MODULAR TRANSFORMATIONS AND VERLINDE FORMULAE
FOR LOGARITHMIC (p+ , p− )-MODELSDAVID RIDOUT AND SIMON WOOD

A BSTRACT. The (p+ , p− ) singlet algebra is a vertex operator algebra that is strongly generated by a Virasoro field ofcentral charge 1 − 6 (p+ − p− )2 /p+ p− and a single Virasoro primary field of conformal weight (2p+ − 1) (2p− − 1).
Here, the modular properties of the characters of the uncountably many simple modules of each singlet algebra areinvestigated and the results used as the input to a continuous analogue of the Verlinde formula to obtain the “fusion
rules” of the singlet modules. The effect of the failure of fusion to be exact in general is studied at the level of Verlinde
products and the rules derived are lifted to the (p+ , p− ) triplet algebras by regarding these algebras as simple current
extensions of their singlet cousins. The result is a relatively effortless derivation of the triplet “fusion rules” that agrees
with those previously proposed in the literature.

1. I NTRODUCTION
The (1, p) singlet and triplet models (for p > 2 a positive integer) are perhaps the most basic known examples
of logarithmic conformal field theories. Introduced in [1], their logarithmic nature was exposed, at least for p = 2,
through a connection to symplectic fermions and bc-ghosts [2, 3]. Further investigations, for example [4–7],
addressed the issue of generalising the tools familiar from rational conformal field theory to these models andthey have remained popular objects of study ever since.
Generalisations of these triplet models, called (p+ , p− )-models, for p+ and p− coprime and positive, wereintroduced in [8]. Because their central charges match those of the Virasoro (p+ , p− ) minimal models when p+
and p− are greater than 1, they are sometimes referred to as logarithmic minimal models. One might hope thatthese logarithmic models could capture the universal features of critical lattice models that are missed by theminimal models (crossing probabilities, fractal dimensions and so on), but this is still contentious. Nevertheless,there has been persistent interest in these models from both theoretical physicists and mathematicians. One reason
for this interest is that the underlying vertex operator algebras are not simple, so these models allow one to explore
the consequences of this non-simplicity in a tractable, though still very challenging, setting. We remark that thesimple quotients are precisely the minimal model vertex operator algebras.
As with other logarithmic models, one of the main difficulties to surmount is that of obtaining a detailed structural understanding of the reducible but indecomposable modules which appear in the spectrum. While a complete
classification of the indecomposables may well be infeasible, a first aim would be to identify the spectrum ofsimple modules and their projective covers. This is expected to be sufficient to construct bulk state spaces withmodular invariant partition functions, for example. However, the current state of knowledge regarding projectives
in non-semisimple module categories over vertex operator algebras is still in its infancy, so much of our intuition
stems from examples like the (p+ , p− )-models.
But even in examples, the rigorous identification of projectives remains a formidable task. Indeed, this has
only been achieved for p+ = 1 [9]. However, there is no shortage of proposals and conjectures for general p+
and p− (with varying degrees of structural detail), see [8, 10–13] for example. These proposals rely on conjectured equivalences of categories, numerical computations within integrable lattice discretisations, and explicit
construction, the latter giving the most information (but requiring the most effort). In this direction, a powerfultool for structural investigations of indecomposables is the celebrated Nahm-Gaberdiel-Kausch algorithm [14–16]
that explicitly constructs (filtered quotients of) the fusion product of two modules.
October 28, 2013.

1

2

D RIDOUT AND S WOOD

Determining fusion rules is, of course, another of the main difficulties one would like to overcome along the pathto understanding a given logarithmic conformal field theory. While the Nahm-Gaberdiel-Kausch fusion algorithm
allows one to construct enough of a fusion product to identify it completely in principle, the calculations are too
computationally intensive for all but the smallest theories, even when performed by computer. Another issue isthat the algorithm in practice only provides an “upper bound” on the fusion product in the sense that the true
result could be, in principle, a proper subspace of what has been deduced. However, both of these issues can be
circumvented by generalising another standard tool from rational conformal field theory to the logarithmic setting:
the Verlinde formula.
The Verlinde formula [17] for rational conformal field theories computes the fusion product of two modules
from the modular transformation properties of their characters. As this formula may be shown to follow fromthe internal consistency conditions that must be satisfied by any conformally-invariant quantum field theory [18],one expects that it should remain valid in some form for more general classes of non-rational theories. In thelogarithmic setting, characters cannot distinguish between reducible but indecomposable modules and the direct
sum of their simple composition factors, hence the Verlinde formula cannot be expected to compute the true
fusion rules, but only tell us which composition factors appear, and with what multiplicity, in a given fusion
product. However, this is already very valuable information. In many cases, one can easily rule out the possibility
that the simple factors combine to form an indecomposable and then the Verlinde formula gives the fusion rules
as in rational theories.1 If an indecomposable can be formed, then this formula provides the character of theindecomposable effortlessly, thus solving the “upper bound” problem. Moreover, it also tells us which fusion
products need to be checked for indecomposability, thus potentially saving computational resources.
Unfortunately, the modular properties of the triplet (p+ , p− )-models (with (p+ , p− ) 6= (1, 1)) are not as niceas one could have hoped for. In particular, the character of the vacuum module transforms under S : τ → −1/τ
into a linear combination of of other characters, but the coefficients depend non-trivially on τ [4, 8]. This would
appear to invalidate a naı̈ve application of the Verlinde formula. Nevertheless, one can arrive at a τ-independent
S-transformation by postulating a non-standard automorphy matrix and a generalised Verlinde formula exploiting
this has been demonstrated for the triplet models with p+ = 1 [7]. This proposed recipe does produce nonnegative integer structure constants which agree with the known (Grothendieck) fusion coefficients. However,
the generalised Verlinde formula itself is significantly more unwieldy than the original and we are not aware of
any attempts to derive its analogues for other logarithmic conformal field theories. Another way of obtaining τindependent coefficients is to enlarge the space of characters to the space of torus 1-point functions, that is, to add
certain linear combinations of characters multiplied by appropriate powers of τ [19], see also [4, 8, 20].
Here, we follow a different path to the Verlinde formula. Instead of working directly with the triplet (p+ , p− )models that have been so thoroughly studied in the literature, we focus our attention on the relatively unexplored
singlet (p+ , p− )-models. Whereas the triplet vertex operator algebras are known to possess a finite number of
inequivalent simple modules [4, 8, 21–24], the singlet algebras admit an uncountable infinity of them. However,this is not a bug, but a feature! We will see that the modular transformation properties of the characters of these
simple singlet modules are very well behaved. Moreover, applying the standard Verlinde formula (but with an
integral replacing the sum) leads again to non-negative integer structure constants. Finally, these results can belifted from the singlet algebra to its triplet cousin using the technology of simple current extensions. In particular,
our results provide an effortless derivation of the (1, p) triplet (Grothendieck) fusion rules without the need fornon-standard automorphy factors and complicated generalisations of the Verlinde formula.
This path to the Verlinde formula is actually a special case of a rather more general formalism that has been
proposed for non-rational conformal field theories in [25]. There, one starts with a continuous spectrum of so-called
standard modules which are typically simple and whose characters have good modular properties. In logarithmic
theories, the atypical standard modules are reducible but indecomposable and the not-so-good modular properties
1We are implicitly assuming here that fusing with a fixed module defines an exact functor on our module category.

MODULAR TRANSFORMATIONS AND VERLINDE FORMULAE FOR LOGARITHMIC (p+ , p− )-MODELS

3of the characters of the simple subquotients may be determined using standard methods of homological algebra.
This approach was developed for logarithmic models based on affine (super)algebras [26–29] where the natural
spectrum is continuous. One of the successes of this approach is the complete resolution of the longstanding
problem of negative fusion coefficients in fractional level Wess-Zumino-Witten models [30, 31].
The application of this general formalism to the (1, p) singlet models is relatively straightforward (see [25,Sec. 3] for the case p = 2 and [32] for more general p). However, the generalisation to all (p+ , p− ) singlet
algebras is rather more interesting because, in the case where p+ and p− are both greater than 1, the fusion productis no longer expected to define exact functors on the (natural) category of vertex operator algebra modules. This
non-exactness was first noted in [11] for the (2, 3) triplet model. Consequently, the Grothendieck group spanned bythe (equivalence classes of) simple modules does not inherit a ring structure from the fusion product. One therefore
cannot expect that the ring structure defined by the Verlinde formula on the span of the characters of the simple
modules — we call the resulting object the Verlinde ring — will coincide with a Grothendieck ring of fusion. The
natural question of how the non-exactness of fusion is manifested in the Verlinde ring is what motivated our workon this problem.2 As we will see, the answer is natural and satisfying, though there are subtleties worth remarking
upon.

We begin in Section 2 with notations and conventions, reviewing the definitions of the singlet and triplet(p+ , p− )-algebras in terms of the Heisenberg algebra and its simple current extensions. The irreducible modulesof both vertex operator algebras are constructed and their classifications are quoted with the necessary structural
aspects of these modules being deferred to an appendix. This material has many sources, for example [34, 35].
Here, we mostly follow the notation of [24].
In Section 3, the modular properties of the characters of the singlet modules are derived. We begin with thestandard modules which are of Feigin-Fuchs type, adding a Heisenberg charge to their characters so that all nonisomorphic simple modules have distinct characters. The S-transformations of these characters are deduced in
the usual way. The algebraic definitions of the remaining (atypical) simple modules then lead to resolutions foreach atypical simple in terms of standards. The resulting character formulae then give the S-transformations of
the atypical characters directly. Of note here is that when p+ and p− are both greater than 1, there exist atypicalsimple modules Lr,s whose S-matrix entries cannot be expressed as functions, but only as distributions. Indeed,this is also the case for the (non-simple) vacuum module.
We then turn to the Verlinde formula and the Verlinde product that it induces in Section 4. Most importantly,
we show that the Lr,s completely decouple in the Verlinde ring and may be consistently set to zero. This lets
us replace, when p+ , p− > 1, the S-matrix entries involving the vacuum module by those involving its maximal
submodule, which happens to be simple. The Verlinde formula is then well-defined, because we no longer needto divide by a distribution, and direct computation ensues. We thereby obtain a completely explicit description in
Equation (4.24) of the Verlinde product of the characters of any two simple singlet modules, excepting the Lr,swhose characters have been set to 0.
This result is then lifted to the triplet models through their realisations as simple current extensions of thecorresponding singlet models. Actually, these realisations remain conjectural in general because we can only
verify the simple current property at the level of the Verlinde rings, not the fusion rings themselves. Nevertheless,
we apply standard simple current technology to deduce the triplet analogues of Equation (4.24). The resulting
Verlinde product rules, reported in Equation (5.9), are then compared favourably with the rules that have been
proposed elsewhere in the literature. We close with a conclusion and discussion of our results.

2A second motivation is to study the modular story for examples of logarithmic theories (see also [33] in this regard) involving indecomposables
that are structurally more complicated than those of the (1, p)-models.

