Fermi-gas interpretation of the RSOS path representation of the superconformal unitary minimal models
aa r X i v : . [ h e p - t h ] A ug Fermi-gas interpretation of the RSOS pathrepresentation of the superconformal unitaryminimal models
P. Jacob , and P. Mathieu ∗ Department of Mathematical Sciences,University of Durham, Durham, DH1 3LE, UKand D´epartement de physique, de g´enie physique et d’optique,Universit´e Laval, Qu´ebec, Canada, G1K 7P4.October 25, 2018
ABSTRACT
We derive new finitized fermionic characters for the superconformal unitary minimal mod-els by interpreting the RSOS configuration sums as fermi-gas partition functions. This ex-tends to the supersymmetric case the method introduced by Warnaar for the Virasoro unitarymimimal models. The key point in this construction is the proper identification of fermi-typecharged particles in terms of the path’s peaks. For this, an instrumental preliminary stepis the adaptation to the superconformal case of the operator description of the usual RSOSpaths introduced recently.
The solution of the Andrews-Baxter-Forrester and the Forrester-Baxter restricted-solid-on-solid (RSOS) models [2, 17] by the corner-transfer matrix method leads to an expression forthe local state probability of the order variable in terms of a configuration sum. Each suchconfiguration sum provides thus a finitization of the character of an irreducible module of thecorresponding minimal model. In this description, every state is represented by a particular ∗ [email protected], [email protected]. These statistical models are related to the minimal models [19, 28, 27]. More precisely, in their scalinglimit, the statistical models correspond to conformal field theories only at criticality. The minimal models areassociated to the transition from regime III to IV. But even off-criticality, the configuration sums describeconformal characters [10, 11, 12]. This remarkable off-critical relationship is explained in [30, 31]. In this way,the characters of the minimal models M ( p ′ , p ) are related to the configuration sums in regime III. L . The usual Virasoro characters are recoveredin the limit L → ∞ .The path description is combinatorial: counting the paths gives directly the characterwithout subtraction. This provides thus a royal road for the derivation of fermionic expres-sions (cf. [23, 24, 26]) of the characters.Various lines of attack for deriving fermionic formulae from configuration sums have beenproposed. In a seminal contribution, Melzer [26] has conjectured many finitized fermionicexpressions for the unitary minimal models (extending significantly the original conjecturesof [24]). The first few conjectured positive multiple-sum formulae were proved by demon-strating that they satisfy the same recurrence relations that characterize the correspondingconfiguration sums. This method of proof has been generalized and made into a powerfultechnique (under the name of telescopic expansion method) in [5], where many fermionic for-mulae (those for modules ( r, s ) with s = 1) for all Virasoro unitary models are demonstrated.(All cases are covered in [29].) Note however that this telescopic expansion method is essen-tially a verification tool that requires a candidate expression for the fermionic form. But in[5], the fermionic expressions are presented as natural q -deformations of state counting prob-lems for a specific (RSOS motivated) truncation of the space of states of the thermodynamiclimit of the quantum XXZ spin chain [4]. This observation has proved to be fruitful. In [6]many new fermionic formulae (i.e., for all minimal models) are conjectured, all motivated bythis counting procedure. These were successively generalized and proved in [7] by verifyingthat they satisfy RSOS-type recurrence relations. Note however that these expressions arerelated to paths in a very indirect way.A frontal attack of the difficult combinatorics of the paths for the generic Forrester-Baxtermodels is considered in [14, 15, 16, 36] (albeit using variables that are characteristics of theXXZ spectrum analysis and whose path interpretation is not immediate). The key preliminarystep is the discovery of a new (manifestly positive definite) characterization of the weight ofa Forrester-Baxter RSOS path (cf. App. A of [15]). From then on, the strategy followed inthese works is to describe a generic path pertaining to the (finitized version of the) minimalmodel M ( p ′ , p ) in terms of successive transformations acting on the unique and trivial pathfor the (formal) M (1 ,
3) model. This relies on two explicit combinatorial transformationsdefined directly on the paths: a Bressoud-type transformation [9] that relates paths within afamily defined by M ( p ′ , p + kp ′ ) for different values of k ≥
1, and a duality transformation[6] that relates M ( p ′ , p ) to M ( p − p ′ , p ). Although these basic transformations are quiteintuitive, the resulting construction turns out to be technically rather involved. Nevertheless,all characters for all the minimal models have been written in fermionic form along this line[36].For the unitary models, the combinatorics of the RSOS paths is considerably simplified. Inthat case, Warnaar has shown that a configuration sum can be regarded as a (grand-canonical)partition function of a one-dimensional gas of fermi-type particles subject to restriction rules[33, 34]. This method leads to the fermionic expression of the characters in a simple, direct andtotally constructive way. Moreover, in this simpler context, the procedure can be formulatedin terms of variables that have a clear path interpretation. For completeness, it should be added that a fermi-gas description has been obtained also for the M (2 , p )models in [32], but not directly from RSOS paths. Moreover, we have shown recently that the non-unitaryminimal models of the type M ( k + 1 , k + 3) do have a path representation similar to that of the unitaryminimal models and thus an analogous fermi-gas-type representation [21]. However, these specific paths do not(yet) originate from a RSOS model and their relation to the known RSOS paths (i.e, by means of a bijection) S M ( k + 2 , k + 4) RSOS paths
A configuration pertaining to the (regime-III) RSOS realization of the finitized SM ( k +2 , k + 4) unitary minimal models is described by a sequence of values of the height variables ℓ i ∈ { , , · · · , k + 3 } . The height index is bounded by 0 ≤ i ≤ L . Adjacent heights are has not been established so far. ℓ i − ℓ i +1 ∈ {− , , } ,ℓ i + ℓ i +1 ∈ { , , · · · , k + 2) } . (1)Each configuration is specified by particular boundary conditions: the values of ℓ and ℓ L . Aconfiguration is weighted by the expression w = L − X i =1 i | ℓ i − − ℓ i +1 | . (2)A path is the contour of a configuration. It is thus a sequence of edges joining the adjacentvertices ( i, ℓ i ) and ( i + 1 , ℓ i +1 ) of the configuration. These edges can be either North-East(NE), South-East (SE) or horizontal (H), corresponding to the cases where ℓ i +1 − ℓ i = 2 , − k + 3 is not allowed: H edges on the boundaries of the rectangular strip delimitatingthe paths are forbidden.The expression (2) for the weight of a configuration applies directly to the correspondingpath. It follows that vertices at extremal positions in the paths (either minima or maxima)do not contribute. Vertices at position i in-between two NE or two SE edges contribute to i , while those in-between an H edge and a non-H edge (in both orders) contribute to i/ The four types of contributing vertices and their weight. In all cases, it is understood that thevertex horizontal position is i . Here and in some of the following figures, the two type of vertices are denotedby black dots (with weight i ) or circles (with weight i/ b b bc bc bc bc i i i/ i/ i/ i/ X ℓ ,ℓ L ( q ) = X paths with fixed endpoints ℓ and ℓ L q w . (3)With the proper relation between ( ℓ , ℓ L ) and the irreducible indices ( r, s ), where 1 ≤ r ≤ k +1and 1 ≤ s ≤ k + 3, this is the finitized version of the superconformal characters.To a large extend, the combinatorics of the paths is independent of the boundary condi-tions. To avoid unnecessary complications, we will thus confine ourself to the analysis of thesimple case where ℓ = ℓ L = 1. These conditions characterize the vacuum module. With thepath extremities fixed in this way, the only allowed heights are the odd numbers between 1and k + 3. It is thus convenient to reduce the vertical scale by a factor of two as shown in Fig.2 where the path describing the vacuum state (for all unitary minimal models) is displayed.Its weight is readily seen to be zero. A generic path for a model with k > The path of lowest energy for all finitized superconformal unitary models SM ( k + 2 , k + 4). Thispath is associated to the vacuum state. Note that here and in all other figures, the vertical axis has beenrescaled by a factor 2. The height of a peak refers to this rescaled height. The path is thus a sequence ofpeaks of height 1 whose number n is fixed by the length L , here equal to 12, via L = 2 n . S M ( k + 2 , k + 4) RSOS paths
