Modular properties of characters of the W3 algebra

In a previous work, exact formulae and differential equations were found for traces of powers of the zero mode in the W3 algebra. In this paper we investigate their modular properties, in particular we find the exact result for the modular transformations of traces of $W_0^n$ for n = 1, 2, 3, solving exactly the problem studied approximately by Gaberdiel, Hartman and Jin. We also find modular differential equations satisfied by traces with a single $W_0$ inserted, and relate them to differential equations studied by Mathur et al. We find that, remarkably, these all seem to be related to weight 0 modular forms with expansions with non-negative integer coefficients.


Introduction
In our recent paper [1], we found formulae for characters of the W 3 algebra with insertions of powers of the zero mode W 0 (see appendix B for the definition of the W 3 algebra). These characters with insertions have been of interest recently since they are involved in counting the states of black holes in (2+1)-dimensional AdS higher-spin gravity. There are two regimes which are related by modular transformation, and it is of particular interest to understand the properties of the characters under modular transformation. In this paper we study the properties of the characters we found in [1] under modular transformations.
We studied two classes of W 3 -algebra representations in [1], Verma modules and minimal model representations. We found that traces over the minimal model representations with a single W 0 insertion satisfied differential equations and in Section 3 here we show that these are modular, that is, they are covariant under modular transformations and this implies that the solutions (the traces) have particular modular properties.
In general, there are many fewer non-zero solutions than there are minimal model representations, and so the non-zero traces transform in a smaller representation of the modular group SL (2, Z) than the full set of minimal model characters, but we show that they are in fact compatible with a general result of Gaberdiel et al [2] that the traces of zero-modes of primary fields transform as modular forms of particular weights with a standard matrix representation.
In Section 4, we use this result of [2], combined with the form of a particular primary field, to find the exact formula for the transform of the trace with the insertion of (W 0 ) 2 for which an approximate result was found in [2]. We verify our result by performing the exact calculation from [2], and further use their method to find the transformation of a trace with (W 0 ) 3 inserted.
Finally, in Section 5 we consider the transform of the trace of a (W 0 ) 2 insertion in the limit where the central charge c becomes large. As we found an expression for this trace in [1], we can calculate this transform directly and, interestingly, we get a different result to that reported in [2]. We find the assumption that leads to this discrepancy and confirm that our direct calculation agrees with the results from Section 4, in two cases where c can be taken to infinity.