4

D RIDOUT AND S WOOD

2. T HE (p+ , p− ) S INGLET AND T RIPLET M ODELSIn this section, we introduce and fix our notation for the singlet and triplet models. These conformal field
theories are parametrised by two coprime positive integers p+ and p− . As one would expect, many of the important
quantities that we will study take a somewhat unwieldy form when expressed in terms of these parameters, so to
partially alleviate this, we introduce the following quantities:
s
s
2p−
2p+
α+ =
,
α− = −,
α0 = α+ + α− ,(2.1a)
p+
p−
p
1−r
1−s
1
αr,s =
α = p+ α+ = −p− α− = 2p+ p− ,
α+ +
α− ,
αr,s;n = αr,s + nα.
(2.1b)
222Here, r, s and n will always be assumed to be integers. Note that the αr,s;n so-defined satisfy
αr∓p+ ,s;n = αr,s;n±1 = αr,s±p− ;n .

(2.2)

We may therefore choose n so that 1 6 r 6 p+ and 1 6 s 6 p− , when convenient.
b (1) with highest weight
2.1. Feigin-Fuchs Modules. Consider the Fock module Fλ of the Heisenberg algebra gl
λ ∈ R. As is well known, the vacuum module F0 carries the structure of a vertex operator algebra. There exists
a continuous family of conformal structures for this vertex operator algebra and we will choose the corresponding
Virasoro algebra so that the central charge is
c = 1 − 3α02

(2.3a)and the highest weight vector of Fλ has conformal weight11
α0 2 α02
∆λ = λ (λ − α0 ) =−λ−.
(2.3b)
2
2
2
8
Restricting to the action of this Virasoro algebra, the Fock modules Fλ become Virasoro modules which we shallalso denote by Fλ . When considering the Fλ as Virasoro modules, we shall refer to them as Feigin-Fuchs modules.
The structure of these Feigin-Fuchs modules was determined by Feigin and Fuchs in [34] (see also [35] for acomprehensive treatment). If λ is not of the form αr,s;n for some r, s, n ∈ Z, then Fλ is simple as a Virasoro module.
If we choose n in what follows so that 1 6 r 6 p+ and 1 6 s 6 p− , then the structure depends only upon whether r
and s are p+ and p− , respectively, and upon the sign of n. We defer the explicit structural details of these Virasoro
modules, in the form of socle filtrations, to Appendix A.
The structure of the Feigin-Fuchs modules may be used to derive the Felder complexes [36]· · · −→ Fr,s;−2 −→ F p+ −r,s;−1 −→ Fr,s;0 −→ F p+ −r,s;+1 −→ Fr,s;+2 −→ · · ·(r 6= p+ ),(2.4a)· · · −→ Fr,s;+2 −→ Fr,p− −s;+1 −→ Fr,s;0 −→ Fr,p− −s;−1 −→ Fr,s;−2 −→ · · ·(s 6= p− ),(2.4b)

where we have simplified our notation by setting Fr,s;n ≡ Fαr,s;n . Indeed, the Virasoro homomorphisms defining
these complexes may be identified with (suitably regularised) powers of screening operators [37]. The complex(2.4a) may be checked to be exact when s = p− . Moreover, it only fails to be exact when s 6= p− at the n = 0 term,in which case the homology is the simple Virasoro module Lr,s whose highest weight vector has conformal weight∆r,s ≡ ∆αr,s . Similarly, (2.4b) is exact for r = p+ and otherwise only has non-zero homology, again given by Lr,s ,at n = 0.
2.2. The Singlet Algebra and its Modules. We define the following (Virasoro) submodules of Fr,s;n for 1 6 r 6
p+ and 1 6 s 6 p− :K+r,s;n = ker Fr,s;n −→ F p+ −r,s;n+1 ,K−r,s;n = ker Fr,s;n −→ Fr,p− −s;n−1 ,−Kr,s;n = K+r,s;n ∩ Kr,s;n ,I+r,s;n = im F p+ −r,s;n−1 −→ Fr,s;n
I−
r,s;n = im Fr,p− −s;n+1 −→ Fr,s;n−
Ir,s;n = I+r,s;n ∩ Ir,s;n(r 6= p+ , s 6= p− ).

(r 6= p+ ),(s 6= p− ),(2.5)

MODULAR TRANSFORMATIONS AND VERLINDE FORMULAE FOR LOGARITHMIC (p+ , p− )-MODELS

5+
If p+ = 1, the K+r,s;n are not defined by (2.5). We will therefore set Kr,s;n = Fr,s;n in this case. Similarly, if p− = 1,

we set K−r,s;n = Fr,s;n . Because of the exactness of the Felder complexes, we have the identifications (r 6= p+ ,s 6= p− )∼ +K+r,p− ;n = Ir,p− ;n ,∼ −K−
p+ ,s;n = I p+ ,s;n(for all n);K•r,s;n ∼= I•r,s;n(for all n 6= 0),(2.6)

where the superscript “•” stands for +, −, or is empty. Working out the Virasoro module structures of the n = 0
modules using the socle series of the Feigin-Fuchs modules (Appendix A), one arrives at the (non-split) short exact
sequences
0 −→ I•r,s;0 −→ K•r,s;0 −→ Lr,s −→ 0

(r 6= p+ , s 6= p− ).

(2.7)

Finally, the definitions (2.5) immediately imply the exact sequences+
0 −→ K+r,s;n −→ Fr,s;n −→ I p+ −r,s;n+1 −→ 0

(r 6= p+ ),−0 −→ K−r,s;n −→ Fr,s;n −→ Ir,p− −s;n−1 −→ 0

(s 6= p− ),(2.8)

for all n, and further contemplation of Virasoro structures (see Appendix A) leads to
0 −→ Ir,s;n −→ I+
r,s;n −→ Ir,p− −s;n−1 −→ 0
0 −→ Ir,s;n −→ I−
r,s;n −→ I p+ −r,s;n+1 −→ 0

(r 6= p+ , s 6= p− ),(2.9)

again for all n, which are likewise exact.
−
Recall that F1,1;0 = F0 carries the structure of a vertex operator algebra. As K+
1,1;0 and K1,1;0 are both kernelsof screening operators acting on this vertex operator algebra, they define vertex operator subalgebras, as does theirintersection K1,1;0 . The vertex operator algebra corresponding to K1,1;0 is called the singlet algebra I p+ , p− . It−
is simple if and only if p+ or p− is 1. We remark that if p+ = p− = 1, then K+
1,1;0 = K1,1;0 = K1,1;0 = F1,1;0 and
we see that the singlet algebra I 1, 1 is nothing but the Heisenberg algebra (with central charge 1). In general, thesinglet algebra is strongly generated by the energy-momentum tensor and a single Virasoro primary of dimension(2p+ − 1) (2p− − 1) [8, 23, 24].
Each of the Fλ , K•r,s;n and I•r,s;n , as well as the Lr,s , become modules for the singlet vertex operator algebra. Acomplete list of simple I p+ , p− -modules is given by• the Fλ with λ 6= αr,s;n for any r, s, n ∈ Z,
• the F p+ ,p− ;n for all n ∈ Z,
• the I+r,p− ;n for all 1 6 r 6 p+ − 1 and n ∈ Z,
• the I−p+ ,s;n for all 1 6 s 6 p− − 1 and n ∈ Z,
• the Ir,s;n for all 1 6 r 6 p+ − 1, 1 6 s 6 p− − 1 and n ∈ Z,
• the Lr,s ∼= L p −r,p −s for all 1 6 r 6 p+ − 1 and 1 6 s 6 p− − 1.
+−This is a straightforward corollary of [24, Thm. D], though the case when p+ or p− is 1 was settled previouslyin [38, 39]. We remark that when p+ = 1, the sets of I+r,p− ;n , Ir,s;n and Lr,s are empty — the only I 1, p− -simples
are the Fλ , the F1,p− ;n and the I−
1,s;n . The story when p− = 1 is similar.
Notice that the Fλ are simple for generic λ . In the formalism proposed in [25] for general (logarithmic)
conformal field theories, the Fλ may be identified as the standard singlet modules. The simple standard modules,
those with λ 6= αr,s;n or with λ = α p+ ,p− ;n , are called typical in this formalism and the remaining simple modules,
the I•r,s;n and Lr,s , are examples of atypical singlet modules. We will use this terminology freely in what follows.

2.3. The Triplet Algebra and its Modules. Just like the singlet algebra I p+ , p− , the triplet algebra W p+ , p−can be defined as a vertex operator subalgebra of a lattice vertex operator algebra V p+ , p− . This lattice algebra
may be characterised as the simple current extension of F1,1;0 by the group of simple currents generated by F1,1;2(or alternatively by F1,1;−2 ). In terms of Feigin-Fuchs modules, we therefore have the decomposition
 M
V p+ , p− =F1,1;2k .
k∈Z(2.10)

6

D RIDOUT AND S WOOD

This lattice vertex operator algebra is known to be rational, meaning that all of its modules are semisimple andthat there are only finitely many inequivalent simple modules. The number of inequivalent simple modules is, inthis case, exactly 2p+ p− and they can be parametrised by two integers, 1 6 r 6 p+ and 1 6 s 6 p− , and a label“±”. We denote these simple V p+ , p− -modules by F±
r,s . They may be decomposed into Feigin-Fuchs modules
as follows:
F+
r,s =

M

Fr,s;2k ,

k∈Z

F−
r,s =

M

Fr,s;2k+1 .

(2.11)k∈Z

Just like the Feigin-Fuchs modules, the lattice modules F±
r,s can also be arranged into Felder complexes connected by (appropriately regularised) powers of screening operators:ε−εε· · · −→ Fεr,s −→ F−εp+ −r,s −→ Fr,s −→ F p+ −r,s −→ Fr,s −→ · · ·(r 6= p+ ),ε−εε· · · −→ Fεr,s −→ F−εr,p− −s −→ Fr,s −→ Fr,p− −s −→ Fr,s −→ · · ·(s 6= p− ).

(2.12a)(2.12b)

Here, ε stands for either “+” or “−”. As with the singlet algebra, the triplet algebra W p+ , p− may also be
defined as an intersection of kernels:
h
i
h
i

−
+
−
W p+ , p− = ker F+
1,1 −→ F p+ −1,1 ∩ ker F1,1 −→ F1,p− −1 .

(2.13)

Again, when p+ = 1 or p− = 1, at least one of the Felder complexes is not defined and its corresponding ker+nel is replaced by the lattice module F+
1,1 . In particular, it follows that W 1, 1 = F1,1 which is well knownb (2) . In general, the triplet vertex operator algebra
to be isomorphic to the level 1 vertex operator algebra sl1
W p+ , p− is strongly generated by the energy momentum tensor and three Virasoro primaries of dimension(2p+ − 1) (2p− − 1) [8, 23, 24]. The singlet algebra I p+ , p− is naturally generated as a vertex operator subalgebra by removing two of these Virasoro primaries.

