The periodic sl(2|1) alternating spin chain and its continuum limit as a bulk Logarithmic Conformal Field Theory at c=0

The periodic sl(2|1) alternating spin chain encodes (some of) the properties of hulls of percolation clusters, and is described in the continuum limit by a logarithmic conformal field theory (LCFT) at central charge c=0. This theory corresponds to the strong coupling regime of a sigma model on the complex projective superspace $\mathbb{CP}^{1|1} = \mathrm{U}(2|1) / (\mathrm{U}(1) \times \mathrm{U}(1|1))$, and the spectrum of critical exponents can be obtained exactly. In this paper we push the analysis further, and determine the main representation theoretic (logarithmic) features of this continuum limit by extending to the periodic case the approach of [N. Read and H. Saleur, Nucl. Phys. B 777 316 (2007)]. We first focus on determining the representation theory of the finite size spin chain with respect to the algebra of local energy densities provided by a representation of the affine Temperley-Lieb algebra at fugacity one. We then analyze how these algebraic properties carry over to the continuum limit to deduce the structure of the space of states as a representation over the product of left and right Virasoro algebras. Our main result is the full structure of the vacuum module of the theory, which exhibits Jordan cells of arbitrary rank for the Hamiltonian.


Introduction
Many applications of conformal field theory to statistical mechanics and condensed matter physics are related to the case of central charge c = 0. These applications include the statistical properties of critical geometrical models such as self-avoiding walks (polymers) or percolation, and the critical properties of non interacting 2 + 1 dimensional disordered electronic systems -for instance, at the transition between plateaux in the integer quantum Hall effect.
Unfortunately, apart from some supergroup WZW models, the only well understood c = 0 conformal field theory is the trivial, minimal and unitary, one, which contains no field but the identity. This corresponds to the existence of the trivial modular invariant partition function Z = 1 at c = 0, which is the result obtained by calculating the n → 0 limit of the partition function of O(n) models or replicated disordered systems, or the Q → 1 limit of the Q-state Potts model (see the recent review [2]). Of course, these geometrical or disordered models have non trivial observables and critical exponents. But to observe them, one needs to understand what is happening "outside" the minimal trivial theory. While this issue was identified rather early [3], it has proven surprisingly hard to control entirely. For instance, despite the huge progress realized in determining exponents for polymers and percolation, including rigorous work and connections with the SLE formalism, there is to this day no agreement -let alone a consistent proposal -on what "the" proper conformal field theoretic description of say percolation clusters could be. This means in particular that very little is known about four point functions of geometrical observables in the bulk, despite the well established existence of measurable, universal quantities [4].
An aspect that got rather quickly understood is that the introduction of non trivial observables in say percolation forces one to consider a non unitary conformal field theory. While this is, on general grounds, a rather small price to pay for the description of observables which are most of the time non local in geometrical terms, the non-unitarity of the c = 0 theory one is after is bound to be rather violent (in contrast say to what happens in the Lee-Yang singularity), and leads in particular to the emergence of logarithmic features [5]. The adjective logarithmic refers to logarithmic dependence of the four point functions on the harmonic ratio, and to logarithmic terms in the OPEs. It does not imply that the field theory is in any way non local, but rather that the representations of the Virasoro algebra which are involved in the description of the Hilbert space of the theory are not fully reducible. In other words, the action of the dilatation operator L 0 is not always diagonalizable [6].
It turns out to be pretty hard to deal with logarithmic conformal field theories in general, and for many years the field has seen but little progress. The difficulty can be traced back to the complexity of Virasoro algebra representation theory once the criterium of unitarity -or semi-simplicity -has been relaxed. In fact, the study of LCFTs appears at first sight at least as difficult as the study of non semi-simple Lie algebras, which is proverbially intricate indeed -and plagued most of the time by wilderness issues [7].
Nevertheless, the study of WZW models on super groups [8,9], the construction of restricted classes of indecomposable modules [10,11,12], and the discovery of deep relations with the theory of associative algebras [13,14,1] have suggested that the problem, however hard, might not be impossible. In the last few years, based in part on the analysis of lattice regularizations and of the deeper role played by quantum groups at roots of unity [15,16], a lot has been understood about boundary LCFTs. There are now reasonable conjectures about the general structure of Virasoro indecomposable modules [17] and the fusion rules [18,14,1,12,19,20,21,22], methods to determine the logarithmic couplings (indecomposability parameters) [23,24], etc., see the recent reviews [25,26]. An important role in these recent developments has been played by algebra, and concepts such as projective and tilting modules, which we review below.
The case of bulk LCFT remains however less understood. Indecomposable features now occur not only within the chiral and antichiral sectors, but also in the way they are mixed, and there is evidence that the relationship between bulk and boundary LCFTs is considerably more intricate than for unitary CFTs [27,28,29]. This paper is the fourth in a series [30,31,32] aimed at extending in the bulk case a lattice approach that was quite successful in the boundary case. Rather than try to build a c = 0 LCFT abstractly, we focus on a well defined, local, lattice model, the sℓ(2|1) Heisenberg spin chain with alternating fundamental and conjugate fundamental representations. This chain is closely related -this is discussed in details below -with the properties of the hulls of percolation clusters. It is gapless, and can be argued to have a conformal invariant continuum limit indeed, which must have c = 0 and be logarithmic. In a nutshell, our strategy is to infer as many properties of this LCFT as possible from our analysis of the lattice model. Despite the fact that we focus on what is, after all, a very simple model, the endeavor remains highly difficult, and our results will be presented in a series of papers.
One of the key ideas at the root of our strategy is that most indecomposable aspects of the LCFT are already present on the lattice, in finite size, the algebra of local energy densities playing the role of a lattice version of the Virasoro (or product of left and right Virasoro) algebra. For the model we consider here (as well as in our previous papers), this algebra is the Temperley Lieb algebra in the boundary case, and the Jones Temperley Lieb algebra in the bulk case. A representation theoretic analysis of these algebras acting on our models is thus a perquisite to our understanding of the logarithmic properties of their continuum limit, and will occupy us in a great deal of the present paper.
We start in section 2 by discussing the percolation problem and how it is related with conformal field theory. We then focus on properties of cluster boundaries -so called hulls. We recall how they are formulated in terms of an alternating sℓ(2|1) super spin chain, which is expected to flow, in the continuum limit, to a super projective sigma model [33,34]. We also remind the reader of basic considerations about modular invariance, observables, and logarithmic features. Section 3 provides reminders on the algebraic description of the sℓ(2|1) spin chain, and some basic representation theoretic properties of the corresponding algebras JTL au 2L (m) and JTL 2L (m), both in the generic case, and in the special case m = 1 of interest here.
In section 4 -which is the most important of this paper -we discuss the decomposition of the sℓ(2|1) spin chain over indecomposable representations of JTL 2L (1). This involves several technical aspects. First, we make the crucial observation -discussed in more detail in appendix B -that the sℓ(2|1) spin chain provides a faithful representation of the algebra JTL 2L (1), in sharp contrast with the gℓ(1|1) spin chain studied in [30,31], where the corresponding representation was not faithful. The continuum limit of this gℓ(1|1) spin chain is the ubiquitous c = −2 symplectic fermions theory, a rather simple example of LCFT, which is quite well understood now: accordingly, the indecomposable modules appearing in this spin chain are of very manageable form. We believe this is due to the non faithfulness, and the fact that the gℓ(1|1) spin chain sees only a small part of the complexity of the full JTL 2L (0) algebra: this complexity would be revealed in the gℓ(2|2) spin chain, whose continuum limit, although also having c = −2, is considerable more involved than symplectic fermions. Note that, in contrast with the periodic version, open gℓ(1|1) and sℓ(2|1) spin chains both provide faithful representations of the the ordinary Temperley-Lieb algebra, and the modules appearing in their continuum limit are of similar complexity.
Faithfulness gives us access to powerful tools in the analysis of the spin chain, especially when combined with the fact that JTL N (m) is a cellular algebra. This requires manipulating several key concepts of the theory of associative algebras, which are explained in section 4.1. In section 4.2, we discuss the structure of projective modules over JTL N (1). In section 4.3, we discuss why, the spin chain admitting a non degenerate bilinear form, the representation of JTL N (1) should be self-dual. This allows us to argue that the building blocks of our spin chains are a special kind of modulescalled tilting -which we introduce in section 4.4. In section 4.6, we put all these ingredients together to obtain the decomposition of the spin chain. Details on the structure of the tilting modules are provided in section 4.7, together with a discussion of the rank of Jordan cells. A remarkable result is that arbitrarily large ranks for the Hamiltonian are encountered as the length of the chain increases.
We then turn to conformal field theory. This is considerably more involved than in the gℓ(1|1) case since the sℓ(2|1) spin chain is not free. To start, we focus in section 5 on the generating functions of energy levels, which contain information about the left-right Virasoro vir ⊕ vir content in the scaling limit. While the chain is not solvable in the usual sense, these generating functions can be obtained by using known results for twisted XXZ spin chains where the same standard modules appear: results for the sℓ(2|1) chain itself are then obtained by using the algebraic analysis. In section 6 we then turn to the discussion of the operator content of simple JTL modules at c = 0. While in the case of the gℓ(1|1) spin chain, simples of JTL led, in the scaling limit, to direct sums of simples of vir ⊕ vir, we find that it is not the case here. We provide the essential features of the left-right Virasoro structure (2,2) (2, 7) (2, 5) (2, 0) (0, 2) (5, 2) (7, 2) (0, 7) (0, 5) (2, 1) (2, 2) (0, 0) (2, 2) (1, 2) (5, 0) (7, 0) (2, 7) (2, 5) (2, 0) (0, 2) (5, 2) (7, 2) (2, 2) Figure 1: The "kernel" part of the vacuum module of our c = 0 bulk theory considered as a vir ⊕ virsubquotient (see section 7.1). The nodes (h,h) denote irreducible vir⊕vir subquotients with conformal weight (h,h). The arrows represent the "irreversible" action of the algebra vir ⊕ vir: if two nodes A and B are connected by an arrow A −→ B, this means that one can go from A to B by acting with vir ⊕ vir, but not the other way around. We show only the positive and negative modes action, i.e., the action of L 0 +L 0 is not shown explicitly.
of simple JTL modules in the scaling limit, and reach in particular the conclusion that the size of Jordan cells in the continuum theory can be even larger than those observed on the lattice. This is a new feature, compared with the boundary and gℓ(1|1) cases. We also discuss the field content of our theory up to level (2,2). In section 7.1, we finally come to the discussion of the indecomposable content of the scaling limit of the tilting modules. We discuss in particular the identity or vacuum module, and the appearance of Jordan cells for L 0 +L 0 of arbitrarily large rank. The last section contains conclusions and directions for future work.
To help the reader we now provide a summary of our main results, with the indication of where they can be found in the text.
First, for the lattice: • The Hilbert space of the spin chain decomposes onto tilting modules T j,P that are glueings of standard modules as in Figs. 15 and 16. The multiplicities can be obtained, see Sec. 4.6 and an example in Sec. 4.6.1, but are complicated and not particularly illuminating at this stage. They correspond to combinations of many representations of sℓ(2|1). For a given tilting, these multiplicities quickly stabilize as N increases, and are the same in the scaling limit.
• The structure of the tilting modules in terms of JTL simples can be obtained. It is also complicated, and depends on N . But patterns as N increases can easily be understood. The most detailed analysis is provided in Fig. 18 for the vacuum "tilting" module.
• A consequence of the structure of the modules is the likely appearance of Jordan cells of arbitrarily large size for the Hamiltonian as N increases. The analysis is described in Sec. 4.7.
(2, 2) 2 −1 , with the corresponding indecomposability parameters b =b = −5 [28]. The irreducible subquotients are simply represented by their conformal weight (h,h), except for the vacuum |0 state which has (h,h) = (0, 0). We also show some of the corresponding quantum fields, including the stress energy tensor T and its logarithmic partner t. Note that the vacuum is unique, and satisfies in particular L −1 |0 =L −1 |0 = 0. Second, in the scaling limit: • In contrast with the gℓ(1|1) case of the open case, simple representations of JTL do not become direct sum of simple representations of vir ⊕ vir in the scaling limit. In particular, the scaling limit should exhibit Jordan cells which are not present on the lattice. Examples are given in equations (6.22) and (6.23).
• The indecomposable structure of the tilting modules under vir ⊕ vir is thus particularly cumbersome. It is worked out in full detail for the vacuum module with the final structure given in Fig. 1 and with explicit left-right Virasoro action up to conformal weights (2,2) in Fig. 2, where for notations see also Sec. 7.1. The complete description of the operator content (including the multi hulls operators) of our theory is in Sec. 6.3. Several important conclusions follow, among which: • The identity field occurs with multiplicity one, and satisfies all the properties expected from a well-defined vacuum of a bulk field theory -in particular, it is invariant under translations.
• There is a Jordan cell of rank two for the fields in (0, 2) and (2, 0) -the stress energy tensor has a single logarithmic partner, with indecomposability parameter b = −5.
• There is a Jordan cell of rank three for the field (2, 2).
• Jordan cells of arbitrarily high rank occur in the scaling limit for large enough conformal weights. These ranks can be calculated, and examples are given in section 7.2.

Notations
To help the reader navigate throughout the paper, we provide a partial list of notations (consistent with all other papers in the series): TL N -the (ordinary) Temperley-Lieb algebra, Finally, we stress that we adopt the convention that when a module is called 'indecomposable module', it is not irreducible.
2 The model and some general observations.

Remarks about percolation
It is important to start by some generalities about geometrical models and conformal field theory. In the case of a critical statistical mechanic model which is defined locally in terms of spins and their interaction, the definition of the conformal field theoretic limit is a priori obvious: what one needs to do is simply consider the scaling limit of local observables in order to obtain scaling fields [35]. Considering for instance the Ising model, the most natural such observables are the spin and the energy, which, together with the identity, constitute in fact the full primary operator content of what is usually considered as 'the' CFT for the Ising model, which is simply the minimal, unitary c = 1 2 CFT. If one tries to apply the same approach to the problem of percolation, one is forced to recognize that the only local observable is the presence or absence of a bond on a given edge. In percolation however, edges are occupied independently of each other. There is thus no trivial scaling limit to this local observable, and the CFT description of the problem is bound to be the trivial one, with only the identity field, and c = 0. The obvious point is that, in order to obtain interesting quantities in percolation, one needs to consider observables which are in fact non local in terms of the basic edges occupancy. Part of the history of the field has been to recognize that these observables could also be described by a local field theory, via maneuvers which trade non-locality for non-unitaritythis will be discussed more below. The important point here is that, once non local observables are introduced, it is not at all clear where to stop, nor is it clear which set of such observables one can hope to describe within a consistent CFT.
In percolation, one can first consider the connectivity of clusters, that is, define correlation functions via the probability that a given set of points belongs to the same cluster. The associated conformal weight is known to be h =h = 5 96 , and can be formally reproduced using the Kac formula with half integer labels: h = h 3/2,3/2 . Three point functions have been studied recently, and found in numerical agreement with a continuation of the formula for Liouville theory to the imaginary (time like) domain [36]. Nothing is known so far for the four point functions, and there is no evidence that the continuation of the Liouville theory itself makes sense as a CFT.
Meanwhile, one can consider refined connectivities, for instance via the probability that a set of points not only belongs to the same cluster, but are connected via two non intersecting paths on this cluster. The corresponding exponent (related to the so called backbone fractal dimension) is not known in closed form. It has been determined numerically to be h =h = .1784 ± 0.0003 by transfer matrix calculation. It can also be obtained, in principle, within the SLE formalism and the numerical solution of a differential equation [37].
To add to the confusion, there are many more geometrical observables one can consider. Of particular interest to us are the properties of percolation hulls. One can indeed consider the probability that two points belong to the boundary of the same cluster: the associated conformal weight is known to be h =h = 1 8 . It is then easy to generalize this to observables O k where k cluster boundaries come together, with the exponent h k =h k = 4k 2 −1 24 [38]. When discussing the possible LCFT description of a problem such as percolation, it is crucial to consider which set of observables one would like to describe, with the understanding that there is probably no single LCFT encompassing them all, since different observables may well not be mutually local. Attempts to define abstractly the fusion algebra of percolation without specifying which observables one is interested in appear as pure nonsense to us.
Although the cluster connectivity is the most natural observable to consider, we have unfortunately not been able to make much progress in its bulk conformal description, for technical reasons that we will discuss later. Instead, this paper will focus mostly on attempts at constructing a LCFT describing the properties of hulls, and in particular the observables O k . Before getting into details, we would like to stress one more confusing fact: while in the bulk, the properties of the insides and hulls of clusters are profoundly different, they coincide near a boundary. This is simply because, near a boundary, a point which belongs to a cluster necessarily also belongs to its hull. Hence, the set of possible geometrical observables in percolation is smaller in the boundary case than in the bulk case, a deep indication of the fact that, for LCFTs, bulk and boundary are not as tightly related as for unitary, rational CFTs [39].

