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Exact effective interactions and 1/4-BPS dyons in heterotic CHL orbifolds

by Guillaume Bossard, Charles Cosnier-Horeau, Boris Pioline

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Submission summary

As Contributors: Boris Pioline
Arxiv Link:
Date submitted: 2018-06-27
Submitted by: Pioline, Boris
Submitted to: SciPost Physics
Discipline: Physics
Subject area: High-Energy Physics - Theory
Approach: Theoretical


Motivated by precision counting of BPS black holes, we analyze six-derivative couplings in the low energy effective action of three-dimensional string vacua with 16 supercharges. Based on perturbative computations up to two-loop, supersymmetry and duality arguments, we conjecture that the exact coefficient of the $\nabla^2(\nabla\phi)^4$ effective interaction is given by a genus-two modular integral of a Siegel theta series for the non-perturbative Narain lattice times a specific meromorphic Siegel modular form. The latter is familiar from the Dijkgraaf-Verlinde-Verlinde (DVV) conjecture on exact degeneracies of 1/4-BPS dyons. We show that this Ansatz reproduces the known perturbative corrections at weak heterotic coupling, including tree-level, one- and two-loop corrections, plus non-perturbative 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 four-dimensional 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 Taub-NUT instantons. We show that instanton corrections from 1/4-BPS black holes are precisely weighted by the BPS index predicted from the DVV formula, including the detailed moduli dependence. We also extract two-instanton corrections from pairs of 1/2-BPS black holes, demonstrating consistency with supersymmetry and wall-crossing, and estimate the size of instanton-anti-instanton contributions.

Ontology / Topics

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BPS dyons Black holes CHL orbifolds
Current status:
Has been resubmitted

Reports on this Submission

Anonymous Report 2 on 2019-4-29 Contributed Report

  • Cite as: Anonymous, Report on arXiv:1806.03330v1, delivered 2019-04-29, doi: 10.21468/SciPost.Report.926


This paper studies a certain six-derivative term in the effective action of three-dimensional 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 six-torus 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 (non-trivial) extension of those ideas. The effective coupling under consideration receives contributions from 1/2-BPS and 1/4-BPS 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 half-plane. Many string theorists would be familiar with the simpler 1/2-BPS version of this formula (which governs a different 4-derivative term) in the context of threshold corrections to couplings in four-dimensional 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 non-holomorphic 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 so-called "non-perturbative Narain lattice" which is roughly-speaking the sum of all 1/4-BPS states in string theory. At a technical level, it is not a priori clear that this formula is well-defined, 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 string-like objects (but containing the full non-perturbative spectrum of string theory), and it is not clear why e.g. there is not a higher-dimensional 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 non-trivial leap.

The authors do spend some time on defining the integral -- the essential problem as I see it are infra-red divergences, which they regulate in some manner. Having done that they can calculate various limits (weak string-coupling, 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 1/4-BPS black holes in the CHL models which carry non-primitive 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 six-derivative effective coupling of the string theory. There are also mathematical spin-offs. 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 as-yet-unknown 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.)

  • validity: -
  • significance: -
  • originality: -
  • clarity: -
  • formatting: -
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Author Boris Pioline on 2019-06-07
(in reply to Report 2 on 2019-04-29)
answer to question

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 "non-perturbative string", but rather as a mathematical device (inspired by string perturbation theory) to produce the relevant automorphic form. Similarly, the 'non-perturbative Narain lattice' should $not$ be thought of as a lattice of non-perturbative charges. We also took this opportunity to add a few references, as well as a footnote 6 on p16 on the issue of instanton-anti-instanton contributions. We hope that these clarifications will make the arguments in our paper easier to follow.

Anonymous Report 1 on 2019-4-19 Invited Report

  • Cite as: Anonymous, Report on arXiv:1806.03330v1, delivered 2019-04-19, doi: 10.21468/SciPost.Report.917


1- Impressive technical computations. The paper combines techniques from string theory, supersymmetry, modular forms in a highly non-trivial way.


1-This 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.


In this paper the authors conjecture the exact coefficient of a certain six-derivative coupling in the low energy effective action of three-dimensional string vacua with 16 supercharges and check that this coefficient reproduces perturbative corrections up to two loops and has the right form of non-perturbative 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 high-level 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 6-derivative 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.

  • validity: top
  • significance: low
  • originality: good
  • clarity: good
  • formatting: good
  • grammar: perfect

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Anonymous on 2019-04-28
answer to question

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 six-derivative 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 four-derivative 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 micro-physics 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 non-generic orbits of the U-duality 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.