## SciPost Submission Page

# 1/8-BPS Couplings and Exceptional Automorphic Functions

### by Guillaume Bossard, Axel Kleinschmidt, Boris Pioline

### Submission summary

As Contributors: | Boris Pioline |

Arxiv Link: | https://arxiv.org/abs/2001.05562v1 (pdf) |

Date submitted: | 2020-01-27 01:00 |

Submitted by: | Pioline, Boris |

Submitted to: | SciPost Physics |

Discipline: | Physics |

Subject area: | High-Energy Physics - Theory |

Approach: | Theoretical |

### Abstract

Unlike the $\mathcal{R}^4$ and $\nabla^4\mathcal{R}^4$ couplings, whose coefficients are Langlands-Eisenstein series of the U-duality group, the coefficient $\mathcal{E}_{(0,1)}^{(d)}$ of the $\nabla^6\mathcal{R}^4$ interaction in the low-energy effective action of type II strings compactified on a torus $T^d$ belongs to a more general class of automorphic functions, which satisfy Poisson rather than Laplace-type equations. In earlier work, it was proposed that the exact coefficient is given by a two-loop integral in exceptional field theory, with the full spectrum of mutually 1/2-BPS states running in the loops, up to the addition of a particular Langlands-Eisenstein series. Here we compute the weak coupling and large radius expansions of these automorphic functions for any $d$. We find perfect agreement with perturbative string theory up to genus three, along with non-perturbative corrections which have the expected form for 1/8-BPS instantons and bound states of 1/2-BPS instantons and anti-instantons. The additional Langlands-Eisenstein series arises from a subtle cancellation between the two-loop amplitude with 1/4-BPS states running in the loops, and the three-loop amplitude with mutually 1/2-BPS states in the loops. For $d=4$, the result is shown to coincide with an alternative proposal in terms of a covariantised genus-two string amplitude, due to interesting identities between the Kawazumi-Zhang invariant of genus-two curves and its tropical limit, and between double lattice sums for the particle and string multiplets, which may be of independent mathematical interest.

### Ontology / Topics

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### Submission & Refereeing History

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## Reports on this Submission

### Anonymous Report 1 on 2020-3-4 Invited Report

- Cite as: Anonymous, Report on arXiv:2001.05562v1, delivered 2020-03-04, doi: 10.21468/SciPost.Report.1552

### Strengths

1-delivers long sought-after result, resolves puzzles

2-amazing structural clarity, good explanations

3-very good introduction in a complex field, with a lot of references

### Weaknesses

1-complexity of the content, if you are not an expert working in this particular field, you will have a lot of work reading the article. It is, however, a very well written article, so suffering is because of complexity of the topic exclusively.

### Report

The article ,,1/8 BPS couplings and exceptional automorphic functions'' by

Guillaume Bossard, Axel Kleinschmidt and Boris Pioline is an interesting and

important contribution towards the determination of coeffiecients of various

couplings in the low-energy effective action of type II string theories. Of

central interest in the current article is the $\nabla^6{\cal R)^4$ coupling.

Results for the calculation of the coefficients of various couplings in

type-II-actions can be obtained by either performing a calculation in

exceptional field theory or as a direct string calculation for the appropriate

genus. While both calculations lead to automorphic forms eventually, they

theories differ by the particle content allowed to run in the ,,loop''

effectively: while particles in the exceptional field theory are confined to

1/2 BPS states, string theory allows in addition for 1/4, 1/8 BPS states as

well as non-BPS ones.

In earlier work of two of the authors, it was already suggested, that a

particular two-loop EFT calculation would deliver the result for the

$\nabla^6{\cal R)^4$ coupling up to some yet to be determined

Langlands-Eisenstein series. This gap is closed with the current article by

calculating the missing terms and interpreting their appearances (and

cancellations) between different loop orders in a string calculation.

Most of the insight relating EFT calculation to the (covariantized) string

answer rests on considering a particular representation of the Kawazumi-Zhang

invariant, which can be represented as a Poincar\'e series over its tropical

limit. This representation allows to express the string integral in terms of a

constrained lattice sum over pairs of vectors in the string multiplet lattice.

Similarly, the EFT calculation can be brought into a form of an integral over a

constrained sum over pairs of vectors/spinors in a particle multiplet lattice.

The comparison can then be performed recognizing identities between the two

lattice sums mentioned before: the central formula to consider. These lattice

identities provide the link between the two formulations and allow the careful

investigation and disection of various contributions in the sections to follow.