The starting point of the analysis is the interpretation of a path as being filled by a particulardistribution of one-dimensional interacting charged particles. The particles are the ‘basicpeaks’ and their charge is related to their height, with the height measured in terms of thereduced vertical scale.The proper charge characterization of an isolated peak, namely, a peak delimitated bytwo points on the horizontal axis, is the following. A peak of charge j has diameter 2 j andheight ⌊ j ⌋ . The charge j can take both integer and half-integer values and it is bounded by1 ≤ j ≤ k/ j corresponds to a peak described by a triangle with j NEand j SE edges. A particle of half-integer charge is described by a flatten triangle with ⌊ j ⌋ NE edges followed by one H edge and ⌊ j ⌋ SE edges; the triangle is thus topped by one (andonly one) H edge. Clearly, the introduction of half-integer charges is forced by the presenceof H links in the paths. The simplest examples are drawn in Fig. 4.The rationale underlying this characterization of the basic particles is the following. Asthe parameter k is increased, one expects the number of basic particles to increase. (For thefollowing discussion, it should always be kept in mind that the vertical axis has been rescaledby 2, that is, the height between 1 and 3 is rescaled to 1). For k = 0, only triangles of height1 are allowed. This is a trivial model with a single state represented by the path zig-zaggingbetween 1 and 3, the state illustrated in Fig. 2. The maximal height is also 1 when k = 1except that now an H edge is permitted at height 1. It is thus natural to associate to this newallowed particle a charge differing from that of the sole particle appearing in the k = 0 case.Therefore, the flatten version of the triangle of height 1 is attributed charge 3/2. As k ischanged from 1 to 2, there is another possibility, which is a particle of charge 2 correspondingto a triangle of height 2. However its flatten version is not allowed since an H edge at the topof the path’s bordering strip is ruled out by the condition (1). As k is changed from 2 to 3, apeak of height 3 is not allowed but the flatten version of the peak of height 2 is now possible;it is given charge 5/2. We thus see that as k increases by one, there is a new allowed chargefor the particles: to the charges 1 , · · · , k/ k + 1) / j of an isolated particle is related to its height h and the number As usual, ⌊ j ⌋ denotes the largest integer smaller than j . A generic path valid for all models with k ≥
4. The procedure for determining its particle contentis explained is section 3. There are three complexes, delimited by the four points on the horizontal axis. Thepath can be decomposed into a sequence of charged particles. The charge of the highest peak of a complex(or the leftmost one if it is not unique) is its height plus 1/2 if it is topped by at least one horizontal (H) edge.For instance, the peak centered on the horizontal position 7/2 has charge 5/2. Note that extra H edges at itstop would not modify its charge. Similarly, the one at position 17 has charge 3 and that at 28 has charge 1.For the others, as explained in section 3.2, the charge is read from the top to the indicated baseline (drawnas a dotted line a bit below its actual position for clarity). This charge assignment has to be adjusted if theparticle incorporates an H edge at the height of the baseline (cf. section 3), in which case this increases itscharge by 1/2. The arrows of the dotted lines delimitated the actual particles; the length of these dotted linesis the particle diameter, which is also twice the particle charge. The remaining H edges are particles of charge1/2. (In order to avoid any ambiguities with our subsequent analysis, note that in the following, we will forbidthe insertion of charge 1/2 particles in-between two particles. In the present context, it means that it is theH edge between i = 5 and 6 that belongs to the charge 3/2 particle and the subsequent H edge represents acharge 1/2 particle inserted within this charge 3/2 particle. Of course, this reinterpretation does not modifythe charge content.) If n j stands for the number of particles of charge j , the particle content of this path is n = n = 3 , n = n = n = n = 1. The length is L = P jn j = 29. j = h + ( H ) /
2. The allowed values of these charges are1 , · · · , k/ ≥
1, we introduce particles of charge 1/2. These,obviously, cannot be isolated by two points on the horizontal axis as the height h of an Hedge must be strictly greater than 1. Moreover, these particles must be distinguished fromthe top H edge of the particles of half-integer charge. In other words, a particle of charge 5 / As in standard RSOS paths [33], identical particles have a hard-core repulsion. The smallestdistance between two particles with the same integer charge is equal to the diameter of oneparticle, i.e., 2 j if they have charge j . The distance is measured from center to center, that is,from one peak to another. This also holds for half-integer charges if the distance is measuredfrom the middle of the particle, namely, the position of the middle of the top H edge. Forinstance, the separation between two 5/2 particles is 5.In contrast, particles of different charges can interpenetrate. This operation must preservethe particle identities. For the penetration of a particle of charge i within one of larger charge6igure 4: The first few particle shapes together with their charge assignment. / / j , this means that no height larger than j should result, taking the particles to be isolated. This implies in turn that there must exist a minimal distance between the two particles(measured from their respective center). This minimal distance is 2 i if the particle i is at theright of the largest particle and 2 i + 1 / i and height h is (up to possiblecontributions of one or two H edges in a way to be explained below) the largest number c ≥ i ′ , h − c ) and ( i ′′ , h − c ) on the path with i ′ < i < i ′′ andsuch that between these two vertices there are no peak of height larger than h and every peakof height equal to h is located at its right [8]. Draw a baseline at vertical distance c from thepeak. If there are H edges on the baseline which are preceded or followed by a peak withheight larger than the one under study, add an extra 1/2 to the height. If there are H edgeson the top of the peak, add 1/2 to the charge. Delimitate the particles using their charge That the height of the path does not become larger than that of the highest peak ensures that thispenetration process is well-defined within a model specified by a given value of k , which sets the upper boundon the allowed peaks height. The various stages of interpenetration of a particle of type i through a particle of type j (with j > i ) for the cases where (a) ( i, j ) = (2 , i, j ) = (2 , / , (c) ( i, j ) = (3 / ,
3) and (d) ( i, j ) = (3 / , / b b b b b b b b b bbc bc bc bc b b b b b b b b b bbc bc bc bcb b b b b b a ) b b b bbc bc b bbc bc bc bc bc bc b b b bbc bc b ) b b b b bc bc b b b bbc bc b b b bbc bc b b b bbc bcb b b bbc bc b b b bbc bc b b b bbc bc c ) b bbc bc bc bc b bbc bc bc bc b bbc bc bc bc b bbc bc bc bcb bbc bc bc bc d )determination, that is, identify those H edges that are part of the particles of charge ≥ / ≥ /
2) Hedge in-between particles. The remaining H edges are interpreted as particles of charge 1/2.For instance, the charge content of the path in Fig. 3 is detailed in the figure caption.
