User:Sylvain Ribault/Bootstat2021/Meineri

Bootstrap without positivity: tools and challenges.

Bounds from positivity
Crossing equations:

\sum_{\Delta,\ell} p_{\Delta,\ell} \vec{v}_{\Delta,\ell} = \vec{T} $$ where $$p$$ are structure constants, and $$\vec{v}$$ are blocks. Unitarity implies that structure constants are positive. This leads to bounds on parameters, excluding some regions.

Generically, OPEs allow infinitely many operators, which form a basis of any finite-dimensional space. This does not mean that crossing symmetry is powerless: it can still constrain structure constants, especially in asymptotics (in space). But this is not what we want to do.

Positivity lost: unitary and non-unitary examples
Reflection positivity: consider a four-point function

\left\langle \phi_1(x_1)\phi_2(x_2)\phi_1(x_3)\phi_2(x_4)\right\rangle \geq 0 $$ for particular configurations of points, from requirement that we can Wick-rotate to a unitary Lorentzian CFT.

We can lose this positivity in unitary theories by considering correlators of different configurations. For example, 4pt functions with unequal operators. Or in wrong channel. Or in the presence of a boundary, depending on the channel.

Non-unitary CFT: Lee-Yang model has been most studied.

Unitary CFT with non-positive 4pt functions: consider matrices of several 4pt functions, so that we get positivity in the sense of matrices. Adding more correlators do not solve all problems, as we should understand OPEs in all involved 4pt functions.

Gliozzi method: minors and singular values
Other names of the method: Expectation that operators with large dimensions do not contribute much to OPEs. Truncate crossing equations to finitely many operators and extract approximate solutions. It is computationally cheap, gives good results (not bounds) in various situations. It can be used to explore part of the landscape, even for reflection positive correlators.
 * Truncation method.
 * Severe truncation method.
 * Determinants method.

The method is more an art than a science: hard to scale up, not easy to estimate the error.

Possible refinements: some special solutions are sparse and can be deformed from unitary to non-unitary region.

The method


\sum_{k=1}^\infty p_k F_k(u_i) = 1 $$ Truncate sum to make it finite. Take $$M$$ derivatives around special point, this give us a matrix equation of size $$M\times N$$. The matrix must have a kernel. Overconstrain the system by choosing $$M\geq N$$. We expect an approximate solution.

Determinants method: compute minor determinants as fct of parameters i.e. dimensions of fields. The zeros of these determinants should intersect in a small region in parameter space. In some examples it works very well, even if the neglected terms are not small, because they are proportional to the non-neglected terms, and can be included by changing the structure constants.

Singular values method: look for small singular values of the matrix. Plot minimum singular value as fct of parameter. This can give good idea of where to look for solutions of crossing. Advantages: looking at only 1 quantity rather than many, looking for minimum rather than zero.

Application
Method works well in 2d Lee-Yang minimal model, with Virasoro fusion rule $$\phi \times \phi = 1 + \phi$$. Quite a sparse spectrum due to null states. Only 6 quasi-primaries up to dimension 6. Need to know how many operators at each spin, from weakly coupled description or some other method.

Estimates of the systematics in Gliozzi's method
In positive bootstrap, the error is systematic and well-understood.

In Gliozzi's method, 3 ways to estimate error:
 * Spread of solutions in determinants method. But solutions can drift when adding operators, the spread does not see this.
 * Add one operator. This adds two unknowns (dimension, structure constant) so we need to add two derivatives, otherwise we can find a family of solutions.
 * Compare different crossing equations: same data can appear in different correlators, leading to independent estimates that can be compared.