Unlike the lattice algebra V p+ , p− , the triplet algebra W p+ , p− is not rational in general,3 because there
exist non-semisimple triplet modules [5, 22]. However, the number of inequivalent simple W p+ , p− -modules isfinite and, in fact, this number is 12 (p+ − 1)(p− − 1) + 2p+ p− [8, 23, 24]. We give a complete list of these simples
along with their decompositions into singlet modules (for later convenience):• W+
p+ ,p− =M

F p+ ,p− ;2k and W−
p+ ,p− =k∈Z• W+r,p− =

M

M−
I+
r,p− ;2k and Wr,p− =

• W+
r,s =

M

I+
r,p− ;2k+1 for all 1 6 r 6 p+ − 1,k∈ZI−p+ ,s;2k

and W−
p+ ,s =k∈Z

M

F p+ ,p− ;2k+1 ,

k∈Z

k∈Z

• W+
p+ ,s =

M

Ir,s;2k and W−
r,s =

k∈Z

M

I−
p+ ,s;2k+1 for all 1 6 s 6 p− − 1,k∈ZM

Ir,s;2k+1 for all 1 6 r 6 p+ − 1 and 1 6 s 6 p− − 1,k∈Z

• Lr,s ∼= L p+ −r,p− −s for all 1 6 r 6 p+ − 1 and 1 6 s 6 p− − 1.
±
Again, this is an easy consequence of the results of [8, 23, 24]. Note that when p+ = 1, there are no W±r,p− , Wr,sor Lr,s in this list — the simples are exhausted by the W±
1,s (with s = p− allowed). Again, the story is similar for
p− = 1.
In the terminology of [25], the W±
p+ ,p− are the typical triplet modules, being direct sums of typical singlet
modules. The W±r,s , with either r 6= p+ or s 6= p− , and the Lr,s are then atypical triplet modules. We remark thatthe decompositions of the triplet simples into singlet simples suggest that the triplet algebra is just a simple current
extension of the singlet algebra. Indeed, the decomposition of W+
1,1 shows that the simple currents responsible forthis conjectured extension are the I1,1;2n . We will verify that this conjecture is consistent with our Verlinde formulacomputations in Section 5.1, though we will see that there are interesting subtleties which prevent the evidence
from being conclusive.
3The single exception is W 1, 1 ∼ sl
= b (2)1 which is well known to be rational and even unitary.

MODULAR TRANSFORMATIONS AND VERLINDE FORMULAE FOR LOGARITHMIC (p+ , p− )-MODELS

7

3. C HARACTERS AND M ODULAR T RANSFORMATIONS FOR S INGLET M ODELSThis section details the derivation of the modular S-transformations of the characters of the simple I p+ , p− modules. The methodology follows the approach proposed in [25] for general non-rational conformal field theories.
Specifically, the characters of the standard I p+ , p− -modules, simple and non-simple, are taken as a (topological)
basis for a vector space which is shown to be preserved by the natural action of S. This space then carries arepresentation of SL 2; Z of uncountably-infinite dimension. Resolutions are then derived for the atypical simple
modules in terms of the non-simple standard modules and this gives expressions for the characters of the former
as infinite alternating sums of the characters of the latter (the basis characters). In this way, we arrive at S
transformations for all simple I p+ , p− -modules.
The computations are straightforward when p+ or p− is 1. However, the case where p+ , p− > 1 is moreinteresting (as expected) because the “S-matrix entries”, that describe the decomposition of the S-transformed
character of Lr,s into standard characters, are no longer functions of the parameters, but must instead be regarded
as distributions. Consequently, the same is true for the S-matrix entries of the vacuum module K1,1;0 , leading toconceptual difficulties in applying the Verlinde formula.

3.1. Characters of Feigin-Fuchs modules and their Modular Transformations. For any Virasoro module M
at central charge c, one defines its character to be the following power series in q = exp(2πiτ): ch M (τ) = trM qL0 −c/24 .

(3.1)

For example, the characters of the Feigin-Fuchs modules Fλ of Section 2.1 are given by22 q(λ −α0 ) /2q∆λ −(1−3α0 )/24ch Fλ (τ) =
=
,
i
η(q)
∏∞
i=1 (1 − q )(3.2)

where η(q) is the Dedekind eta function. As one can see from this formula, the Feigin-Fuchs modules Fλ andFα0 −λ have identical characters. In order to disambiguate these characters [25], we generalise them by adding an
extra formal variable z = exp(2πiζ ):2 q(λ −α0 /2) /2 zλ −α0 /2ch Fλ (τ, ζ ) =
.
η(q)

(3.3)

Here, one can think of z as keeping track of the eigenvalue of the Heisenberg zero-mode (shifted by −α0 /2).
The modular S-transformation of characters is the map4  ch M (τ, ζ ) 7→ ch M (− τ1 , ζτ ) .

(3.4)

The characters of the Feigin-Fuchs modules satisfy the transformation formulae
Z h
i  
 
ch Fλ (− τ1 , ζτ ) = S Fλ → Fρ ch Fρ (τ, ζ )dρ,

(3.5)



where the S-matrix coefficients S Fλ → Fρ are given byhi
S Fλ → Fρ = exp [−2πi(λ − α0 /2)(ρ − α0 /2)] .

(3.6)

R

These coefficients follow from a straightforward gaussian integral, convergent for Im τ > 0 hence |q| < 1. In
particular, for λ = αr,s;n , the S-matrix coefficients specialise tohi
S Fr,s;n → Fρ = eiπrα+ (ρ−α0 /2) eiπsα− (ρ−α0 /2) e−iπnα(ρ−α0 /2) .

(3.7)

4Technically, one needs an additional transformation variable to absorb the so-called automorphy factor. We refer to [25, Sec. 1.2] for the detail(in the case α0 = 0).

8

D RIDOUT AND S WOOD

3.2. Characters of Singlet Modules and their Modular Transformations. With Equations (3.6) and (3.7), wehave determined the modular transformations of the standard I p+ , p− -modules. The Felder complexes of Section 2.1 and the exact sequences of Section 2.2 now determine resolutions (or coresolutions or two-sided resolutions) of the singlet modules K•r,s;n , I•r,s;n and Lr,s in terms of Feigin-Fuchs modules. These, in turn, allow us tocalculate the characters of the atypical singlet modules in terms of characters of the standard (Feigin-Fuchs) singlet
modules.
It is important to note that the maps defining the Felder complexes and exact sequences are not Heisenberg
algebra homomorphisms in general. The interpretation of z in Feigin-Fuchs characters as tracking the eigenvalueof the Heisenberg zero-mode therefore does not lift to the character formulae for the atypical singlet modules that
we shall derive. There is no Heisenberg zero-mode in the singlet algebra, so the singlet characters should in fact
be computed with z = 1. However, if one does this, one immediately encounters the problem that non-isomorphic
singlet modules can have identical characters. We will therefore keep z as a formal variable in the singlet character
formulae that follow. Its function remains to naturally facilitate the distinguishing of characters of non-isomorphic
modules, though it no longer appears to have any (obvious) interpretation in terms of eigenvalues of zero-modes.
3.2.1. The I+r,s;n modules. As long as we avoid the non-exact parts of the Felder complexes (2.4), we can use themto give (co)resolutions of the I+r,s;n in terms of Feigin-Fuchs modules, which in turn allow us to derive character
formulae and S-matrix coefficients. When n > 0, we can iteratively splice the first exact sequence of (2.8) withitself, using the isomorphisms (2.6). The result is a resolution for I+
r,s;n . For n < 0, the same method results insteadin a coresolution. We therefore obtain, for 1 6 r 6 p+ − 1 and 1 6 s 6 p− , the following (co)resolutions of theI+r,s;n :
· · · −→ F p+ −r,s;n−3 −→ Fr,s;n−2 −→ F p+ −r,s;n−1 −→ I+
r,s;n −→ 0(n 6 0),(3.8a)

0 −→ I+r,s;n −→ Fr,s;n −→ F p+ −r,s;n+1 −→ Fr,s;n+2 −→ · · ·(n > 1).

(3.8b)

Note that both of these sequences will be exact for all n ∈ Z if and only if s = p− . From these (co)resolutions, we
can read off the character formulae






ch F p+ −r,s;n−2k−1 − ch Fr,s;n−2k−2

∑



k>0
ch I+




r,s;n =
 ∑ ch Fr,s;n+2k − ch F p+ −r,s;n+2k+1

if n 6 0,(3.9)
if n > 1

k>0


and the S-matrix coefficients are obtained by adding and subtracting the S-matrix entries S F p+ −r,s;n−2k−1 → Fρ






and S Fr,s;n−2k−2 → Fρ , for n 6 0, or S Fr,s;n+2k → Fρ and S F p+ −r,s;n+2k+1 → Fρ , for n > 1:−2i sin[πrα+ (v − α0 /2)]eiπsα− e−iπ(n−2)α(ρ−α0 /2) ∑ e2πikα(ρ−α0 /2) if n 6 0,

h
i 
S I+
r,s;n → Fρ =k>0iπsα −iπnα(ρ−α0 /2)∑ e−2πikα(ρ−α0 /2)
+2i sin[πrα+ (v − α0 /2)]e − e(3.10)

if n > 1.

k>0

The infinite sums in this formula are geometric series at the boundaries of their radius of convergence. Nevertheless, we will replace these geometric series by their analytic continuations5−x−1∑ x+2k 7−→ x − x−1 ,k≥0x∑ x−2k 7−→ x − x−1 .

(3.11)

k≥0

With these replacements, the S-matrix coefficients simplify to the common form (for all n)hiiπsα− (ρ−α0 /2) −iπ(n−1)α(ρ−α0 /2) sin[πrα+ (ρ − α0 /2)]
S I+
e(r 6= p+ ),r,s;n → Fρ = esin[π p+ α+ (ρ − α0 /2)]
where we have used the identity α = p+ α+ in the denominator.
5We refer to [32] for an explicit example of how to regularise these sums in the case that p or p equal 1.
+−(3.12)

MODULAR TRANSFORMATIONS AND VERLINDE FORMULAE FOR LOGARITHMIC (p+ , p− )-MODELS

93.2.2. The I−r,s;n modules. The derivation of (co)resolutions, character formulae and S-matrix coefficients for the+
singlet modules I−
r,s;n is completely analogous to the derivation for the Ir,s;n .

For 1 6 r 6 p+ and 1 6 s 6 p− − 1, the following sequences are exact and define (co)resolutions of I−r,s;n :
· · · −→ Fr,p− −s;n+3 −→ Fr,s;n+2 −→ Fr,p− −s;n+1 −→ I−
r,s;n −→ 0
0 −→ I−r,s;n −→ Fr,s;n −→ Fr,p− −s;n−1 −→ Fr,s;n−2 −→ · · ·These (co)resolutions lead to the character formulae






ch Fr,p− −s;n+2k+1 − ch Fr,s;n+2k+2

∑



k>0
ch I−




r,s;n =
 ∑ ch Fr,s;n−2k − ch Fr,p− −s;n−2k−1(n > 0),(3.13a)(n 6 −1).

(3.13b)if n > 0,(3.14)
if n 6 −1,

k>0

which in turn imply the S-matrix coefficientshisin[πsα− (ρ − α0 /2)]
S I−→
F
= eiπrα+ (ρ−α0 /2) e−iπ(n+1)α(ρ−α0 /2)ρr,s;nsin[π p− α− (ρ − α0 /2)]

(s 6= p− ),(3.15)

for general n. Here, we have used the identity α = −p− α− in the denominator as well as the analytic continuations(3.11).

