Global conformal symmetry in Virasoro minimal models

In two dimensions, a diagonal minimal model exists for any two coprime integers $2\leq p<q$, but it is unitary only if $q=p+1$. There is also a generalized minimal model for any generic complex value of the central charge. Studying minimal models from the point of view of global (rather than local) conformal symmetry can help understand the structure of more general CFTs, in particular non-unitary conformal field theories.

Signs of structure constants

It is possible to normalize fields such that all three-point structure constants are real, even in the non-unitary cases. Non-unitarity manifests itself by the negativity of some two-point structure constants. We distinguish Virasoro structure constants, which are coefficients of Virasoro conformal blocks, from global structure constants, which are coefficients of global conformal blocks.

Virasoro structure constants

In a field normalization such that three-point structure constants are real, two-point structure constants are

$B(P)={\frac {\Upsilon _{\beta }(2\beta -\beta ^{-1})\Upsilon _{\beta }(\beta ^{-1})}{\prod _{\pm }\Upsilon _{\beta }(\beta \pm 2P)}}$

where $\beta$ is related to the central charge by $c=13-6\beta ^{2}-6\beta ^{-2}$ and $P$ to the conformal dimension by $\Delta ={\frac {c-1}{24}}+P^{2}$. The relevant values of the momentum are $2P_{(r,s)}=r\beta -s\beta ^{-1}$ where $1\leq r\leq q-1$ and $1\leq s\leq p-1$ are Kac table indices, with $\beta ^{2}={\frac {p}{q}}$. The function $\Upsilon _{\beta }(x)$ is a type of w:multiple Gamma function. We have normalized the two-point structure constant such that $B(P_{(1,1)})=1$.

Using the known behaviour of $\Upsilon _{\beta }(x)$ under shifts of its argument by $\beta ,\beta ^{-1}$, we find

$\operatorname {sign} B(P_{(r,s)})=(-1)^{\left\lfloor (q-p){\frac {r}{q}}\right\rfloor +\left\lfloor (q-p){\frac {s}{p}}\right\rfloor +\left\lfloor {\frac {q}{p}}-1\right\rfloor }$

In particular, this is always positive if $q=p+1$ or $(r,s)=(1,1)$. In any non-unitary minimal model, some primary fields have negative conformal dimensions i.e. violate the unitarity bound, and some primary fields have negative two-point structure constants, while all irreducible representations are non-unitary. For example, here is the Kac table of the model with $\beta ^{2}={\frac {3}{5}}$, where the dimensions in red are for primary fields with negative two-point structure constants:

${\begin{array}{c|cccc}2&\color {red}{\frac {3}{4}}&\color {red}{\frac {1}{5}}&-{\frac {1}{20}}&0\\1&0&-{\frac {1}{20}}&\color {red}{\frac {1}{5}}&\color {red}{\frac {3}{4}}\\\hline &1&2&3&4\end{array}}$

For a given pair of indices $(r,s)$, we have

$\max \left({\frac {r-1}{r}},{\frac {s}{s+1}}\right)<\beta ^{2}\leq 1\implies B(P_{(r,s)})>0$

This interval of positivity includes all unitary minimal models whose Kac tables contain $(r,s)$. There are other intervals of positivity for lower values of $\beta ^{2}$.

Global structure constants

Any diagonal Virasoro primary field of dimension $\Delta$ has infinitely many descendants that are global primary fields. Their structure constants are of the type

$B_{\Delta }d_{\Delta ,n}d_{\Delta ,{\bar {n}}}$

where $d_{\Delta ,n}$ is a coefficient in the decomposition of a w:Virasoro conformal blocks into global blocks:

${\mathcal {F}}_{\Delta }(z)=\sum _{n=0}^{\infty }d_{\Delta ,n}{\mathcal {G}}_{\Delta +n}(z)$

The global block ${\mathcal {G}}_{\Delta +n}(z)$ is a hypergeometric function. To compute Virasoro blocks in minimal models is in general a problem in itself. It is however easy when one of the four fields is degenerate at level two, in which case the blocks are hypergeometric functions (which however differ from global blocks). In unitary minimal models, the signs of $d_{i,n}$ are positive for four-point functions of the type $\left\langle V_{1}V_{2}V_{2}V_{1}\right\rangle$. (Remember that two-point functions of fields with odd spins are negative in unitary CFTs.)

In numerical experiments with non-unitary minimal models, we find that the signs of $d_{\Delta ,n}$ are surprisingly regular as functions of $n$: they oscillate before becoming constant for $n\geq n_{0}$. This regular behaviour however translates into a somewhat erratic behaviour for signs of two-point functions of global primaries $B_{\Delta }d_{\Delta ,n}d_{\Delta ,{\bar {n}}}$: we have to multiply left-moving signs with right-moving signs, and to intercalate contributions from different channels $\Delta$. As a result, the signs of global primaries in our four-point functions do not become constant as the dimension increases.

In some cases, all coefficients $d_{\Delta ,n}$ are positive. This may seem paradoxical, as they can be computed as sums over states which do not all have positive squared norms. However, there typically appear several global primaries at a given level, so that each coefficient is the sum of the contribution of several primaries. Negative contributions may well be compensated by positive contributions.

For example, we consider the four-point function $\langle \phi \phi \phi \phi \rangle$ in the $(5,2)$ Yang-Lee minimal model. We ignore odd indices $n$, which do not contribute to this four-point function due to permutation symmetry. We find the signs

$d_{0,n}=+-0+++++++\cdots \quad ,\quad d_{-{\frac {1}{5}},n}=+0+++++++\cdots$

which means $d_{0,0}=+,d_{0,2}=-$ etc. Taking into account the Virasoro primaries' signs $B_{0}>0$ and $B_{-{\frac {1}{5}}}<0$, we find the following signs for the first few Virasoro primaries, labelled by total dimensions and spins i.e. $(2\Delta +n+{\bar {n}},n-{\bar {n}})$:

$(0,0)+,(-{\tfrac {2}{5}},0)-,(2,2)-({\tfrac {18}{5}},4)-,(4,0)+,({\tfrac {28}{5}},6)-,(6,6)+,({\tfrac {38}{5}},8)-,({\tfrac {38}{5}},0)-,(8,8)+,(8,4)-$

The signs will keep oscillating, mainly because of the alternance between the negative $\phi$ sector and the mostly positive identity sector, and also because of the negative w:Regge trajectories $n=2$ and ${\bar {n}}=2$ in the identity sector.

When the $d_{\Delta ,n}$ are non-negative, this trivially implies that the Taylor coefficients of the Virasoro block are non-negative as well. There is a generalization of this statement in higher dimensions when correlators must be expanded in global conformal blocks which no longer factor into $z$ and ${\bar {z}}$ dependent parts. Nevertheless, unitary theories lead to all $z^{m}{\bar {z}}^{n}$ appearing with positive coefficients, a fact which played a role in the causality proof of the conformal collider bounds. Also, in two dimensions it is possible to expand in powers of the elliptic nome $q$ instead of $z$. Unitarity is again sufficient for positive coefficients in this case which is interesting from the point of view of bulk locality.

It may be interesting to study not only the signs but also the magnitudes of the structure constants, and to see which contributions dominate at large dimensions.