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Exact effective interactions and 1/4BPS dyons in heterotic CHL orbifolds
by Guillaume Bossard, Charles CosnierHoreau, Boris Pioline
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Authors (as registered SciPost users):  Boris Pioline 
Submission information  

Preprint Link:  https://arxiv.org/abs/1806.03330v1 (pdf) 
Date submitted:  20180627 02:00 
Submitted by:  Pioline, Boris 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
Motivated by precision counting of BPS black holes, we analyze sixderivative couplings in the low energy effective action of threedimensional string vacua with 16 supercharges. Based on perturbative computations up to twoloop, supersymmetry and duality arguments, we conjecture that the exact coefficient of the $\nabla^2(\nabla\phi)^4$ effective interaction is given by a genustwo modular integral of a Siegel theta series for the nonperturbative Narain lattice times a specific meromorphic Siegel modular form. The latter is familiar from the DijkgraafVerlindeVerlinde (DVV) conjecture on exact degeneracies of 1/4BPS dyons. We show that this Ansatz reproduces the known perturbative corrections at weak heterotic coupling, including treelevel, one and twoloop corrections, plus nonperturbative effects of order $e^{1/g_3^2}$. We also examine the weak coupling expansions in type I and type II string duals and find agreement with known perturbative results. In the limit where a circle in the internal torus decompactifies, our Ansatz predicts the exact $\nabla^2 F^4$ effective interaction in fourdimensional CHL string vacua, along with infinite series of exponentially suppressed corrections of order $e^{R}$ from Euclideanized BPS black holes winding around the circle, and further suppressed corrections of order $e^{R^2}$ from TaubNUT instantons. We show that instanton corrections from 1/4BPS black holes are precisely weighted by the BPS index predicted from the DVV formula, including the detailed moduli dependence. We also extract twoinstanton corrections from pairs of 1/2BPS black holes, demonstrating consistency with supersymmetry and wallcrossing, and estimate the size of instantonantiinstanton contributions.
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Reports on this Submission
Anonymous Report 2 on 2019429 (Contributed Report)
 Cite as: Anonymous, Report on arXiv:1806.03330v1, delivered 20190429, doi: 10.21468/SciPost.Report.926
Report
This paper studies a certain sixderivative term in the effective action of threedimensional string theory with 16 supercharges. Specifically, the paper deals with heterotic string theory compactified on CHL orbifolds times a circle. (The CHL orbifolds are freely acting orbifolds of a sixtorus which preserve 16 supersymmetries.)
The basic idea of the paper was presented in an earlier paper by the same authors for the heterotic string on $T^6 \times S^1$, and the present paper is a (nontrivial) extension of those ideas. The effective coupling under consideration receives contributions from 1/2BPS and 1/4BPS objects running in loops, and the conjecture of the paper is that the coupling is given by an integral of a certain Siegel modular form over the Siegel upper halfplane. Many string theorists would be familiar with the simpler 1/2BPS version of this formula (which governs a different 4derivative term) in the context of threshold corrections to couplings in fourdimensional string compactifications  the corresponding formula involves an integral over the fundamental domain of the modular group, and the integrand is a product of a certain meromorphic modular form which is a counting function of BPS states (the elliptic genus of the internal manifold) and a nonholomorphic lattice sum over momenta and windings. Here the structure is formally similar: the integrand is a meromorphic modular form which counts BPS states times a lattice sum.
The details, however, seem to involve conceptual as well as technical leaps. At the conceptual level, while the string threshold corrections were derived from the fundamental string theory, the corresponding integral here is a conjecture made by the authors, and involves a sum over a socalled "nonperturbative Narain lattice" which is roughlyspeaking the sum of all 1/4BPS states in string theory. At a technical level, it is not a priori clear that this formula is welldefined, nor that it suffices, and its success points to a very detailed check of the amazing internal consistency of string theory. Another way of stating this point is that integral under question is over the moduli space of stringlike objects (but containing the full nonperturbative spectrum of string theory), and it is not clear why e.g. there is not a higherdimensional moduli space.
Recommendation: If this conceptual point (why only moduli space of 2d surfaces) has been addressed previously then it would be worth commenting on this in the paper, else it could be highlighted so that it is clear to the reader that there is a nontrivial leap.
The authors do spend some time on defining the integral  the essential problem as I see it are infrared divergences, which they regulate in some manner. Having done that they can calculate various limits (weak stringcoupling, decompactification to 4d) of the integral from which they recover known results. There are some new predictions in other directions, e.g. for the exact degeneracies for 1/4BPS black holes in the CHL models which carry nonprimitive charges of a type that has not been studied before, which in principle can be checked independently.
Recommendation: the regulator could be highlighted a little more in the main text because it is important. At the very least, Appendix B.2.4 could be referred to in Sections 1 and/or 2. (At the moment the reference first appears in Section 3 as far as I can see.)
