Virasoro Central Charges for Nichols Algebras

Abstract

A Virasoro central charge can be associated with each Nichols algebra with diagonal braiding in a way that is invariant under the Weyl groupoid action. The central charge takes very suggestive values for some items in Heckenberger’s list of rank-2 Nichols algebras. In particular, this might be viewed as an indication of the existence of reasonable logarithmic extensions of W ₃≡WA ₂, WB ₂, and WG ₂ models of conformal field theory. In the W ₃ case, the construction of an octuplet extended algebra—a counterpart of the triplet (1,p) algebra—is outlined.

Appendices

Appendix A: Virasoro Algebra

In CFT, the Virasoro algebra

$$ [L_m,L_n]=(m-n) L_{m+n}+ \frac{c}{12}(m-1)m(m+1),\quad m,n\in \mathbb {Z}$$

standardly appears in the guise of an energy–momentum tensor $T(z)=\sum_{n\in \mathbb {Z}}L_{n} z^{-n-2}$—a (chiral) field on the complex plane that satisfies the OPEs

$$ T(z) T(w)=\frac{c/2}{(z-w)^4} + \frac{2T(z)}{(z-w)^2} + \frac{\partial T(z)}{z-w}. $$

The c parameter (understood to be multiplied by the unit operator whenever necessary) is called the central charge.

For θ bosonic fields φ(z)=(φ ¹(z),…,φ ^θ(z)) with the OPEs

$$ \varphi^i(z)\varphi^j(w)= \delta^{ij}\log(z-w), $$

(16)

which are also frequently used in calculations in the form

$$ \partial\varphi^i(z) \partial\varphi^j(w)= \frac{\delta^{ij}}{(z-w)^2}, $$

the energy–momentum tensors are parameterized by $\xi\in \mathbb {C}^{\theta}$,

$$ T_{\xi}(z)= \frac{1}{2}\partial\varphi(z).\partial \varphi(z) + \xi.\partial^2\varphi(z). $$

(17)

The corresponding central charge is

$$ c_{\xi}=\theta - 12\xi.\xi . $$

(18)

The OPE of T _ξ (z) with a vertex operator e ^μ.φ(z) is

$$ T_{\xi}(z) e^{\mu.\varphi(w)}= \frac{\Delta e^{\mu.\varphi(w)}}{(z-w)^2}+ \frac{\partial e^{\mu.\varphi(w)}}{z-w},\quad \Delta = \frac{1}{2} \mu.\mu -\xi.\mu $$

A screening operator is, by definition, any expression ∮V(⋅), where V(z) is a field of dimension Δ=1 (and the contour integration is essentially equivalent to taking a residue “after the action of V(z) is evaluated”). For θ=2, any two exponentials e ^α.φ(z) and e ^β.φ(z) with noncollinear $\alpha,\beta\in \mathbb {C}^{2}$ define screening operators with respect to the energy–momentum tensor

$$\begin{aligned} T(z) =& \frac{1}{2}\partial\varphi(z).\partial\varphi(z) \\ &{}- \frac{(2 + \alpha.\beta - \alpha.\alpha) \beta.\beta - 2 \alpha.\beta}{2 \delta^2} \partial^2\varphi_{\alpha}(z) \\ &{}- \frac{(2 + \alpha.\beta - \beta.\beta) \alpha.\alpha - 2 \alpha.\beta}{2 \delta^2} \partial^2\varphi_{\beta}(z), \end{aligned}$$

where ∂φ _α (z)=α.∂φ(z) and ∂φ _β (z)=β.∂φ(z), and δ ²=(α.α)(β.β)−(α.β)². This gives formula (4) for the central charge.

Next, I show that the central charge of the θ-boson energy–momentum tensor that centralizers θ screenings $\oint e^{\alpha_{i}.\varphi(z)}$, 1≤i≤θ, with linearly independent momenta is invariant under Weyl reflections (8) if Eqs. (7) hold.

Given the α _i , 1≤i≤θ, the condition that all the exponentials $e^{\alpha_{i}.\varphi(z)}$ have dimension 1 is expressed by the system of equations for ξ

$$ \frac{1}{2} \alpha_i.\alpha_i - \xi. \alpha_i = 1, \quad 1\leq i\leq\theta. $$

With ξ written as $\xi=\sum_{j=1}^{\theta}x_{j} \alpha_{j}$, this becomes a system for the x _j ,

$$ \frac{1}{2} \alpha_i.\alpha_i - \sum_{j=1}^{\theta}x_j \alpha_j.\alpha_i = 1,\quad 1\leq i\leq\theta, $$

(19)

uniquely solvable if the α _i are linearly independent.