Modular transformations, characters and differential equations
We are concerned in this paper with the modular properties of traces over W-algebra highest weight representations. It will be helpful first to review some general facts.
A conformal field theory has a chiral algebra spanned by the modes of holomorphic fields. This algebra has representations, L i , and characters (sometimes called "specialised characters") χ i (q) = Tr L i (q L 0 −c/24 ) . (2.1) The factor q −c/24 is necessary for the characters to transform nicely under the modular transformation q →q, q = exp(2πiτ ) ,q = exp(−2πi/τ ) . (2.2) If the conformal field theory is "minimal", then the set of representations is finite-dimensional and there is a "standard" modular S-matrix S ij , running over all the representations of the chiral algebra, and the characters satisfy If a representation is not self-conjugate, then both the representation and its conjugate will have the same character, so the space of characters is of smaller dimension than the space of representations.
Perhaps the simplest example is the Wess-Zumino-Witten model based on a 2 at level 1, with symmetry algebra the affine algebra (a 2 ) 1 . The highest weight representations of (a (2.4) (2.5) The standard modular S-matrix is [7] 6) but the characters themselves transform in the two-dimensional representation (2.7) The fact that the characters span a two-dimensional space which is invariant under the modular transformation is reflected in the fact that they satisfy a particular differential equation, 8) where E 2n (q) are Eisenstein series, whose definitions and properties are collected in appendix A. These series have well-defined modular transformation properties which ensure that the differential equation (2.8) is invariant under q →q, that is, if χ(q) satisfies eqn (2.8), then the same function χ(q) satisfies the differential equation with q replaced byq, that is The particular combinations q(d/dq) − (r/12)E 2 (q) act as covariant derivatives, mapping modular forms of weight r to forms of weight (r + 2), and we shall denote them by D (r) -see appendix A for details. Differential equations of the form (2. 10) have been studied intensively [3][4][5][6] as the typical defining equations for characters of conformal field theories with only two independent characters, and the solutions can be found as hypergeometric functions and (as a consequence) the modular transformation matrix found explicitly.
3 Modular differential equations for T r(W 0 ) in minimal models In [1], we found that the traces Tr L W 0 q L 0 , where L is one of the irreducible representations of the W 3 algebra at c = 4/5, satisfied a second-order differential equation in q. We can rewrite the equation in [1] as a different equation for Tr L W 0 q L 0 − c 24 and we find that it becomes This equation automatically implies that the traces transform as weight 3 modular forms. As a consequence, the combinations f L (q) = η(q) −6 Tr L W 0 q L 0 − c 24 transform as regular weight 0 modular forms (where η(q) is the Dedekind eta function), and satisfy the differential equation It comes as a bit of a surprise that the particular equation (3.2) is in fact exactly one of the equations of the form (2.10) studied in [3], corresponding to the Wess-Zumino-Witten model for the exceptional algebra f 4 at level 1, with affine Kac-Moody algebra symmetry (f  Tr [11;12] (W 0 q L 0 −c/24 ) = w [11; 12] Tr [11;13] (W 0 q L 0 −c/24 ) = w[11; 12] η 6 Tr [00001] (q L 0 −c/24 ) . (3.4) This means that the modular properties of the W 0 -traces in the 3-state Potts model are already known, as are exact expressions for these W 0 -traces in terms of the Kac-Weyl character formula [7], as well as hypergeometric functions and contour integrals, using directly the results from [3].
This remarkable coincidence is repeated for the other W-algebra minimal model with two independent traces: The W-minimal model WM (3,8) with central charge −23 has 7 representations, of which 3 are uncharged and there are two pairs of charged representations. Since the traces Tr L W 0 q L 0 − c 24 over self-conjugate representations are zero and over conjugate representations differ just by a sign, these traces are spanned by two independent functions. Just as in the 3-state Potts model, there is a null state at level 7 in the vacuum representation which leads to the traces satisfying the differential equation (3.5) or, equivalently, This is the same differential equation satisfied by the characters of the WZW model (a 1 ) 1 . This model has central charge c = 1 and two representations L [0] and L [1] ; again the traces in the W-algebra are proportional to the characters of the affine algebra, with the same proportionality constant w [11; 13]:χ 0 (q) ≡ Tr [11;13] (W 0 q L 0 −c/24 ) = w [11; 13] η 6 Tr [0] (q L 0 −c/24 ) , χ 1 (q) ≡ Tr [11;12] (W 0 q L 0 −c/24 ) = w [11; 13] where we definedχ i (q) for convenience. The modular transformation of the traces is Finally, the simplest of all is the minimal model WM (3, 7) which has 5 representations and only two conjugate representations with non-zero charge. We find their traces Tr with explicit solutions Tr [11;12] where 1 can be interpreted as the trace over the one-dimensional representation of the Virasoro algebra at c = 0. As there are only two charged representations of WM (3,7), and these are conjugate, the S ′ 'matrix' is 1 × 1 and we have Tr [11;12] (W 0q L 0 −c/24 ) = w [11; 12] η(q) 6 = w [11; 12] iτ 3 η(q) 6 = iτ 3 Tr [11;12] (W 0 q L 0 −c/24 ) (3.12) i.e. S ′ = i. 1 We have checked further and so far in every case, the functions η(q) −6 Tr L W 0 q L 0 − c 24 in the W-minimal models satisfy modular differential equations of the appropriate order and can be normalised (with the same normalisation factor for each representation L in a given model) so that they each have expansions with non-negative integer coefficients. This last point seems a very surprising fact, but they do not seem to be easily identifiable with the sets of characters of known conformal field theories, so that the identifications with the characters of (f 4 Exact results for T r(W n 0 ) for n = 1, 2, 3 The results from the previous section are surprising, but the fact that the traces Tr L W 0 q L 0 − c 24 transform as weight 3 modular forms is not, as is explained in [2]. It is shown there that the trace of the zero mode a 0 of a holomorphic primary field a(z) of weight h transforms as a modular form of weight h with the "standard" S-matrix.
Recall from the previous discussion that if the irreducible representations of the symmetry algebra are L i and the characters are χ i (q) = Tr L i (q L 0 −c/24 ), then We can apply equation (4.2) to the field W (z) of weight 3, we get or Note that this is the "standard" S-matrix; in our case, the traces over self-conjugate fields will be zero and over conjugate fields will differ by just a sign, so the actual dimension of the representation will be smaller.
For example, in the case of WM (3, 8) mentioned above, there are 7 representations so that S ij is a 7 × 7 matrix, but there are only two independent tracesχ i (q) as in (3.7), transforming with a 2 × 2 matrix S ′ given in equation (3.8). This is entirely consistent with the "standard" transformation properties as conjugation is an automorphism of the Hilbert space and so one can restrict the modular S-matrix to non-self-conjugate representations and still obtain a representation of the modular group.
We can also check the result (4.3) directly. In appendix D we show this holds for the modular transformation of traces over Verma module representations with h > (c − 2)/24.