The construction of the generating function proceeds in various steps [33] which are detailedin the following subsections:1. For a fixed particle content, identify the ordering of the peaks that minimizes the weightand evaluate the weight of this minimal-weight configuration. With n j denoting thenumber of particles of charge j , a fixed particle content means a fixed set of values { n j } .2. Identify all possible ways of modifying the ordering of the peaks of charge ≥ / ≥ / n j compatible with the fixed length L , with L = 2 k/ X j =1 / j n j , (4)where the summation is incremented by steps of 1/2 (and it will always be clear fromthe context whenever this is the case, in sums or products). The configuration of minimal weight with specified and fixed value of the n j is the following:all the particles of charge ≥ / /
2. Then the path is terminated by the sequence of particles of charge 1. An example isdisplayed in Fig. 6.Figure 6:
The minimal-weight configuration with particle content: n = n = 1, n = n = 2, n = 1 and n = 3. Note that among the four H edges between 24 and 27, the first one is a constituent edge of the charge3/2 particle. r and then a sequence of these. For a peakcentered on the point r + x , the weight is r − X j =1 ( j + x ) − ( r + x ) = 2( r − r + x ) . (5)In this expression, we have summed the contribution of all vertices in-between x and 2 r + x (recall that 2 r is the diameter of the particle) and subtract the non-contributing middlepoint, the position of the maximum. Consider next the contribution of n r adjacent particlesof charge r : rn r X j =1 ( j + x ) − n r X j =1 ( jr + x ) = 2( r − n r ( rn r + x ) . (6)9ere we add all integer points x such that x < x ≤ x + 2 rn r and remove those points thatcorrespond to extrema, at positions jr + x . In the minimal-weight configuration, these peaksof charge r are preceded by the sequence of all higher charge peaks. This fixes the value of x to x = k/ X j = r +1 / jn j . (7)Consider next the contribution of a particle of half-integer charge r centered at the position ⌊ r ⌋ + x + 1 /
2. It is described by a flatten triangle of height ⌊ r ⌋ . The weight is r − X j =1 ( j + x ) − ( ⌊ r ⌋ + x ) − ( ⌊ r ⌋ + 1 + x ) + 12 ( ⌊ r ⌋ + x ) + 12 ( ⌊ r ⌋ + 1 + x ) = 2( r − r + x ) . (8)Here we have subtracted the contributions of the two top corners of the flatten triangle fromthe sum and then added their contribution to the weight (which is half that of the otherpoints) separately. Since 2 ⌊ r ⌋ + 1 = 2 r , we end up with the same expression as in the integercharge case. The weight of n r peaks is also given by (6).Putting these results together, the energy of the ordered sequence of particles of charge ≥ / k/ X i,j =3 / n i B ij n j , (9)where B is the matrix whose entries are given by B ij = B ji and B ij = 2( i − j for i ≤ j. (10)The insertion of the H edges describing the n / particles of charge 1/2 displaces the SE edgeof the last particle (whose charge is the lowest charge ≥ / n / units. This increases the weight by ( n / ) /
2. Finally, the particles of charge 1 appended tothe end of the path do not contribute to the weight.The weight of the minimal-weight configuration w mwc with fixed particle content, that is,fixed values of all n j , is thus w mwc = k/ X i,j =3 / n i B ij n j + 12 n / . (11) Starting from the minimal-weight configuration, we now consider all possible successive dis-placements of the particles of charge k/ , · · · , , / ≥ / i can be inserted ina particle of charge j > i . The four possible parities of 2 i and 2 j need to be treated separatelybut in all cases we find that the number of configurations is 4( j − i ) + 1.10onsider first the case where i and j are both integers. In the light of the deformationprocess described previously, there are generically two allowed configurations at each insertionpoint. There are ( j − i ) such points on the straight-down side of the particle of charge j and( j − i ) in its straight-up side. However, in the latter case, at the insertion point closest tothe top of the large peak, only one configuration is distinct. This gives a total of 4( j − i ) − i is either before orafter the larger one, for a total of 4( j − i ) + 1. The same counting holds when i and j areboth half-integer. For i integer and j half-integer, we have a total of 2( ⌊ j ⌋ − i ) + 1 + 1configurations where the initial vertex of the particle i is inserted in the straight-down partof the larger particle and 2( ⌊ j ⌋ − i −
1) + 1 configurations where the last vertex of the particle i is inserted in its straight-up portion. For the reversed parities, the numbers are respectively2( j − ⌈ i ⌉ ) + 1 + 1 and 2( j − ⌈ i ⌉ ) + 1. (The cases where the particles are actually separatedare taken into account with the final +1 in each expressions). In both cases, the total is4( j − i ) + 1. All cases are illustrated in Fig. 5. For n i and n j particles of each type, this isgeneralized to the combinatorial factor (cid:18) n i + 4( j − i ) n j n i (cid:19) . (12)It remains to weight the different steps. We start from the initial configuration wherethe two particles are separated, with j at the left of i (which is their relative position in theminimal-weight configuration). The starting distance between the particles is thus j + i . Thesuccessive configurations describe the interpenetration of i within j such that, at each step,the separation distance decreases by 1 /
2. This is pursued until their minimal separation isreached. From there on, the particle identities are interchanged and the subsequent displace-ments are performed from the leftmost peak, so that the distance increases by 1/2 at eachstep. Each such displacement induces a weight difference of 1. This is illustrated in Fig. 5for number of special cases.In order to demonstrate this in general, one compares two successive configurations (withparticle separation differing by 1/2) and observes that, for both parities of 2 i , only fourvertices are modified in their weight contribution. For a penetration in the straight-downpart of the larger particle, the different possibilities are pictured in Fig. 7 for the case of i integer. The four vertices whose weight are affected are diamond-shaped. By comparing thetotal weight of these four vertices in the two successive configurations, one readily verifiesthat the difference is 1 in each case (cf. the sample calculation in the figure caption). Thenet effect of this weight change is that the binomial factor (12) is simply q -deformed: (cid:20) n i + 4( j − i ) n j n i (cid:21) , (13)where (cid:20) ab (cid:21) q = ( ( q ) a ( q ) a − b ( q ) b if 0 ≤ b ≤ a, , (14)with ( z ) a ≡ ( z ; q ) a = (1 − z ) · · · (1 − zq a − ) . (15)The generalization to more complex configurations is immediate. The whole number of q -weighted configurations is thus given by k/ Y i =3 / (cid:20) n i + m i n i (cid:21) q (16)11igure 7: Three successive configurations describing the interpenetration of a particle of integer charge i inside a larger particle. Here only four vertices are displayed and their horizontal positions, l − , l, l + i − , l + i ,are unchanged from one configuration to the other. Only these four indicated vertices have their weightmodified in these processes. The weight of the four vertices in these three configurations is respectively( l − , , l + i − , ( l − , l, l + i − , l + i ) and (0 , l, , l + i ). The sums of the corresponding four weightsare 2 l + i − , l + i − l + i . The difference is thus 1 at each step. The figure for the case where i ishalf-integer is similar but the four vertices whose weight is changed are not the same in the two steps. ld ld ld ld ld ld ld ld ld ld ld ld l l + i where m i = k/ X j = i +1 / j − i ) n j ( i > . (17) The q -weighted combinatorial factor accounting for the different possible insertions of the n particles of charge 1 within those of higher charge, with the charged 1/2 particle stillmaintained fixed, is given by the q -binomial factor (cid:20) n + Mn (cid:21) q , (18)with M defined by M = k X j =1 jn j +1 . (19)The analysis is similar to that of the previous subsection. The main difference is that a charge1 particle cannot be deformed: this affects both the combinatorics and the weight change.Consider first the number of insertions of a particle of charge 1 within a particle of charge j .For j integer, there are 2 j − j − j half-integer, the counting is similar: there are 2 ⌊ j ⌋− j −
