The inheritance of energy conditions: Revisiting no-go theorems in string compactifications

Dear Anonymous Referee,

Thank you again for reading the paper and for your constructive comments toward increasing the paper's applicability scope. We reply to your comment (copied below) about flux contribution as follows:

*The authors have addressed all points in a satisfactory way, except the first one. In their derivation of the energy conditions for a configuration of Op planes and Dp branes, they use explicitly (Eqs 38 to 42) the tadpole cancelation condition without fluxes. Furthermore, since fluxes result in positive charges, the inequality in the right hand side of Equation (42) is precisely the opposite, the number of Dp branes has to be less than 2p−5×NOp+ (assuming there are no Op− planes). The authors should still clarify how the conclusions of this section are valid without using (38)-(41).*

In section 2.2, we have shown that whether an extended object will satisfy the energy conditions or not depends on the sign of its tension. For D-branes, the tension is positive, and the conditions are satisfied. For a single $Op_+$-planes, the tension is negative, and the energy conditions are not satisfied. But if one considers a configuration containing sets of parallel D-branes and O-planes it might happen that the total tension is positive and the energy conditions are satisfied. However, the configuration of extended objects is constrained by the tadpole cancellation condition, which is a constraint on the charge of the system. Since in string theory the charge and tension of extended objects are related, this begs the question of how the tadpole cancellation condition affects the expression of the total tension of the configuration, and in section 2.3 we investigated precisely that.

If we consider a parallel configuration containing D-branes, anti-D-branes, O-planes, and anti-O-planes (including different orientations), the tadpole cancellation condition as coming from the minimal WZ coupling with the corresponding p-form (see last term in equation (37)) reduces to equation (40) and constrains the number of extended objects of each kind. The total tension of the system is given by the first line of equation (41). Upon using (40), the total tension reduces to the second line of equation (41). Thus, we conclude that, if we want the total tension to be positive for the energy conditions to hold, then we need the inequality in (42) to be satisfied.

Now let's discuss which of our results would change if we consider the "non-minimal", general coupling of extended objects to the p-forms
\begin{equation}
S_{\text{WZ}} = \mu_p \int_{\Sigma_{p+1}} e^{B_2 + 2\pi \alpha' F_2}\wedge \sum_i A_i.
\end{equation}
The exponential is a formal expression to collect terms with the required rank to match with $A_i$ and the dimension of the extended source. Note that the minimal coupling considered in the paper corresponds to the first term after expanding the exponential. The action above *does not* depend on the metric and, as such, it *will not* contribute to the energy-momentum tensors of the p-forms and of the extended objects. Such energy-momentum tensors were explicitly used in the discussions of sections 2.1 and 2.2. Therefore, we conclude that *the WZ coupling $S_{\text{WZ}}$ will not change the results of sections 2.1 and 2.2 because it will not modify the energy-momentum tensors of p-forms and/or extended objects*. This is the reason why we wrote footnote 4.

In the tadpole cancellation condition (40), only the first term in $S_{\text{WZ}}$ was considered, and so, the other (non-minimal) terms will modify that condition. We comment about it in our conclusions (see the top of page 16). Discussing the energy conditions validity for a general configuration of extended sources and their full coupling with all p-forms in the theory considered is beyond the scope of the paper, so we simply comment on how the fluxes will modify (42): there will be an extra term in equation (40) and (42), coming from higher order terms in $S_{\text{WZ}}$. We agree with the referee that the new flux contribution will appear as a *positive* term on the left-hand side of equation (40). Hence, it will induce a positive term in the last line of (41) and a negative one on the right-hand side of the inequality (42), without changing the inequality sign, i.e., (42) would be modified to
\begin{equation}
N_{\text{D}p}\geq 2^{p-5}(N_{\text{O}p_+}- N_{\text{O}p_{-}}) - N_{\text{flux}}/\mu_{\text{D}p}.
\end{equation}
In summary, the conclusions of sections 2.1 and 2.2 on the validity of energy conditions for p-forms and extended objects are not modified by the extra WZ flux couplings, and the conclusions in section (2.3) will be slightly modified because of extra terms in equations (39)-(42). The conclusion that the tadpole cancellation has to be considered for the validity of the energy conditions is unchanged, but the condition for positive total tension is modified if fluxes are included. We decided not to discuss these modifications further because (i) they are way beyond the scope of the paper and (ii) they are more suitable for a full analysis of more complex D and O-planes configurations.

We hope this addresses the referee's concern and our work can be published.

Sincerely,
Heliudson Bernardo (on behalf of all authors)