BPS Chaos

Referee 1:

- “1. On page 23, the author makes the assumption: "The Thouless time of the special banded matrix we are studying can be bounded below by the Thouless time of a banded random matrix of a similar shape." The author should clarify what is meant by a “special banded matrix.” This bound cannot universally apply to all banded matrices, as any Hermitian matrix can be transformed into a banded matrix through diagonalization using a unitary transformation.”

We agree with the referee that additional clarification is needed. The referee is correct to observe that even a typical member from random matrix ensemble can be brought into banded form. However, we note that if we transform a random matrix into banded form, it will have distinctive features that separate it from the kind of projected operators we can get in the 1/2-BPS subspace.

To address this question, we removed the vague adjective “special” from the sentence and added a paragraph below (3.33) to discuss the intended meaning of “special” in more detail, which we copy here:

“One might question our assumption that the banded matrix we have is less chaotic than a banded random matrix. After all, one can always transform a matrix, for instance a typical member of a random matrix ensemble, into a banded form. Such a banded matrix will then display stronger chaos than a banded random matrix.\footnote{One should not confuse the notion of “banded random matrix" with a random matrix that is transformed into banded form.} However, we note that if we transform a typical random matrix into banded form, the resulting matrix is expected to carry distinctive features which separate them from the matrices we have. For example, consider the banded matrix being simply diagonal. The diagonal form of a random matrix has its eigenvalues as entries, which are highly random numbers satisfying level repulsion, which differs from $\hat{O}$ in (3.26) whose entries are given by simple expressions.

As a more non-trivial example, we can consider the tridiagonal form $M_{tri-diagonal}$ of the Gaussian orthogonal ensemble. It was shown in \cite{Dumitriu_2002} that in the ensemble of transformed matrices, the entries in the matrices are drawn independently from specific probability distributions. The diagonal elements of $M_{tri-diagonal}$ are drawn from normal distribution with the same standard deviation, while the $n$-th off-diagonal element is drawn from the $\chi_n$ distribution, resulting in a distinctive growing pattern along the off-diagonal. We expect the LMRS operators $\hat{O}$ in the $1/2$-BPS subspace to not display such features in the cases where they are tri-diagonal. Even though here we only discussed the special cases where the banded matrix is diagonal or tri-diagonal, we expect the distinctions between our matrices and random matrices that were transformed into banded form to exist more generally.”

- “2. In Section 6, the author applies the LMRS diagnostic to the half-BPS states in the T^4 symmetric orbifold, projecting the twist operator (6.26) onto the subspace of half-BPS states. Since (6.26) is the operator that deforms the D1-D5 CFT away from the free orbifold point, the author should clarify the distinction between their computation and the computation of the anomalous dimension and mixing matrix in the half-BPS sector. This clarification is important because the half-BPS sector is protected under such a deformation.”

The operator \Sigma in (6.26) that we are considering differs from the actual marginal operator \Sigma_{marginal} by \Sigma_{marginal} \sim G_{-1/2} \bar{G}_{-1/2} \Sigma. As the referee correctly pointed out, the matrix elements of \Sigma_{marginal} in half-BPS states vanish. However, the matrix elements of \Sigma do not vanish.

There is also an analogy with N=4 SYM. The twist operator we consider is analogous to Tr Z^2 in SYM. The marginal deformation in D1-D5 is a super-descendant of the twist operator, and similarly in SYM, the marginal deformation (the SYM interaction) is a super-descendant of Tr Z^2. But as is very familiar in SYM, there are non-zero 3-pt functions between Tr Z^2, Tr Z^k, and Tr Z^l.

We added a sentence after (6.26) for clarification:
“Note that the marginal operators $\Sigma_{marginal}$ which deform the theory away from the orbifold point are the superconformal descendants of $\Sigma$, i.e. $\Sigma_{marginal} \sim G_{-\frac{1}{2}}\bar{G}_{-\frac{1}{2}}\Sigma$, whose matrix elements between half-BPS states vanish due to non-renormalization theorems.”

Referee 2:

- “1. This is not necessary for publication, but it may be useful if the authors can provide clarification on the following. The authors seem to find that, for certain simple operators in both the 1/2 and 1/4-BPS sectors of N=4 SYM, the statistics are not only not chaotic, that is Wigner-Dyson, but Poisson. While this is of course perfectly consistent with the absence of strong chaos, it seems to raise the question of whether is there a notion of integrability in these sectors of the non-planar theory? Is there a reason to believe this is a merely a mirage?”

In our Figure 4, we showed an example in the 1/2-BPS sector where the projected operator whose nearest-neighbor spacings satisfy a distribution close to the Wigner surmise. We view this as evidence that the projected operators in general are not integrable, but only weakly chaotic. It does not rule out the possibility that for a special class of simple operators, such as that considered in Section 3.2, the projected operator can be integrable. We think it is an interesting future question to understand the notion of integrability in projected operators.

Other small changes:
1. Above equation (1.6), we modified misleading statement “In general, BPS states will be superconformal primaries, so we can …” into “In the situation where the BPS states are given by superconformal primaries, we can …”.