(2) which as an abstract group (i.e. forgetting its realization as a complex reflection group) is maps (4.37) into the standard presentation of G 4 . However its explicit realization as a subgroup of SU(2) does not realize it as a complex reflection group. To get a complex reflection group we must twist the standard quaternionic degree 2 Klein realization Q of T by the character χ of one of its two non-trivial one-dimensional representations (the two choices producing equivalent results 26 ). Thus, G 4 is the subgroup of U(2) generated by the two matrices 4.39) or, explicitly, where ω, ǫ are primitive roots of unity of order, respectively, 3 and 8. The irreducible representations of G 4 may be then read directly from the character table of the binary tetrahedral group T . In the same fashion, G 8 is a subgroup of U(2) which is a central extension by Z/2Z of the binary octahedral group O, i.e. of the subgroup of SU(2) associated to the affine graph E with ǫ as before. Again we may obtain the irreducible representations of G 8 by twisting the character table of O. We claim that G 8 is a central extension by Z( To show our claim, the only thing we have to prove is that the quotient of G 8 by its center Z(G 8 ) is the group P SL(2, Z/4Z). Now (4.44) 26 The two choices are related by interchanging the defining representation and its conjugate R ↔ R. 27 T is the 2 × 2 matrix defined in (2.65). In eqn.(4.43) we use that L −1 is conjugate to T in P SL(2, Z). (4.45) The affine diagrams g (1) in the second column are the McKay graphs of the double cover of the P SL(2, Z/nZ) group which is a finite subgroup of SU (2). It is amusing that the three Lie algebras E 7 and E 8 appear again in the game, now as quotients of the modular group by principal congruence subgroups Γ(n). As abstract groups, the dihedral group of order 6 is isomorphic to S 3 , while the tetrahedral one is isomorphic to A 4 .

The E
(1,1) 6 model Let us look case by case. For p = 3 we have dim V 1 = dim V 2 = 3. Comparing the presentation of G 4 in table 3 with eqn.(4.28) we conclude that V 1 , V 2 carry a representation of G 4 ≃ SL(2, Z/3Z). We decompose V s (s = 1, 2) into irreducible representations of G 4 . First we rule out that V s decomposes into the direct sum of three one-dimensional representations since, in that case we would get (T L) 3 = 1 (by eqn.(4.35)), while we know that (T L) 3 is the non-trivial involution −π 12 ; its action on V s has eigenvalues (−1, −1, 1). We also rule out the irreducible three dimensional representation of G 4 -which coincides with the 3-dimensional irrepresentation of T -which, being real, cannot be a complex reflection subgroup of U(3). We remain with the defining two-dimensional representation R in eqn.(4.40) and its conjugate R. For the defining representation R (with the standard choice ω = e 2πi/3 ) we have eigenvalues of T , L in R = (e −2πi/3 , 1), (4.46) and we conclude that where 1 stands for the trivial representation. As a further check, note that in the representation R, (T L) 3 = −1 so that the eigenvalues of (T L) 3 acting on V s are (−1, −1, +1) as 28 The case n = 5 in eqn.(4.45) is never used in our present analysis: we listed it merely to give the complete statement of Klein's fundamental result. The case n = 6 is radically different, since the triangle group (2, 3, 6) yields a tessellation of the Euclidean plane, and hence is an infinite group, while all groups of interest here must be finite by the argument around eqn. (2.25). We have already pointed out that the case of the E (1,1) 8 model, corresponding to p = 6, is slightly different from the other ones: using the Chinese remainder theorem, P SL(2, Z/6Z) is written as the product of two triangle groups instead of a single triangle group as for p = 2, 3, 4, cfr. eqn.(1.6). they should.