1) + 1 possibilities.In both cases (both parities of 2 j ), for n j particles of type j , there are 2( j − n j + 1 distinctconfigurations and when there are n particles of charge 1, this becomes (cid:18) n + 2( j − n j n (cid:19) . (20)Again, for a more general charge content, the factor 2( j − n j is changed into M .12t remains to check that every move toward the left of a particle of charge 1 withina sequence of larger particles induces a weight increase of 2. This operation amounts todisplace by two units toward the right either a (NE,NE) or a (SE,SE) vertex, which indeedaugments the weight by 2. This is illustrated in Fig. 8, for the insertion of a charge 1into a charge 5/2 particle. The result also follows from the adaptation of the analysis of thepreceding subsection with i = 1. However, because the charge 1 particle is not deformable, theintermediate configurations that previously accounted for a weight difference of 1 are missing,so that every allowed move changes the weight by 2. As a result, (20) is not q -deformed butrather q -deformed.Figure 8: The interpenetration of a particle of charge 1 through a particle of charge 5 /
2. The weight of thesuccessive configurations differs by 2. This is easily seen here since either a black dot is moved by two unitstoward the right (as in the first and third steps) or two circled dots are similarly moved by one units each (asin the second step. b bbc bc b bbc bc b bbc bc b bbc bc
The particles of charge 1/2 and 1 do not mix together, that is, they cannot interpenetrate.Indeed, an H edge cannot be inserted within a charge 1 particle since that would transform itinto a particle of charge 3/2. Phrased differently, the minimal distance between them must be1, which prevents any interpenetration. The combinatorial analysis of all possible insertionsof these peaks of charge 1/2 and 1 within the higher charged particles can thus be madeindependently.The insertion of the n / particles of charge 1/2 within those of charge ≥ /
2, taking intoaccount the weight increase, is given by the q -binomial factor " n − Mn q , (21)with M defined in (19). In dealing with such a q -binomial, we would need to impose also therequirement that (cid:20) − (cid:21) q = 1.The above result can be justified as follows. The number of possible distinct insertionsof an H edge within a particle of charge j > j − j . For j integer, the H edge can be placed within any pair of (NE,NE) or(SE,SE) edges and there are 2( j −
1) such vertices. However, the H edge cannot be locatedon the top, at the height j , since that would modify the resulting composite into a particleof charge j + 1 /
2. Phrased differently, this is ruled out by the minimal distance condition –which is 1 in this case –, since a top H edge would give a separation of 1/2.If j is half-integer, an H edge can be inserted between any pair of (NE,NE) or (SE,SE)edges; there are 2( ⌊ j ⌋ −
1) = 2 j − n j particles of charge j gives 2( j − n j possibilities. When the number of H edges is n / , this is generalized to (cid:18) n / − j − n j n / (cid:19) . (22)That the n / H edges representing the particles of charge 1/2 must be placed inside oneof the particle of type j from the start, is the source of the factor −
1. Note also that theH edges cannot be placed in-between two particles: they must be inserted within a specificparticle. This combinatorial factor is illustrated in Fig. 9 with n / = n = 1.A straightforward generalization of this counting argument for the insertion of H edgeswithin a set of particles of different charges accounts for the combinatorial factor (cid:18) n / − Mn / (cid:19) . (23)which is (21) when q = 1.To verify the correctness of the q -deformation of this binomial factor, we note that eachdisplacement of an H edge inside a particle, from right to left, increases the weight by 1.Indeed, the effect of moving the inserted H edge toward the left within the particle is equiva-lent to displace the position of a vertex of type (SE,SE) or (NE,NE) by two units toward theright while displacing two corners (which contribute half their position) by two units towardthe left , so that ∆ w = 2 − /
2) = 1. This is also true when the H edge located at the first(NE,NE) vertex of a particle is moved to the last (SE,SE) vertex of the adjacent particle atits left.Figure 9:
The various displacements of a particle of type 1 / b b b bc bc b b bbc bc b b bbc bc b b bbc bc Let us indicate a compatibility test of our procedure: The insertion of the 1/2 particleswithin particles of charge ≥ / The possible insertion points, indicated by crossed circles, for a particle of charge 1 / /
2. The number of possible insertionsis independent of the interpenetration pattern of the two higher charge particles. bc bc bc bc bc + + + + + bc bc bc bc bc + + + + + bc bc bc bc bc + + + + +
Within the framework of a fermi-gas description of the path (that is, irrespective of theoperator formulation to be spelled out later), this is basically the argument legitimatingthe rule for the distribution of the particles of charge 1/2 within mixtures of higher chargeparticles. This in turn justifies the way we have allowed the particles to be slightly deformedin the penetration process: this ensures that all possible configurations are reached. All configurations can be generated from the minimal-weight configurations by mixing theparticles the way it has been described in the previous three subsections. The generatingfunction for all the paths with ( ℓ , ℓ L ) = (1 ,
1) with the different n j fixed is thus given by theproduct of all these combinatorial factors q w mwc " n − Mn q (cid:20) n + Mn (cid:21) q k/ Y i =3 / (cid:20) n i + m i n i (cid:21) q . (24)The generating function for all paths of length L is then obtained by summing over all thenumbers n j that are compatible with the fixed length condition and this gives ∞ X n ,n , ··· ,n k =0 n +2 n + ··· +( k +2) n k = L q nBn + n " n − Mn q (cid:20) n + Mn (cid:21) q k/ Y i =3 / (cid:20) n i + m i n i (cid:21) q , (25)with B , M , and m i given respectively in (10), (19) and (17). For L even, this generates onlyinteger powers of q . Since the vacuum is in the NS sector, there are also states at half-integer Note that with our rules, the insertion of the H edge in-between the two particles in the second configu-ration of Fig. 10 amounts to change the charge of the particle, from 3/2 to 2. This configuration is effectivelytaken into account but when summing over the charge content. L odd. Therefore, to obtain the fullfinitized version of the vacuum character, one needs to add the contribution of L and L + 1. Consider the infinite length limit obtained by setting the sum n / +2 n → ∞ or equivalently,by setting P = n / + n → ∞ . Note that the variables n / and n enter in the aboveexpression only in the first two q -binomial factors. Taking the infinite length limit amountsthen to evaluate the sum (with n / = n and n = P − n ) S = lim P →∞ P X n =0 q n/ (cid:20) n − Mn (cid:21) q (cid:20) P − n + MM (cid:21) q , (26)where M is defined in (19). Thanks to the converging prefactor q n/ (with the tacit assump-tion that | q | < S = 1( q ; q ) M ∞ X n =0 q n/ (cid:20) n − Mn (cid:21) q , (27)(recall the definition (15)). The summation is then recognized as the expansion of the inverseof the q -factorial ( q / ) M (cf. Theorem 3.3 in [1]): S = 1( q ; q ) M ( q / ) M = ( − q / ) M ( q / ) M ( q ) M ( q / ) M = ( − q / ) M ( q ) M . (28)By redefining the summation variables as n i/ = p i − , so that the multiple-sum indicesare now p , · · · , p k , with M = P ki =1 ip i , we have χ , ( q ) = X p , ··· ,p k ≥ q pB ′ p ( − q / ) M ( q ) M k − Y i =1 (cid:20) p i + P k − ij =1 jp j + i p i (cid:21) q , (29)where B ′ is the symmetric matrix with entries B ′ ij = i ( j + 2) / i ≤ j . This is the vacuumcharacter of the superconformal minimal model SM ( k + 2 , k + 4).For the first three models ( k = 1 , , SM (3 , SM (4 ,
6) and the SM (5 ,