3.2.3. The Ir,s;n modules. For 1 6 r 6 p+ − 1 and 1 6 s 6 p− − 1, the singlet modules Ir,s;n can be resolved by
iteratively splicing the exact sequences (2.9) to obtain++· · · −→ I+
r,p− −s;n+3 −→ Ir,s;n+2 −→ Ir,p− −s;n+1 −→ Ir,s;n −→ 0,(3.16a)−−· · · −→ I−
p+ −r,s;n−3 −→ Ir,s;n−2 −→ I p+ −r,s;n−1 −→ Ir,s;n −→ 0.

(3.16b)We therefore arrive at two seemingly different character formulae:
 



 +

ch Ir,s;n = ∑ ch I+r,p− −s;n+2k+1 − ch Ir,s;n+2k+2(3.17a)

 

 −

−
ch
I
.
= ∑ ch I−
p+ −r,s;n−2k−1r,s;n−2k−2(3.17b)k>0

k>0

However, the resulting S-matrix coefficients are identical:hisin[πrα+ (ρ − α0 /2)] sin[πsα− (ρ − α0 /2)] −iπnα(ρ−α0 /2)e.
S Ir,s;n → Fρ =sin[π p+ α+ (ρ − α0 /2)] sin[π p− α− (ρ − α0 /2)]

(3.18)

One can also coresolve the Ir,s;n in terms of the I±r,s;n with the same result.
3.2.4. The Lr,s Modules. As noted in Section 2.1, the cohomology of the Felder complexes (2.4) is trivial everywhere except when 1 6 r 6 p+ − 1, 1 6 s 6 p− − 1 and n = 0, where it is the simple Virasoro module Lr,s . By the
Euler-Poincaré principle, the Felder complex (2.4a) implies that the character of the simple singlet module Lr,s isgiven by





+
ch Lr,s = ∑ ch Fr,s;2n − ch F p+ −r,s;2n+1 ,(3.19a)whereas the Felder complex (2.4b) gives the character of Lr,s as

−




ch Lr,s = ∑ ch Fr,s;2n − ch Fr,p− −s;2n+1 .

(3.19b)n∈Zn∈Z

It is not clear that these coincide. However, we remark that we should only expect that these character formulae
coincide once we remove the z-dependence (by setting z = 1). Recall that we included z, somewhat artificially,in our definition of singlet characters so as to be able to distinguish simple modules that would otherwise haveidentical characters.

10

D RIDOUT AND S WOOD

Now, unlike the character formulae for the Ir,s;n considered above, the character formulae (3.19) yield different
S-matrix coefficients:hi+
S Lr,s → Fρ = 2i sin[πrα+ (ρ − α0 /2)]eiπsα− (ρ−α0 /2) ∑ e−2πinα(ρ−α0 /2) ,(3.20a)hi−
S Lr,s → Fρ = 2i sin[πsα− (ρ − α0 /2)]eiπrα+ (ρ−α0 /2) ∑ e−2πinα(ρ−α0 /2) .

(3.20b)n∈Zn∈Z

The superscript “±” serves to remind us which Felder complex was used in the derivation. We note that thesums in these formulae do not define functions but must be interpreted as distributions (see Section 4.1). We note
∼ L p −r,p −s .
in addition that the character formulae (3.19) also do not appear to respect the isomorphism Lr,s =+−Again, the disambiguation variable z is to blame. We shall see shortly that this non-uniqueness problem solves
itself rather naturally in the setting of the Verlinde algebra, though there is of course a price that has to be paid.

3.2.5. The Kr,s;0 modules. For 1 6 r 6 p+ − 1 and 1 6 s 6 p− − 1, the exact sequences (2.8) and the formulae(3.19) give two distinct character formulae for the Kr,s;0 singlet modules:

±

±


ch Kr,s;0 = ch Lr,s + ch Ir,s;0 .

(3.21)

These in turn yield two distinct S-matrix coefficients:hi+
S Kr,s;0 → Fρ = 2i sin[πrα+ (ρ − α0 /2)]eiπsα− (ρ−α0 /2) ∑ e−2πinα(ρ−α0 /2)n∈Z

sin[πrα+ (ρ − α0 /2)] sin[πsα− (ρ − α0 /2)] −iπnα(ρ−α0 /2)+e,sin[π p+ α+ (ρ − α0 /2)] sin[π p− α− (ρ − α0 /2)]

(3.22a)hi−
S Kr,s;0 → Fρ = 2i sin[πsα− (ρ − α0 /2)]eiπrα+ (ρ−α0 /2) ∑ e−2πinα(ρ−α0 /2)n∈Z+

sin[πrα+ (ρ − α0 /2)] sin[πsα− (ρ − α0 /2)] −iπnα(ρ−α0 /2)e.
sin[π p+ α+ (ρ − α0 /2)] sin[π p− α− (ρ − α0 /2)]

(3.22b)We remark that the vacuum module is K1,1;0 , so this non-uniqueness is a potential problem for the Verlinde computations below. This is on top of the problem that one will seemingly have to make sense of dividing by a distribution
in order to apply the Verlinde formula. Again, we will resolve these issues in Section 4.1. Note however that theabove formulae are not valid for K1,1;0 when p+ = 1 or p− = 1. In these cases, K1,1;0 ∼= I1,1;0 and the vacuum
S-matrix coefficients are instead given in Equation (3.18).

3.3. Addendum. At this point, an alert reader might object that the character formulae for the I±r,s;n have been
derived from (co)resolutions which avoid the part where the Felder complexes (2.4) fail to be exact. Is it possiblethat we will obtain different formulae if we (co)resolve so as to cross the non-exact piece of the complexes? Now
that we have the characters of the Lr,s , derived from each Felder complex, it is easy to attend to this objection. We
will consider the resolution (3.8a) for I+
r,s;n with r 6= p+ and n > 1, the analysis in the other cases being almost
identical.
First, the splicing of the exact sequences (2.6) stops when the third label (n) of the modules drops to 0 because+
K+r,s;0 and Ir,s;0 are not isomorphic. Instead, we arrive at the long exact sequence+
0 −→ K+∗,s;0 −→ F∗,s;0 −→ · · · −→ Fr,s;n−2 −→ F p+ −r,s;n−1 −→ Ir,s;n −→ 0,(3.23)

where the subscript “∗” stands for r, if n is even, and p+ − r, if n is odd. Because of the exact sequence (2.7),
continuing the splicing to obtain (3.8a) does not yield a resolution of I+r,s;n (for n > 1) because the sequence has
non-trivial cohomology Lr,s when the third label is 0. No matter — Euler-Poincaré says that the character formulaMODULAR TRANSFORMATIONS AND VERLINDE FORMULAE FOR LOGARITHMIC (p+ , p− )-MODELSn(3.9) derived earlier for the I+r,s;n with n 6 0 should be corrected, for n > 0, by (−1) ch L∗,s :








if n > 1 is even, ∑ ch F p+ −r,s;n−2k−1 − ch Fr,s;n−2k−2 + ch Lr,s
 +  k>0
ch Ir,s;n =








 ∑ ch F p+ −r,s;n−2k−1 − ch Fr,s;n−2k−2 − ch L p+ −r,s if n > 1 is odd.

11

(3.24)

k>0+
Because the K+r,s;0 and Ir,s;0 are defined using the Felder complex (2.4a), we must use the character formula (3.19a)for the Lr,s that was derived from this complex. When we substitute this into the above character formulae, wefind that the result precisely reproduces (3.9) for n > 1. In summary, the character formula does not depend upon
whether we use a (not quite exact) resolution or a coresolution.

4. T HE V ERLINDE R ING FOR S INGLET M ODELS
Everything is now in place to use the obvious continuum analogue of the Verlinde formula to compute “fusion
coefficients” for the singlet theories. Roughly speaking, this will define a ring structure on an abelian group
generated by (some of) the simple characters. We will refer to this ring as the (singlet) Verlinde ring, denoting it by
V I p+ , p− . As we will see, the computations are relatively straightforward, although there is the issue of the
distributional nature of the vacuum S-matrix entries to surmount. However, the interpretation of the computations
is not: For I 1, 1 , we do indeed recover the fusion coefficients because of the semisimplicity of the module
category. The Verlinde ring V I 1, 1 is the fusion ring of the Heisenberg algebra. For I 1, p− and I p+ , 1 , nonsemisimple modules exist, so the best we can expect is that the Verlinde ring will coincide with the Grothendieckring of fusion. This, in turn, requires that fusion defines an exact functor on the singlet modules which we expect
to be true (it is true for the triplet algebras W 1, p− and W p+ , 1 [40]). However, fusion is not expected to be
exact for p+ , p− > 1 and, indeed, non-exactness has been demonstrated for the corresponding triplet algebras [11].
So, we can only expect that the Verlinde ring may be identified with a quotient of the Grothendieck group uponwhich fusion restricts to an exact product.
Before we start calculating the structure constants of the Verlinde ring, it is convenient to extend the range ofthe indices of the Ir,s;n to include r = p+ and s = p− . Let 1 6 r 6 p+ − 1 and 1 6 s 6 p− − 1 and then define
I p+ ,s;n := I−
p+ ,s;n ,Ir,p− ;n := I+r,p− ;n ,I p+ ,p− ;n := F p+ ,p− ;n .

(4.1)

For 1 6 r 6 p+ and 1 6 s 6 p− , the S-matrix coefficients (3.7), (3.12), (3.15) and (3.18) can now be compactly
written in the unified formhisin[πrα+ (ρ − α0 /2)] sin[πsα− (ρ − α0 /2)] −iπnα(ρ−α0 /2)e.
S Ir,s;n → Fρ =sin[π p+ α+ (ρ − α0 /2)] sin[π p− α− (ρ − α0 /2)]

(4.2)

4.1. The Decoupling of the Lr,s . For two singlet modules M and N, the Verlinde formula for “fusion products”
states that there is a product (on certain equivalence classes of modules) given by    Z 
M × N = NM N Fν Fν dν.

(4.3)

R

The “fusion coefficents” NM N Fν are defined in terms of the S-matrix coefficients by

 
 
∗
Z
S M → Fρ S N → Fρ S Fν → FρFνNM N =dρ,S K1,1;0 → Fρ
R

(4.4)

where “∗” denotes complex conjugation. Consider the denominator of the above integrand. According to Equation (3.22), this S-matrix coefficient has two possible forms:
h
i±
h
i±
h
i
S K1,1;0 → Fρ = S L1,1 → Fρ + S I1,1;0 → Fρ

± !
h
i
S L1,1 → Fρ .
= S I1,1;0 → Fρ1+ 
S I1,1;0 → Fρ

(4.5)

12

D RIDOUT AND S WOOD

The quotient of the S-matrix coefficients of L1,1 and I1,1;0 can be evaluated using (3.20a) and (4.2):
+
S L1,1 → Fρsin[π p+ α+ (ρ − α0 /2)] sin[π p− α− (ρ − α0 /2)]
 =sin[πα+ (ρ − α0 /2)] sin[πα− (ρ − α0 /2)]
S I1,1;0 → Fρ
· 2i sin[πα+ (ρ − α0 /2)]eiπα− (ρ−α0 /2) ∑ e−2πiα(ρ−α0 /2)kk∈Z

sin[π p− α− (ρ − α0 /2)]
= eiπα− (ρ−α0 /2)· 2i sin[πα(ρ − α0 /2)] ∑ e−2πiα(ρ−α0 /2)k .
sin[πα− (ρ − α0 /2)]
k∈Z

(4.6)

The product of the last two factors on the right-hand-side can be identified with zero because
2i sin[πα(ρ − α0 /2)] ∑ e2πiα(ρ−α0 /2)k = eiπα(ρ−α0 /2) (1 − e−2πiα(ρ−α0 ) ) ∑ e−2πiα(ρ−α0 /2)k
k∈Z

k∈Z

=e

iπα(ρ−α0 /2)∑ (e−2πiα(ρ−α0 /2)k

− e−2πiα(ρ−α0 /2)(k+1) ) .