Under a Weyl groupoid operation $\mathfrak{R}^{(k)}$ in (8), the scalar products change and the solution (x _j ) also changes. The “old” and “new” central charges are

$$ c=\theta-12\sum_{\ell,j=1}^{\theta} x_{\ell} x_j\alpha_{\ell}.\alpha_j \quad\text{and}\quad \mathfrak{R}^{(k)}(c)= \theta-12\sum _{\ell,j=1}^{\theta}\tilde{x}_{\ell} \tilde{x}_{j} \mathfrak{R}^{(k)}(\alpha_{\ell}. \alpha_j), $$

where the $\tilde{x}_{j}$ solve the system “$\mathfrak{R}^{(k)}$((19)).” With $\tilde{x}_{j}=x_{j}+y_{j}$, this system becomes

$$\begin{aligned} &\frac{1}{2}\alpha_i.\alpha_i - a_{k,i}\alpha_i.\alpha_k +\frac{1}{2} a_{k,i}^2\alpha_k.\alpha_k \\ &\quad\quad{}- \sum_{j=1}^{\theta}(x_j+y_j) (\alpha_j.\alpha_i - a_{k,i} \alpha_j.\alpha_k- a_{k,j} \alpha_k.\alpha_i + a_{k,j} a_{k,i}\alpha_k. \alpha_k)\\ &\quad=1, \quad 1\leq i\leq\theta \end{aligned}$$

(for a chosen k). The claim is that this system of equations for the “deformation” of the original solution is solved by the ansatz y _j =δ _j,k y. To see this, substitute such y _j and use (19) in the resulting equations, which then become

$$ a_{k,i} \biggl(\frac{1}{2}\alpha_k. \alpha_k( a_{k,i}+1)-\alpha_i. \alpha_k-1 \biggr) + \Biggl(y+\sum_{j=1}^{\theta}x_j a_{k,j} \Biggr) (\alpha_k.\alpha_i- a_{k,i}\alpha_k.\alpha_k) = 0, $$

where 1≤i≤θ. Remarkably, if (7) holds, then the above equations are indeed solved by

$$ y=1 - \frac{2}{\alpha_k.\alpha_k} - \sum_{j=1}^{\theta}x_j a_{k,j}. $$

(20)

It remains to find the new central charge. With $\tilde{x}_{j} = x_{j} + \delta_{j,k}y$,

$$\begin{aligned} \sum_{\ell,j=1}^{\theta}\tilde{x}_{\ell} \tilde{x}_{j} \mathfrak{R}^{(k)}(\alpha_{\ell}. \alpha_j) =& \sum _{\ell,j=1}^{\theta} x_{\ell} x_j(\alpha_{\ell}.\alpha_j - 2 a_{k,\ell} \alpha_j.\alpha_k + a_{k,l} a_{k,j} \alpha_k.\alpha_k) \\ &{}+ 2 \sum_{j=1}^{\theta} y x_j( a_{k,j}\alpha_k.\alpha_k - \alpha_k.\alpha_j) + y^2 \alpha_k.\alpha_k \end{aligned}$$

and yet another use of (19) shows that this is

$$ {}=\sum_{\ell,j=1}^{\theta} x_{\ell} x_j\alpha_{\ell}.\alpha_j + \Biggl(y + \sum _{j=1}^{\theta}x_j a_{k,j} \Biggr) \Biggl(y \alpha_k.\alpha_k + 2 - \alpha_k.\alpha_k + \sum_{\ell=1}^{\theta} x_{\ell} a_{k,\ell}\alpha_k.\alpha_k \Biggr), $$

where the last factor vanishes by virtue of (20). The central charge is invariant.

Appendix B: W ₃ Logarithmic Octuplet Algebras

With the two screenings as in case 2.1 (the “regular” solution there, with central charge (9) and the W(z) field (10)), I propose a W ₃ counterpart of the (1,p) algebra [28–31] by closely following the constructions in [18].

An octuplet of primary fields is generated from the field e ^γ.φ(z) with $\gamma\in \mathbb {C}^{2}$ such that γ.α ^∨=p and γ.β ^∨=p, i.e., from the field

(which is in the kernel of the two screenings F _α =∮e ^α.φ and F _β =∮e ^β.φ). This is a Virasoro primary field of dimension Δ=3p−2, that is,

and, moreover, a W ₃ primary: as is easy to verify, the modes of the dimension-3 field $W(z)=\sum_{n\in \mathbb {Z}}W_{n}z^{-n-3}$ in (10) act on such that

Then the long screenings (12) generate the octuplet

Here, , , and so on, and ; the dashed arrows represent maps to the target field up to a nonzero overall factor ($\frac{(-1)^{p}}{2}$). All the fields in the diagram are W ₃-algebra primaries, with the same Virasoro dimension. All fields below the top are of the form , where the momenta μ _• are immediately read off from the diagram as μ _α =γ−α ^∨, μ _αβ =μ _βα =γ−α ^∨−β ^∨=0, and so on, and the are differential polynomials in ∂φ _α (z) and ∂φ _β (z), of the orders , , , and .

Calculations in particular examples show the OPE

with nonzero coefficients (and no dimension-3 W(w) field), and the OPEs and that start very similarly. The adjoint-sℓ(3) nature of the octuplet manifests itself in the OPEs such as