The SUSY spin chain
As anticipated in the foregoing discussion, we will discuss in this paper the LCFT description of the hulls of percolation cluster. The main reason for this is that the hulls, whose definition is initially non local, can be described by a fully local lattice model involving spins with nearest neighbor interaction. The drawback of this model is that it is non unitary -the Boltzmann weights are not positive definite. The spins take values in representations of the superalgebra gℓ(2|1), and the model enjoys the corresponding (target space) supersymmetry. The first step to obtain our model is to trade the description of percolation in terms of clusters for a description in terms of loops. Geometrically, these loops are obtained via the so called polygon decomposition of the medial lattice in Fig. 3. For bond percolation on the square lattice -to which we restrict here -the loops live on another square lattice, which they cover entirely. There is a one to one correspondence between loops and clusters, see Fig. 4. At criticality, occupied and empty edges occur with probability p = 1 2 . This translates into the fact that all loop configurations are equiprobable, and that the loops must all be counted with a fugacity equal to one. The partition function of the model with doubly periodic boundary conditions is trivial, and can be taken to be Z = 1 with the proper normalizations. Observe in particular that the operators O k have a natural interpretation in terms of k loops joining two points, see Fig. 5.
The next step is to consider the loops as Feynman diagrams expressing contraction of what will turn out to be supergroup variables. The idea is that vertices of the medial lattice represent interactions, while the fugacity of the loops is obtained by a simple counting argument, the resulting weight being the number of bosonic minus the number of fermionic colors (or degrees of freedom).
To give details, it is more convenient to imagine calculating the partition function or some correlation function using a transfer matrix formalism [33,34]. We consider for instance vertical propagation on our figures, and imagine that every edge carries a color c i which can be bosonic or fermionic. When two edges meet, either there is no interaction, or a contraction takes place. This is described by the two possible diagrams in Fig. 6 and in the second case, the 'Boltzmann weight' is a sign factor (−1) |c i | , where |c i | is zero if the color is bosonic, and unity if the color is fermionic. It is easy to see that this reproduces the required statistics of the loops on the medial graph. The elementary interaction encoded by the second diagram in Fig. 6 corresponds to the action of a Temperley-Lieb algebra generator denoted by e i . This will be discussed in more detail below. First, we need to reformulate the problem slightly in order to make the underlying supersymmetry manifest. To do so, we associate with each edge of the medial lattice a Z 2 graded vector space of dimension 2|1, that is, a bosonic (resp.   fermionic) space of dimension 2 (resp. 1). We choose these vector spaces to alternate: we choose the fundamental V of the Lie superalgebra gℓ(2|1) for even edges, and the dual V * on odd ones, see App. A for definitions. The transfer matrix then acts on the graded tensor product H = (V ⊗ V * ) ⊗N . The distinction between odd and even edges can be interpreted as a choice of orientation for the loops, see Fig. 7 . Figure 7: Example of dense loop configuration obtained as an expansion of the partition function of supersymmetric vertex models. We also show the equivalent percolating clusters. The lattice consists of alternating arrows going up for i odd and down for i even, where i = 1, . . . , N = 2L corresponds to the horizontal (space) coordinate. The alternating ,¯ representations correspond to a lattice orientation, conserved along each loop. The system has periodic boundary conditions in both spacial and imaginary time directions. Each closed loop carries a weight str I = 1.
To proceed, it is convenient to describe the representations V and V * using oscillators. For i even 2), and fermion operators . Notice the minus sign in the last anticommutator; our convention is that the † stands for the adjoint, this minus sign implies that the norm-square of any two states that are mapped onto each other by the action of a single f iα or f α † i have opposite signs, and the "Hilbert" space has an indefinite inner product (with the respect to the adjoint operation). The space V is now defined as the subspace of states that obey the "one-particle per cite" constraints The sums here and below are over a = 1, 2, and α = 1.
The generators of the Lie superalgebra gℓ(2|1) acting on each site of the chain are the bilinear i f iα for i odd, which for each i have the same (anti-)commutators as those for i even.
The next step is to build, in this language, the proper interaction to reproduce the statistical properties of percolation hulls. First, we note that for any two sites i (even), j (odd), the combinations are invariant under gℓ(2|1). Introduce now The interaction at a vertex where edges i and i + 1 meet is then simply given, by the elementary transfer matrix The complete transfer matrix is Finally, we note that, although we have formulated everything so far in terms of an isotropic system, it is well known that the same universality class is obtained by choosing different probabilities of occupancy for horizontal and vertical edges. This corresponds to the more general choice of elementary transfer matrices with p A + p B = 1. It is in particular possible to chose a very anisotropic limit p A → 0 where the transfer matrix description is replaced by a hamiltonian description according to T ≈ e −p A H with

Algebra
The interpretation of the elementary vertices in terms of contractions suggests the simple algebraic nature of the interaction in our model. Indeed, the elementary generators e i satisfy special relations, and provide in fact a representation of a well known algebra, the Jones Temperley Lieb algebra. We assume the reader is familiar with the basics: more details will be given in the next section. An important point is that, while we have restricted so far to gℓ(2|1), a more general gℓ(n + 1|n) model can be introduced, simply by allowing everywhere the labels a, b now to run from 1 to n + 1 and α, β from 1 to n. Because each loop comes weighted with the super trace of the fundamental (or dual fundamental), it still will come with a factor 1. There is thus, in fact, a multiplicity of spin chains related with percolation. As discussed in [14] these chains all describe the same geometrical objects, but the associated field theoretic observables come with different multiplicities. This means that there is in fact an infinite family of LCFTs at c = 0, with larger and larger degeneracies as n increases. We will, in this paper, restrict to the case n = 1.

The sigma model description.
This spin chain formulation can be used to obtain a sigma model description of the low energy excitations: see references [40,41] for non supersymmetric examples, and [42,43] for supersymmetric examples in the context of disordered systems. The target space -which appears in the construction of the coherent state path integral -should be the symmetry supergroup (here U (2|1)) modulo the isotropy supergroup of the highest weight state: in our case, this gives the complex projective superspace CP 1|1 = U(2|1)/(U(1) × U(1|1)). The mapping could be fully controlled by using a 'higher spin' generalization of the spin chain, obtained by taking larger irreducible representations, represented by the value of the highest weight S, with 2S integer (with S = 1 2 for the fundamental) and their duals on alternate sites. For an appropriate family of such representations, the sigma model can be obtained with a bare coupling constant g 2 σ ≈ 1/S. On top of this, for the hamiltonian (2.9), there is a topological angle with bare value θ = 2πS mod 2π. The Lagrangian of the sigma model involves a multiplet of fields with complex bosonic components z a (a = 1, 2) and fermionic components ξ α (α = 1). These fields satisfy the constraint equation z † a z a + ξ † α ξ α = 1, modulo U (1) gauge transformations, so they provide a parametrization of CP 1|1 . In terms of these fields, the euclidian Lagrangian density reads is a gauge potential and D µ = ∂ µ + ia µ is the covariant derivative. Finally, the beta function for the model is where l = ln L is the logarithm of the length scale at which the coupling is defined. The beta function is independent of θ, and the beta function for θ is zero, in perturbation theory. The beta function being positive at weak coupling, it is expected that the same fixed point CFT will be reached in the universal, large length scale, low energy limit for all spin chains with 2S an odd integer, and in particular for S = 1 2 . This fixed point theory is the LCFT we are trying to build in this paper. While this theory should have U (2|1) symmetry, it is not expected to be a WZW theory, as confirmed by the early analysis of the spectrum in [34]. This is because the general arguments promoting conformal invariance plus continuous group symmetry into a current algebra symmetry fail in non unitary cases, where, typically, logarithms can appear in the OPE of the currents [44].

A note on modular invariance
The partition function of our model on a torus is Z = 1. This is the only modular invariant we will associate with our model. This does not mean that the operator content is trivial of course. The point is, that doubly periodic boundary conditions for the geometrical model corresponds to periodic boundary conditions for the bosonic and fermionic degrees of freedom. Turning to a transfer matrix description, the partition function is thus the supertrace of the appropriate power of the transfer matrix, which itself acts on a system of periodic bosons and fermions. While the Hilbert space in which the transfer matrix acts is non trivial, since it is of dimension (2n + 1) 2 L , this same space has a simple superdimension SdimH = [Sdim(V )Sdim(V * )] L = 1.
(2.12) This means, as we shall see below, that all operators but the identity appear in non trivial representations of the supersymmetry [45]. Now, in order to identify the operators present in the model together with their associated representation content, it is convenient to consider a modified partition function [34]. This partition function is defined simply by taking, instead of the supertrace, the ordinary trace. Of course one has and now each level of the transfer matrix/hamiltonian will be counted with multiplicity one. The modified partition function is not modular invariant, and there is no reason why it should be. There is also no reason why it should be part of a bigger theory encompassing both periodic and antiperiodic boundary conditions in the (imaginary) time and space directions. The model as we define it on the lattice is perfectly local with periodic boundary conditions, and should lead, as it is, to a local field theory in the scaling limit.

Observables
Having introduced the supersymmetric formulation of the loop model, it is fair to ask what kind of observables we can now describe. The answer, ultimately, will be provided by the detailed analysis of the spectrum and the operator content given below. It is easy however to see that one should expect first all the multihull operators O k . The two point function of the O 2 operators for instance is expressed geometrically as the probability that there exists (at least) two lines connecting a pair of neighboring edges in r 1 and a pair of neighboring edges in r 2 . In order to select the appropriate diagrams in the sum over all configurations, all one has to do is to insert the proper terms to prevent contractions of the lines. In other words, the two-point function can be expressed as (2.14) Figure 9: A pair of non contractible loops (a percolation 'hull') winding around the axis of the cylinder.
where the labels are shown on Fig. 8. Consider for instance a diagram such as the one shown on figure 8(b). The contraction of the lines forces d 1 = d 2 (while of course c 3 = c 1 and c 4 = c 2 ). For each such diagram, the insertion of 1 − δδ in the two point function subtracts the diagram where the lines are contracted in r 2 . The sum over the labels with the (−1) |d 1 | inserted gives the loop thus formed a weight equal to one, so, summing over the rest of the system, the contributions arising from Figs. 8(b) and 8(c) exactly cancel out. Similar reasonings show that the only diagrams that survive the sum are those where the lines in r 1 , r 2 are never contracted as in 8(a) but simply go through the system. This means in turn that the two points belong to the hull of the infinite percolation cluster. It is convenient to think of this after a conformal map onto the cylinder. Two point functions are then evaluated in the transfer matrix language. By translation invariance, we see that the eigenvalue in the sector with 2k through lines should then give the exponent (h k , h k ) (zero spin). It is also clear that lines connecting r 1 , r 2 can obviously wind around the axis. On the other hand, it is well known, within the Coulomb gas formalism, that primary fields can be obtained, for a given number of through lines, by inserting at either extremity of the cylinder additional charges, whose effect is to give an additional phase factor z ±1 for every line going clockwise/counterclockwise like in Fig. 9.
Setting z ≡ e iπα (2.15) the exponents for 2k through lines are A given value of z leads to many choices for α mod 1 (the sign of z is not relevant since in the model only pairs of lines propagate), and all the corresponding operators are present in the spectrum. Another important fact, which occurs only for the percolation model, is that it is impossible to define higher O k operators without some amount of 'mixing' with lower ones. This is because of the fact that the object preventing contractions among three lines: cannot be normalized to become a projector. In other words, one has It follows that mixtures between the various sectors with different numbers of non contractible lines must occur producing in the end complicated representations of the periodic TL algebra. This part will be discussed more below.

The algebraic description: generalities
The following two subsections contain material discussed already in our earlier work on the subject, which we prefer to reproduce here for clarity, completeness and in order to establish notations. The reader familiar with one of our previous papers [30,31,32] can go directly to Sec. 4 where the crucial aspect of faithfulness is discussed.

The Temperley-Lieb algebras in the periodic case
We define here finite-dimensional quotients of the affine Temperley-Lieb algebra [46,47,48,49,51] denoted here by T a N (m) and spanned by particular diagrams on an annulus. A general basis element in the algebra of diagrams corresponds to a diagram of N sites on the inner and N on the outer boundary of the annulus (we will restrict in what follows to the case N even, and parametrize N = 2L.) The sites are connected in pairs, and only configurations that can be represented using lines inside the annulus that do not cross are allowed. We will often call all such diagrams affine diagrams. Examples of affine diagrams are shown in Fig. 10 for N = 4, where we draw them in a slightly different geometry: we cut the annulus and transform it to a rectangle, which we call framing, with the sites labeled from left to right. An important parameter is the number of through-lines, which we denote by 2j, with j = 0, 1, . . . , L, connecting 2j sites on the inner and 2j sites on the outer boundary of the annulus; the 2j sites on the inner boundary we call free or non-contractible. Multiplication a · b of two affine diagrams a and b is defined in a natural way, by joining an inner boundary of a to an outer boundary of the annulus of b, and removing the interior sites. Whenever a closed contractible loop is produced when diagrams are multiplied together, this loop must be replaced by a numerical factor m. We also note that the diagrams in this algebra allow winding of through-lines around the annulus any integer number of times, and different windings result in independent algebra elements. Moreover, in the ideal of zero through-lines, any number of non-contractible loops (like in the fourth diagram in Fig. 10) is allowed.
In terms of generators and relations, the algebra T a N (m) is generated by the e i 's together with the identity, subject to the usual relations [46,47,51] e 2 j = me j , e j e j±1 e j = e j , e j e k = e k e j (j = k, k ± 1), (3.1) where j = 1, . . . , N and the indices are interpreted modulo N , and by generators u and u −1 of translations by one site to the right and to the left, respectively. The following additional defining relations are then obeyed, and u ±N is a central element. The algebra generated by the e i and u ±1 together with these relations is usually called the affine Temperley-Lieb algebra T a N (m). We will consider translations by an even number of sites only, i.e., restrict to powers of u 2 , and replace a non-contractible loop by a numerical factor m, as for the contractible loops. This constraint (see 4.2.2 and 4.3.1 in [46]) together with taking a quotient by the ideal generated by u N − 1 makes the algebra finite dimensional. We call the resulting object the augmented Jones-Temperley-Lieb algebra JTL au 2L (m), where recall we have set N = 2L. This algebra is slightly bigger than the one used in [14,31], called the Jones-Temperley-Lieb algebra, which we denote by JTL 2L (m). The difference is entirely in the sector/ideal with zero through-lines. The algebra JTL au 2L (m) introduced here contains in this ideal all affine diagrams while the algebra JTL 2L (m) has only planar (or usual TL) diagrams in this sector 1 . In other words, in JTL 2L (m), one only keeps track of which points are connected to which in the diagrams, while in JTL au 2L (m), one also keeps information of how the connectivities wind around the annulus (the ambiguity does not arise when there are through-lines propagating). Formally, we have a covering homomorphism (surjection) of algebras which acts non-trivially only in the zero through-lines subalgebra and maps the diagrams as 3) The first algebra JTL au 2L (m) has dimension (as follows from dimensions of generically irreducible modules described below) We will only be concerned in this paper with the case m = 1 for which the algebra JTL au 2L (m) is non semi-simple; in the following we usually suppress all reference to m.