(4.7)

k∈Z


−
A similar calculation gives the same conclusion for S L1,1 → Fρ . We may therefore replace the vacuum S±matrix coefficient S K1,1;0 → Fρin the Verlinde formula by S I1,1;0 → Fρ , the S-matrix coefficient of its
simple submodule I1,1;0 . In this way, the non-uniqueness of the vacuum S-matrix coefficients is neatly bypassed.

±

±
We remark that the S-matrix coefficients S Lr,s → Fρ , and hence the S Kr,s;0 → Fρ as well, can only beunderstood as distributions:hi+1πkr −iπks/p−
S Lr,s → Fρ = + ∑ 2i sineδ (ρ − α0 /2 − k/α) ,
α k∈Z
p+
h
i−
1
πks +iπkr/p+
S Lr,s → Fρ = − ∑ 2i sineδ (ρ − α0 /2 − k/α) .
α k∈Z
p−

(4.8a)(4.8b)

From this point of view, the vanishing of the quotient (4.6) and its “−” version is manifest because the factors
sin[π p± α± (ρ − α0 /2)] may be replaced by ± sin[kπ] = 0 when brought into the sum over k, so the coefficient ofeach delta function is zero.
We next argue that at the level of the Verlinde products, the minimal model modules Lr,s decouple, meaning that
F

that for any singlet module N, we have NLr,s N ρ = 0. This follows from an obvious generalisation of the argument
above for (r, s) = (1, 1) to the quotients±
S Lr,s → Fρ.
S I1,1;0 → Fρ

±


F
It therefore follows from (4.4) and the replacement S K1,1;0 → Fρ → S I1,1;0 → Fρ that NLr,s N ρ = 0. Summarising, we have seen that the alarming non-uniqueness and distributional nature of the S-matrix coefficients that
we observed for the Lr,s is not at all troublesome for Verlinde computations as these modules decouple completely.

In other words, the Lr,s may be naturally identified with zero in the Verlinde ring: Lr,s = 0.
4.2. Verlinde Products. We are now ready to compute the “Verlinde products” defined by the Verlinde formula.
   First, we will compute the product Fλ × Fµ of standard singlet module characters for λ , µ ∈ R.6 The Verlinde
coefficient we seek is
N F F µ Fν =

Z

λ

Z

=


 
 
∗
S Fλ → Fρ S Fµ → Fρ S Fν → FρdρS I1,1;0 → FρR
sin[π p+ α+ (ρ − α0 /2)] sin[π p− α− (ρ − α0 /2)] −2πi(λ +µ−ν−α0 /2)(ρ−α0 /2)edρsin[πα− (ρ − α0 /2)]
R sin[πα+ (ρ − α0 /2)]

(4.9)

which, by means of the trigonometric identity
sin[px] eipx − e−ipx p−1 i(p−1−2 j)x
= ix
= ∑e
,
sin[x]e − e−ix
j=0
6In what follows, we will drop the “ch” from the character chM  of a module M for brevity and to make contact with Equation (4.3).

(4.10)

MODULAR TRANSFORMATIONS AND VERLINDE FORMULAE FOR LOGARITHMIC (p+ , p− )-MODELS

13

simplifies top+ −1 p− −1 Z

N F F µ Fν = ∑λ∑e−2πi(λ +µ−ν+ j+ α+ + j− α− )(ρ−α0 /2) dρ

j+ =0 j− =0 R

p+ −1 p− −1= ∑∑ δ (ν − λ − µ − j+ α+ − j− α− ) .

(4.11)

j+ =0 j− =0

The Verlinde formula (4.3) therefore yields the following product:   p+ −1 p− −1
Fλ × Fµ = ∑ ∑ Fλ +µ+ j+ α+ + j− α− .

(4.12)

j+ =0 j− =0  
We next turn to the product Ir,s;n × Fµ for 1 6 r 6 p+ , 1 6 s 6 p+ , n ∈ Z and µ ∈ R. The remaining
Verlinde products will then be calculated by applying this product to the appropriate (co)resolutions. This time,
the Verlinde coefficient to be computed isNIr,s;n Fµ

Fν

Z

=

sin[πrα+ (ρ − α0 /2)] sin[πsα− (ρ − α0 /2)] −iπnα(ρ−α0 /2) −2πi(µ−ν)(ρ−α0 /2)eedρR sin[πα+ (ρ − α0 /2)] sin[πα− (ρ − α0 /2)]

r−1 s−1= ∑∑ δ (ν − αr,s;n − µ − j+ α+ − j− α− ) .

(4.13)

j+ =0 j− =0

The Verlinde formula therefore gives   r−1 s−1 Ir,s;n × Fµ = ∑ ∑ Fαr−2 j+ ,s−2 j− ;n +µ .

(4.14)

j+ =0 j− =0

As the Verlinde formula is defined entirely in terms of characters of modules, it cannot differentiate between an
indecomposable module and the direct sum of its simple composition factors. We therefore obtain the following
identifications, for 1 6 r 6 p+ − 1 and 1 6 s 6 p− − 1, from Equations (2.6), (2.8) and (2.9):
 +  
 


 
 

Ir,s;n = Ir,s;n + Ir,p− −s;n−1 ,F p+ ,s;n = I p+ ,s;n + I p+ ,p− −s;n−1 , −  
 


 
 

Ir,s;n = Ir,s;n + I p+ −r,s;n+1 ,Fr,p− ;n = Ir,p− ;n + I p+ −r,p− ;n+1 .

(4.15)

Here, we recall the definitions (4.1) that we have already made for r = p+ or s = p− . By applying the Verlinde
product (4.14) to the (co)resolutions (3.8a), (3.8b), (3.13a), (3.13b), (3.16a) and (3.16b), we obtain the following
products involving the I1,1;m , I2,1;0 and I1,2;0 :

 
 

I1,1;m × Ir,s;n = Ir,s;m+n ,I2,s;nif r = 1,



 


 

I2,1;0 × Ir,s;n =Ir−1,s;n + Ir+1,s;n
if 1 < r < p+ ,









 I
if r = p+ ,1,s;n−1 + 2 I p+ −1,s;n + I1,s;n+1Ir,2;nif s = 1,


 
 
 

I1,2;0 × Ir,s;n =if 1 < s < p− ,Ir,s−1;n + Ir,s+1;n


 




 I
if s = p− .
r,1;n−1 + 2 Ir,p− −1;n + Ir,1;n+1
We note for future use that the linear Z-span of the Ir,s;n is closed under the Verlinde product.
(4.16a)(4.16b)(4.16c)To illustrate how to apply (4.14) to (co)resolutions, we present the details of the derivation of the product 
I2,1;0 × I p+ ,s;n , with s 6= p− . First, we recall that I p+ ,s;n = I−
p+ ,s;n , so the resolution (3.13) (or rather thecorresponding character formula (3.14)) allows us to write7

 




 

I2,1;0 × I p+ ,s;n = ∑ I2,1;0 × F p+ ,p− −s;n+2k+1 − F p+ ,s;n+2k+2
k>0

7Note that (3.14) gives two character formulae, one for n > 0 and the other for n 6 −1. We assume here that n > 0 for clarity. One can easily
check that we get the same answer for the Verlinde product when we use the formula for n 6 −1 instead.

14

D RIDOUT AND S WOOD

=∑



 

F p+ +1,p− −s;n+2k+1 + F p+ −1,p− −s;n+2k+1

k>0


 

− F p+ +1,s;n+2k+2 − F p+ −1,s;n+2k+2(using Equation (4.14) and αr+p+ ,s;n = αr,s;n−1 )
=∑



 
 

F1,p− −s;n+2k − F1,s;n+2k+1 + I−
p+ −1,s;n

k>0


  −

= I−
1,s;n−1 + I p+ −1,s;n(using Equation (3.14) again)



 

= I1,s;n−1 + 2 I p+ −1,s;n + I1,s;n+1 ,(4.17)

the last equality following from (4.15). The remaining Verlinde products of (4.16) are similarly derived.


 



4.3. Presentations and the Verlinde Ring. It is clear from Equation (4.16) that I1,1;±1 , I2,1;0 and I1,2;0
generate all of the Ir,s;n , for 1 6 r 6 p+ , 1 6 s 6 p− and n ∈ Z, by repeatedly taking Verlinde products with one
another. Mathematically, this implies that there is a ring homomorphism φ from the polynomial ring Z[X,Y, Z, Z −1 ]to the subring of the singlet Verlinde ring V I p+ , p− that is spanned by the Ir,s;n . We will denote this subring
by Vatyp. I p+ , p− , referring to it as the atypical Verlinde ring for the singlet algebra (the only typical simples
involved are the I p+ ,p− ;n ≡ F p+ ,p− ;n ). The homomorphism φ is defined by
φ : Z[X,Y, Z, Z −1 ] −→ Vatyp. I p+ , p− ;






X 7−→ I2,1;0 ,Y 7−→ I1,2;0 ,Z ±1 7−→ I1,1;±1 .

(4.18)

This map is surjective by construction.








If we restrict the Verlinde products to those of I2,1;0 with Ir,1;0 (or I1,2;0 with I1,s;0 ), then we observethe familiar sl 2 -structure that may be formalised in terms of Chebyshev polynomials of the second kind, at least
for r < p+ (s < p− ). Recall that these Chebyshev polynomials are defined recursively by
U−1 (X) = 0,

U0 (X) = 1;

Un+1 (X) = XUn (X) −Un−1 (X)(n > 0)(4.19)and that they satisfy the simple multiplication formulae,
Uk (X)U` (X) =k+`0∑ Um (X),(4.20)

m=|k−`|

where the primed summation means that the label increases in steps of 2, not 1. From the Verlinde products (4.16)
and the recursion relations (4.19), it is not hard to see that the ring homomorphism φ acts as
φ (Ur−1 (X)Us−1 (Y )Z n ) = Ir,s;n .

(4.21)

However, the kernel of φ is non-trivial as is evidenced by the Verlinde products with r = p+ and s = p− . They
imply thatφ (U p+ (X) −U p+ −2 (X) − Z − Z −1 ) = φ (U p− (Y ) −U p− −2 (Y ) − Z − Z −1 ) = 0.