7) model respectively, the explicit form of the vacuum character reads: ∞ X p =0 q p ( − q / ) p ( q ) p , ∞ X p ,p =0 q p +4 p +4 p p ( − q / ) p +2 p ( q ) p +4 p (cid:20) p + 2 p p (cid:21) q , ∞ X p ,p ,p =0 q p +4 p + p +4 p p +5 p p +10 p p ( − q / ) p +2 p +3 p ( q ) p +4 p +6 p (cid:20) p + 2 p + 4 p p (cid:21) q (cid:20) p + 2 p p (cid:21) q . (30)16 .10 Comparison with known fermionic formulae The vacuum character of the SM ( k + 2 , k + 4) given in [3, 29] takes the form χ , = X ℓ , ··· ,ℓ k +1 ≥ ℓ i even for i ≥ q ℓCℓ ( q ) ℓ k +1 Y i =1 i =2 (cid:20) ( ℓ i − + ℓ i +1 ) ℓ i (cid:21) q , (31)(with the understanding that ℓ = ℓ k +2 = 0). Here C is the A k +1 Cartan matrix ( C i,i =2 , C i − ,i = C i,i +1 = −
1) and ℓCℓ = k +1 X i,j =1 ℓ i C ij ℓ j . (32)To relate (29) and (31), we first use the identity( − q / ) M = M X j =0 q j / (cid:20) Mj (cid:21) q (33)Then the comparison of the two q -factorials in the denominator and the products of q -binomials lead to the following relation between the two sets of variables { ℓ i } and { j, p i } : ℓ = M − j, ℓ = 2 M and ℓ i = 2 k − i +1 X j =0 ( j + 1) p j + i − for i ≥ . (34)(For instance, for k = 4, ℓ = 2 p +4 p and ℓ = 2 p .) Note that the variables ℓ i so defined aresuch that ℓ is a non-negative integer while ℓ , · · · ℓ k +1 are all even. It is then straightforwardto verify the equality of the q -exponent in the numerators:14 ( ℓCℓ − j ) = pB ′ p. (35)Note that this change of variables involves a summation mode j that has no interpretation interms of the path. This hints that there is no direct relation between the finitized versions of[3, 29] and ours. It appears thus that the formula given in (25) is new. Further support for In order to see the structural difference between the two finitized expressions in the simplest context,consider the expression for the finite version of the vacuum SM (3 ,
5) character. In [3, 29], it reads:˜ χ (3 , , ( q ; L ) = X ℓ ,ℓ ≥ L + ℓ , ℓ even q ( ℓ + ℓ − ℓ ℓ ) » ℓ ℓ – q » ( ℓ + L ) ℓ – q . This is to be compared with˜ χ (3 , , ( q ; L ) = ∞ X n ,n ,n =0 n +2 n +3 n = L q n + n " n − n n q » n + n n – q . In both cases, with L even (odd), we recover integer (half-integer) powers of q , hence the tilde on the character(reminding that this is not the full finitized character). These formulae give identical expansions (for L ≤ q ) ℓ appearing in (31) is the infinite length limit ofthe q -binomial 1( q ) ℓ = lim L →∞ » ( ℓ + ℓ + L ) ℓ – q . (36)This q -binomial factor captures the whole dependence of the finite form upon L . q → q − ) is shown to lead a new form of the parafermonic Z k +2 characters. The differencewith the usual expression [25] – which can be obtained by duality for the finite charactersof [3, 29] as demonstrated in [29] –, indicates a difference between the two finite versionssince in the dual case, the infinite length limit is taken differently, that is, via n k/ → ∞ .In particular, that our finite character involves a q -binomial factor reflects itself in thecorresponding parafermionic character by the presence of a non-usual ( q ; q )-factorial term. In [22], we have derived an operator interpretation for the paths describing the unitaryminimal models (cf. [13] for an earlier approach along these lines). The idea is to associate toall contributing vertices, the action of an operator acting at the corresponding (horizontal)position. This position fixes the mode index of the operator. These operators are subjectto certain rules that are readily lifted from the mere characteristics of the path. These rulesconstitute the basis relations.The same approach can be followed here. One introduces two types of operators, a and a ∗ . The local part of their action is described as follows. Let us again indicate the vertexin-between edges of type A and B as (A,B). Locally, a acts on vertices of type (NE,SE) or(NE,H), to transform them respectively into (NE,H) and (NE,NE), that is: a : (NE,SE) → (NE,H)(NE,H) → (NE,NE). (37)We denote by a i the operator acting at i ; it has weight i/
2. Two operators can act at i if itis the position of a maximum: a i transforms then (NE,SE) into (NE,NE) (and it has totalweight i ). Note however that three operators or more cannot act at the same point: a ni = 0for n > a ∗ action is as follows: a ∗ : (SE,NE) → (SE,H)(SE,H) → (SE,SE). (38)Again, its mode index indicates the position where it acts, which is twice its weight.Like their non-supersymmetric counterparts [22], these operators actually act in a verynon-local way: their action affects the path from the point of application to its right extremity.The action of a i on the ground-state amounts to change all edges at the right of i onto Hedges, that is, it transforms the zig-zag path tail into a straight line. A further action of a i undo the flattening of the path by reinserting the zig-zag pattern but lifted upward byone unit compared to its ground-state position. The full action of a i on the ground-state isthus to create a NE edge at i , translate the tail of the path by one unit both vertically andhorizontally and remove the last edge. This is akin to the action of the operator b defined in[22]. The operator a ∗ i acts similarly but in the opposite vertical direction. Sample actions onthe ground (or vacuum) state are pictured in Fig. 11. The present a i is similar (in its action on the ground state) to the operator b i in [22]. The successive actions of the sequence of operators a ∗ a ∗ a on the ground-state path displayed inthe first line. The action of each operator is manifestly non-local, modifying the path from its insertion pointto its right extremity. bc + + + + + + + + bb bc + + + + b bc bc + a a a ∗ a a ∗ a ∗ a Any path can thus be represented by a string of operators acting on the vacuum ground-state path. It will be understood that these are ordered in increasing values of their modesfrom left to right. Observe that in order to preserve the boundary conditions of the ground-state path, the string must contain an equal number of a and a ∗ operators. For instance, thepath of Fig. 3 is associated to the following sequence of operators: a ∗ · · · a ∗ a a ∗ a ∗ a a a .As already indicated, the path characteristics capture all the conditions that need tobe imposed on sequences of operators in order to define a basis. An analysis of this typehas already been presented in [22] and it can be adapted to the present context with minormodifications. Instead of following this line of presentation, we will proceed in a more informalway by treating the first few models explicitly. From these considerations, the general patternwill emerge naturally. Note also that the operator method, taken independently of its pathorigin, is not naturally finitized. In a first step, we thus avoid this finitization constraint andbuild up directly the L → ∞ expression of the characters. Our main objective is not muchto be complete and systematic concerning the operator construction but rather to indicatehow it makes natural our previous particle identification within a path. SM (3 , model For k = 1, the first excitation in the vacuum module corresponds to a peak of charge 3/2centered at the position 3/2. This value of the charge is also the maximal allowed heightfor a peak in this model. It is understood that the path is infinite, being completed by aninfinite sequence of peaks of charge 1. In our operator terminology, this peak of charge 3/2 isequivalent to the string of operators a ∗ a (of weight 3/2, half the sum of the indices), actingon the ground state. The various states that can be reached from this elementary sequenceare simply those obtained by increasing the modes (or displacing them, using a pictorialterminology that carries a path flavor) in all possible ways by respecting their ordering. Theorder needs to be preserved since, from the definition of its action, a ∗ cannot act first, thatis, directly on the vacuum state. This fixed-ordering condition is interpreted as a hard-corerepulsion between the two modes. 19et us describe explicitly the different configurations that can be obtained from a ∗ a andderive their relative weight. The a ∗ mode can be increased by any integer or displaced byany number of steps and each unit displacement increases the weight by 1/2. In the pathterminology, a move of a ∗ by m steps amounts to insert m particles of charge 1/2 (or m Hedges) within the particle of charge 3 /
