$T \overline{T}$-Like Flows and $3d$ Nonlinear Supersymmetry

We are grateful to the anonymous referee for giving feedback on our work.

We agree with the reviewer's point that the analysis of our paper is entirely classical, which was also commented upon in the first referee report. It would indeed be quite intriguing if one could extend some of our results to the quantum theory. For instance, it could be interesting to compute the partition functions of the $3d$ theories considered in this work on $\mathbb{R} \times S^2$ or $S^1 \times S^2$ and see whether our deformations are solvable at the quantum level. If one could extend the flows in our paper from $\mathcal{N} = 1$ to $\mathcal{N} = 2$ supersymmetry, it may even be possible to apply localization techniques in this setting.

However, we feel that considering classical flow equations can already serve as a useful first step, especially given that classical aspects of the more well-known two-dimensional $T \overline{T}$ deformation -- such as the fact that it yields the gauge-fixed Nambu-Goto string action, or that solutions to the deformed classical equations of motion can be generated from a field-dependent diffeomorphism -- exhibit a rich structure that has been investigated in many works.

Below we have included more specific responses to the two suggested improvements.

(1) We have added two paragraphs in Section 2.2, at the bottom of page 18 and top of page 19, commenting on the issue of S-matrix integrability and the relationship between our results and those of 2308.12189.

This connection is very interesting; since our $3d$ operator produces the $3d$ membrane action, and the results of Seibold and Tseytlin show that the dimensional reduction of this theory to $2d$ does not have an integrable S-matrix (even at tree level), it indeed follows that the dimensional reduction of our $3d$ operator cannot preserve integrability. However, this fact does not lead to any contradiction, since the dimensional reduction of our $3d$ operator necessarily differs from the usual two-dimensional $T \overline{T}$ operator, which does preserve integrability.

(2) The missing $\phi$ in equation (2.34) has been corrected.

We thank the referee for these remarks, especially for pointing out 2308.12189, and we hope that the revised version of our manuscript will now be suitable for publication in SciPost Physics.

Sincerely,

Christian Ferko, Yangrui Hu, Zejun Huang, Konstantinos Koutrolikos, and Gabriele Tartaglino-Mazzucchelli

We thank the referee for reading our manuscript and for his or her comments.

Let us first address the two points raised in the ``Weaknesses'' section of the referee report, which are in fact related. All of the observables which the referee mentioned (namely partition functions, energy spectra and S-matrices) are defined in the quantum theory, and therefore one would first need to define a quantum-mechanical version of our deforming operators in order to derive flow equations for any of these quantities. The issue of defining such a quantum operator is raised in the referee's second point. We should mention that the purely bosonic combination introduced in our equation (2.14) does not give rise to a well-defined operator by point-splitting, even in a conformal field theory. The reason for this is that the trace $T^a_{\; \; a}$ vanishes in such a theory, so only the first term in our deforming operator survives, while in a $d$-dimensional conformal field theory one has

\begin{align}
\langle T_{ab} ( x ) T^{ab} ( 0 ) \rangle = \frac{d - 2}{x^{2d}} + \text{regular} \, .
\end{align}

The first term only vanishes in $d = 2$ but not in any higher dimension. Because of this position dependence, one cannot take the limit $x \to 0$ to define a local operator in $d = 3$, as one does in two dimensions.

Although defining a quantum $T \overline{T}$-like operator in $3d$ is outside of the scope of this paper, one might speculate that introducing supersymmetry might be necessary in order to do so. The reason for this is that supersymmetric extensions of the $T \overline{T}$ operator, such as those considered in our Section 3, contain contributions from additional fields in the stress tensor multiplet. One might hope that, in theories with sufficient supersymmetry, a certain combination which includes both $T_{ab} T^{ab}$ and also additional terms involving these extra fields in the multiplet might give rise to a position-independent two-point function that could then be used to define a local operator by point-splitting. Again, we did not consider this question in the present work, but we feel that an analysis of supersymmetric extensions of the $T \overline{T}$ operator in $d > 2$ such as the one in this manuscript might be a necessary first step towards solving this problem.

In response to the three items in the ``Requested changes'' section of the report:

(1) Although we did not comment on it explicitly in the supersymmetric discussion, the same sign choice for $\mathfrak{t}$ or $x_1$ appears in this setting, since truncating these supersymmetric flows to the bosonic sector reproduces the flows of Section 2, and therefore all of the same sign choices must apply.

As the referee points out, strictly speaking, we have only explained how to deform part of the phase space (e.g. $| \vec{E} |^2 < | \vec{B} |^2)$ of the Maxwell theory into the Born-Infeld theory. However, it is straightforward to deform the other part of the phase space by choosing slightly different coefficients in the definition of our deforming operator (as we briefly mention below equation (2.11)).

One could therefore consider a piecewise-defined version of the deformation, which acts differently in the two parts of the phase space, to handle both cases at once. Note in particular that this piecewise definition is continuous across the transition point, where $\mathfrak{t} = 0$ and therefore the operator $R$ vanishes, so either choice of $\pm R$ is equivalent.

We have added a sentence below equation (2.11) in the revised manuscript to clarify this point.

(2) We have clarified the discussion around equation (2.18) and added comments about how it relates to equation (2.58). What we meant to emphasize is that equations (2.19) and (2.20) only hold up to total derivative ambiguities; we did not mean to suggest that these can always be ignored, since as we see around equation (2.58), these total derivative terms can be important in some contexts.

We agree with the referee that, in the quantum theory, the spectrum of local operators (including the stress tensor) is part of the definition of the theory, and there is no freedom to perform improvement transformations. However, although it is outside the scope of the present work, it is possible to argue that the trace flow equation for the $T \overline{T}$ operator holds at the quantum level, without resorting to the classical argument we have presented that suffers from total derivative ambiguities. (For instance, one way that this trace flow equation has been justified in the literature is by performing a gravity analysis in the holographic dual to a $T \overline{T}$ deformed CFT.)

(3) We agree that this rewriting of the flow in terms of a relevant operator is interesting, and we note that the same rewriting can be done for the $2d$ deformation that produces the Nambu-Goto action, as pointed out in 2301.10411. In the $2d$ setting, the irrelevant form of the deformation is of course well-defined at the quantum level, although the interpretation of the relevant form is still not clear.

Although a relevant deformation is not expected to lead to the phenomena observed under a $2d$ $T \overline{T}$ deformation, as the referee mentioned, we should comment that the form of this relevant operator is quite complicated and involves division by a non-analytic combination of stress tensors. Even if one expands around a constant (non-zero) stress tensor background in order to Taylor expand this combination, it leads to an infinite series of stress tensor operators. This is a very complicated object from the perspective of the low energy (undeformed) theory, but it might be more natural if it could be expressed in terms of other objects in the finite-$\lambda$ deformed theory, which is believed to be non-local and therefore does not possess local operators.

One might be tempted to interpret the possibility of performing this RG-relevant rewriting as a signal of UV/IR mixing. Such mixing is a hallmark of non-commutative field theories, and various authors have speculated that a $T \overline{T}$-deformed $2d$ QFT might share some properties with a non-commutative field theory (for instance, both $T \overline{T}$ and non-commutativity lead to CDD-factor-like modifications of the S-matrix). Unfortunately, we do not yet have any precise statements to make on this subject.

We believe that these modifications have improved the overall quality of the paper, and we again thank the referee for these suggestions.

Sincerely,

Christian Ferko, Yangrui Hu, Zejun Huang, Konstantinos Koutrolikos, and Gabriele Tartaglino-Mazzucchelli