(4.22)

In fact, one can show that the kernel of φ is generated, as an ideal, by the above polynomials, hence that we havethe following explicit presentation of the atypical Verlinde ring:
Vatyp. I p+ , p− ∼=Z[X,Y, Z, Z −1 ].
hU p+ (X) −U p+ −2 (X) − Z − Z −1 ,U p− (Y ) −U p− −2 (Y ) − Z − Z −1 iThis presentation can be used to easily compute the Verlinde product of two Ir,s;n modules:(4.23)

MODULAR TRANSFORMATIONS AND VERLINDE FORMULAE FOR LOGARITHMIC (p+ , p− )-MODELS

a+ (r+r0 )0Ir,s;n × Ir0 ,s0 ;n0 =

a− (s+s0 )
0

∑0

∑0



I j+ , j− ;n+n015



j+ =|r−r |+1 j− =|s−s |+1+

a+ (r+r0 )0

s+s0 −1−p− 

0

∑0

∑

 
 

I j+ , j− ;n+n0 −1 + I j+ ,p− − j− ;n+n0 + I j+ , j− ;n+n0 +1

j+ =|r−r |+1 j− =b− (s+s0 )+a− (s+s0 )0

r+r0 −1−p+
0

∑

∑0



 
 

I j+ , j− ;n+n0 −1 + I p+ − j+ , j− ;n+n0 + I j+ , j− ;n+n0 +1

j+ =b+ (r+r0 ) j− =|s−s |+1
r+r0 −1−p

+

∑

0

+

s+s0 −1−p− 

0

∑

 
 



I j+ , j− ;n+n0 −2 + I j+ ,p− − j− ;n+n0 −1 + I p+ − j+ , j− ;n+n0 −1 + 2 I j+ , j− ;n+n0

j+ =b+ (r+r0 ) j− =b− (s+s0 )


 
 
 

+ I p+ − j+ ,p− − j− ;n+n0 + I j+ ,p− − j− ;n+n0 +1 + I p+ − j+ , j− ;n+n0 +1 + I j+ , j− ;n+n0 +2 , (4.24a)
where


t − 1a± (t) = p±p − 1±

if t − 1 − p± 6 0,
if t − 1 − p± > 0 is even,b± (t) =if t − 1 − p± > 0 is odd,

1

if t − 1 − p± is odd,

2

if t − 1 − p± is even.

(4.24b)Finally, we remark that the sums involving b± (t) in this result should be understood to vanish whenever t − 1 −p± 6 0. This formula follows directly from the multiplication formulae (4.20) and the easily derived relations
U p+ +k (X) = U p+ −2−k (X) +Uk (X)(Z + Z −1 )mod ker φ(0 6 k 6 p+ − 2),

U p− +k (Y ) = U p− −2−k (Y ) +Uk (Y )(Z + Z −1 )mod ker φ(0 6 k 6 p− − 2).

(4.25)

5. T HE V ERLINDE R ING FOR T RIPLET M ODELS
Having determined explicit formulae for the Verlinde products of simple singlet modules, we now consider
analogous formulae for the triplet modules. We note that there is almost nothing in the literature devoted tofusion rules for singlet models, but that there are many sources where triplet fusion rules have been conjectured
or computed [5, 7, 8, 12, 40–42]. Comparing these results with the triplet Verlinde products that we will deduce
therefore gives very strong consistency checks of both our results and those in the literature.

5.1. Simple Currents for the Singlet Verlinde Ring. As previewed in Section 2.3, the I1,1;n are simple currentsin the Verlinde ring V I p+ , p− , that is, they are units of the Verlinde ring that act as permutations on the set of
all simple modules. Indeed, the Verlinde products (4.14) and (4.16a) give   


 
 

I1,1;n × Fµ = Fµ+nα/2 ,I1,1;n × Ir,s;n0 = Ir,s;n+n0 .
(5.1)






In particular, I1,1;−n is the inverse, with respect to the Verlinde product, of I1,1;n (because I1,1;0 is the identity). We note that when p+ = 1 and p− > 1 (p− = 1 and p+ > 1), these simple currents are identified with the −   + I1,1;n ( I1,1;n ). For p+ = p− = 1, the identification is rather with the F1,1;n .
When p+ = p− = 1, so the singlet algebra coincides with the Heisenberg algebra, the F1,1;n are well known tob (2) . We conjecture
be simple currents in the fusion ring. Indeed, extending I 1, 1 by F1,1;2 leads to W 1, 1 ∼= sl
1that this generalises so that the I1,1;n define simple currents, in a sense that we will shortly describe, with respectto the fusion product of the singlet algebra I p+ , p− . As remarked in Section 2.3, this conjecture is alreadysuggested by the decomposition of the simple triplet algebra modules into singlet modules, at least for n even.
Let us consider the case where p+ = 1 and p− > 1 (the case p− = 1 and p+ > 1 is analogous). As we have
noted above, there is then no Felder complex (2.4b), hence the list of simple I 1, p− -modules given in Section 2.2
truncates to the typical Fλ and the atypicals I1,s;n = I−
1,s;n with 1 6 s 6 p− − 1 and n ∈ Z (the analogous modules
with s = p− are typical: I−
1,p− ;n = I1,p− ;n = F1,p− ;n ). In particular, there are no problematic modules Lr,s toworry about and we have a bijective correspondence between the simple I 1, p− -modules and their (linearly

16

D RIDOUT AND S WOOD

independent) representatives in the Verlinde ring. We therefore claim that the Verlinde products (5.1) lift to fusion
products as follows:
−
−
(5.2)
I−
I−
1,1;n × I1,s;n0 = I1,s;n+n0 .
1,1;n × Fµ = Fµ+nα/2 ,Note that I−
1,1;0 is the vacuum module of I 1, p− . These singlet fusion products are consistent with the triplet

fusion products that have appeared in the literature, though we will only verify this here at the level of the Verlinde
ring. Essentially, we claim that the Verlinde products guarantee that these fusion products are simple, hence that
there are no possible ambiguities concerning their structure. This is equivalent to fusion being exact on I 1, p− modules and the Verlinde ring, as defined above, coinciding with the Grothendieck ring of fusion.
If we accept these arguments supporting the I−
1,1;n being simple currents, then it is a simple matter to determine
the (simple) spectrum of the simple current extension. We will do this for the group of simple currents corre
sponding to n = 2 in order to compare with the known spectrum of the triplet algebra W 1, p− . As the simple
currents act freely (under the fusion product) on the simple singlet modules, the simple extended algebra modules
are realised by summing over the orbits of the simple current group:F+[λ ] =Mk∈Z

Fλ +kα ,F−[λ ] =

M

Fλ +(k+1/2)α ;

W+
1,s =

k∈Z

M

I−
1,s;2k ,

W−
1,s =

M

k∈Z

I−
1,s;2k+1 .

(5.3)

k∈Z

Note that F+= F−, so we may restrict λ to the real interval 0 6 λ < α/2. We remark that because the simple[λ ][λ +α/2]
currents I−
1,1;n , with n even, have integer conformal weights, it is natural to restrict to the untwisted extended algebra
modules upon which the simple current fields act with integer moding. Referring to the conformal weights listed±
in Appendix A, we quickly find that all of the W±
1,s are untwisted, but that the simple F[λ ] are only untwisted when

λ = α1,p− . Defining±W±
1,p− = F[α1,p− ](5.4)

and comparing with the decompositions of the simple triplet modules given in Section 2.3, we conclude that thesimple untwisted modules of the simple current extension of I 1, p− by the group generated by I−
1,1;2 may beidentified with the simple W 1, p− -modules. We view this as extremely strong evidence for the claim that the
triplet algebra is just the simple current extension of the singlet algebra by I−
1,1;2 .
The general case, where p+ , p− > 1, is somewhat more delicate because of the simple I p+ , p− -modules Lr,swhich are set to zero in the Verlinde ring. In particular, we must allow for the possibility that a given fusion productmay have composition factors of the form Lr,s which are not visible in the Verlinde product. This spoils any chance
of a bijective correspondence between simple I p+ , p− -modules and their representatives in V I p+ , p− . The
connection between the Verlinde product and the fusion product is therefore correspondingly weaker. The best
we could hope for then is that fusion turns out to define a product on the quotient of the Grothendieck group ofI p+ , p− -modules by the Lr,s and that the Verlinde ring coincides with this quotient. We shall assume this in what
follows.
Despite the expected lack of exactness and ambiguities concerning the Lr,s , we conjecture that the fusion products corresponding to the Verlinde products (5.1) are
I1,1;n × Fµ = Fµ+nα/2(µ 6= αr,s;n ),I1,1;n × Lr,s = 0, K• ∗ if r 6= p , s 6= p and n + n0 = 0,
+
−
r,s;0
•
I1,1;n × Ir,s;n0 = I•otherwise,
0

(5.5)

r,s;n+nwhere “•” stands for +, − or is empty, and “∗” denotes the contragredient dual. The appearance of the contragredient modules when n + n0 = 0 is suggested by the W 2, 3 fusion rules computed in [11] and is consistent with

the W p+ , p− fusion rules proposed in [12, 42, 43]. An interesting consequence of this is that the simple currents∗I1,1;n do not generate a group under fusion because K•1,1;0 is not the vacuum module, but its contragredient.
Instead, they generate semigroups (this possibility seems to have been first noticed in [44]).
We therefore conjecture that the triplet algebra W p+ , p− , with p+ , p− > 1, is the simple current extension of
the singlet algebra I p+ , p− by the semigroups generated by I1,1;±2 . This is easily checked to be consistent with

MODULAR TRANSFORMATIONS AND VERLINDE FORMULAE FOR LOGARITHMIC (p+ , p− )-MODELS

17

the decompositions of the simple triplet modules given in Section 2.3 and, again, we identify the triplet modules
with the untwisted modules of the simple current extension. Evidently, this subtlety of semigroups is irrelevant at
the level of the Verlinde ring, so we will assume the standard simple current machinery in what follows for all p+
and p− .
5.2. Verlinde Products. In this section, we will calculate the Verlinde products of the simple triplet modules

under the assumption that the triplet algebra W p+ , p− is the simple current extension of the singlet algebraI p+ , p− . The results are then compared with the physics literature, in particular with the products proposedin [8, 12, 42, 43].

If this assumption is valid, then the Verlinde products of the triplet Verlinde ring V W p+ , p− can be computed in terms of singlet Verlinde products by regarding each triplet module as a direct sum over an orbit of singlet
modules under the action of the simple current (semi)group, choosing arbitrary representatives of each orbit, computing the Verlinde product of the representatives and, finally, determining the orbit of the resulting product. It is
not hard to see that this general procedure reduces to the following simple rules:
 +  ε    −  ε   Wr,s  Wr0 ,s0 = Ir,s;0 × Wεr0 ,s0 ,Wr,s  Wr0 ,s0 = Ir,s;1 × Wεr0 ,s0 .
(5.6)


Here, we distinguish the Verlinde product () of the triplet Verlinde ring V W p+ , p− from that (×) of its
singlet counterpart V I p+ , p− . It is clear now that the triplet Verlinde products can be directly read off from
those of the singlet Verlinde ring V I p+ , p− . For example, −   + 
 
 



W1,1  Wr,s = I1,1;1 × W+
r,s = I1,1;1 × ∑ Ir,s;2nn∈Z= ∑ Ir,s;2n+1 =






W−
r,s



.