The E
(1,1) 7 model By comparing presentations, we see that for p = 4 the space V s (s = 1, 2, 3) carry a representation of the Shephard-Todd group G 8 . We have Next we consider the space V 2 which is spanned over C by the three strict-sense flavor charges of the QFT model. The eigenvalues of T , L in V 2 are (+1, +1, −1) so G matter acts on V 2 as a real reflection group, i.e. a Coxeter group. The central element (T L) 3 acts as 1.
Since L 2 and T 2 act trivially, the group which acts effectively on V 2 is P SL(2, Z)/N(2) ≡ P SL(2, Z)/Γ(2) ≡ P SL(2, Z/2Z) ≃ S 3 . (4.51) We have two possibilities: either V 2 is two copies of the trivial representation of the symmetric group S 3 plus the sign one-dimensional representation σ, or it is the trivial representation plus the irreducible degree 2 representation of S 3 , W . To distinguish the two possibilities, note that in the first case T L acts as the identity, and then (in particular) Using eqn.(2.85), the above sum becomes theory are permuted by the action of the S-duality group which acts through its S 3 ≡ P SL(2, Z/2Z) factor group, cfr. eqn. (1.13). This fact was already noted in refs. [39].

The E
(1,1) 8 model For p = 6 we have (4.55) The central element (T L) 3 acts as multiplication by −e πis on V s . On the one-dimensional representations V 1 , V 5 , the generators L, T act as multiplication by e −πi/3 and e πi/3 , respectively. We denote these two characters by χ and χ.
The two conjugate representation V 2 and V 4 are irreducible by criterion (4.35). Acting on these representations we have  Again, we have two possibilities: either (i) V 3 is the direct sum of the trivial and the sign representations, V 3 = 1 ⊕ σ, or (ii) V 3 is the irreducible two-dimensional representation W of S 3 . Again, the first possibility implies T L = 1. We repeat the diagnostics in eqn.(4.52) Explicitly the sum has the form while the last sum does not vanish mod rad q. Again, we conclude that In this appendix, we write the explicit matrices which give the action of T and L on the canonical basis i.e.
where {L a } n a=1 are the line bundles in eqn.(2.42) such that φ a = [L a ]. From section 4 we have where E is the upper triangular matrix in eqn. (A.14) A.3 E  15) which satisfy the identities:

B Cluster-tilting
In this appendix we give some more details on the relation between the present approach to the BPS spectra of the four SCFT D -which is based on the Abelian category coh X p -and the standard quiver approach [1,26] which is based on the module category of the Jacobian algebra CQ/(∂W) of the quiver Q with superpotential W.
We write T for the direct sum of all sheaves of the canonical basis (2.42) We already know that the canonical algebra Λ p having the same weight type (p 1 , · · · , p s ) as the line X p is given by (here H p = coh X p ) However, Λ p does not coincide with the Jacobian algebra CQ/(∂W) for any choice of (Q, W) in its mutation class. What is true [3] is that there exists a Q in the class which is the completion of the quiver Q can of Λ p : Q contains (s − 2) extra arrows going from the sink of Q can to its source and Q is endowed with a superpotential linear in the new arrows η a so that the Jacobian relations ∂W/∂η a give back the original relations of Λ p . For instance, for E (1,1) r we add just one new arrow η and the superpotential becomes W = η(X p 1 1 + X p 2 2 + X p 3 3 ). (B. 3) The modules of Λ p are then identified with the class of modules of the Jacobian algebra with η = 0. One may wonder whether our treatment 'forgets' the modules with η = 0. The answer is that these modules are already properly taken into account thanks to the properties of the cluster-category C p .
The category C p has the same objects as D b H p and morphism spaces and hence it is equivalent [17] to the category H p having the same objects as H p and Z 2graded morphism spaces Hom Hp (X, Y ) = Hom Hp (X, Y ) ⊕ Ext 1 Hp (X, τ −1 Y ), (B.5) with the appropriate composition law [17]. The tilting object T ∈ H p in eqn.(B.1) is also a cluster-tilting object [37] for C p (for a review see [38]). With respect to H p , the category H p In particular, the indecomposable objects of the category H p ≡ coh X p are the indecomposable modules of the Jacobian algebra together with the n ≡ r(g) + 2 line bundles O( a + ω) with a ∈ C (cfr. eqn.(B.1)).