Overall it is an interesting paper, it informs an important program in string theory (finding the structure of all the effective action in various vacua), and certainly deserves to be published. The main point is that there is a new (conjectural) formula for a certain sixderivative effective coupling of the string theory. There are also mathematical spinoffs. For example the $R \to \infty$ limit has a mysterious piece $G^{(I\bar{I})}$ required by supersymmetry, but has not been calculated directly by the unfolding technique. Supersymmetry thus gives a constraint for the asyetunknown unfolding technique for meromorphic Siegel modular forms.
My main criticism is as follows. The paper is quite long, and in spite of a very good effort by the authors to maintain clarity (without which it would have been very difficult to read), the nature of the paper is that it has become quite dense with technology that are not familiar to most string theorists.
Recommendation: one specific idea to make it more readable is to have a title like "Main results" where the main results are presented. (It is already a bit unusual that the statements of main results are spread between Sections 1 and 2, and as a result they are a little diffused.)
Author: Boris Pioline on 20190607 [id 536]
(in reply to Report 2 on 20190429)We appreciate the careful reading, positive comments and suggestions for improvements. In order to help the reader, we have inserted a list of the main new results in the introducrtion (top of p6 in revised version). We also included several sentences in the opening paragraphs of Sec 2, 3, 4, 5 with pointers to the relevant subsections where a discussion of the physical implications of our computations can be found, and pointers at appropriate places (e.g. below 2.19) to Section B.1.3 and B.2.4 where the regularization prescription is explained. As for the referee's specific technical comment, we included a footnote 1 on page 8 emphasizing that the Siegel fundamental domain should $not$ be thought of as parametrizing the worldsheet of a physical "nonperturbative string", but rather as a mathematical device (inspired by string perturbation theory) to produce the relevant automorphic form. Similarly, the 'nonperturbative Narain lattice' should $not$ be thought of as a lattice of nonperturbative charges. We also took this opportunity to add a few references, as well as a footnote 6 on p16 on the issue of instantonantiinstanton contributions. We hope that these clarifications will make the arguments in our paper easier to follow.
Anonymous Report 1 on 2019419 (Invited Report)
 Cite as: Anonymous, Report on arXiv:1806.03330v1, delivered 20190419, doi: 10.21468/SciPost.Report.917
Strengths
1 Impressive technical computations. The paper combines techniques from string theory, supersymmetry, modular forms in a highly nontrivial way.
Weaknesses
1This works contains an enormous amount of work for very small payback in terms of physics. While it is impressive that the authors managed to do this computation, it is not clear what qualitative new physics we learn from it.
Report
In this paper the authors conjecture the exact coefficient of a certain sixderivative coupling in the low energy effective action of threedimensional string vacua with 16 supercharges and check that this coefficient reproduces perturbative corrections up to two loops and has the right form of nonperturbative corrections. The results are also “decompactified” to four dimensions.
The 155 pages of this paper are packed with technical details, and this work certainly demonstrates highlevel technical virtuosity. However, one is left to wonder: “why do we care?” Why is this specific coupling worth of 155 pages of hard work? I don’t think the authors provide any convincing argument for that. The introduction nicely summarizes how they came about to study this question, starting from questions regarding precision counting of BPS black holes. However, the paper contains no conclusions or outlook.
There is no discussion of what we have actually gained from this study, in the bigger scheme of things. Of course, we now have a good guess for this 6derivative coupling, but what qualitative new physics did we learn that we didn’t already know?
While this paper is technically impressive I believe it is of marginal interest for the community. It should be an editorial decision whether SciPost would like to publish papers of this type.
Requested changes
1 The authors should add Conclusions and outlook.
Anonymous on 20190428 [id 503]
It appears that the referee has not fully perceived the motivations and main results of our work. We have chosen not to include a conclusion because the paper is indeed rather technical and very long, and instead opted for a review of our main results in Section 2, after some putting them into context. We now see that the introduction of Section 2 fails to clearly state that this section reviews the main results of the paper, which are then derived and explained in more detail in later sections.
To address the referee's question "why do we care", it may be helpful to summarize again our main achievements: as explained in Section 2.3,
we determine the exact value of a particular sixderivative coupling in the low energy effective action, which is related by supersymmetry to all other couplings at the same order in the derivative expansion. Along with the fourderivative coupling derived in our previous work SciPost Phys. 3 (2017) no.1, 008, it therefore determines the complete low energy effective action up to six derivatives for this class of string theory vacua. Admittedly, these string vacua are not relevant for phenomenology, but they serve as a very useful testing ground for a quantitative understanding of black hole microphysics and holography. In particular, BPS indices relevant for supersymmetric black holes are fully captured by the type of protected coupling considered in our work. As explained in section 2.2, equation (2.14) provides the helicity supertrace counting 1/4 BPS black holes states for arbitrary primitive charges in CHL models, which were previously known only for electromagnetic charges in certain nongeneric orbits of the Uduality group. We believe that this type of quantitative information is very valuable and justifies the effort reported in this paper. We shall be glad to improve our manuscript to make it easier for the reader to navigate through the profusion of technical details and appreciate the main new physics points.