(5.7)

n∈Z



This shows, of course, that W−
1,1 is a simple current in the triplet Verlinde ring — this is the residual simple
current symmetry after extending the singlet Verlinde ring by I1,1;±2 .8
The general product formulae for the triplet Verlinde ring V W p+ , p− now follow directly from the decompositions of Section 2.3 and the Verlinde products of the singlet Verlinde ring given in (4.24a). The orbit of the singlet module I1,1;0 gives rise to the identity of V W p+ , p− : W+
1,1 . As the orbits of the singlet generators++
I1,1;±1 , I2,1;0 and I1,2;0 define the triplet modules W−
1,1 , W2,1 and W1,2 , respectively, Equation (4.16) implies their

Verlinde products:  ε   −ε (5.8a)
W−
1,1  Wr,s = Wr,s ,


ε


if r = 1,
 W2,s +   ε  
W2,1  Wr,s =(5.8b)
Wεr−1,s + Wεr+1,s
if 1 < r < p+ ,







−ε
2 Wε
if r = p+ ,p+ −1,s + 2 W1,s
 Wεr,2
if s = 1, +   ε 
 ε   ε 
W1,2  Wr,s =(5.8c)W+ W
if 1 < s < p− ,  r,s−1  r,s+1




−ε
2 Wε
if s = p− .
r,p− −1 + 2 Wr,1
 −   + 
 + 


The W1,1 , W2,1 and W1,2 therefore generate V W p+ , p− . The general formula for the Verlinde product

in the triplet Verlinde ring is similarly obtained from (4.24a):


Wεr,s







0
Wεr0 ,s0=a+ (r+r0 )0∑0

a− (s+s0 )
0

∑0



0

Wεε
j+ , j−

j+ =|r−r |+1 j− =|s−s |+1

8The fact that W− fuses with itself to give the contragredient of the vacuum module, at least for small p and p [11, 12], is our main reason+−
1,1

for proposing the appearance of the contragredient modules in (5.5).

18

D RIDOUT AND S WOOD+

a+ (r+r0 )0∑

s+s0 −1−p− 0∑j+ =|r−r0 |+1 j− =b− (s+s0 )

+

2



0
W−εε
j+ , j−

r+r0 −1−p+ s+s0 −1−p− 
0
0

∑

∑

j+ =b+ (r+r0 ) j− =b− (s+s0 )

+



0
Wεε
j+ ,p− − j−+r+r0 −1−p+0∑

a− (s+s0 )0∑

j+ =b+ (r+r0 ) j− =|s−s0 |+1  εε 00 
2 W−εε+W
p+ − j+ , j−j+ , j−  εε 0 −εε 0
 −εε 0
0, (5.9)4 Wεεj+ , j− + 2 W j+ ,p− − j− + 2 W p+ − j+ , j− + W p+ − j+ ,p− − j−where a± (t) and b± (t) were defined in (4.24b). These formulae reproduce the Grothendieck fusion rules conjectured in [8] from a Kazhdan-Lusztig-like correspondence with a certain quantum group, and are consistent withthe fusion rules proposed in [42,43] from lattice considerations, and those computed for certain small values of p+
and p− in [12] using the Nahm-Gaberdiel-Kausch algorithm.
6. D ISCUSSION AND C ONCLUSIONS
We have seen above that the Verlinde ring of the singlet algebra I p+ , p− may be straightforwardly derived
from the modular transformation properties of the simple singlet modules and a continuous version of the Verlindeformula. Moreover, the Verlinde ring of the triplet algebra W p+ , p− then follows from some basic simple current
technology and the result compares favourably with what is known of the triplet fusion ring. Indeed, it appears that
this rather effortless approach captures pretty much all the information about the fusion ring that can be divinedfrom the simple characters alone. The most difficult step was, in a sense, understanding the representation theoryof the singlet algebra in the first place.
On a heuristic level, we can understand the good behaviour of the modular properties of the singlet characters,
as compared with those of the triplet characters, as stemming from the uncountable nature of the spectrum ofsimple singlet modules. For the singlet, the parametrisation of standard modules defines a countable set of points— points of atypicality let us say — at which the standard singlet modules become reducible. For the triplet,with its finite spectrum of simple modules, one finds instead that the majority correspond to atypical points (the
exception is W 1, 1 of course). Now, observe that the S-matrix elements derived for the atypical simple singlet
modules have poles at atypical parameter values. Consequently, we see that these poles need not cause problems
when integrating over a continuous spectrum as we do for the singlet (the poles form a set of measure zero afterall) but that they will definitely cause problems if one tries to perform a discrete sum as one would like to do forthe triplet.
It was first discovered in [11] that the fusion product of the triplet algebra W p+ , p− , with p+ and p− greaterthan 1, does not necessarily map exact sequences to exact sequences. However, this failure of exactness was always
observed to involve the simple modules Lr,s . As one can see from the fusion products proposed in [12, 43], the
Lr,s form an ideal with respect to the fusion product in the category of W p+ , p− -modules. It was therefore
conjectured in [24] that if one takes the quotient of the category of W p+ , p− -modules by the ideal generated by
the Lr,s , then the fusion product on this new quotient category, there referred to as the Whittaker category, will be
exact. It seems natural to expect that this quotient category is equivalent to the category of quantum group modules
that has been studied under the name “Kazhdan-Lusztig correspondence” [45].

We can of course form the same kind of quotient category for the singlet algebra I p+ , p− . This quotient
appears to be exactly what the singlet Verlinde ring V I p+ , p− sees, since the formalism we introduced above
naturally sets all the Lr,s to zero. We therefore conjecture that the I1,1;n form simple currents of this quotientcategory. Settling these conjectures would appear, to us, to be the natural next step to tackle in understanding
the (p+ , p− )-models. In particular, we would like to see results concerning rigidity and projectivity, which havereceived some attention for the triplet algebras, being generalised to the singlet algebras. It would be interesting
to know if there is some variant of a Kazhdan-Lusztig correspondence that applies to singlet models, given thatthese theories are arguably more fundamental in the sense that the singlet theories are more closely related to otherlogarithmic theories [46–49] than the triplet theories.
Finally, let us remark upon some small overlap of our results with those of the recent paper [32]. There, thefocus is on the (1, p) singlet algebras and the relation between atypical singlet modules and the modular properties

MODULAR TRANSFORMATIONS AND VERLINDE FORMULAE FOR LOGARITHMIC (p+ , p− )-MODELS

19(or lack thereof) of certain variants of Jacobi theta functions that are known in number-theoretic circles as false andpartial theta functions. The idea is to regularise these functions and analyse modular aspects of the regularisation.
In this respect, their regularisation parameter ε plays the same role, roughly speaking, as our formal variable z.
They finish by computing a regularised Verlinde formula for (1, p) singlet models that agrees with our results. It
would be interesting to study their regularisation procedure for general (p+ , p− ) singlet algebras to see whether
any of the “bad” features of these models, such as non-exactness of fusion, can at all be ameliorated.
ACKNOWLEDGEMENTS
DR’s research is supported by an Australian Research Council Discovery Project DP1093910. SW’s work
is supported by the World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan; theGrant-in-Aid for JSPS Fellows number 2301793; and the JSPS fellowship for foreign researchers P11793.
A PPENDIX A. S TRUCTURAL DATAIn this appendix, we quote the structural results concerning Feigin-Fuchs modules that are required in the text.
This material may be found in [34, 35]. We also need the corresponding results for the simple modules of thesinglet algebras. Our presentation follows [24] rather closely.
Recall that a Heisenberg weight λ ∈ R and its corresponding conformal weight ∆λ are related by Equation (2.3b):11
α0 2 α02
∆λ = λ (λ − α0 ) =
−
λ−
.
2
2
2
8

(A.1)

For λ = αr,s;n , we suppress the α in ∆αr,s;n and write∆αr,s;n ≡ ∆r,s;n =(p− r − p+ s − 2np+ p− )2 − (p+ − p− )2.
4p+ p−(A.2)

We will denote the simple Virasoro module generated by a highest weight state of central charge c = 1 − 3α02 andconformal weight ∆ by L(∆).
The socle of a Virasoro module M is its maximal semisimple submodule. The socle series of M is the ascending
series0 = M0 ⊂ M1 ⊂ M2 ⊂ · · ·(A.3)

of submodules of M for which Si (M) = Mi /Mi−1 (for i > 1) is the socle of Mi+1 /Mi−1 . Socle series are unique
if they exist. Recalling the notation Fr,s;n ≡ Fαr,s;n , there are five different possibilities for the socle series factors
Si (Fλ ) of the Feigin-Fuchs modules Fλ :(1) For 1 6 r < p+ , 1 6 s < p− and n ∈ Z, we have
S3 (Fr,s;n ) =ML(∆ p+ −r,s;|n|+2k+1 ),k>0

S2 (Fr,s;n ) =ML(∆r,s;|n|+2k ) ⊕k>a

S1 (Fr,s;n ) =ML(∆ p+ −r,p− −s;|n|+2k ),

k>1−a

M

(A.4)

L(∆r,p− −s;|n|+2k+1 ),k>0where a = 0 if n > 0 and a = 1 if n < 0.
(2) For 1 6 s < p− and n ∈ Z, we have
S2 (F p+ ,s;n ) =ML(∆ p+ ,s;|n|+2k ),k>a

S1 (F p+ ,s;n ) =M
k>0

where a = 0 if n > 1 and a = 1 if n < 1.

L(∆ p+ ,p− −s;|n|+2k+1 ),

(A.5)

20

D RIDOUT AND S WOOD

(3) For 1 6 r < p+ and n ∈ Z, we have
S2 (Fr,p− ;n ) =ML(∆ p+ −r,p− ;|n|+2k−1 ),k>aM

S1 (Fr,p− ;n ) =L(∆r,p− ;|n|+2k ),

(A.6)

k>0

where a = 1 if n > 1 and a = 0 if n < 1.
(4) For n ∈ Z, the Feigin-Fuchs module F p+ ,p− ;n is semisimple as a Virasoro module:
S1 (F p+ ,p− ;n ) =ML(∆ p+ ,p− ;|n|+2k ).

(A.7)

k>0

(5) Finally, for λ ∈ R, with λ 6= αr,s;n for any 1 6 r 6 p+ , 1 6 s 6 p− and n ∈ Z, the Feigin-Fuchs module Fλ issimple as a Virasoro module:
S1 (Fλ ) = L(∆λ ).
(A.8)

We will also need socle factors for the simple I p+ , p− -modules decomposed as Virasoro modules. The Fλ ,with λ 6= αr,s;n for any 1 6 r 6 p+ , 1 6 s 6 p− and n ∈ Z, are simple singlet modules, as are the F p+ ,p− ;n . Their
socle factors were given above. As the result for the simple singlet modules Lr,s is obvious, it only remains to list
the factors for the I•r,s;n :(1) For 1 6 r < p+ , 1 6 s < p− and n ∈ Z,
Ir,s;n = S1 (Fr,s;n ) =ML(∆r,p− −s;|n|+2k+1 ).