Case n = 2
In the case n = 2, there is no primary field M (z) such that Tr and so we cannot apply (4.2) directly to calculate the modular transform of Tr L W 2 0 q L 0 − c 24 . However, we can apply equation (4.2) to the field M (z) that corresponds to the state (4.5) 2 As T is not primary, this expression is not valid for L0. Instead, the insertion of L0 in a trace is equivalent to the action of the differential operator L0(q) = q d dq + c 24 and we can transformq → q directly, which gives This state is a Virasoro highest weight state for the choices 3 a = − 776 + 1978c + 225c 2 2 (−1 + 2c) (22 + 5c) (68 + 7c) , In this case, M (z) is a Virasoro primary field of weight 6 and consequently we can apply (4.2) to obtain Using the results and methods of [1], we obtain and Simplifying this expression and using the modular transformation properties of L 0 and E 2n , we eventually obtain This is the exact result for the modular transform. The leading c approximation was found by Gaberdiel et al in [2] (note that our algebra generators are related to those of [2] by W GHJ = √ 10 · W IW ). We can also calculate the leading order term from (4.10). Under the assumption that the leading contribution to the sum on the right hand side of (4.10) comes from the vacuum character, then the leading term is given from the action of D (2) D on the vacuum character and we find agreement with the calculation of [2], (4.11) 4. 3 The GHJ calculation revisited: n = 2 again, n = 3 As stated above, Tr r W 2 0q L 0 − c 24 was calculated in [2] to leading order in c and under the assumption that the dominant contribution comes from the vacuum. However, their techniques do allow the calculation to be performed without such assumptions, which we do here. We find precise agreement with (4.10). Of course, once the appropriate limits are taken, we recover (4.11).
In [2] the key tool for this calculation was developed: a recursion relation for traces containing zero modes, which for the weight-3 W (z) field of interest here is Here we have used 'bracketed modes' W [m], which are related to operator modes on the torus [10] rather than the complex plane, and (slightly non-standard) Weierstrass functions P m : The expansion coefficients C hjm (not to be confused with the charge conjugation matrix) are defined through (4.14) The modular transformation is then applied by After some lengthy calculations, we find 4.16) and where we have used that, inside a trace, L 0 = D (r) + rE 2 /12 + c/24 for any r. Note that (4.16) is exactly the result we found using our method, given in (4.10).
We have listed some of the intermediate results needed for checking (4.16) and (4.17) that are not given explicitly in [2] in appendix E.
We have performed a variety of checks, all of which support these results. Firstly, we have checked that applying the modular transformations twice gives back the original trace (that is, the modular transformation squares to the identity). This is a non-trivial check as cancellations are needed between different terms.
Further, for minimal models with small numbers of representations it is possible to calculate the traces numerically (using the formulae given in [1] as sums over the Weyl group of su (3)). We have checked that the formulae agree numerically, using the W 3 S-matrices which can be found in [9]. Next, we assumed that the equations take the form given in (4.16) and (4.17) but with an unspecified S-matrix and fitted the values of the S-matrix using various different values of τ , recovering the correct S-matrix. We have also used the same method to fit the coefficient of the E 4 terms, as these are not constrained by the requirement that the modular transformation squares to the identity, and again recovered the correct results with excellent numerical agreement. We are therefore satisfied that these formulae are correct.