Standard and co-standard modules
It is useful to go back for a little while to the case of the full affine Temperley Lieb algebra T a N (m). Set m = q + q −1 . For generic q = 1, the irreducible representations we shall need are parametrized by two numbers. In terms of diagrams, the first is the number of through-lines 2j, with j = 0, 1, . . . , L. Using the natural action (by stacking affine diagrams) of the algebra discussed in the previous subsection, we now decide that the result of this action is zero whenever the affine diagrams obtained have a number of through lines less than 2j. Furthermore, for a given non-zero value of j, it is possible using the action of the algebra, to cyclically permute the free sites: this gives rise to the introduction of a pseudomomentum K. Whenever 2j through-lines wind counterclockwise around the annulus l times, we can decide to unwind them at the price of a factor e 2ijlK ; similarly, for clockwise winding, the phase is e −i2jlK [46,48]. This action gives rise to a generically irreducible module, which we denote by W j,e 2iK . In the parametrization (t, z) chosen in [49], this corresponds to t = 2j and the twist parameter The dimensions of these modules W j,e 2iK over T a 2L (m) are then given bŷ Note that the numbers do not depend on K (but representations with different e iK are not isomorphic). These generically irreducible modules W j,e 2iK are known also as standard (or cell) T a N (m)modules [49].
Keeping q generic, degeneracies in the standard modules appear whenever The representation W j,q 2j+2k then becomes reducible, and contains a submodule isomorphic to W j+k,q 2j . The quotient is generically irreducible, with dimensiond j −d j+k . The degeneracy (3.7) is well-known [48,49] 2 . When q is a root of unity, there are infinitely many solutions to the equation (3.7), leading to a complex pattern of degeneracies to which we turn below. The case j = 0 is a bit special. There is no pseudomomentum, but representations are still characterized by another parameter, related with the weight given to non contractible loops. Parametrizing this weight as z + z −1 , the corresponding standard module of T a 2L (m) is denoted W 0,z 2 and has dimension 2L L .
We now specialize to the Jones-Temperley-Lieb algebra JTL N (m). In this case, the rule that winding through-lines can simply be unwound means that the pseudomomentum must satisfy jK ≡ 0 mod π [47]. All possible values of the parameter z 2 = e 2iK are thus j-th roots of unity (z 2j = 1, [51]). The kernel of the homomorphism ψ described by (3.2) and (3.3) (and the ideal in T a N (m) generated by u N − 1, in particular) acts trivially on these modules if j > 0. In what follows, we will thus use the same notation W j,z 2 , with j > 0, for the standard JTL N (m)-modules. We note that two standard JTL N -modules having only different signs in the z parameter are isomorphic.
If j = 0, requiring the weight of the non contractible loops to be m as well leads to the standard JTL au N (m)-module W 0,q 2 which is reducible even for generic q -it contains a submodule isomorphic to W 1,1 . Meanwhile, on the standard module W 0,q 2 the kernel of the homomorphism ψ is non-trivial: the standard module over JTL N (m) for j = 0 is obtained precisely by taking the quotient W 0,q 2 /W 1,1 as in [49]. This module is now simple for generic q, has the dimension In the full construction of direct summands of our spin-chains -the tilting modules -we shall also need a concept of so-called co-standard modules. They are defined as the duals W j,P * or vector spaces of linear functionals on W j,P endowed with the JTL au N action by where · * is an anti-automorphism on JTL au N (m) defined by interchanging the inner and outer boundaries of the affine diagram. Equivalently, we can say that the basis in co-standards W j,P * is defined by reflecting the framing in a horizontal line. Then, the value of v * on W j,P is given by the bilinear form defined in [49,Sec 2.6]. It was shown in [49] that W j,z * is generically isomorphic to W j,z −1 .
At critical values of the parameters, special pseudomomenta K and roots of unity values of q, there is a non-trivial homomorphism from W j,z * to W j,z −1 controlled by the bilinear form.

The structure of the standard JTL au N (m) modules
A central point in representation theory of the algebras JTL au N (m) and JTL N (m) is their cellularalgebra structure and the full power of cellular algebras technique [52,49] can be used for studying many important indecomposable modules -including the so-called projective modules. We now review shortly these important concepts.

Cellular algebras and (co)cell modules
We first give a brief review of cellular algebras and general aspects of their representation theory. These algebras were initially introduced in [52]. Roughly, cellular algebras are defined as those with a special basis having particularly nice properties under multiplication. To be more precise, a cellular algebra A over C is an associative algebra equipped with a finite partially ordered set S (the set of weights with an order on it) and a finite set W (λ), for any λ ∈ S, such that the algebra has a basis C λ i,j , where λ ∈ S and i and j run through all elements in W (λ), with the following two properties: (i) for each a ∈ A the product aC λ i,j can be written as is a linear combination of basis elements with upper index µ strictly smaller than λ and where the coefficients c a (i, k) do not depend on j; (ii) there exists an anti-automorphism · * on A such that its square is the identity, and that it sends C λ i,j to C λ j,i . The diagram algebras such that the ordinary Temperley-Lieb algebras TL N (m) are examples of cellular algebras. The special basis C λ i,j in the latter case is given by Temperley-Lieb diagrams of arcs and through lines without crossings. Elements from the set S of weights are just numbers of through lines in the diagrams and the partial order is just the ordinary order on natural numbers. The algebras JTL au N (m) are just another example of a class of cellular algebras. Now the weights λ are the pairs of numbers (j, P ), with 2j is the number of through lines and P is the exponent of the pseudomomentum. The anti-automorphims · * from the definition is just the reflection of a diagram through the horizontal axis. The proof of cellularity is rather straightforward following the definition of the multiplication and the basis in JTL au N (m) given in the previous section.
A cornerstone of cellular algebras theory is the notion of cell modules. These modules are parametrized by elements λ from the set S of weights. A cell (or standard) module W λ over a cellular algebra A is a vector space with a basis {C j | j ∈ W (λ)}, with the action of any a ∈ A given by aC j = k∈M (λ) c a (j, k)C k , where the numbers c a (j, k) are those from the definition of the cellular algebras. Similarly, we introduce co-cell modules as duals to the cell modules, and defined with the use of the algebra anti-automorphism · * . The modules W j,P , with P being all j-th roots of unity, introduced in the previous section give all the cell modules for JTL au N (m). The co-cell modules are given by the corresponding costandards described in the end of Sec. 3.2.
A cellular algebra itself and its projective modules are filtered by the cell modules, or in other words, can be constructed as appropriate glueings or extensions between cell modules. Precise statements about projective modules will be given below. Now, we go into details for the class of the diagram algebras JTL au N (m).

3.3.2
The partial order on cell JTL au N -modules weights We now turn to the details of the representation theory of JTL N (1), which we will then apply to the study of our super-spin chain. We first analyze the representation theory for the augmented algebra JTL au N (1) -which is technically easier -and then restrict ourselves to the JTL N (1) algebra. The subquotient structure of the cell JTL au N (1) modules can be easily obtained using results of [49]. The set of weights for the cell JTL au We introduce a (weak) partial order (due to [49]) on this set of weights such that (j 1 , P 1 ) (j 2 , P 2 ) if j 2 − j 1 = k for a non-negative integer k and the pairs (j 1 , P 1 ) and (j 2 , P 2 ) satisfy Note that the partial order generates equivalence classes on the set of the weights (j, P ) -two weights are in the same equivalence class if and only if they are in the relation . The result of [49] is that there exist non-trivial homomorphisms only between cell (or standard) modules having weights from the same equivalence class. The idea thus is that simple JTL modules can be glued with each other (or extended by each other) non-trivially only if their weights are from the same equivalence class. There are non-trivial classes (containing two or more weights) only when q is a root of unity. In this case, many cell modules are reducible but indecomposable.
We give now several examples of the partial order on the set of weights. For the case q = i, the equivalence classes (generated by the partial order ) are given by the disjoint oriented graphs on the diagram on Fig. 11, where it is apparent that there are two non-trivial classes, denoted by arrows of different types. For the case q = e iπ 3 , the equivalence classes are given in the diagram on Fig. 12. In this case, we also have only two non-trivial equivalence classes containing two and more weights, while all the other classes (that is, all the nodes without in-and out-going arrows) are trivial, and contain only one weight. For a trivial class, the corresponding cell module is simple. The non-trivial class with dashed arrows on the figure contains all weights (j, 1) with j mod 3 = 0 and each node has a single in-going or out-going arrow. We call this class singly critical. The second class (with solid arrows) involves the weights (0, q 2 ) and (j, 1) with j mod 3 ∈ {1, 2} and (j, q ±2 ) whenever   [0, e   j mod 3 = 0 and j > 0. Nodes in this class have double arrows and we call it doubly critical. As we shall see below, the cell modules from the singly critical class are of chain type while the cell modules having weights from the doubly critical class have a "braid-type" subquotient structure.
Each cell module W j,P is known to be indecomposable: we denote its top simple subquotient -the quotient by its maximal submodule -by X j,P ; we will also use the short hand notation [j, P ]. Using the partial order on S described earlier, the simple-module content of the cell JTL au N (1)-modules can be deduced [49] [W j,P ] = where the notation [W j,P ] means that the corresponding module, as a vector space, is given by the right hand side, and that it has the simple subquotients denoted by [j, P ] that appear in the sum. We have thus described the structure of W j,P up to glueings or arrows between simple subquotients. The information about arrows can be also deduced from results of [49]: the cell modules over JTL au N (1) have a sequence of embeddings (see the description of injective homomorphisms from Thm. 3.4 and also the proof of Thm. 5.1 in [49]) for any sequence of weights (j 1 , P 1 ) (j 2 , P 2 ) . . . (j n , P n ) in Fig. 12. This sequence of embeddings tells us that modules corresponding to the doubly-critical class have a braid-type subquotient structure, that can be extracted from Fig. 13, showing the subquotient structure for the JTL au N (1)module W 0,q 2 . The other W j,P , with P = 1 for j mod 3 ∈ {1, 2} and P = q ±2 when j mod 3 = 0, are obtained as the submodules 'emanating' from the corresponding simple subquotient. The cell modules W j,1 , with j mod 3 = 0, from the singly critical class are also presented in Fig. 13 and are of chain type.
We note now that the cell modules over the quotient JTL N (1) (and not JTL au N (1)) for j > 0 are given by the same diagrams in Fig. 13 and we use the same notation W j,P for them. The only difference is for j = 0. The cell JTL N (1)-module with the weight (0, q 2 ) is given by the quotient W 0,q 2 /W 1,1 , as we discussed previously, and has only two irreducible subquotients: We conclude this subsection with a comment on the structure of the costandard modules W j,P * introduced at the end of Sec. 3.2. Generically, these modules are isomorphic to W j,P −1 while for our choice of q and parameters P we have only homomorphisms ψ j,P : W j,P −1 → W j,P * that map the irreducible head X j,P −1 to the (unique) irreducible submodule in W j,P * and the kernel of ψ j,P can be studied using the bilinear form in [49,Sec 2.6]. We thus have that the diagrams for W * j,1 (with j mod 3 equal 1 or 2) have in the bottom the irreducible submodule X j,1 and all the arrows reversed while modules W * j,q ±2 (with j mod 3 = 0) have in the bottom the irreducible submodule X j,q ∓2 .

Dimensions of simple JTL modules
The subquotient structure of the standard modules allows us to compute the dimensions of all simple JTL modules. It will be convenient in what follows to introduce the notation and to recall the notationd j from (3.6) for dimensions of the cell modules. Dimensions of the corresponding simples will be denoted asd . . . . . .
. . . . . . 1] . . . . The two modules on the left are cell modules from the doubly critical class: on the first one L i = L − 2 + i and P i = 1 when L mod 3 = 2, and L i = L, P 1 = q 2 , and P 2 = q −2 when L mod 3 = 0; the second module is for the case L mod 3 = 1; the rightmost module corresponds to the doubly critical class (with j mod 3 = 0 andL = L − L mod 3). We denote simple subquotients by their weights [j, P ].
Using the subquotient structure from Fig. 13, we can write immediately the dimension of the simple associated with j = 0 and e 2iK = e 2iπ/3 : where d j =d j −d j+1 , with j > 0, and d 0 =d ′ 0 −d 1 =d 0 are the dimensions of the cell modules for the ordinary Temperley-Lieb algebra. We thus see thatd From the diagrams in Fig. 13 one can obviously derive the following more general results: The final result can be thus obtained recursively. Comparing these relations with those in the open TL case [1] we see thatd where d 0 j are dimensions of simple TL modules corresponding to tops of the cell modules with 2j through lines.