In short: the EFT calculation is able to provide certain subparts of the string

calculation, whose weak-coupling and decompactification (one of the

compactifications becomes large) limits allow to quantify parts of the

remaining ingredients to the final answer. In particular do the results allow

to canonically attribute properly regulated expressions to particular

coefficients in a U-duality compatible way resulting in the final formula

(1.32) (which is commonly called non-perterbative to underline the U-duality

properties).

Section 2 is devoted to the explanation (and partial derivation) of the central

lattice sum identity mentioned in the previous paragraphs. The two sides of the

identities are investigated considering appropriate Laplace identities to be

satisfied by the EFT lattice sum and the string lattice sum. Furthermore,

tensorial differential equations extending the Laplace equations are considered

and interpreted in the language of nilpotent orbits, which in turn can be used

for classification of Fourier coefficients of autormorphic forms. The equality

of the two lattice sums is then finally established by integrating over a

suitable Maa\ss eigenform and evaluating the result using functional relations

for the resulting Langlands-Eisenstein series. The discussion is complemented

by a subsection about convergence of the used expressions and a subsection

extending the lattice sums on the string side of the central formula (1.23) to

spinor representations.

In sections 3 and 4 (which are of equal structure), the weak coupling limit and

the decompactification limit of the EFT expression are considered. The analysis

relies on splitting the appropriate double theta series into different so-called

layers, which correspond to a decomposition of the lattice sum into suitable

subsets of charges, each of which allows for evaluation separately.

Calculations here are rather involved, in particular if it comes to regulating

the integrals. Several details of the calculation are referred to various

appendices, which is absolutely necessary: even after doing so the exposition

is clear but very dense.

In section 5, the various results are combined, various further expansions are

determined and regulations left out in previous sections are provided. Again,

the section is divided into considerations originating in the weak-coupling

limit and the decompactification limit. For subsection 5.2 (weak-coupling

limit), the overview of what has been done hides in a paragraph on page 63,

which I would have loved to read at the beginning of this subsection. The

natural appearance of the logarithmic term already mentioned in the

introduction is properly derived and explained afterwards. For subsection 5.3 a

similar discussion is performed, which culminates in the final (yet careful)

claim (1.31) is indeed the correct (non-perturbative) coefficient in front of

the $\nabla^6{\cal R}^4$ term.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Refereeing this article was an ardous task. While definitely rather close to the

research field of the referee, the actual tools used in this particular

subfield are very special and require a rather thorough knowledge of

automorphic forms and the underlying group theory/representation theory.

Moreover, it is many particular identities, special cases, nifty special

functions, particular special cases in various dimensions, which the referee

had to look up or doublecheck. I did so in most of the cases.

The first section of the article provided an excellent guideline towards a

rather steep entrance in the field. Having followed the recent developments

loosely, it was easily possible to assemble all previous information and

results consistently, mostly based on other publications of (subsets of) the

authors. The balance between restating previous results, explanations

of their origin on the one-hand side and the justification and motivation for

the investigations to follow as well as corresponding results on the other hand

is very carefully kept. Still, the complexity of arguments as well as the tools

used is rather high. In particular, a lot of work presented in the article is

not explained but referenced: this makes it sometimes difficult to follow and

cumbersome to check calculations.

Careful reading of the article required a couple of days, which I enjoyed to

the very end of the main part. The outcome -- beyond the recommendation for

publication -- is a couple of comments at the end of this report. In addition,

I have been consulting the appendices several times, however, did not check

every single argument carefully.

The article is very well written, of overall amazing structural clarity (with a

couple of points for improvement in section 5, see above). The article is

almost free of typos and great care has been taken to format long equations in

a nicely reeadible and accessible way. It is impossible within the timeframe

for refereeing such an article to check all calculations: this would amount to

several weeks of work. However, I checked some calculations explicitly. Part of

this (cross-) checking work is also done by the authors already: any hint and

crosscheck from U-duality and other results from different calculations is

carefully taken into account, explained and (positively) checked.

Thus, the result is certainly correct and constitutes the probably final step

towards the determination of the coefficient of the $\nabla^6{\cal R)^4$ term.

In total, I strongly recommend this article for publication in SciPost and

would recommend to consider some small corrections related to the suggestions

at the end of this report.

One thing I would like to mention finally is the fact, that for each of the

statements the authors have taken great care in stating whether it is proven,

conjectured, a claim. Several calculations, in particular related to analytic

continuation when regulating are probably correct, but not yet proven: I

appreciate that these uncertainties are clearly indicated (and discussed).

### Requested changes

Here is a list of things, which might help to improve readibility of the

article. The list is neither weighted nor complete or should pose requests: it

is a mere collection of suggestions having apppeared to the referee

during reading:

0. General

%%%%%%

- maybe a new paragraph in the abstract (before ,,Here'') would divide between previous results and new results from the current article.