The large c limit
The result (4.10) on the modular transformation of Tr L W 2 0 q L 0 − c 24 is exactly correct. The leading c behaviour can be extracted from this formula and agrees with that in [2], assuming that the summation over representations j is discrete and that the leading term in the sum comes from the vacuum representation. However, we can also investigate the modular properties of Tr L W 2 0 q L 0 − c 24 directly and we find a different behaviour.
In the c → ∞ limit, a generic unitary representation will be free of null states and so the Verma modules are irreducible, with characters We also have the exact formula for Tr V (W 2 0 q L 0 − c 24 ) over a Verma module so we can examine its modular transformation properties directly. We reproduce here this result from [1]: This can be written in terms of Eisenstein series and the η function as Clearly, for fixed w and h, the leading c behaviour comes from the term Taking the ratio of this trace with the character (5.1) of the Verma module, we have Applying the modular transformation, in the τ → +i∞ (i.e. E 2n → 1) limit this becomes which differs from the result (4.11) by a factor of 5c/2.
The difference between (4.11) and (5.7) comes from the assumption that the leading contribution to the modular transformation (4.10) comes from the vacuum representation. This is not the case in either the c → +∞ or c → −∞ limits.

The c → ∞ limit
One way to consider the c → ∞ limit in the context of a well-known model is through real coupling Toda theory. In real coupling a 2 Toda theory, the central charge satisfies c > 98. Some properties are given in appendix C, and in particular the spectrum has a minimum value of h, h min = (c−2)/24. Now that h ∼ c, the leading c behaviour of (5.4) comes from (5.8) however this still gives so the behaviour of (5.7) still holds -particularly, there is still a factor of 5c/2 difference from (4.11).
Putting this minimal value of h into (4.10) we see that the dominant contribution now comes from the term β iπ cE 4 1440 ∼ 1 450 πi , (5.10) entirely in agreement with (5.7).

The c → −∞ limit
While it is not possible to reach c → ∞ for unitary minimal models, it is certainly possible to reach c → −∞ for nonunitary minimal models. To take a concrete series WM (3, 3p + 1) has central charge In these models there is also a minimum value of h which is less than 0, which is 12) and so again the leading term from (4.10) comes from the (β/πi)(cE 4 )/1440 term and is in agreement with the analysis from the Verma module, (5.7).
The conclusion is that taking the c → ∞ limit is slightly trickier than one might imagine.