Indecomposable modules: general definitions
We collect in this subsection the definitions of several important mathematical concepts that are needed in order to fully appreciate the rest of this paper, such as projectiveness, self-duality, and tilting modules.
We begin with a definition of what is called an injective hull of a simple module. It is the maximal indecomposable that can contain this simple module as a submodule. Then, any injective module is by definition a direct sum of injective hulls and if a submodule in a bigger module is injective then it is a direct summand. In contrast, projective modules are defined as direct sums of projective covers, where the projective cover of a simple module is the unique (for finite-dimensional algebras) indecomposable module of maximal dimension that can cover the simple module, i.e., the projective cover contains it as a top subquotient. Then, if a subquotient of a bigger module is projective then it is a direct summand. Note that the distinction between the notions subquotient and submodule is crucial here. Therefore, the projectiveness property does not necessarily imply injectiveness and vice-versa.
Projective modules appear as direct summands of free modules like the regular representation of an algebra, but for spin chains -and thus presumably LCFTs -the direct summands are, more generally, tilting modules which are defined and discussed in details below. In some cases, the tiltings are also indecomposable projective, but this is not necessarily the case. In general, there are tilting modules which are not projective, and projective modules which are not tilting.
In the theory of cellular algebras [52,53], there is a general theorem that allows one to obtain the subquotient structure of projective covers knowing the subquotient structure of the standard (cell) modules. The essential part of this theorem can be expressed as a reciprocity property of projectives. Let [W : X] and [P : W] denote the number of appearance of X in a diagram for a standard module W and the number of appearance of W in a diagram for the projective cover P, respectively. Then, the reciprocity property reads i.e., the projective cover P that covers X is composed of those standard modules W that have the simple module X as a subquotient. This statement can be deduced from the proof of Thm. 3.7 in [52].
Having an indecomposable (and reducible) module M , we call socle its maximum semisimple submodule -in terms of nodes and arrows in the subquotient diagram for M , the socle is the direct sum of all nodes having only ingoing arrows. Similarly, the top of a module M is the maximal subquotient with respect to the property that a quotient of M is a semisimple module, i.e., it is the subquotient of M having only outgoing arrows.
The spin-chains we consider have a non-degenerate bilinear form given explicitly, for example, in terms of spins. These spin-chains provide a special class of representations with two essential properties: (i) they are filtered by standard modules and (ii) they are self-dual, i.e., H * Direct summands in such representations are called tilting modules.
We define a tilting module over a cellular algebra as a module that has a filtration by standard modules -these are W j,P in our case -and an inverse filtration by the corresponding duals -the costandard modules W j,P * , which have reversed arrows in their subquotient diagram. Note that both standard and costandard modules are introduced in the context of the JTL algebras in Sec. 3.2 and their structure is described in Sec. 3.3. We recall that a filtration of an A-module M by A-modules W i , with 0 ≤ i ≤ n − 1, is a sequence of embeddings such that the quotient M i+1 /M i of "neighbor" submodules (called the ith section) is isomorphic to W i , or in simple words we can say that M is a glueing of W i 's. The tilting modules are thus self-dual by our definition. Several explicit examples will be given below.
We will also show that tilting JTL modules satisfy a uniqueness property: one can introduce the tilting module T j,P generated from a standard module W j,P as the indecomposable tilting module containing this standard module as a submodule. This property of having a standard submodule uniquely defines the tilting module, up to an isomorphism. One should replace each simple subquotient of this standard module by a costandard module having this simple module in its socle -the unique simple subquotient that has only incoming arrows. The result is then automatically a tilting module, by construction.
Before discussing spin-chains and the reasons why tilting modules are more important objects for applications, we give some results about projective modules for the JTL algebra.

The projective modules over JTL N (1)
We first describe the cell content of projective covers for all simple modules over JTL N (1). Let be the partial order on the set S of weights of cell modules introduced above and let greek letters (λ, µ, ν) denote the weights [j, P ] for simplicity. The projective cover P λ of a simple module X λ has the cell content where by [W ν : X λ ] we denote the multiplicity of the appearance of the simple module The cell content (4.3) of the projective covers is actually a consequence of the general reciprocity result [W ν : X λ ] = [P λ : W ν ] discussed above. Note that using the subquotient structure of the cell modules given in Fig. 13 we see that all the numbers [W ν : X λ ] are zero or one. Moreover, the projective module P λ has a filtration by cell modules respecting the cell-filtration of the cellular algebra, i.e. the cell modules with the lowest weight (w.r.t to the order ) are submodules; quotienting by them gives a module with cell submodules corresponding to the next-to-lowest weight and so on. This general property of the cell filtrations of projective covers allows now to describe subquotient structure of the projective covers P λ in terms of cell modules. We assume for simplicity that N mod 3 = 0. Then, the subquotient structure of P L,q ±2 , where we set N = 2L, in terms of cell modules, is as simple as on the left of Fig. 14. Further, the cell filtration of the projective covers P j,P , with j = 0, 2, is obtained from Fig. 14 (taking N = 2L ≥ M ) as the submodule generated from W j,P . Now, it is easy to see that the projective covers have a sequence of embeddings (opposite of the one in (3.12)) for any sequence of weights (j 1 , P 1 ) (j 2 , P 2 ) . . . (j n , P n ) in Fig. 12 with j k = 0, 2. The projective JTL N -modules P 1,1 and P 0,q 2 are isomorphic to the cell modules W 1,1 and W 0,q 2 , respectively; the module P 2,1 is given in the middle diagram in Fig. 14.
The diagrams allow us to conclude that the indecomposable projectives P j,P (excepting P 2,1 for N = 6) are not self-dual because their socles contain the direct sum [L, 1]⊕[L, 1] when L mod 3 = 1 or 2[L, 1]⊕2[L−1, 1] when L mod 3 = 2 or the sum 2[L, q 2 ]⊕2[L, q −2 ] when L mod 3 = 0. Therefore, the projective modules are not injective hulls and can be embedded into larger (of course decomposable) modules. Figure 14: The cell-filtraion of projective JTL N (1)-modules P j,P . The two modules on the left are projective covers from the doubly critical class: a projective cover P j,P , with j = 0, 2, is the submodule generated from W j,P on the left-most diagram given for 2L mod 3 = 0; the second module is P 2,1 ; the right-most projective module corresponds to the singly critical class with j mod 3 = 0.
We will see below that the spin-chain representation is a self-dual JTL module and is decomposed onto a special class of so-called tilting modules. On the other hand it is a faithful representation and thus all projective covers should appear in the spin-chain, though not as direct summands. It turns out that the projective covers are embedded into a direct sum of tilting (self-dual) modules.

Self-duality of the spin chain representation H N
In this section, we show that the faithful representation of JTL N (1) on the periodic sℓ(2|1) spin chain H N is in addition self-dual. To show this we use a non-degenerate bilinear form (·, ·) on H N × H N , which can be given explicitly in the Fock space realization [14]. Recall that the representation and even a large family of representations π m,n of JTL N (m) are explicitly defined in App. B using this Fock realization. Our case corresponds to π 1,1 which we will denote simply by π. The generators e j of JTL N (1) are self-adjoint with respect to the bilinear form, i.e. π(e j ) † = π(e j ). The adjoint of the translation operator π(u 2 ) is the inverse π(u N −2 ). Together with non-degeneracy of the bilinear form, this means that the representation π is isomorphic to the dual one on the space H * N of linear functionals. Indeed an isomorphisms ψ between JTL N (1)-modules H N and H * N is given by ψ : N , and · stands for an argument. The non-degeneracy of the bilinear form implies that the kernel of ψ is zero. The statement on the self-duality is obviusly true in the general case of π m,n representations.
Self-duality of the module H N implies that the subquotient structure (with simple subquotients) is not affected by reversing all the arrows representing the JTL N -action. On the other hand, the faithfullness of the representation of JTL N (1) implies that all the projective covers of simples should be present in the spin-chain decomposition. As we saw above, the projective JTL N (1)-modules (those which are not simple) are not self-dual and therefore they are not injective modules and can in principle be non-direct-summand submodules in some 'bigger' modules. These bigger and self-dual JTL Nmodules indeed exist: they turn up to be tilting modules, and we show below that the projectives can be embedded into a direct sum of tilting modules. We can thus say that, for our needs at least, the tilting modules are more fundamental objects than projective modules in the sense that tilting modules are the building blocks (direct summands) of the spin-chains.

Tilting modules over JTL N (1)
Recall that a tilting module T is a module (over a cellular algebra) with a filtration by cell modules W and with the dual filtration by the duals to the cell modules -the co-cell (or costandard) modules W * . The tilting modules are thus self-dual modules by this definition, i.e. T * ∼ = T .
For any weight (j, P ) ∈ S, we define a tilting JTL au N -module T j,P generated from the cell module W j,P as the indecomposable tilting module containing this cell module as a submodule. We will see that if such a tilting module exists this property indeed fixes it uniquely, see Thm. 4.5 below. In most cases, our results on the structure of the tilting modules generated from a cell module can be expressed by very simple rules: to construct T j,P we take the cell module W j,P and each simple subquotient X j ′ ,P ′ (in the diagram for W j,P ) replace by its co-cell module W * j ′ ,P ′ such that different co-cells are glued following the pattern for the diagram for the original cell module W j,P . The result is then obviously a module with a filtration by the duals to cell modules. We then check the dual filtration and the module is what we call the tilting module T j,P .
It will be shown below that tilting modules T j,P indeed exist and exhaust all possible indecomposable tilting modules (of course up to an isomorphism). We will now describe our results on the subquotient structure. The diagrams in Fig. 15 describe cell and co-cell filtrations of the 'biggest' tilting modules T 1,1 and T 2,1 . The structure of T 1,1 is given just by the substitution 2 → 1. The two tilting modules correspond to the cases when L mod 3 equals 2 or 0: in the first case L i = L − 2 + i and P i = 1, in the second -L i = L, P 1 = q 2 , and P 2 = q −2 . The tilting module T 2,1 in the case L mod 3 = 1 is obtained by identifying the two nodes W L 1 ,P 1 and W L 2 ,P 2 with W L,1 (with their two arrows) at the top on the left and similarly for duals at the bottom on the right.
An important consistency check for the existence of these tilting modules (in addition to their self-duality) is that the projective modules P j,P defined above in Fig. 14 cover any cell-subquotient in the tiltings, where the kernel of the projection is isomorphic to W 1,1 . The tiltings T 1,1 and T 2,1 themselves can be alternatively obtained as a quotient of the projective module P L 1 ,P 1 ⊕ P L 2 ,P 2 .
A tilting module T j,P with j > 2 and appropriate P can be extracted also from Fig. 15. It is the submodule generated from the subquotient W * j,P on the right diagram or, dually, it is the corresponding quotient containing all W j ′ ,P ′ , with [j ′ , P ′ ] [j, P ], on the left diagram. The tilting modules from the singly critical class with j mod 3 = 0 and P = 1 are constructed in a similar way and they are of chain type.
A peculiarity happens with the tilting JTL N -module containing the cell submodule W 0,q 2 . If we were to proceed with the construction used so far, we would get a non self-dual module with the subquotient structure W * 0,q 2 → W * 2,1 , so it does not work in this case (meanwhile, the construction would still work for the former algebra JTL au N (1)). However, there turns out to be a universal construction of tilting modules [54,55] that requires knowledge of extension groups associated with indecomposable Figure 15: The structure of the tilting JTL N -module T 2,1 with the co-cell modules filtration on the right. The two isomorphic modules correspond to L i = L − 2 + i and P i = 1 if L mod 3 = 2, and L i = L, P 1 = q 2 , and P 2 = q −2 when L mod 3 = 0; the tilting module T 2,1 in the case L mod 3 = 1 is obtained by identifying the two nodes W L 1 ,P 1 = W L 2 ,P 2 with W L,1 and the two arrows at the top on the left and their duals at the bottom on the right. The structure of T 1,1 is given just by the substitution 2 → 1.
modules. This construction suggests to begin with a glueing (an extension) corresponding to an exact non-split sequence 0 → W 0,q 2 → M → W 2,1 → 0, (4.6) i.e., the first step of the universal construction produces a module that has a submodule isomorphic to W 0,q 2 and as a top W 2,1 with multiplicity given by dimension of Ext 1 JTL N (W 2,1 , W 0,q 2 ). We know from the structure of the projective cover for W 2,1 (see above) that this dimension equals 1. In the second step, we should compute the extension group Ext 1 JTL N (W 3,q ±2 , M ). We checked on the first few values of N that a JTL N -module having M as a submodule and with a cell filtration containing sections isomorphic to W 3,q ±2 can not be self-dual, and actually can not have a filtration by duals to cell modules. For example, it is quite easy to see that at N = 6 a JTL N -module with the subquotient structure (n + W 3,q 2 ⊕ n − W 3,q −2 ) → M is self-dual if and only if n ± = 0 (see also the example below). Hence, we can conjecture that the extension groups Ext 1 JTL N (W 3,q ±2 , M ) are trivial. It turns out that for N = 8, the only self dual module having cell and co-cell filtrations is the one having the subquotient structure W 4,1 → M (see examples for N = 8 below). We call this module M ′ . Then, we repeat our analysis for N = 10 and we get a module M ′′ = W 5,1 → M ′ , etc. So, the only way we see that the tilting JTL N -module T 0,q 2 containing W 0,q 2 as a submodule might have cell as well as co-cell filtrations is the one given on Fig. 16. We checked for N ≤ 18 that a module with this cell filtration is self-dual and we believe that its existence can be proved by taking a quotient of the projective module P L 1 ,P 1 ⊕ P 2,1 ⊕ P L 2 ,P 2 (see definition of L i , P i above). Note also that this module contains the projective cover P 2,1 , which is an important module for our spin-chain.
Finally, we claim that the tilting modules just described exhaust all indecomposable tilting modules over JTL N (1). To show this we use the important result about the JTL N (1) algebra that it is a quasi-hereditary algebra. We first recall the corresponding definition [56,57,58].
Definition Let A be a finite dimensional associative algebra over C, S an indexing set for the isomorphism classes of simple A-modules with correspondence λ → X λ , and ≤ a partial order on S. We say that (A, ≤) or simply A is a quasi-hereditary algebra if and only if for all λ ∈ S there exists a left A-module, W λ , called a standard module such that • there is a surjection W λ → X λ and the composition factors (subquotients), X µ , of the kernel satisfy µ < λ.
• the indecomposable projective cover P λ of X λ maps onto W λ via a map ψ : P λ → W λ whose kernel is filtered by modules W λ with µ > λ.
In our setting, the cell modules W j,z 2 for JTL N (1) are the standard modules with the weight λ = (j, z 2 ). Using our results on the projective JTL modules described above and choosing for the quasi-hereditarity partial order ≤ the one opposite to the cellular partial order , i.e., if ≤= , we readily see that JTL N (1) is a quasi-hereditary algebra (it is actually quasi-hereditary for any q = i). Then, as for any quasi-hereditary algebra [54,55], we have a key theorem on tiliting JTL modules 3 : • For any weight [j, P ] ∈ S, there is a unique indecomposable tilting module T j,P such that [T j,P : W j,P ] = 1 and W j,P is a submodule in T j,P , and (4.7) • Any indecomposable tilting module is isomorphic to some T j,P .
This theorem guarantees existence and uniqueness of indecomposable tilting JTL N -modules generated from a cell module. It also gives a good criteria on whether a given cell module appears in the cell filtration of a tilting module: if [j ′ , P ′ ] ≺ [j, P ] then the multiplicity T j,P : W j ′ ,P ′ = 0. Using also the subquotient structure of projective covers for cell modules, it is easy to see that the numbers T j,P : W j ′ ,P ′ are less than 2 (0 or 1) if [j ′ , P ′ ] ≻ [j, P ]. Further, the structure for T j,P modules we proposed above does give a filtration by cell and duals to the cell modules, as indicated in Fig. 15 and Fig. 16. To prove that these modules for j > 0 are indeed indecomposable we observe, using the universal construction as we did for T 0,q 2 above, that if one of the multiplicities T j,P : W j ′ ,P ′ is 0, for [j ′ , P ′ ] ≻ [j, P ], the corresponding module would not be able to have a filtration by co-cell modules. We thus have the criteria (4.7) with the stronger condition "if and only if". By uniqueness we finally obtain that the modules with the subquotient structure proposed are indecomposable 4 .