- in the table of contents, formul\ae{} are not bold face

- there are a couple of occurrences of ,,2-loop'' in the text (e.g. page 12), deviating from the general strategy of writing ,,two-loop''

- pages 12 and 63 and in general: zero-th -> zeroth

1. Introduction

%%%%%%%%%%%%%%%

- page 1, first paragraph, ,,Combined with the ... from supersymmetry'': something is missing in this sentence.

- page 1: allows one to determine -> allows to determine

- the hat in eqn. (1.6) is not explained (only later an explanation shows up on page 12)

- the standard decomposition of \tau into \tau_1 and \tau_2 should be provided at the bottom of page 6 and before eqns.(1.14) and (1.15).

- the argument of {\cal I}_d in eqn.(1.17) contains \phi, which is only defined in the text above eqn.(1.1). Maybe one could repeat it here.

- I would suggest to make the importance of eqn. (1.23) more clear when stating it. It is one of the most referred equations in the following.

- page 12: contribution -> contributions

- page 12: cancel each others -> cancel each other

2. From particle to ...

%%%%%%%%%%%%%%%%

- page 14: below eqn.(2.2): I hope I didn't miss on a definition, but when D=10-d, then D+2=12-d and not 8-d.

- same paragraph: is there a good reference for ,,convenient convention'' and maybe as well for ,,top degree space'' and the other group theory concepts mentioned in the following paragraphs?

- eqn.(2.10) has been mentioned in the introduction already. Maybe one could refer back to the introduction here? I am not sure about a general strategy, as this occurs several times later.

- page 17: the space after a.k.a. appears to be too large.

- page 19, eqn. 2.19: H^{(U)}_N is probably the Hecke operator. (It is mentioned after 2.23, though in a different notation. ).

- page 19, eqn. 2.21: please define K_{s-\frac{1}{2}} (Bessel function). It shows up at several places below and I wouldn't consider that common knowledge.

- page 21: after eqn.(2.30) you mention the symmetry of the (finally) Riemann zeta function. Why should the rhs be invariant und this symmetry? Please explain.

- page 21: The statement right above subsection 2.5 appears strange as it stands: one better doesn't put d=3 in the expressions for the maximal weights, as this would lead to $\Lambda_2=0$. However, in the footnote on the same page the replacements appear to be correct: what did I miss?

- I would like to suggest a reference for relation (2.35): maybe 1511.04265 is good here, maybe also for (2.36).

3. Weak ....

%%%%%%%%%%%%

- page 25: at one point (probably in the introduction) the notion of ,,non-perturbative'' should be related to U-duality. It is probably clear to everybody in the field, but still.

- page 25: standard constant formulas -> standard constant term formulas ?

- page 25: it follows from _this_, ... -> it follows from the above equation, ....

- page 26: in your recapitulation you left out the argument $L$ in eqn.(3.7) as well as in the text. But in the next sentence you write about the $L$-dependent counterterm. I would suggest to stick to the original way of writing the integral.

- page 36: a specific choice of triality assignments ... here I would have loved a reference.

- page 39: the calculation of the Fourier coefficients is really tough to read. However, I wouldn't know how to improve, unfortunately.

- please, provide a reference for the matrix variate Bessel functions above (3.88)

4. Decompactification limit

%%%%%%%%%%%%%%%%%%%%%%%%%%%

- in the first paragraph: does the enhanced symmetry for d=5 actually lead to a crosscheck or a simplification?

- Frequently you use the phrase ,,One recognizes''. You do, for sure. I did sometimes, in many situations I had to look it up. It would pave the way to rather use something along the lines: as shown in ....

5. Regularisation and divergences

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

- page 58: after eqn (5.2): what is J_3? Again a Bessel-type function?

- page 60, eqn.(5.17) needs a full stop.

- page 63: second paragraph: cancel -> cancels

(in reply to Report 1 on 2020-03-04)

We are grateful to the referee for taking up the daunting task of reviewing this admittedly arduous paper, for his/her kind remarks and for numerous suggestions for improving the manuscript. We understand the subsection on Fourier coefficients is very dense, but we have not found a better way to present it so we have kept its original form. We have implemented most of the referee's suggestions, but since they are minor, it is probably not useful to list them all.

One more significant change is that we have spelled out the definition of the Hecke operator above Eq. (2.19), and corrected a misprint in its action on the Fourier modes in the line below 2.24 (a factor of $d^{-1}$ was missing). We have also dropped the comment below eq 2.30, which was incorrect as stated and unnecessary.

To answer one of the referee's query, $J_3$ below eq 5.2 denotes the projection of the angular momentum along the z-axis in the rest frame of the particle, i.e. the helicity. We did not think that it was necessary to spell it out.