Conclusions
We have studied the modular transformations of traces of various powers of W 0 . We have found that these have nice explicit forms. We have studied these using both the methods of Gaberdiel et al in [2] as well as using a new method based on the explicit forms of these traces found in [1].
Firstly, we have shown that Tr L W 0 q L 0 − c 24 are vector-valued modular forms of weight 3. In the case of minimal models, we have also shown how the trace Tr L W 0 q L 0 − c 24 obeys nice modular differential equations.
Next we found the exact modular transformation law of Tr L W 2 0 q L 0 − c 24 and Tr L W 3 0 q L 0 − c 24 using the methods of [2] and checked our results extensively. The calculations are somewhat cumbersome and we have not, as yet, extended them to higher powers. We also found a new method (based on Virasoro primary fields which exist for all values of c) and applied this to Tr L W 2 0 q L 0 − c 24 and found agreement with the result using the method of [2]. This new method can be generalised to any trace Tr L W n 0 q L 0 − c 24 . It is also somewhat cumbersome but (we think) conceptually simpler.
The results we have found are rather nice and it looks as though it might be possible to find the general term or exponentiate the results. They are more complicated that the analogous results for the trace of J n 0 in representations of an affine algebra, for example the trace of J 2 0 given in [2], but still simpler than might have been the case.
We have also shown that c → ∞ limit is tricky. Applying the modular transformation to the trace over a single W 3 representation, we do not get agreement with the result in [2]. The c → ∞ limit considered in [2] is relevant to the application to state counting in black holes, and their calculation agrees with the gravity calculation. The reason is that there is an assumption in [2] that the vacuum representation dominates the modular transform; the limit is modified if the lowest conformal dimension in the spectrum is not zero. We have shown how in both the c → ∞ limit (in affine Toda field theory) and the c → −∞ limit (for non-unitary minimal models), using the correct representation of lowest conformal weight we can find agreement with the direct analysis of our results. This has implications for the space of states in a gravity dual: it must be more complicated than a single W 3 model such as affine Toda field theory. We will be looking into this in more detail.
In the future, it should be possible to study the modular transformation properties using the numerical evaluation of the traces in the minimal models. We only used this as a check of the results we found, but it should be possible to determine (numerically) the transformations of traces of higher powers, and check whether there is a pattern which could lead to a general formula.

A Eisenstein series, the η function etc
The first few Eisenstein series, E 2 , E 4 and E 6 , are For n > 1, E 2n is a modular form of weight 2n while E 2 is a holomorphic connection [8]. This means that under a modular transformationq = exp(−2πi/τ ) → q = exp(2πiτ ), and so the combination is a modular covariant derivative which maps forms of weight r to forms of weight (r + 2). We also define D ≡ D (0) = q(d/dq).
The Dedekind η function is defined as and is a modular form of weight (1/2), satisfying The W 3 algebra, its representations and minimal models The W 3 algebra is generated by modes W m , L m with commutation relations The minimal models of the W 3 algebra are well-understood. They occur for values of c such that 6) where t = p/p ′ is rational with p and p ′ co-prime integers, both greater than two. The W 3 algebra at this central charge has N representations, where The possible modular invariant partition functions for minimal models have been classified by Beltaos and Gannon in [9]. We are not interested in the particular partition functions, but just the representations that arise, and so we shall simply call these WM (p, p ′ ), for convenience.
The highest weight weight representations are parametrised by (h, w) which are the eigenvalues of L 0 and W 0 on the highest weight state |h, w .
The highest weight representations in the minimal model WM (p, p ′ ) are parametrised by two weights of affine a 2 , µ and µ ′ , at levels p−2 and p ′ −2 respectively, modulo the action of Z 3 . The representations are conventionally labelled [rs; r ′ s ′ ] where µ = pΛ 0 + (r − 1)Λ 1 + (s − 1)Λ 2 is a weight of (a (1) 2 ) p−2 , and µ ′ = p ′ Λ 0 + (r ′ − 1)Λ 1 + (s ′ − 1)Λ 2 is a weight of (a (1) 2 ) p ′ −2 (these are not the same Λ as in (B.5) -see [7] for details of the weights of affine algebras). These W 3 algebra representations have conformal weight and W 0 eigenvalue The vacuum representation of the W-algebra is hence labelled [11; 11]. The Z 3 symmetry is given by [rs; The advantage of this labelling is that the representation [rs; r ′ s ′ ] has independent singular vectors at levels rs and r ′ s ′ . Conjugation of a representation takes (h, w) → (h, −w); only representations with w = 0 (also referred to as uncharged representations) are self-conjugate.