Centralizer and formal decomposition
In our previous works [30,31,32] on the gℓ(1|1) spin chain, the analysis of the spin chain and the scaling limit properties was based on the structure of the centralizer of the JTL N (0) algebra, dubbed U odd q sℓ (2). The centralizer was found in a rather straightforward way as a proper subalgebra of the open-case centralizer -U q sℓ(2) at q = i. The subalgebra U odd q sℓ(2) was identified with appropriate polynomials in the finite number of generators of U q sℓ (2). Unfortunately, the centralizer A 2|1 (N ) for the open sℓ(2|1) spin-chains is not explicitly described in terms of generators and defining relations: it is not U q sℓ(2) at q = e iπ/3 but only a Morita equivalent algebra. At best, we have only its cellular basis description [14] and it is complicated to identify in a straightforward way a subalgebra in A 2|1 (N ) that commutes only with elements from JTL N (1). The lack of suitable description of the open-case centralizer thus makes the centralizer approach used in the gℓ(1|1) case less promising in the present case of faithful representations of JTL algebras.
Although the faithfulness makes, at first glance, the analysis of the periodic spin chains much harder than in the non-faithful case of gℓ(1|1) spin chains, it also provides many advantages. First of all, we know what kind of 'complexity' of JTL modules we might expect in the decompositions. They should be as complicated as the projective covers P j,z 2 described above. So, the structure of the periodic sℓ(2|1) spin chain is apparently much more involved than the one described in [31]. Further, we have seen that projective covers are still not good candidates for direct summands in the spin chain, since they are non-self dual. The best candidates are the tilting JTL modules which contain the projective covers as submodules. Once again, each indecomposable tilting module is self-dual and 'smaller' than a projective cover but we can always find two non-isomorphic tilting modules that contain the projective cover in their direct sum, as follows from their structure. It is thus quite reasonable to expect that the periodic spin chain, as a self-dual JTL N -module, is decomposed onto tilting modules.
We should emphasize that the important missing step in this new analysis of periodic spin chains based on the theory of tilting modules is a proof 5 that our spin-chain is a full tilting module. To show this point one should prove that the periodic sℓ(2|1) spin chain indeed has a filtration by cell modules: 4 We have to note that our definition for tilting modules is slightly stronger than the one for quasi-hereditary algebras [54,55]. Tiltings are usually not required to have necessary dual or opposite filtration by co-standrard modules, just any filtration by co-standards. It turned out that our tiltings do have a filtration by co-standards opposite to the one by the standard or cell modules. 5 We believe that one could repeat steps in the Martin's paper [60] for the mirror spin-chains which are representations of the blob algebra (also some quotient of affine TL). The crucial technical problem is that one should find a proper embedding of JTL au N into JTL au N+1 for any N .
using the fact that all e j 's are self-adjoint operators, it would be then straightforward to show that the full spin chain is a full tilting module. There are strong arguments suggesting that the spin chain has a filtration by cell modules. In particular, the spin chain is deeply related with a generic loop model, which is defined for arbitrary values of q [34]. In this model, non contractible loops get the weight m if they wind around the small (space) cycle, and a modified weight q ′ + q ′−1 if they wind around the long (imaginary time) cycle (the particular case we are interested in corresponds to m = 1 and q ′ + q ′−1 = 3). For generic values of q, it is possible to show that the generating function of levels expands only on traces over cell modules, so the corresponding 'spin chain' has a filtration by cell modules indeed. However, this argument cannot be made rigorous because, for q generic, there is in fact no such spin chain, and the generating function of levels is only a formal object. Hence the existence of a filtration by cell modules cannot be proven by 'analytic continuation'.
Assuming this crucial assumption about cell-modules filtration however, we see that the spinchain itself should be a full tilting module, and the decomposition, up to multiplicities, follows from the foregoing discussion of these modules.
The next step in our analysis is to obtain the multiplicity of each tiliting module. Since we know the structure of all tilting modules in terms of standard modules, and taking into account Thm. 4.5, we could compute each of these multiplicities iteratively if we knew the numbers [H N : W j,z 2 ] of appearance of each JTL standard module in the full spin-chain. For example, the multiplicity of T 0,q 2 equals [H N : W 0,q 2 ] because the subquotient isomorphic to W 0,q 2 is contained only in T 0,q 2 . Then the multiplicity of T 2,1 equals [H N : W 2,1 ] − [H N : W 0,q 2 ] because subquotients isomorphic to W 2,1 were already counted in the T 0,q 2 modules [H N : W 0,q 2 ] times, etc.
The numbers [H N : W j,z 2 ] can in fact be computed assuming the possibility of an analytical continuation from semi-simple cases where these numbers are known [14]. We have an infinite family of super-symmetric spin chains defined in the same way as in the sℓ(2|1) case but with the sℓ(n+m|n) symmetry such that m > 2 and n ≥ 1, see App. B. The nearest neighbour coupling or the projection onto a singlet for these spin chains also defines a faithful representation of the JTL N (m) algebra with q + q −1 = m > 2 6 . The decomposition of the spin chains for these cases can be written as where we have used the notationD ′ j,K with K = πp/M for the dimensions of the irreducible representations of the centralizerÂ n+m|n (2L). This centralizer is discussed in [14] for the semi-simple cases. We will discuss it briefly below, and for now just recall the dimensionŝ where j ∧ r is the greatest common divisor, j ∧ 0 = j and w(j, d) = q 2d + q −2d δ j/d≡0 + q ′2d + q ′−2d δ j/d≡1 , (4.10) with congruences being taken modulo 2 and we set q ′ + q ′−1 = m + 2n.
Let us give a few examples of these multiplicities. For the case 2L = 6, the decomposition of the semisimple Hilbert space reads We can actually think about the expression (4.9) formally as a polynomial in the complex variable q. We note then that the multiplicities are well defined in the critical cases when q + q −1 equals 0, 1 or 2, i.e., the polynomialsD ′ jK give positive integer numbers for q = i, e iπ/3 and 1 as well. Of course, we will not have a direct sum decomposition as in (4.11) at these critical values of q. Each direct summand will become reducible but indecomposable and it should be glued with other direct summands in (4.11) in order to make a self-dual module (recall that our space of states H N is always a self-dual module). We should thus think about the numberD ′ j,K as the number of appearance of (subquotients isomorphic to) W j,e 2iK in the full space of states H N . In other words we assume the equality [H N : W j,e 2iK ] =D ′ j,K . (4.12) It will be shown below (by a numerical analysis) that this assumption indeed gives correct results on the number of higher-rank Jordan cells, in particular. So, in the sℓ(n + 1|n) case we get the following filtration by cell modules W j,z 2 , setting l ≡ 2n + 1: where by H N we denote a formal decomposition on cell modules modulo glueings (this is why we do note use here the direct sum '⊕' symbol) 7 . For l = 3 meanwhile (that is, the sℓ(2|1) spin-chain) we get H 6 = W 0,q 2 + 8 W 1,1 + 23 W 2,1 + 24W 2,−1 + 112 W 3,1 + 105 W 3,e 2iπ/3 + W 3,e 4iπ/3 . (4.14) Having these filtrations (or formal decompositions) by cell modules, we already see that the tilting module T 0,q 2 appears always with the multiplicity one (it is the trivial sℓ(2|1) module) which is quite important for our analysis -it means that we have only one ground state. The sℓ(2|1) content of the other multiplicities in front of T j,z 2 will be discussed below.
We now give examples for N = 6, 8, 14 detailing the subquotient structure (with simple subquotients) of the tilting modules T j,P .

N = 6 example
We begin with a simple example on N = 6 sites. We give first the structure of indecomposable tiltings, using Fig. 15 and Fig. 13.
• [2,1] where different types of nodes denoted by • and • show the cell-filtration of the tilting modules: symbols are assigned to cell modules that appear on the left from equalities and all simple subquotients (on the right) from a particular cell module are denoted by the corresponding symbol. Here, the notation 2(1) means that any of the two numbers 2 or 1 can be the first index in the subscript. All other tiltings over JTL 6 (1) are irreducible. We thus get the following dimensions Using the diagrams we also obtain that the only nontrivial homomorphisms between the tiltings are Hom(T 2,1 , T 0,q 2 ) = C, Hom(T 0,q 2 , T 2,1 ) = C, Hom(T 2,1 , T 3,q ±2 ) = C, Hom(T 1,1 , T 3,q ±2 ) = C, (4.16) Hom(T 3,q ±2 , T 2,1 ) = C, Hom(T 3,q ±2 , T 1,1 ) = C. (4.17) Next, using the filtration of H N by cell modules given in (4.14) and the subquotient structure of tiltings we deduce the spin-chain decomposition for N = 6 over the tilting modules T j,P : Together with (4.15)-(4.17), this gives the dimensions of simple modules over the centralizer of JTL 6 (1) and its indecomposable tiltings. Because the centralizer is bigger than the sℓ(2|1) symmetry, the multiplicities in (4.18) arise in general from direct sums of atypical/typical representations of sℓ(2|1). Let us describe the sℓ(2|1) content of these multiplicities here. Using the formulas in [61,62], we find the following decomposition of the Hilbert space over sℓ(2|1) (4.19) where we use the notations of appendix A. Recall that the dimension of typicals {b, j}, with b = ±j, is 8j, while the projective sℓ(2|1)-modules P ± (j) have dimension 16j + 4 for j > 0 and 8 for j = 0.
First of all, it is clear that the multiplicity 1 in front of T 0,q 2 in (4.18) corresponds to the trivial atypical sℓ(2|1)-module {0, 0}. This is because T 0,q 2 contains the groundstate of the Hamiltonian which transforms trivially with respect to sℓ(2|1). This accounts for the direct summand {0, 0} in the sum (4.19) plus the 5 copies of the top of P (0) and the 5 copies of the (bottom) submodule {0, 0} ⊂ P (0) -we have thus counted 11 copies which is precisely the dimension of T 0,q 2 , see (4.15). The multiplicity 8 of T 1,1 corresponds to the adjoint sℓ(2|1) representation {0, 1}: we shall see in the following that the scaling limit of this tilting module contains the Noether currents associated with the sℓ(2|1) symmetry. This accounts for 17 out of the 18 {0, 1} modules in the decomposition (4.19).
Interpreting the multiplicity 22 in front of T 2,1 is a bit more difficult. We use the decomposition on 4 sites, 20) where this multiplicity also appears in front of T 2,1 , and observe that it is enough to look at the bimodule for JTL and its centralizer -this gives many constraints for the sℓ(2|1) content. The bimodule can be easily constructed using the subquotient structure of the tilting modules and information about possible non trivial Hom spaces between them, as those in (4. where the last pieces come from breaking up the projectives P ± (1/2 Note that we obtained that both multiplicities 75 correspond to isomorphic sℓ(2|1) modules; they are non-isomorphic only as modules over the JTL's centralizer.

General structure of tilting modules and Hamiltonians's Jordan cells
In this section, we give more examples of the subquotient structure of tilting modules, and provide finally the corresponding general pattern. We also discuss some of our results on Jordan cells for the spin chain Hamiltonian H. Remarkably, we not only observe the Jordan cells of rank higher than 2, but in fact show that the maximum rank in the spin chain grows with the number of sites.

N = 8 example
We analyze first the more interesting case of 8 sites, where Jordan cells of rank 3 appear for the first time. They involve now states in the simple subquotients X 4,1 . To justify this, we describe the structure of indecomposable tiltings with simple subquotients, using Fig. 15 and Fig. 13.
T 2(1),1 : • [4,1] T 0,q 2 : • [2,1] • [4,1] where different types of nodes denoted by , • and • show the cell-filtration of the tilting modules: symbols are assigned to cell modules that appear on the left from equalities while all simple subquotients (on the right) from a particular cell module are denoted by the corresponding symbol. The notation 2(1) means that any of the two numbers 2 or 1 can occur as the first index in the subscript. Given this structure for the tilting modules T 2(1),1 , we expect the Hamiltonian H to show rank-3 Jordan cells involving (generalized) eigenstates from simple subquotients [4,1]. Note that this is not a rigorous result, since the indecomposability does not necessarily mean that all elements of the algebra have Jordan cells -only that some might do. Nevertheless, there is overwhelming evidence that, from an algebraic point of view, the Hamiltonian behaves in a very 'generic' fashion, as we now verify numerically. On N = 8 sites for instance, the module X 4,1 is one-dimensional and corresponds to the eigenvalue 0. Since the tilting modules We have actually checked the presence of the 30 rank-3 Jordan cells in the whole spectrum. Note that we have also carefully analyzed the multiplicities of the rank-2 Jordan cells on small sizes, and found a perfect agreement with our algebraic results.

N = 14 example
We next analyze the case of 14 sites where Jordan cells of rank 4 should appear for states from simple subquotients X 7,1 . The first time rank 4 Jordan cells for the Hamiltonian might appear is actually for N = 12 but diagrams for subquotient structure in the case of N = 14 look somewhat nicer and we have decided to discuss this case instead. As above, using the structure of tilting modules in Fig. 15 given in terms of cell modules from Fig. 13, we can describe the structure of indecomposable tiltings in terms of simple subquotients. For this, one should also use self-duality arguments. An example is given in Fig. 17 for T 3,q ±2 . We also notice in the diagram a pattern of appearance of isomorphic simple subquotients. From the previous case we learned that these subquotients at different sections (levels) of the diagram are connected by the action of our Hamiltonian. For example, simple subquotients X 7,1 indicated in bold appear 6 times but at 4 levels. So, we expect that these modules allow Jordan cells for the Hamiltonian H of (at least) rank 4.
Next, the tilting JTL N -modules T 2,1 and T 1,1 for N = 14 allow Jordan cells of rank 5 for the Hamiltonian H acting on states from simple subquotients X 7,1 . The rank is actually stabilized, e.g., for all N > 14 the maximum rank on the whole spin chain (and at least for first low lying states) for subquotients X 4,1 is 3, for X 7,1 is 5, etc.
Finally, we give the structure of the "vacuum" (we call it so since it contains the vacuum state of the spin chain) tilting module T 0,q 2 at N = 14  where •'s denote simple subquotients from the standard modules W 0,q 2 and W 7,1 , •'s denote subquotients from W 2,1 and W 6,q ±2 , and 's are for the standard modules W 4,1 and W 5,1 . Using these notations, one can easily see the filtration by standard modules proposed before. One can also note that the structure of T 0,q 2 is essentially (but not completely) fixed by the filtration proposed and the self-duality requirement. The structure (4.23) for the tilting module T 0,q 2 is obviously invariant under the duality operation. We drew only the minimum number of arrows between the simple subquotients -those fixed by the duality -though there might be additional arrows, for example, between X 7,1 and X 6,q ±2 . One can also check that the projective covers found in previous sections, see Fig. 14, indeed cover any submodule in T 0,q 2 . For example, the top node/subquotient X 7,1 is covered by the projective module P 7,1 in the following way: the kernel of the map P 7,1 → T 0,q 2 contains the submodule W 1,1 , the maximum proper submodule of W 2,1 (considered itself as a submodule in P 7,1 , see Fig. 14) and a linear combination of the maximal proper submodules in the two subquotients W 3,q 2 and W 3,q −2 as they are presented in Fig. 14.
We finally note in this example, that the vacuum tilting module T 0,q 2 has two irreducible topsthe subquotients X 7,1 marked by '•' and X 2,1 marked by '•'. This happens because we can not have arrows connecting the nodes [6, q ±2 ], lying higher in the diagram, with the node [2, 1] -otherwise the projective modules P 6,q ±2 could not cover submodules growing from these nodes [6, q ±2 ] which is a contradiction. Similarly, one can exclude many other arrows and end up with the diagram we present in (4.23). This property that the states from X 2,1 and the vacuum state from X 0,q 2 are somehow disconnected from the rest of the module is peculiar to the vacuum tilting module. All other indecomposable tiltings for N = 14 have a unique top subquotient.