2. The generating function for all the displacementsof the a ∗ mode is (1 − q / ) − . The a mode can also be increased but this must be twinedwith a similar augmentation of the a ∗ mode, i.e., in its displacement, a drags the a ∗ mode.Moreover, the path interpretation of the move shows that is must be done by steps of twounits: the path cannot start with an H edge, so a full particle of charge 1 has to be insertedin front of the particle of charge 3/2. The generating function for these displacements is thus(1 − q ) − . Taking care of both type of displacements, as well of the weight of the lowest-weight configuration, the generating function for all states described by a single pair of a ∗ a operators is q (1 − q )(1 − q ) . (39)In order to see how this gets generalized to the case where there are n pairs of a ∗ a operators, consider the case where n = 2. Again, the modes increase must preserve theoperator ordering since an interchange would either lead to a a ∗ acting on the vacuum orgenerate a peak higher than 3/2. The displacements of the last two modes are describedas before. The leftmost a ∗ mode can be moved by any integer and each unit displacementdrags the pair of a ∗ a that acts after by one unit each so that the total weight increase is3 /
2. The generating function for displacements of this type is thus (1 − q / ) − . Finally, thefirst a mode can be moved by steps of two units, with a weight change of 4 each times (sinceits motion drags the subsequent three modes) and these are generated by (1 − q ) − . Thelowest-weight string is a ∗ a a ∗ a , with weight 6. The generating function for all states withtwo pairs of a ∗ a is thus: q (1 − q / )(1 − q )(1 − q / )(1 − q ) . (40)Proceeding along this way, it is rather immediate to see that the generalization to n pairs is q n n Y i =1 − q i − )(1 − q i ) . (41)Using the identity n Y i =1 − q i − )(1 − q i ) ! = ( − q ) n ( q ) n , (42)this reduces to q n ( − q ) n ( q ) n . (43)The full generating function, which is nothing but the vacuum character of the SM (3 , n : χ (3 , , ( q ) = ∞ X n =0 q n ( − q ) n ( q ) n . (44)This matches the first expression in (30), with n = p .20 .3 The operator description of the SM (4 , model For the case where k = 2, the are two types of contributing peaks, those with charge 3/2and 2. In terms of operators, these correspond to the two basic structures: a ∗ a and a ∗ a .These are called 1- and 2-blocks respectively. Denote by p and p their respective number.The sequence of lowest-weight is the one with modes as closely packed as possible and indecreasing value of the block content.Let us consider first the analysis of the elementary 2-block a ∗ a (of weight 4) and thevarious displacements of its constituents. One a ∗ can be moved by any units, with a weightincrease given by half its displacement. The generating factor is (1 − q / ) − . Similarly, thetwo a ∗ modes can be moved together but by two units each step (which amounts to inserteach time a peak of charge 1 in between the a ∗ and a modes). The weight change is 2 eachstep, leading to the generating factor (1 − q ) − . Next, a a mode can be displaced by steps ofone unit, with each displacement dragging the two a ∗ operators: the weight difference is 3/2for each step and the generating factor is (1 − q / ) − . Finally, the displacements of the two a modes are generated by (1 − q ) − . The generating function of all states obtained form one2-block is q (1 − q )(1 − q )(1 − q )(1 − q ) . (45)The extension to the case of p q p p Y i =1 − q i − )(1 − q i ) = q p ( − q ) p ( q ) p . (46)Let us then turn to the case where both types of blocks are present. By considering firstthe case p = p = 1, one can easily figure out the weight generating factor that correspondsto the various displacements that maintain the operator order fixed and it is: q (1 − q )(1 − q )(1 − q )(1 − q )(1 − q )(1 − q ) . (47)For a generic number of 1- and 2-blocks, the minimal-weight configuration is readily foundto be 4 p + 4 p p + 3 p /
2. The generating factor for the order-preserving displacements is: q p +4 p p + p ( − q ) p + p ( q ) p +2 p . (48)But this is not quite the complete result: although the ordering of the terms withineach block is immutable, it remains to take into account the mixing of blocks, that is, theimmersion of the smaller block into the larger one. We argue, in the following subsection,that this gives rises to the multiplicative factor (cid:20) p + 2 p p (cid:21) . (49)The product of (48) and (49) summed over all values of p and p gives the full character.The resulting expression is found to be identical with the second expression in (30).Granting the result (49), we see from these two examples, that the operator constructionprovides a rather efficient way of generating the fermionic form of the characters.21 .4 Block mixing It is convenient to consider the block-mixing operation in general terms, by considering themixing of a i -block within a j -block, with j > i . A j -block is of the form a ∗ j a j and its chargeis j/ This prevents the production of higher-type blocks in the immersion process. Equivalently,it ensures that the blocks preserve their identity. The center of a j -block is in-between the a ∗ and a modes. Assign charge 1 to a and − a ∗ : q ( a ) = − q ( a ∗ ) = 1. Introduce the ℓ -thpartial charge of a sequence of operators c i m · · · c i , where c is either a or a ∗ , to be the sumof the charge of the ℓ first terms: q ℓ = ℓ X s =1 q ( c i s ) . (50)Then, for a sequence of operators describing a j -block followed by a i -block (with j > i ), thepartial charge q ℓ must satisfy 0 ≤ q ℓ ≤ j for all values of ℓ . This is the basic requirement tobe imposed in block mixing to ensure the block identities [22]. Again, identical configurationsare not counted twice.The insertions of the i -block within the substring a ∗ j lead to configurations of the form: a ∗ n ( a ∗ i a i ) a ∗ j − n a j (0 ≤ n ≤ j ) , (51)(where the inserted block is delimitated by parentheses). The largest partial charge q j − n + i is i + n and in order to respect the bound q j − n + i ≤ j , we need to have i + n ≤ j . Thisexcludes the insertion at the center of the j -block ( n = j , which would correspond to a singleblock of type i + j ) and its near vicinity ( j − i < n < j ). The deepest penetration of the i -block within the a ∗ -side of the j -block corresponds to the configuration with n = j − i : a ∗ j − i ( a ∗ i a i ) a ∗ i a j (52)There are thus j − i +1 distinct insertions within the substring a ∗ j , counting the one with n = 0(which is the original configuration where the i -block follows the j -block). The insertions ofthe i -block within the substring a j are of the form a ∗ j a j − n ′ ( a ∗ i a i ) a j − n ′ (0 ≤ n ′ ≤ j ) . (53)The strongest constraint on the partial charge is q j − n ′ + i = j − n ′ + i ≤ j , which forces n ′ ≥ i .There are also j − i + 1 possible insertions on this a side. However, the case n ′ = i : a ∗ j a j − i ( a ∗ i a i ) a j − i (54)when dropping the unessential parentheses, leads to a configuration identical to one alreadyconsidered, cf. (53), and it should not be counted twice. The total number of distinct casesis thus 2( j − i ) + 1.The successive displacements of the i -block within the j -block are considered from theleft to the right. Once the closest-to-the-center penetration configuration is reached, whichis (53), it is exchanged with its symmetrical version (54) before the next displacement is The following discussion parallels the one already presented for paths. However, it is spelled out again inthis novel context since its naturalness is our original rationale for the special particle deformation previouslyintroduced in the interpenetration process. i -block withinthe j -block increases the weight by 1. This is rather obvious in the operator formalism (cf.