(A.9)

k>0

The minimal conformal weight of the states of Ir,s;n is therefore ∆r,p− −s;|n|+1 .
(2) For 1 6 s < p− and n ∈ Z,
I−
p+ ,s;n = S1 (F p+ ,s;n ) =ML(∆ p+ ,p− −s;|n|+2k+1 ).

(A.10)

k>0

The minimal conformal weight of the states of I−
p+ ,s;n is therefore ∆ p+ ,p− −s;|n|+1 .
(3) For 1 6 r < p+ and n ∈ Z,
I+r,p− ;n = S1 (Fr,p− ;n ) =ML(∆r,p− ;|n|+2k ).

(A.11)

k>0

The minimal conformal weight of the states of I+r,p− ;n is therefore ∆r,p− ;|n| .
Of course, the minimal conformal weight of the states of F p+ ,p− ;n is ∆ p+ ,p− ;|n| and that for Fλ is ∆λ .
R EFERENCES[1] H Kausch. Extended Conformal Algebras Generated by a Multiplet of Primary Fields. Phys. Lett., B259:448–455, 1991.
[2] V Gurarie. Logarithmic Operators in Conformal Field Theory. Nucl. Phys., B410:535–549, 1993. arXiv:hep-th/9303160.
[3] H Kausch. Curiosities at c = −2. arXiv:hep-th/9510149.
[4] M Flohr. On Modular Invariant Partition Functions of Conformal Field Theories with Logarithmic Operators. Int. J. Mod. Phys.,
A11:4147–4172, 1996. arXiv:hep-th/9509166.
[5] M Gaberdiel and H Kausch. A Rational Logarithmic Conformal Field Theory. Phys. Lett., B386:131–137, 1996. arXiv:hep-th/9606050.
[6] M Gaberdiel and H Kausch. A Local Logarithmic Conformal Field Theory. Nucl. Phys., B538:631–658, 1999. arXiv:hep-th/9807091.
[7] J Fuchs, S Hwang, A Semikhatov, and I Yu Tipunin. Nonsemisimple Fusion Algebras and the Verlinde Formula. Comm. Math. Phys.,
247:713–742, 2004. arXiv:hep-th/0306274.
[8] B Feigin, A Gainutdinov, A Semikhatov, and I Yu Tipunin. Logarithmic Extensions of Minimal Models: Characters and Modular Transformations. Nucl. Phys., B757:303–343, 2006. arXiv:hep-th/0606196.
πi[9] K Nagatomo and A Tsuchiya. The Triplet Vertex Operator Algebra W (p) and the Restricted Quantum Group U q (sl2 ) at q = e p . Adv.
Stud. Pure Math., 61:1–49, 2011. arXiv:0902.4607 [math.QA].
[10] J Rasmussen. W-Extended Logarithmic Minimal Models. Nucl. Phys., B807:495–533, 2009. arXiv:0805.2991 [hep-th].
[11] M Gaberdiel, I Runkel, and S Wood. Fusion Rules and Boundary Conditions in the c = 0 Triplet Model. J. Phys., A42:325403, 2009.
arXiv:0905.0916 [hep-th].

MODULAR TRANSFORMATIONS AND VERLINDE FORMULAE FOR LOGARITHMIC (p+ , p− )-MODELS

21[12] S Wood. Fusion Rules of the W (p, q) Triplet Models. J. Phys., A43:045212, 2010. arXiv:0907.4421 [hep-th].
[13] D Adamović and A Milas. An Explicit Realization of Logarithmic Modules for the Vertex Operator Algebra Wp,p0 . J. Math. Phys.,
53:073511, 2012. arXiv:1202.6667 [math.QA].
[14] M Gaberdiel. Fusion in Conformal Field Theory as the Tensor Product of the Symmetry Algebra. Int. J. Mod. Phys., A9:4619–4636,
1994. arXiv:hep-th/9307183.
[15] W Nahm. Quasirational Fusion Products. Int. J. Mod. Phys., B8:3693–3702, 1994. arXiv:hep-th/9402039.
[16] M Gaberdiel and H Kausch. Indecomposable Fusion Products. Nucl. Phys., B477:293–318, 1996. arXiv:hep-th/9604026.
[17] E Verlinde. Fusion Rules and Modular Transformations in 2D Conformal Field Theory. Nucl. Phys., B300:360–376, 1988.
[18] G Moore and N Seiberg. Polynomial Equations for Rational Conformal Field Theories. Phys. Lett., B212:451–460, 1988.
[19] M Miyamoto. Modular Invariance of Vertex Operator Algebras Satisfying C2 -Cofiniteness. Duke Math. J., 122:51–91, 2004.
arXiv:math/0209101 [math.QA].
[20] M Flohr and M Gaberdiel. Logarithmic Torus Amplitudes. J. Phys., A39:1955–1968, 2006. arXiv:hep-th/0509075.
[21] W Eholzer, A Honecker, and R Hübel. How Complete is the Classification of W Symmetries? Phys. Lett., B308:42–50, 1993.
arXiv:hep-th/9302124.
[22] D Adamović and A Milas. On the Triplet Vertex Algebra W (p). Adv. Math., 217:2664–2699, 2008. arXiv:0707.1857 [math.QA].
[23] D Adamović and A Milas. On W -Algebras Associated to (2, p) Minimal Models and Their Representations. Int. Math. Res. Not.,
2010:3896–3934, 2010. arXiv:0908.4053 [math.QA].
[24] A Tsuchiya and S Wood. On the Extended W -Algebra of Type sl2 at Positive Rational Level. arXiv:1302.6435 [hep-th].
[25] T Creutzig and D Ridout. Logarithmic Conformal Field Theory: Beyond an Introduction. arXiv:1303.0847 [hep-th].
[26] L Rozansky and H Saleur. Quantum Field Theory for the Multivariable Alexander-Conway Polynomial. Nucl. Phys., B376:461–509,
1992.
[27] T Quella and V Schomerus. Free Fermion Resolution of Supergroup WZNW Models. JHEP, 0709:085, 2007. arXiv:0706.0744 [hep-th].
[28] T Creutzig and D Ridout. Relating the Archetypes of Logarithmic Conformal Field Theory. Nucl. Phys., B872:348–391, 2013.
arXiv:1107.2135 [hep-th].
[29] T Creutzig and D Ridout. Modular Data and Verlinde Formulae for Fractional Level WZW Models I. Nucl. Phys., B865:83–114, 2012.
arXiv:1205.6513 [hep-th].
b (2)[30] D Ridout. sl−1/2 : A Case Study. Nucl. Phys., B814:485–521, 2009. arXiv:0810.3532 [hep-th].
[31] T Creutzig and D Ridout. Modular Data and Verlinde Formulae for Fractional Level WZW Models II. Nucl. Phys., B875:423–458, 2013.
arXiv:1306.4388 [hep-th].
[32] T Creutzig and A Milas. False Theta Functions and the Verlinde Formula. arXiv:1309.6037 [math.QA].
[33] A Babichenko and D Ridout. Takiff Superalgebras and Conformal Field Theory. J. Phys., A46:125204, 2013. arXiv:1210.7094[math-ph].
[34] B Feigin and D Fuchs. Representations of the Virasoro Algebra. Adv. Stud. Contemp. Math., 7:465–554, 1990.
[35] K Iohara and Y Koga. Representation Theory of the Virasoro Algebra. Springer Monographs in Mathematics. Springer-Verlag, London,
2011.
[36] G Felder. BRST Approach to Minimal Models. Nucl. Phys., B317:215–236, 1989.
[37] A Tsuchiya and Y Kanie. Fock Space Representations of the Virasoro Algebra – Intertwining Operators. Publ. RIMS, Kyoto Univ.,
22:259–327, 1986.
[38] D Adamović. Classification of Irreducible Modules of Certain Subalgebras of Free Boson Vertex Algebra. J. Alg., 270:115–132, 2003.
arXiv:math/0207155 [math.QA].
[39] D Adamović and A Milas. Logarithmic intertwining operators and W(2,2p-1)-algebras. J. Math. Phys., 48:073503, 2007.
arXiv:math/0702081 [math.QA].
[40] A Tsuchiya and S Wood. The Tensor Structure on the Representation Category of the W p Triplet Algebra. arXiv:1201.0419 [hep-th].
[41] P Pearce, J Rasmussen, and P Ruelle. Integrable Boundary Conditions and W -Extended Fusion in the Logarithmic Minimal Models
LM (1, p). J. Phys., A41:295201, 2008. arXiv:0803.0785 [hep-th].
[42] J Rasmussen. Polynomial Fusion Rings of W-Extended Logarithmic Minimal Models. J. Math. Phys., 50:043512, 2009. arXiv:0812.1070[hep-th].
[43] J Rasmussen. Fusion of Irreducible Modules in W LM (p, p0 ). J. Phys., A43:045210, 2010. arXiv:0906.5414 [hep-th].
[44] C Dong, H Li, and G Mason. Simple Currents and Extensions of Vertex Operator Algebras. Comm. Math. Phys., 180:671–707, 1996.
arXiv:q-alg/9504008.
[45] B Feigin, A Gainutdinov, A Semikhatov, and I Yu Tipunin. Kazhdan-Lusztig Correspondence for the Representation Category of the
Triplet W -Algebra in Logarithmic CFT. Theo. Math. Phys., 148:1210–1235, 2006. arXiv:math/0512621.
(1)[46] D Adamović. A Construction of Admissible A1 -Modules of level − 34 . J. Pure Appl. Alg., 196:119–134, 2005. arXiv:math/0401023[math.QA].
b (2)[47] D Ridout. sl−1/2 and the Triplet Model. Nucl. Phys., B835:314–342, 2010. arXiv:1001.3960 [hep-th].

22

D RIDOUT AND S WOODb (1|1). In V Dobrev, editor, Proceedings of the 9-th International Workshop ”Lie[48] T Creutzig and D Ridout. W-Algebras Extending gl
Theory and Its Applications in Physics”, volume 36 of Springer Proceedings in Mathematics and Statistics, pages 349–368, Varna, 2011.
arXiv:1111.5049 [hep-th].
[49] T Creutzig, D Ridout, and S Wood. Coset Constructions of Logarithmic (1, p)-Models. arXiv:1305.2665 [math.QA].
(David Ridout) D EPARTMENT OF T HEORETICAL P HYSICS , R ESEARCH S CHOOL OF P HYSICS AND E NGINEERING ; AND M ATHEMATI CAL S CIENCES I NSTITUTE ; AUSTRALIAN NATIONAL U NIVERSITY, ACTON , ACT 2600, AUSTRALIA

E-mail address: david.ridout@anu.edu.au(Simon Wood) K AVLI I NSTITUTE FOR THE P HYSICS AND M ATHEMATICS OF THE U NIVERSE , T HE U NIVERSITY OF T OKYO , 1-5,
K ASHIWANOHA 5-C HOME K ASHIWA - SHI , C HIBA 277-8583, JAPANE-mail address: simon.wood@ipmu.jp