General structure for T 0,q 2 and Jordan cells for H
We now discuss the general structure of the vacuum tilting module in terms of irreducible subquotients. It turns out that this structure is best formulated through the example of N = 18. Using the same ideas as before, we obtain the diagram for the structure of T 0,q 2 presented in Fig. 18, where •'s denote simple subquotients from the standard modules W 0,q 2 , W 7,1 and W 8,1 , the •'s denote subquotients from W 2,1 and W 6,q ±2 , and 's are for the standard modules W 4,1 , W 5,1 and W 9,q ±2 . Using these notations, one can easily see the filtration by standard modules proposed before. We see that in general we have two cones: one consisting of •'s, and the other reflected in the horizontal line containing X 0,q 2 . The vacuum irreducible subquotient X 0,q 2 lives in the intersection of the cones while the boundaries of these cones give bounds for the appearance of simple subquotients X j,P , with j > 2. We denote this particular structure of the vacuum tilting module the "Eiffel tower" structure. Now for larger number of sites N , the vacuum tilting module T 0,q 2 has essentially the same pattern of nodes and arrows between them 8 , the increasing values of N simply giving rise to longer and longer 'ladders' on the left and right sides of the corresponding "Eiffel tower".
Finally, we give a short comment on the other tilting modules T j,P with j > 0 that belong to the doubly critical class, i.e., those corresponding to the weights [2,1], [3, q ±2 ], etc. Though it is not easy to write down a general diagram for their subquotient structure in terms of irreducible subquotients (one can imagine a "ladder-of-ladders" structure, of course), the most important is actually their general diagram in Fig. 15 in terms of cell modules.

On Jordan cells.
Having now a better understanding on how the diagrams for tilting modules are organized in terms of simple subquotients (at least for the vacuum tilting module) we are ready to discuss the possible structure of Jordan cells for the spin chain Hamiltonian H. Indeed, following the discussion at the end of Sec. 4.7.1 and Sec. 4.7.2, and in agreement with numerical experimentation, the rank of the Hamiltonian Jordan cells involving states of a particular (generalized) eigenvalue can be estimated by counting, in the diagram for a given module, the horizontal levels that contain the simple subquotients to which these states belong. So, for general values of N , we believe that the Hamiltonian Jordan cells involving states from X j,P subquotients in T 0,q 2 have rank at least as large as the number of appearance of X j,P on the left (or right) part of the corresponding diagram of T 0,q 2 . Once again, to obtain the diagram for larger values of N > 18 one should extend the one in Fig. 18 in a rather obvious way, and, for values N < 18, just remove nodes for subquotients with the j index greater than N/2. We also note that we do not assume that the Hamiltonian is diagonalizable in each simple JTL. There are recent results [63] stating that the Hamiltonian H = N i=1 e i has Jordan cells of rank 2 in cell (standard) modules for the periodic Temperley-Lieb algebra, which is rather surprising comparing to the open case. It could thus be that counting the horizontal levels in the diagram only gives lower bounds on the rank Jordan cells. We give our conjecture on these bounds below.
Using the general diagram in Fig. 15 and taking into account the fact that an irreducible subquotient [j ′ , P ′ ] appears in a cell module W j,P only if j ′ > j and with multiplicity one, see Fig. 13, Figure 18: Subquotient structure "Eiffel tower" of the vacuum tilting JTL 18 -module T 0,q 2 . Here, •'s denote simple subquotients from the standard modules W 0,q 2 , W 7,1 and W 8,1 , the •'s denote subquotients from W 2,1 and W 6,q ±2 , and 's are for the standard modules W 4,1 , W 5,1 and W 9,q ±2 .
we can estimate lower bounds for the rank of the Hamiltonian Jordan cells in each tilting module. We should just count the number of levels (also called Loewy layers) where a particular irreducible subquotient [j ′ , P ′ ] appears. So, at large enough N the Jordan cell rank for states from [j ′ , P ′ ] in T j,P with j ′ ≥ j is bounded by the number of (horizontal) levels counted from the level of the diagram in Fig. 15 containing the node W j,P up to the level containing the node W j ′ ,P ′ . One can easily compute this number. For example, the highest rank Jordan cells for j ′ > 2 should be in T 2,1 and T 1,1 and states from [j ′ , P ′ ] subquotients are expected to be involed into Jordan cells of H of rank at least given by the following number: Further, using the structure of T 0,q 2 presented in Fig. 16, we also expect that the lower bound for states from X 2,1 subquotients is 2, and for states from X 3,q ±2 is 1. For the other subquotients with j ′ > 3 we have that the value for the lower bound of the Jordan cells rank differs by one from that in (4.24). It is because the diagram for T 0,q 2 in terms of cell modules has no nodes W 3,q ±2 and thus, the number of corresponding horizontal levels or Loewy layers is lowered by one unit. Tilting modules in the singly critical case where the corresponding cell modules have a chain structure can be studied in a similar fashion. We do not describe explicitly their structure here and we refer the interested reader to [17] where very similar modules were encountered in the context of the blob algebra.

Taking the continuum limit of the sℓ(2|1) spin chain
We now turn to the scaling (continuum) limit of the sℓ(2|1) spin chain: our goal is to infer from the foregoing algebraic analysis results about the representations of the product of left and right Virasoro algebras that act on the low energy states, and ultimately, all the properties of the corresponding LCFT. This is a difficult task, which we will only begin in this paper.
A major difference with the case of gℓ(1|1) studied in [30,32] is that the sℓ(2|1) spin chain is not free. It cannot be diagonalized using free fermions (or a combination of free fermions and free bosons), and therefore, we have much less control on the (generalized) eigenstates and eigenvalues, and consequently, on the scaling limit.
Many properties of the sℓ(2|1) spin chain can nevertheless be obtained exactly, by combining the algebraic analysis with the Bethe ansatz. It is important at this stage to stress that our chain is not the integrable alternating spin chain one would obtain [45] from the general inverse scattering construction. To obtain, for instance, the (generalized) eigenvalues of the Hamiltonian, what one must do is observe that, since the chain provides a representation of the Jones-Temperley-Lieb algebra, the eigenvalues in each standard module can be obtained using the results from another, Bethe ansatz solvable chain, where the same modules appear -in our case, the twisted XXZ spin chain.
Focussing now on the continuum limit, we are interested in the generating function of gaps for each standard JTL module. Since we know the decomposition of the spin chain over tilting modules, and each summand consists of a glueing of many standard JTL modules, the generating function for gaps in our spin chain -which will then give information on the Virasoro content -will be obtained using results of the previous section. To start, we thus describe the scaling limit of the twisted XXZ models, where each spin sector is isomorphic to a standard module over the affine TL at generic values of parameters and the asymptotics (at large N ) of the generating functions is known.

Twisted XXZ spin chain and continuum limit
It is well known that the 6-vertex model (or the XXZ spin-chain) provides a natural representation of the affine Temperley-Lieb algebra, where the generators read acting on (i, i + 1)th tensor components or spins {↑↑, ↑↓, ↓↑, ↓↓} of the "Hilbert" space H XXZ = (C 2 ) ⊗2N and 1 ≤ i ≤ 2N − 1. In the basis of the last and the first spins, the last generator e 2N is where of course it is implied that e 2N acts as the identity operator on all the other spins. The resulting Hamiltonian H = − i e i reads, up to an irrelevant constant We shall refer to this model as the twisted XXZ spin chain. The choice of the twist e iφ = e 2iK allows to select specific generically irreducible representations of the affine TL algebra. For j = 0, the Hilbert space of this model in the sector with the total spin S z = j is isomorphic to the standard module W |j|,e ±2iK , where '+' is for positive j and '−' is for negative value of j. This can be easily checked using the translation generator u of the affine TL represented in the twisted chain as where the s i 's are the permutations of the ith and (i + 1)th sites. The scaling limit of each sector can be inferred using the basic fact [64] that the generating function of the energy and momentum spectra is related to conformal spectra (for the critical Hamiltonian at |q| = 1) as where H and P are the lattice hamiltonian (normalized such that the velocity of sound is unity) and momentum, e 0 is the ground state energy per site in the thermodynamic limit, we also set q(q) = exp − 2π N (β R ± iβ I ) with β R,I real and β R > 0, and N is the length of the chain. The trace on the left is taken over the states of the spin chain in a given sector, and the trace on the right over the states occurring in this sector in the continuum limit.
The traces of the scaling hamiltonian in the generic irreducible representations are easily worked out using the XXZ hamiltonian, to which methods such as the Bethe ansatz or Coulomb gas mappings can be readily applied [50]. Let us introduce the usual notations for the central charge and the conformal weights where with this parametrization q = e iπ/(x+1) . For x = 2 or c = 0 the conformal weights h r,s are arranged in the Kac table in Fig. 19.
The trace on the left hand side of (5.5) taken over the subspace H j of spin projection S z = j, with −L ≤ j ≤ L, in the XXZ chain of length N = 2L has the limit [50,65] where K = π M p and F j,e 2iK = q −c/24q−c/24 P (q)P (q) Recall that W j,e 2iK modules over the JTL N algebra are irreducible at generic values of q, where the pseudomomenta e 2iK are now taken as jth roots of unity, and we have an isomorphism H j ∼ = W j,e 2iK for j > 0 and the choice of the twist φ = 2K. Therefore, the generating function of levels in the scaling limit of the JTL modules W j,e 2iK , can be simply written as where we used same notations for the modules W j,e 2iK and their scaling limits 9 . The character formulas for the scaling limit of the JTL simple modules at generic q shows that the scaling limit of the JTL N algebra should be an operator algebra containing vir ⊕ vir as a proper subalgebra.
The case j = 0 requires more care as one has to be careful about the loops that wrap around the spatial direction due to the periodic boundary conditions which would get a weight 2 if φ = 0 in (5.2). One then needs to introduce a twist e iφ = q 2 to account for this 10 . In this case, the sector S z = 0 corresponds to the standard module 11 H 0 ≃ W 0,q 2 = W 0,q 2 → W 1,1 . The trace over the scaling limit of the representation W 0,q 2 thus reads 12) or introducing the character of the Kac representation we have Finally, we quote a formula that will play a fundamental role later: with, recall, q = e iπ/(x+1) . Of course, (5.14) is just a particular case where j = 0 and k = 1. We note that the pseudomomenta (jth roots of unity) for our JTL modules appear in this formula only at integer values of x.
The generating functions of standard modules in (5.8) can be obviously written in terms of the characters k r,s of Virasoro Verma modules. Even at generic values of x or the central charge c, it does not mean, of course, that the scaling limit of the corresponding JTL modules is a direct sum of products of Verma modules. For example, the Feigin-Fuchs (FF) module F r,s has the same character k r,s as the Verma module of weight h r,s , but is not Verma (see a review on FF modules in App. D). Actually, it is known that XXZ chains are closely related to the Coulomb Gas model [50]. Given that FF modules are constructed directly from a free boson picture, it is rather natural to expect our characters to describe the product of FF modules instead of Verma modules. We will actually see later directly at c = 0 that assuming (a filtration by) Virasoro Verma modules in the scaling limit of the JTL standards will be in contradiction with the algebraic structure, while Feigin-Fuchs modules give a consistent picture.
We also note that it is not clear whether at generic values of q the scaling limit of each JTL standard (also simple) module W j,e 2iK considered as a vir ⊕ vir module is self-dual. This is related with the question of whether loop models at generic values of q (or x) are "physical", that is, described by consistent local bulk CFTs 12 . The answer is not clear, because for generic q, there exists no 10 It is indeed easy to check that within our twisted XXZ representation, non-contractible loops carry a weight 2 cos φ 2 . 11 We note that choosing the twist as e iφ = q −2 gives H0 as the costandard module W * 0,q 2 = W1,1 → W 0,q 2 . 12 Recall that one of consistency conditions for bulk CFTs is a non-degeneracy of the two-point function that is defined as the Virasoro-algebra invariant bilinear form on the space H of all non-chiral fields. The non-degeneracy condition implies that the space of linear forms on H is isomorphic to H itself, as vir ⊕ vir modules, or in other words H has to be self-dual.
(supersymmetric) spin chain formulation, no self-dual 'Hilbert space', etc. Self-duality of the limits of W j,e 2iK , as vir ⊕ vir modules, at generic values of q on the other hand would have very important practical consequences: this will be discussed elsewhere.

Operator content of simple JTL modules
We start our analysis by discussing the vir ⊕ vir content of the simple JTL N (1) modules in the continuum limit for c = 0. By this, we mean the representation content of the states that contribute to the scaling limit in the JTL N modules. It is convenient for this purpose to first evaluate the generating functions of levels in some of the JTL N modules we have encountered previously.
The characters of the corresponding JTL simples X j,P in the limit are obtained by a series of subtractions and additions, just like in the computation of dimensions for finite chains: the only difference is that the series is infinite -but convergent. As an example, we consider We now use the basic identities together with, for l = 1, 2, where the χ's are as usual characters of the Virasoro algebra simples. By straightforward manipulations we find first as was expected because the dimension of X 0,q 2 is one for any even number of sites. Note also that χ 12 = χ 11 = 1.