[22]) but since this has already been established from the paths, it will not be proved again.The combinatorial factor for p i i -blocks inserted in p j j -blocks, keeping track of the relativeweight, is (cid:20) p i + 2( j − i ) p j p i (cid:21) . (55)For i = 1 , j = 2, this is the announced result (49).Note that once the generating factor is obtained for a given block content, including thosefactors that account for the block mixing, we need to sum over all possible values of the p j ,with 1 ≤ j ≤ k . This justifies preserving the block identities in the mixing process. As previously claimed, the successive block insertions match the successive steps of particlepenetration, with the shape of the smaller particle alternating between its deformed andundeformed version. This is most neatly illustrated by means of an example. For instancethe block representation of the various paths illustrating the penetration of a particle 3/2within a particle 5/2 are as follows:( a ∗ a ) a ∗ a ∗ a ∗ aaa, a ∗ ( a ∗ a ) a ∗ a ∗ aaaa, a ∗ a ∗ ( a ∗ a ) a ∗ aaa, a ∗ a ∗ a ∗ aa ( a ∗ a ) a, a ∗ a ∗ a ∗ aaa ( a ∗ a ) . (56)These are in respective correspondence with the paths illustrated in Fig. 5d. In particular,in the path version of the second and the fourth block configurations above, the 3/2 particlehas its top H edge moved to its front or its rear. As in [22], it is not difficult to write the conditions defining an operator basis in terms ofconstraints to be imposed on successive modes of operators acting on the vacuum state. Theconditions can be summarized as follows: c i m c i m − · · · c i : c is either a or a ∗ , with q ( a ) = − q ( a ∗ ) = 1,0 ≤ q ℓ ≤ k − , with q = 1 and q m = 0 , for q s even : i s +1 − i s = 1 + 2 n + 12 (1 − q ( c i s +1 ) q ( c i s )) , for q s odd : i s +1 − i s = n + 12 (1 − q ( c i s +1 ) q ( c i s )) , (57)where n is any non-negative integer and q ℓ is defied in (50). The vacuum character is thegenerating function of all the sequences of operators subject to (57). We have presented a fermi-gas description of the RSOS path representation of the super-conformal unitary minimal models along the lines of [33]. The finitized fermionic charactersobtained here are new. However, their infinite length limit reproduce the formulae previously23ound in [3, 29]. Since our interest lies essentially in the latter and that our method does notproduce novel expressions for the conformal characters, we have restricted our presentationto paths with the simplest boundary conditions, those pertaining to the vacuum module. The main interest of the present work is methodological: it shows that the fermi-gastechnique – which is fully constructive – can be extended to models other than the Virasorounitary ones. The resulting positive multiple-sum character is thus formulated in terms ofdata that have a clear particle meaning in the path context.The generalization of the fermi-gas method from the Virasoro to the super-Virasoro case isnot quite straightforward, however, since the particle interpretation is not completely obviousin presence of horizontal edges within the path. The key hint in that regard comes fromthe (non-local) operator representation of the path displayed in section 4, which generalizesour recent work [22]. Retrospectively, this application is the best concrete motivation forintroducing these operators.The unitary Virasoro and super-Virasoro minimal models are the first two diagonal cosetsin the series c su (2) k ⊕ c su (2) n / c su (2) k + n , corresponding to the cases n = 1 , n > A From the finitized RSOS characters to the parafermioniccharacters
The finitized characters in regime III are related to those of regime II by the duality trans-formation q → q − [29] (see also [33, 14, 21]). The conformal characters obtained by dualitycorrespond to the limit L → ∞ but now evaluated with n k/ → ∞ . The dual charactersare those of the Z k +2 parafermionic theories. However, before this limit is taken, a correcting L -dependent factor needs to be introduced. Actually, the transformed character must bemultiplied by q kL / k +2) . With these transformations, we then recover the character of theparafermionic vacuum module, denoted χ ( k +2)0 : χ ( k +2)0 ( q ) = lim L →∞ q k k +2) L ˜ χ ( k +2 ,k +4)1 , ( q − ; L ) . (58)The resulting expression is (with n i/ replaced by m i ): χ ( k +2)0 ( q ) = ∞ X m , ··· ,m k +1 =0 q N + N + ··· + N k +1 − k +2 N + ( m − m ) ( q ) m ( q ; q ) m ( q ) m · · · ( q ) m k +1 (59) See however App. A for an interesting consequence of the novelty of these finite characters in the contextof parafermionic conformal theories. N i = m i + · · · + m k +1 and N = k +1 X i =1 im i . (60)This is to be compared with the standard form [25] (see also [20]): χ ( k +2)0 ( q ) = ∞ X m , ··· ,m k +1 =0 q N + N + ··· + N k +1 − k +2 N ( q ) m ( q ) m ( q ) m · · · ( q ) m k +1 . (61)The equivalence of these two forms boils down to the identity ∞ X m ,m =0 q Λ+ ( m − m ) ( q ) m ( q ; q ) m = ∞ X m ,m =0 q Λ ( q ) m ( q ) m , (62)where Λ = 1 k + 2 (cid:0) ( k + 1) m + 2 k ( m m + m ) (cid:1) + ( m + 2 m ) ∆ , (63)for any ∆.The direct verification of this identity would be another validation of our expression forthe finite characters. Warnaar [35] offers us a simple but clever proof of (62). The first stepamounts to transform the identity into a form that makes transparent its independence uponboth k and ∆. For this, we replace q − / ( k +2) by z and q ∆ by y and get: ∞ X m ,m =0 q ( m + m ) + m + ( m − m ) z ( m +2 m ) y m +2 m ( q ) m ( q ; q ) m = ∞ X m ,m =0 q ( m + m ) + m z ( m +2 m ) y m +2 m ( q ) m ( q ) m . (64)This can be rewritten as ∞ X ℓ =0 z ℓ y ℓ X m ,m ≥ m +2 m = ℓ q ( m + m ) + m + ( m − m ) ( q ) m ( q ; q ) m = ∞ X ℓ =0 z ℓ y ℓ X m ,m ≥ m +2 m = ℓ q ( m + m ) + m ( q ) m ( q ) m , (65)which reduces to X m ,m ≥ m +2 m = ℓ q ( m + m ) + m + ( m − m ) ( q ) m ( q ; q ) m = X m ,m ≥ m +2 m = ℓ q ( m + m ) + m ( q ) m ( q ) m . (66)Eliminating m and renaming m as m give the following finite-sum identity: ℓ/ X m =0 q ℓ ( ℓ +1) − m ( q ) ℓ − m ( q ; q ) m = ℓ/ X m =0 q ℓ − ℓm +2 m ( q ) ℓ − m ( q ) m . (67)For the demonstration of this identity, we will proceed in two steps, by considering bothparities of ℓ separately.Take first the case where ℓ is even. Set ℓ = 2 k , multiply both sides by q − k and thenreplace m → k − m . This yields: k X m =0 q m ( q ) m ( q ; q ) k − m = k X m =0 q m ( q ) m ( q ) k − m . (68)25t his point, the strategy of the proof is to try to reinterpret this as an identity for suitablyspecialized basic hypergeometric series. With this goal in mind, we multiply both sides by z k and sum k over all integers ≥
0. We then interchange the summations order and use thefact that 1 / ( q ) n = 0 if n < k to m . The two k -summations can then be performed explicitly using Euler’s q -exponential sum (cf. [1] eq.(2.2.5) or [18] eq. (II.1)): ∞ X m =0 t n ( q ) n = 1( t ) ∞ . (69)The result is 1( z ; q ) ∞ ∞ X m =0 ( zq ) m ( q ) m = 1( z ) ∞ ∞ X m =0 z m q m ( q ) m , (70)which we can rewrite under the form: ∞ X m =0 ( zq ) m ( q ) m = ( z ; q ) ∞ ( z ) ∞ ∞ X m =0 z m q m ( q ) m = 1( zq ; q ) ∞ ∞ X m =0 z m q m ( q ) m . (71)We next rewrite the left-hand-side in terms of of the φ ( a, b ; c ; q ; z ) series defined as [18]: φ ( a, b ; c ; q ; z ) = ∞ X n =0 ( a ) n ( b ) n z n ( c ) n ( q ) n ≡ ∞ X n =0 ( a ; q ) n ( b ; q ) n z n ( c ; q ) n ( q ; q ) n . (72)Using the simple identity: ( q ) m = ( q ; q ) m ( q ; q ) m , we see that ∞ X m =0 ( zq ) m ( q ; q ) m ( q ; q ) m = φ (0 , q ; q ; zq ) . (73)The identity (71) follows from a special Heine transformation, namely (see [18] eq. (III.3)): φ ( a, b ; c ; q ; z ) = ( abz/c ; q ) ∞ ( z ; q ) ∞ φ ( c/a, c/b ; c ; q ; abz/c ) . (74)In the present context, we must set a, b → a → ( q/a ) n a n = ( − n q n . (75)Hence, the right-hand-side of (74), with a, b → , c = q, z → zq and q → q , yields directlythe second member of (71). This completes the proof of the identity (67) when ℓ is even.Consider next the case where ℓ = 2 k + 1. Once this substitution is done in (67), wemultiply the two sides by q − k − k and let again m → k − m to obtain: k X m =0 q m ( q ) m +1 ( q ; q ) k − m = k X m =0 q m ( m +1) ( q ) m +1 ( q ) k − m . (76)We then proceed as before: multiply both sides by z k , sum over k ≥
0, and use (69). Thatgives ∞ X m =0 ( zq ) m ( q ) m +1 = 1( zq ; q ) ∞ ∞ X m =0 ( zq ) m q m ( q ) m +1 . (77)26ext, we use another simple identity: ( q ) m +1 = (1 − q ) ( q ; q ) m ( q ; q ) m , and cancel thefactor (1 − q ) from both sides to get ∞ X m =0 ( zq ) m ( q ; q ) m ( q ; q ) m = 1( zq ; q ) ∞ ∞ X m =0 ( zq ) m q m ( q ; q ) m ( q ; q ) m . (78)But this is again a special case of Heine’s transformation (74): φ (0 , q ; q ; zq ) = 1( zq ; q ) ∞ lim a → φ ( q /a, q /a ; q ; q ; a z/q ) . (79)This completes the proof of (67), hence of (62). ACKNOWLEDGMENTS
We thank O. Warnaar for generously providing us with the proof of (62) that has beenpresented in App. A and for clarifications concerning the identity (28). This work is supportedby NSERC.