C Real coupling a 2 Toda theory
Quantum conformal Toda field theories can be associated to any finite-dimensional Lie algebra g and are theories constructed from r = rank(g) bosonic scalar fields which depend on a coupling constant (denoted variously β [11] and b [14]) and which have W-algebra symmetries [11]. When the coupling constant is imaginary, the central charge takes values c ≤ r, but for real coupling the central charge can become large and positive. The W 3 algebra is a symmetry of a 2 Toda theory, and the central charge is The states in the a 2 Toda field theory are parametrised by a "momentum", denoted variously ω [12], m [13] and α [14]. If we use the notation of [14], then we can restrict the momenta to the values where (a 1 , a 2 ) are two real numbers. With our normalisations, these are h = a 2 1 + a 2 a 2 + a 2 2 3 + (c−2) 24 , w = 2 3 (a 1 −a 2 )(2a 1 +a 2 )(a 1 +2a 2 ) 9 34 + 15u + 15/u .
In this theory, the spectrum of fields is continuous but the conformal dimensions are bounded below by h min = (c − 2)/24. These representations are all irreducible (they are the delta-normalised states of [13]) with character χ h , and the partition function is given by the partition function for a pair of uncompactified free bosons, This is clearly modular invariant.

D Modular properties for traces over Verma modules.
We can now discuss the modular properties of Toda theory and of a possible reduction.
We start with the modular S-matrix for two uncompactified free bosons. Consider the momentum state |p with p = (p 1 , p 2 ) of conformal weight h(p) = 1 2 p 2 . The character is Using the Fourier transform We can now use this to find the modular transform of the characters of the Virasoro algebra that appear in Toda theory. If we take the momentum to be This means we can rewrite (D.3) as a modular transformation of the specialised W-algebra charcters, The integral in (D.6) is over all (b 1 , b 2 ) and this overcounts the representations of the W algebra. The reason is that the weights (C.3) are invariant under the Weyl group of a 2 , generated by ω 1 : (a 1 , a 2 ) → (−a 1 , a 1 + a 2 ) , ω 2 : (a 1 , a 2 ) → (a 1 + a 2 , −a 2 ) , (D. 8) Under the action of the Weyl group, the space of momenta splits into six Weyl chambers, with the fundamental Weyl chamber being {b i ≥ 0}, so that we can rewrite equation (D.6) as = S (a 1 ,a 2 ),(b 1 ,b 2 ) + S (a 1 ,a 2 ),(−b 1 ,b 1 +b 2 ) + S (a 1 ,a 2 ),(b 1 +b 2 ,−b 2 ) + S (a 1 ,a 2 ),(b 2 ,−b 1 −b 2 ) + S (a 1 ,a 2 ),(−b 1 −b 2 ,b 1 ) + S (a 1 ,a 2 ),(−b 2 ,−b 1 ) , (D.10) We can now change variables from (a 2 , a 2 ) to (h, w), and rewrite (D.9) as (D.12) where the region D is given by 13) and the Jacobian is (D.14) and consequently the S-matrix is S (h,w),(h ′ ,w ′ ) = and so finally obtain (D.19) exactly in agreement with (4.3).
E Some results needed for calculations in section 4.3.
Here we state some useful intermediate results obtained in the calculation of Tr i W n 0q L 0 − c 24 for n = 2 (4.16) and n = 3 (4.17).

E.1 n = 2
After one application of the recursion relation (4.12), we reach the expression (where here and below we use the shorthand notation i..
(E.1) so we need where in the last line we have used L 0 = D (r) + rE 2 /12 + c/24 inside a trace. If we now put these results into (E.1), we obtain ( . (E.6e) Finally, putting these results into (E.4) and again using L 0 = D (r) + rE 2 /12 + c/24 when inside a trace, we recover (4.17).