Left-right Virasoro structure of simple JTL modules
We stress that so far we only computed characters F j,e 2iK of the (scaling limit of) the simple JTL modules X j,e 2iK : an additional analysis is required to see whether each of the modules X j,e 2iK is a direct sum of simples over left-right Virasoro vir ⊕ vir, as was in the case of gℓ(1|1) spin chains [32]. Unfortunately, we give strong arguments below that JTL simples (those belonging to an indecomposable block of JTL modules) in the case of the sℓ(2|1) spin chain are not, in general, direct sums of Virasoro simples -and involve instead reducible but indecomposable modules. This suggests that the analysis of the scaling limit should involve a bigger algebra than just vir ⊕ vir -in other words, that the scaling limit of the JTL algebra is more than vir ⊕ vir. This will be discussed in the conclusion, and in subsequent work.
We begin our analysis with the standard module W 0,q 2 . We note that this JTL module is welldefined for generic q or x. For such values of x, using the character (5.14) and the fact that weights h r,1 do not differ by an integer for different integer values of r, the scaling limit of the vacuum standard module, which we will denote by the same symbol W 0,q 2 as for a finite lattice, is decomposed over the left-right Virasoro onto the direct sum of simple vir ⊕ vir-modules where K r,s are Virasoro modules with the Kac characters K r,s given in (5.13) and we also introduce the corresponding anti-chiral modulesK r,s . These modules are simple at generic central charges c or generic values of our parameter q. Moreover, there cannot be glueings/extensions among the modules K r,1 in such generic cases, so the limit of W 0,q 2 must be a direct sum. Note that the first term K 1,1 ⊠K 1,1 contains the identity (or the vacuum state of dimension (0, 0)), for any x, and its descendants with respect to left-right Virasoro, while the next term K 2,1 ⊠K 2,1 is spanned by the descendants of the primary field of conformal dimension 3+x 4x , 3+x 4x (or ( 5 8 , 5 8 ) at c = 0), the term K 3,1 ⊠K 3,1 is spanned by the primary field of dimension 2+x x , 2+x x (which equals (2, 2) at c = 0) and its descendants, etc.
In contrast, when q = e iπ/3 or c = 0, the vacuum standard module (scaling limit of W 0,q 2 ) is not a direct sum of simple Virasoro modules anymore. The subsequent analysis requires, for the time being, a series of natural assumptions that we spell out below. These assumptions are checked a posteriori by the consistency of the full picture.
When q = e iπ/3 , or c = 0, each module K r,1 , for odd values of r, becomes indecomposable but reducible with the subquotient structure K r,1 : and each tensor product K r,1 ⊠K r,1 in (6.7), assuming the decomposition for q = e iπ/3 involves the 'continuation' of the modules present for q generic, becomes the following left-right Virasoro module K r,1 ⊠K r,1 : Moreover, these diamonds could be in principle glued with each other: what we can say at this stage of our analysis is that the vacuum standard module has a filtration by the left-right Virasoro modules K r,1 ⊠K r,1 (recall our definition of the filtration around (4.2)). This means that K 1,1 ⊠K 1,1 is a Virasoro submodule (as it contains the vacuum and we expect the trivial action of positive modes on the identity and energy-momentum tensors), that there might be arrows due to positive Virasoro modes action from K 3,1 ⊠K 3,1 into K 1,1 ⊠K 1,1 , and so on. Modulo the arrows connecting different diamonds we have the structure of the full vacuum module W 0,q 2 : W 0,q 2 : ) ⊕ ( Note once again that these diamonds are just products of the indecomposable Kac modules K r,1 at the logarithmic c = 0 point, and that the existence of a filtration by these products of Kac modules follows from the generic point decomposition (6.7). All the modules K r,1 can be equivalently obtained as quotients/submodules of the Feigin-Fuchs modules introduced in App. D. We see from this filtration that the scaling limit of the simple JTL module X 2,1 is not a direct sum of simples over the Virasoro algebra but a direct sum of indecomposable but reducible modules with subquotients marked by '•', and of those having fractional conformal dimensions. Note that the module X 0,q 2 is always one dimensional; it is spanned by the vacuum state, and its scaling limit is marked by '•' and corresponds to the trivial Virasoro representation. The two states generating subquotients (2, 0) and (0, 2) are respectively the chiral T and anti-chiralT energymomentum tensors, and the state generating (2, 2) is their product TT . Note finally that we could assume a filtration of W 0,q 2 by products of modules dual to the Kac ones, i.e. by those with reversed arrows in their subquotient structure (the character would be the same). In our case however, the unique vacuum state would be a descendent of the energy-momentum tensors T andT . Of course, this is not allowed.
We now discuss the scaling limit of all other JTL standards W j,z 2 and the corresponding simples. For these purposes it is technically easier at x = 2 (or c = 0) to introduce intermediate modules which are quotients similar to W 0,q 2 : Now, each module W j,q 2j+2k is a glueing of two JTL simples. For example, we have 14) etc. In terms of these modules, we can now describe the structure of all JTL standards by a chain structure W 2,1 = W 2,1 → W 3,q 4 → W 5,1 → . . . .

(6.15)
The important thing is that the intermediate JTL modules W j,z 2 have nice scaling limit properties. To start, we can consider the characters with the right hand side a sum of products of (Virasoro) Kac modules characters. For example, the generating functions for the scaling limit of W 1,1 , W 2,1 , and W 3,q 2 obey We can again say here that the scaling limit of the modules W 1,1 , W 2,1 , W 3,q 2 , etc., is filtered by the product of indecomposable Virasoro modules K r,n having the character equal K r,n (the Kac character), i.e., the first term in the sums corresponds to a submodule, the third term is glued by positive Virasoro modes action to the first one, and so on. The only difference from the case W 0,q 2 is that the anti-chiral modulesK r,n , with n > 3, (they are generically irreducible) are more complicated in the case c = 0. To obtain the character K r,n , the singular vector in the corresponding Verma module to be set to zero is at the level nr. For example, the module K 3,4 has the same highest weight h 3,4 = 0 as K 1,1 , but its subquotient structure has six nodes (not two) corresponding to Virasoro irreducibles. The module K 5,7 has the same highest weight and its structure is even more complicated, and so on. The other problem is that the limit K r,n of generically irreducible Virasoro modules might not be a quotient of the corresponding Verma modules. Indeed, we will see below that they are actually quotients/submodules of Feigin-Fuchs modules 13 . Note also that the first component in the sum (6.16) is always K r,1 or K r,2 and the corresponding modules have two subquotients only and they are indeed Kac modules (quotients of the Verma modules) and at the same time quotients of the Feigin-Fuchs modules.
To show the complexity of the scaling limit of the quotient JTL modules W j,z 2 we describe the left-right Virasoro structure of several terms in the character sum (6.18) corresponding to W 2,1 . The first term K 1,1K1,5 has the diamond structure on the left part of Fig. 20, i.e., four subquotients, a situation encountered before; the next term K 2,1K2,5 corresponds to the left-right Virasoro module of a chain type ( 5 8 , 5 8 ) → ( 5 8 , 21 8 ). The first interesting term is K 3,1K3,5 . If we assumed that the corresponding module at c = 0 was the product of two Kac modules (quotients of Verma) then the corresponding structure would be • (2,12) • (7,12) • (6.20) where the down-left arrows describe the chiral Virasoro action and down or down-right ones are for the antichiral part. We also denote simple Virasoro subquotients contributing to the scaling limit of X 2,1 by '•' and those from the limit of X 3,q 2 by '•'. We used here the corresponding Virasoro character expressions in (6.5) and (C.4). As we can see from the diagram in (6.20), there would be a problem then -there are arrows • → • mapping states from X 3,q 2 to states in X 2,1 and this contradicts to the structure of W 2,1 : X 2,1 → X 3,q 2 . To solve this problem we should actually assume thatK 3,5 , instead of a Kac module, is the quotient of a Feigin-Fuchs module, where half the arrows in the corresponding Verma module are reversed. We then obtain the consistent structure shown on the right of Fig. 20 We see once again from this analysis that the scaling limit of JTL simple modules is a direct sum of Virasoro indecomposable but reducible modules.
of the Feigin-Fuchs modules, or of the duals to them. So, there is a sort of stabilization in the structure of diagrams. Nevertheless, the diagrams for terms having the chain-type structure seem to be growing: the module K 4,1K4,5 has the structure ( 33 8 where we again used the Feigin-Fuchs pattern of arrows, the module K 6,1K6,5 has 4 subquotients and so on. As a first interesting result of our analysis, we get the Virsoro structure of the simple JTL module X 2,1 in the scaling limit by comparing its filtrations, in (6.10) and the one described in Fig. 20. Note that in the two diagrams the nodes (2, 0), (0, 2) and twice (2,2) labeled by •'s are connected in different ways (both diagrams do not contradict to each other; they would if we had used Verma and not Feigin-Fuchs structure). Therefore, all these arrows should be present in the structure for X 2,1 . Further, to complete the diagram for X 2,1 we should recall that the only known existing indecomposable Virasoro module involving irreducible subquotients with the weights 2 twice, 0, and 7 is the staggered module [66]  (0, 2) (2, 0) (2, 7) (7, 2) . . .
plus a direct sum of terms with non-integer (rational) weights. The dots in the diagram mean higher terms like (7, 7), etc. Following this structure, we also conclude that there is a Jordan cell of rank 2 for the Hamiltonian in the scaling limit of X 2,1 . We note that both the indecomposability parameters b andb equal 5 6 for the states with conformal weights (2,2).
Using similar analysis, we can in principle obtain vir ⊕ vir subquotient structure for the scaling limit of all other JTL simples. For example, the structure for X 3,q 2 is X 3,q 2 : (2,5) . . . . . .

(6.23)
Similarly, the structure for X 3,q −2 is obtained just by replacing (h,h) by (h, h).
Our conclusion in this section is that the scaling limit of the JTL simples involved in the doubly critical class (containing X 2,1 , X 3,q ±2 , X 4,1 , etc) gives vir ⊕ vir-modules with non-trivial Jordan cells of rank 2 for the Hamiltonian L 0 +L 0 . It may have escaped the reader at the end of this long discussion that these Jordan cells are not present on the lattice 14 and they only arise in the scaling limit. This is definitely unpleasant, although perfectly possible: the existence of Jordan cells for all system sizes indicates a Jordan cell in the continuum theory as well, but the converse does not have to be true. Nevertheless, such phenomenon did not occur for boundary theories, or for the gℓ(1|1) spin chain. It implies, in particular, that the lattice algebraic analysis by itself can only provide lower bounds to the size of the Jordan cells.

The field content up to level (2, 2)
Before turning to the indecomposable structure of the full LCFT, we describe the operator content of our theory up to level (h,h) = (2, 2) [34], and analyze the multiplicities with respect to the sℓ(2|1) supersymmetry, using the analysis of section 4.6. The results are gathered in Tab. 1.
The one-hull operators O 1 , with conformal weights (h,h) = ( 1 8 , 1 8 ), form an adjoint representation as well, so they appear with multiplicity 8. More interesting are the two-hulls operators O 2 with (h,h) = ( 5 8 , 5 8 ) as one of these fields is the logarithmic partner of the energy operator -the relevant thermal perturbation of our critical theory. These fields appear twice in T 0,q 2 and once in T 2,1 , they therefore transform as 24 = {0, 2} ⊕ P (0). The energy field lives at the bottom of the indecomposable module P (0), it is invariant under sℓ(2|1).
The three-hull operators O 3 with conformal weights (h,h) = ( 35 24 , 35 24 ) appear once in the standard module W 3,1 -or in the tilting module T 3,1 . They thus come with a multiplicity 112 in the full Hilbert space our theory, and transform as (6.24) under sℓ(2|1).   2). We show the multiplicities of each fields in the various tilting modules, together with the total multiplicity with which they appear in the Hilbert space. The way they transform with respect to sℓ(2|1) is also given in the last column.

Content of indecomposable tilting modules: the full LCFT
In the previous section, we discussed left-right Virasoro content in the scaling limit of simple JTL modules and we learned that the simples correspond to (a direct sum of) indecomposable Virasoro modules. We also learned that (reducible) quotients W j,P of JTL standard modules are filtered in the limit by products (of quotients or submodules) of Feigin-Fuchs modules and they have quite simple Virasoro character expressions (6.16). To proceed, it is thus useful to express the structure of tilting modules in the scaling limit in terms of the quotient modules W j,P . We begin with the structure of the tilting module T 2,1 presented in Fig. 21. For simplicity and readability of the diagram, we do not show all the arrows -only the minimum number needed for consistency (for example, there might also be arrows like • and , and so on). In other words, one can translate all arrows on the top surface of the "cube" in Fig. 21 down along the lattice to recover all the arrows for the limit of T 2,1 . The structure for T 1,1 is obtained by replacing the central line in the diagram by W 1,1 → W 3,q 2 → W 4,1 → . . . . All the other tilting modules T j,P with the relations (j, P ) (2, 1) can be similarly presented by the "cubic" diagrams.
Finally, the structure of the vacuum tilting module T 0,q 2 in the limit in terms of W j,P looks slightly different and it can be obtained using the cell filtration in Fig. 16   applications, we give below a more detailed analysis of its structure in terms of irreducible Virasoro algebra subquotients.

The indecomposable tilting module for the stress tensor
We now give a more detailed analysis for the vacuum tilting module T 0,q 2 in the scaling limit N → ∞.
The subquotients structure in terms of JTL simples can be obtained in this limit by continuing the "Eiffel tower" diagram from Fig. 18 where the general pattern is quite clear: one should simply continue "ladders" on the left and right parts of the diagram without restrictions on N . Of course, each JTL simple in the limit is a direct sum of complicated indecomposables over the left-right Virasoro algebra vir ⊕ vir at c = 0 (as we just discussed previously), and it is actually very hard to give a full picture. Nevertheless, to describe the field content for the first few excited levels it is enough to consider only the "kernel" part of the vacuum tilting module from Fig. 18. We depict this part in terms of JTL simples as It is important to note that there are no arrows in T 0,q 2 coming to [2,1] • and no arrows going from the bottom [2,1] • , as we learned from the lattice analysis, for any N , and thus it should be true in the limit. We also note that the full field content up to the level (2, 2) is contained in this kernel part. Now, the idea is to use vir ⊕ vir-module diagrams for irreducible JTL subquotients and compose these pieces into a crucial part of the vacuum tilting module -the part that would contain energy momentum tensors and all their logarithmic partners and all the necessary vir ⊕ vir suquotients that admit the indecomposability parameters b =b = −5 measured in [28]. We recall that the vir ⊕ vir structure of these pieces (scaling limit of JTL simples) was discussed in Sec. 6.2. Combining the diagrams for subquotients in (7.1) (represented in (6.22) and (6.23)) as vir ⊕ vir-modules, joining nodes from different pieces 15 we obtain in Fig. 1 the subquotient structure of a part (of the whole vacuum tilting module) that we call the physically crucial part. We empasize that a vir ⊕ vir-module corresponding to this diagram is not a submodule or a quotient of the scaling limit of T 0,q 2 but it is an (non-irreducible) self-dual vir ⊕ vir-subquotient in the full vacuum tilting module. Therefore, in the full picture, there should be additional arrows coming from above to and going out of this diagram. We do not draw them for simplicity.
We also note that Ridout's "Ockham's razor" for a non-chiral staggered vir ⊕ vir-module [29] with indecomposability parameters b =b = −5 is only a submodule in Fig. 1. This submodule is generated by states from the two nodes (2, 0) and (0, 2) at the fourth layer if counted from below (second layer counted from the top).
The structure for the "kernel" part of the vacuum tilting module involving fields of dimensions up to (2, 2) deserves a more detailed analysis. This structure is described diagrammatically on Fig. 2.
We note that at our level of analysis it is hard to state whether this part of the vacuum module is indecomposable or not. Most probably the rightmost subquotient (2, 2) and its descendants like (2,7), etc., are decoupling from the vacuum module. After glueing two indecomposables, the final module can in principle be a direct sum of non-trivial indecomposables (even infinitely many of them if we talk about Virasoro algebra modules). Arrows coming in and out of this node (2, 2) are thus depicted in a dotted style. Recall that T andT denote states corresponding to chiral and anti-chiral energy momentum tensors while t andt are their logarithmic partners, respectively. The state ψ is for the descendant At =Āt, with the operator A = L −2 − 3 2 L 2 −1 . Following this diagram we thus expect a Jordan cell of rank 2 for the fields in (0, 2) and (2, 0), and of rank 3 for those from (2,2). In particular, the field TT should be involved into a Jordan cell of rank 3. This will be discussed in more detail in a subsequent paper.