References [1] G.E. Andrews,
The theory of partitions , Cambridge Univ. Press, Cambridge, UK, (1984).[2] G. E. Andrews, R. J. Baxter and P. J. Forrester,
Eight-vertex SOS model and generalizedRogers–Ramanujan-type identities , J. Stat. Phys. (1984), 193–266.[3] E. Baver and D Gepner, Fermionic Sum Representations for the Virasoro Characters ofthe Unitary Superconformal Minimal Models
Phys. Lett.
B372 (1996) 231-235.[4] V.V. Bazhanov and N.u. Reshetikhin,
Critical RSOS models and conformal field theory
Int. J. Mod. Phys. A4 (1989) 115-142.[5] A. Berkovich, Fermionic counting of RSOS-states and Virasoro character formulas forthe unitary minimal series M ( ν, ν + 1) . Exact results , Nucl. Phys. B431 (1994) 315-348.[6] A. Berkovich, B. M. McCoy,
Continued fractions and fermionic representations for char-acters of M(p,p’) minimal models , Lett. Math. Phys. (1996) 49-66.[7] A, Berkovich, B. M. McCoy, A. Schilling, Rogers-Schur-Ramanujan type identities forthe M ( p, p ′ ) minimal models of conformal field theory , Commun. Math. Phys. (1998)325-395[8] A. Berkovich and P. Paule, Lattice paths, q -multinomials and two variants of theAndrews-Gordon identities , Ramanujan J. (2002) 409–425.[9] D. Bressoud, Lattice paths and Rogers-Ramanujan identities , in
Number Theory, Madras1987 , ed. K. Alladi, Lecture Notes in Mathematics (1987) 140-172.[10] E. Date, M. Jimbo, A. Kuniba, T. Miwa and M. Okado,
Automorphic properties of localheight probabilities for integrable solid-on-solid models , Phys.Rev.
B35 (1987) 2105-2107.[11] E. Date, M. Jimbo, A. Kuniba, T. Miwa and M. Okado,
Exactly solvable SOS models:local height probabilities and theta function identities , Nucl. Phys.
B290 (1987) 231-273.2712] E. Date, M. Jimbo, T. Miwa and M. Okado,
Solvable lattice models , Proceedings ofSymposia in Pure Math. (1989) 295-331.[13] G. Feverati and P. A. Pearce, Critical RSOS and minimal models I: Paths fermionicalgebras and Virasoro modules , Nucl. Phys.
B663 (2003) 409-442.[14] O. Foda and T. Welsh,
Melzer’s identities revisited , Contemp. Math. (1999) 207-234.[15] O. Foda, K.S. M. Lee, Y. Pugai and T. A. Welsh,
Path generating transforms , in q-Seriesfrom a contemporary perspective , Contemp. Math. 254, Amer. Math. Soc., Providence,RI, 2000, 157–186.[16] O. Foda and T. A. Welsh,
On the combinatorics of Forrester-Baxter models , Prog. Comb. (2000) 49-103.[17] P. J. Forrester and R. J. Baxter,
Further exact solutions of the eight-vertex SOS modeland generalizations of the Rogers-Ramanujan identities , J. Stat. Phys. (1985) 435–472.[18] G. Gasper and M. Rahman, Basic hypergeometric series , Cambridge Univ. Press, 2nded., (2004).[19] D.A. Huse,
Exact exponents for infinitely many new multicritical points , Phys. Rev.
B30 (1984) 3908-3915.[20] P. Jacob and P. Mathieu,
Parafermionic quasi-particle basis and fermionic-type charac-ters
Nucl. Phys.
B620 (2002) 351-379.[21] P. Jacob and P. Mathieu,
New path description for the M ( k + 1 , k + 3) models and thedual Z k graded parafermions , J. Stat. Mec. (2007) P11005, 43 pages.[22] P. Jacob and P. Mathieu, Nonlocal operator basis for the path representation of theunitary minimal models , arXiv:0802.4298.[23] R. Kedem, T.R. Klassen, B. M. McCoy and E. Melzer,
Fermionic quasi-particle repre-sentations for characters of ( G (1) ) × ( G (1) ) / ( G (1) ) , Phys. Lett. B304 (1993) 263-270[24] R. Kedem, T.R. Klassen, B. M. McCoy and E. Melzer,
Fermionic sum representationsfor conformal feld theory characters , Phys. Lett.
B307 (1993) 68-76.[25] J. Lepowsky and M. Primc,
Structure of the Standard Modules for the Affine Lie Al-gebra A (1)1 , volume 46 of Contemporary Mathematics . American Mathematical Society,Providence, 1985.[26] E. Melzer,
Fermionic Character Sums and the Corner Transfer Matrix , Int. J. Mod.Phys. A9 (1994) 1115-1136.[27] T. Nakanishi, Nonunitary minimal models and RSOS models , Nucl. Phys.
B 334 (1980)745-766.[28] H. Riggs,
Solvable lattice models with minimal and nonunitary critical behavior in two-dimensions , Nucl. Phys.
B 326 (1989) 673-688.[29] A. Schilling,
Polynomial Fermionic Forms for the Branching Functions of the RationalCoset Conformal Field Theories c su (2) M × c su (2) N / c su (2) M + N , Nucl. Phys. B459 (1996)393-436. 2830] H. Saleur and M. Bauer,
On Some Relations Between Local Height Probabilities AndConformal Invariance , Nucl .Phys.
B320 (1989) 591.[31] K. A. Seaton and B. Nienhuis,
Surface critical exponents and cylindrical partition func-tions for the CSOS model , Nucl. Phys.
B384 (1992) 507-522.[32] S.O. Warnaar
The Andrews-Gordon identities and q -multinomial coefficients Comm.Math. Phys. (1997) 203-232.[33] S. O. Warnaar,