Higher rank Jordan cells for
Here, we discuss Jordan cells of ranks higher than 3. These cells involve fields of higher conformal dimensions from the vacuum tilting module. Recall first the structure of T 0,q 2 with irreducible subquotients in Fig. 18 and its continuation to N → ∞. It has infinite 'ladders' on the left and right sides of the corresponding "Eiffel tower". Using this diagram and the finite lattice analysis in Sec. 4.7 on the structure of Hamiltonian's Jordan cells (see in particular the discussion at the end of Sec. 4.7.1), we can conjecture lower bounds (of the maximally 16 possible rank in a given subquotient) for Jordan cells of the Hamiltonian L 0 +L 0 in the corresponding bulk LCFT. These bounds are given by the number of appearance of the subquotients [j, P ] on the left or right part. And note that this number is finite for a fixed j and was already computed from the lattice (since it stabilizes at large enough N ) at the end of Sec. 4.7.3.
We also mentioned above that due to additional degenerations in the Hamiltonian's eigenvalues at N → ∞ we have additional Jordan cells of rank 2 in the scaling limit of JTL simples, at least for those from the doubly critical class. Actually, using rather natural assumptions we were able to see these additional Jordan cells directly in the scaling limit for states having integer conformal weights. We also saw in Fig. 2 that the two Jordan cells are combined into a Jordan cell of maximum rank 3 and not 4. This happens because the bottom part of the upper cell is at the same (Loewy) layer as the top of the lower Jordan cell and they thus can not be connected by vir ⊕ vir action. So, we obtain that the maximum rank of Jordan cells for states from X 2,1 is 3. Similarly, we expect that states from X 3,q ± 2 in T 0,q 2 are in rank 2 maximum (and not 1, as one could expect using only the lattice analysis), states from X 4,1 are in rank 3 (and not 2), states from X 6,q ± 2 are in rank 4 maximum (and not 3), etc. In other words, we expect the ranks of the lattice Hamiltonian Jordan cells discussed in Sec. 4.7.3 (see the results (4.24) and below (4.24)) to be increased by one in the scaling limit, giving rise to the following conjecture on the rank of Jordan cells for L 0 +L 0 in the vacuum tilting module. Conjecture 7.2.1. In the scaling limit of JTL tilting modules T 0,q 2 , states from (limits of ) irreducible X j,P subquotients should be involved in Jordan cells of the Hamiltonian L 0 +L 0 of maximum rank given by the following number: 3 , j mod 3 = 0 and P = q ±2 , j > 0 2 j−2 3 + 1, j mod 3 = 1 or 2 and j > 2 and P = 1. Note that the statement here is about maximum rank because there are states of non-integer conformal dimensions in X j,P subquotients which might be in cells of lower rank. The rank for those states in the vacuum tilting module that have integer conformal dimensions is expected to be given precisely by (7.2).
Since the vir ⊕ vir characters of the scaling limit of the JTL simples X j,P are known (Sec. 6.1 and App C), this conjecture gives possible values of ranks of Jordan cells involving corresponding fields of conformal weights (h,h).
Note finally, that similar results on ranks of Jordan cells can be formulated for other tilting modules. For example, states from (limits of) X j,P subquotients in T 2,1 , for j > 2, should be involved in Jordan cells of maximum rank greater given by those in (7.2) plus one.

Conclusion
The first obvious conclusion is that the situation seems more complicated than one may have expected. If the lattice analysis makes the understanding of titling modules possible, the fact that JTL simples do not correspond to direct sums of vir ⊕ vir simples forces a very delicate discussion, and a proliferation of arrows of doubtful use. We believe this simply means simply that vir ⊕ vir is not the proper object to analyze the continuum limit of our spin chain -and probably LCFTs in general. The proper object should be the full scaling limit of the JTL algebra, which contains vir ⊕ vir, but extends it, giving rise to what we called in [32] the interchiral algebra. We did discuss this algebra in the case of gℓ(1|1), showing that it was generated by the additional inclusion of the field Φ 2,1Φ2,1 of weights h =h = 1. We expect that an analogous interchiral algebra appears in the scaling limit of JTL(1) represented in the sℓ(2|1) spin chain, and is probably generated by the field Φ 2,1Φ2,1 again. Note however that now this field has non integer dimensions h =h = 5 8 . It is likely that each simple JTL module goes over, in the scaling limit, to a simple module over this interchiral algebra, and that the analysis in terms of these modules simplifies considerably. We leave the corresponding discussion for a subsequent paper however. Our algebraic results also have interesting physical consequences: for instance, the field T T with conformal weights (2, 2) is found to lie at the bottom of a Jordan cell of rank 3. It would be very interesting to understand how this Jordan cell arises in the c → 0 limit of operator product expansions.
We also would like to briefly comment that the LCFT obtained as the scaling limit of the sℓ(2|1) spin chain differs fundamentally from previous proposals at c = 0: in both [67] and [27], the vacuum indeed appears with multiplicity greater than one. Moreover, a detailed analysis of the conformal weights and their multiplicities shows that the operator contents of these proposed theories are not at all compatible with the one we obtained here. It is not clear to us whether there might be other lattice models whose scaling limit would correspond to the theories in these references, of whether the corresponding LCFTs are really consistent 17 .
It is probably useful to reiterate here that the only modular invariant our theory is associated with is the trivial invariant Z = 1. See the discussion in Sec. 2 of this paper.
There are certainly many aspects we did not discuss much. Among these is the centralizer and the corresponding bimodule structure. Why this played a crucial role in our analysis of the gℓ(1|1) spin chain, it turned out to be not so important here since the sℓ(2|1) provides a faithful representation of the JTL algebra, and other tools could then be used. But the nature of the centralizer (the 'symmetry' of the theory) in the sℓ(2|1) spin chain and what it becomes in the continuum limit remains to be understood.
We also note that, according to our analysis, indecomposable tilting module T j,P can be considered in the scaling limit, under the algebra vir ⊕ vir, as a (complicated) glueing of an infinite number of Feigin-Fuchs modules. This is reminiscent of the construction in [68], where full LCFTs are obtained as glueings of many copies of free-boson theories via the introduction of extra zero modes. In general, understanding the "naturalness" of the (scaling limit of) the tilting modules and how to obtain them in terms of some free field representation would be, we believe, a great progress. This would probably require a thorough numerical examination.
A The Lie superalgebra sℓ(2|1) and some of its representations In this appendix, we gather some well-known results about the Lie superalgebra sℓ(2|1). In particular, we give an explicit Fock space formulation of the fundamental and dual representations used to construct our Temperley-Lieb spin chain. We also recall some properties of several finite-dimensional representations that shall be used throughout this paper. We follow here [61].

A.2 Fundamental and Dual representations in Fock space
Three-dimensional representations of this superalgebra are readily obtained using creation and annihilation operators. To construct what we will refer to as fundamental representation , we introduce two boson operators [b σ , b † σ ′ ] = δ σ,σ ′ , where σ ∈ {↑, ↓}, and one fermion {f, f † } = 1. The generators read 18 B These generators furnish a representation of sℓ(2|1) in the space The generators act as Note also that the operator is the projector onto the singlet in the tensor product representation ⊗¯ . This expression will be used as a definition of the Temperley-Lieb generator.

A.3 Finite dimensional representations
We also recall some usual notations for the finite dimensional representations of sℓ(2|1). We begin with the irreducible representations. Except for the trivial representation {0} of dimension 1, the irreducible atypical representations {j} ± are labeled by the half-integer j = 1 2 , 1, . . . ; they have dimension 4j + 1. There are also typical representations {b, j} (with dimension 8j and b = ±j) where b is a U (1) charge, they are also projective. When b = ±j, the modules {±j, j} become indecomposable. Using these notations, the fundamental and dual representations are = { 1 2 } + and = { 1 2 } − ; and the adjoint representation is {0, 1}. We will also be interested in atypical projective covers P ± (j) (with dimension 16j + 4 for j = 0 and dimension 8 if j = 0). The projective covers P ± (j), for j = 0, have the subquotient structure by the left diagram whereas P (0) is given by the right one: 18 We use the same notation for the generators and their representation in the Fock space.
The arrows represent here the action of the generators of sℓ(2|1). We shall not describe the tensor product of all these representations here, and refer the interested reader to Ref. [61]. Using those results, one can decompose the Hilbert space H = ( ⊗¯ ) ⊗N onto projective representations only (except for the fundamental). In particular, we have ⊗¯ = {0} ⊕ {0, 1}.

B Faithfulness of JTL N (m) representations on super-spin chains
We discuss in this appendix the more general case of periodic spin-chains with sℓ(m + n|n) symmetry. The sℓ(m + n|n) super-spin chain [14] is the tensor product H N = ⊗ N j=1 V j , with V j ∼ = C m+n|n , which consists of N = 2L sites labelled j = 1, . . . , 2L with the fundamental representation of sℓ(m + n|n) on even sites and its dual on odd sites. We choose a basis v i , 1 ≤ i ≤ m + 2n in each V j such that the range 1 ≤ i ≤ m+n corresponds to the grade-0 (bosonic) subspace and the v i with m+n < i ≤ m+2n span the fermionic subsapce.
We consider the following representation of JTL au 2L (m), which we denote by π m,n : JTL au N (m) → End C (H N ), with the matrix elements for e j (for even j, at least), with 1 ≤ c j , d j ≤ m + 2n, π m,n (e j ) Here, we use the basis in H N spanned by the monoms v c 1 ⊗ v c 2 ⊗ . . . ⊗ v c N , with v c j being the basis in each V j ∼ = C m+n|n . The representation π m,n is equivalent to the one in [14] for any m, n ≥ 0. In particular, for m = 1, the equation (B.1) gives the representation defined in (2.4) and (2.5).
Obviously, the representation π m,n defines a homomorphism of JTL au N (m) to (a representation of) the Brauer algebra. The sector of affine diagrams with N through lines is spanned by u 2j (with the relation u N = 1) and the corresponding image in the Brauer algebra is spanned by 'planar' diagrams, where 2j through-lines on the right (of the fundamental rectangle) are going from the right to the left-top crossing the other lines and thus picking up a sign factor (−1) 2j−1 l=0 |c N−l | N−2j k=1 |c k | as in (B.2). Modulo multiplication by appropriate even powers of u, linearly independent affine diagrams in the ideal of JTL au N (m) with number of through lines less than N contain only arcs and through lines connecting the inner boundary with the outer one and without 'winding' around the annulus (this guarantees that the corresponding diagram in the Brauer algebra have no intersecting through lines). Therefore, any such diagram, sandwiched between appropriate even powers of u, is represented by a matrix with elements obtained by the following rules: The representation π m,n is therefore non-faithful by trivial reasons -the kernel of π m,n contains affine diagrams in the ideal J 0 (without through lines) that are not planar, i.e. the image π m,n (J 0 ) has the dimension 2L L − 2L L−1 2 and is generated by (the representation of) the diagrams that are in bijection with the usual Temperley-Lieb diagrams without through lines (which represented faithfully [14] for n ≥ 1 or m > 1 and n = 0). We next give a proof that non-planar diagrams in J 0 ⊂ JTL au N (m) exhaust the kernel of π m,n , for m, n > 0 or n > 1. The case m = 0 and n = 1 is known to be highly non-faithful.
Proof. Our proof is essentially an adaptation of the original proof of V. Jones [47] in the bosonic case (n = 0) to the super-symmetric case.
We have already shown above that the kernel of π m,n contains the non-planar diagrams 19 from the ideal J 0 ⊂ JTL au N (m) and π m,n J 0 ∩ TL N (m) has no additional linear relations among the planar diagrams due to the faithfullness of π m,n restricted to the subalgebra TL N (m). The non-planar diagrams in the kernel of π m,n span an ideal in J 0 which we denote as J n 0 . In the following, we will thus consider only diagrams µ from the quotient JTL = JTL au N (m)/J n 0 (we use the simplified notation for the algebra JTL N (m)) by this ideal.
Suppose the π m,n has a non-trivial kernel in JTL. It means there are linear relations in End C (H N ) among the basis diagrams µ ∈ JTL, i.e. there exists a linear combination A = µ∈JTL a µ π m,n (µ) ≡ 0, (B .3) where some a µ ∈ C are non-zero for a diagram with through-lines. Consider a diagram µ (which we denote as µ j,k ) with non-zero a µ and containing a through-line connecting, say, the jth site on the inner boundary of the annulus with the kth site on the outer. Matrix elements of this diagram are given either by the ones for π m,n (u 2r ), with an integer r = |j − k|/2, in (B.2) or, following the rules 1.-3. given above, π m,n (µ j,k ) ...d k ...d N ...c j ...c N = δ(c j , d k ) for appropriate pairs {(j ′ , k ′ )}, with j ′ = j and k ′ = k, corresponding to end points of all the other through lines, and pairs {(i, l)}, {(i ′ , l ′ )} corresponding to end points of arcs at the inner and outer boundaries, respectively. Consider a projector P j from End C (H N ) with the kernel of P j being the bosonic (grade-0) subspace in the jth tensorand V j and the fermionic (grade-1) subspace in each V j ′ with j ′ = j (the image of P j is thus spanned by v c 1 ⊗ v c 2 ⊗ . . . ⊗ v c j ⊗ . . . ⊗ v c N with m + n + 1 ≤ c j ≤ m + 2n and 1 ≤ c j ′ ≤ m + n for j ′ = j). Then, we have the equalities (we recall the definition (B.3)) 0 = P k AP j = µ∈JTL a µ P k π m,n (µ)P j = µ j,k a µ j,k P k π m,n (µ j,k )P j (B.5) while P k π m,n (µ j,k )P j = 0, (B.6) where the last sum in (B.5) is taken over all possible affine diagrams having a through-line connecting the jth site with the kth site. The last equality in (B.5) and inequality (B.6) follow from a direct simple calculation using (B.4). We note that some of the a µ j,k in the last sum in (B.5) are non-zero by our construction and the set of diagrams {µ j,k } is in a bijection with planar diagrams from TL N −1 ; the bijection is obtained by cutting the annulus along the through line connecting the jth site with the kth one. We thus obtain that P k π m,n (µ j,k )P j (after multiplying it with translation operators by j and k sites on the right and left, respectively) represents a planar diagram from TL N −1 (m + n) on the (n copies of) sℓ(m + n)-invariant spin-chain of length N − 1. This representation is well-known to be faithful [47], in contradiction with (B.5). This finishes the proof.
Remark B.1.1. As a corrollary to the previous theorem we conclude that the representation π m,n can be restricted to the quotient-algebra JTL N (m) introduced before (3.5) and this representation is faithful on the alternating super-spin chains with sℓ(m + n|n) symmetry for m, n > 0. Faithfullness in the case m = 0 and n > 1 can be proved in a similar way. The orthogonal projectors P j can be chosen such that they project onto a one-dimensional subspace in the tensorand V j and onto the complementary subspace in each tensorand V j ′ , with j ′ = j.

D.1 Feigin-Fuchs modules
For 1 ≤ r ≤ p + , 1 ≤ s ≤ p − , and n ∈ Z, let F r,s;n denote the Fock module of the Heisenberg algebra generated from (the state corresponding to) the vertex operator V r,s;n (z). The zero mode ϕ 0 = 1 2iπ dz∂ϕ(z) acts in F r,s;n by multiplication with the number We write F r,s ≡ F r,s;0 . For convenience of notation, we identify F 0,s;n ≡ F p + ,s;n+1 and F r,0;n ≡ F r,p − ;n−1 . Let Y r,s;n with 1 ≤ r ≤ p + , 1 ≤ s ≤ p − , and n ∈ Z denote the Virasoro module that coincides with F r,s;n as a linear space, with the Virasoro algebra action given by (D.3) (see [69]). As with the F r,s;n , we also write Y r,s ≡ Y r,s;0 .
The Fock spaces introduced above constitute a free-field module It can be regarded as (the chiral sector of) the space of states of the Gaussian Coulomb gas model compactified on the circle of radius √ 2p + p − . Note that the central charges we consider correspond to p + = x and p − = x + 1. Figure 22: Subquotient structure of the Feigin-Fuchs module Y r,s . The notation is as follows. The cross × corresponds to the subquotient J r,s , the filled dots • to J r,p − −s;2n+1 with n ∈ N 0 , the triangles to J p + −r,s;2n+1 with n ∈ N 0 , the open dots • to J r,s;2n with n ∈ N, and the squares to J p + −r,p − −s;2n with n ∈ N. The notation [a, b; n] in square brackets is for subquotients isomorphic to J a,b;n . The filled dots constitute the socle of Y r,s .