CPHT-RR019.032022

Perturbative higher-derivative terms
in ${\cal N}=6$ asymmetric orbifolds

Massimo Bianchi1, Guillaume Bossard2, Dario Consoli1

1Dipartimento di Fisica, Università di Roma “Tor Vergata” & Sezione INFN
Roma2,
Via della ricerca scientifica 1, 00133, Roma, Italy
2Centre de Physique Théorique, CNRS, Institut Polytechnique de Paris
91128 Palaiseau cedex, France

We analyse the perturbative four-point amplitudes in the simplest string
theory examples of T-fold backgrounds, which enjoy ${\cal N}=6$
supersymmetries in four dimensions. There are two theories defined as
asymmetric orbifolds of order $2$ and $3$, respectively. The perturbative
spectrum and the one-loop four-point amplitudes are shown to be invariant
under an arithmetic symplectic group defined over the Hurwitz (respectively
Eisenstein) quaternions. The supersymmetry constraints on the low energy
effective couplings are determined and we conjecture the U-duality group of
the theory and the non-perturbative completion of the six-derivative coupling
as a rank one theta series. We discuss the non-perturbative spectrum of BPS
states in the light of our analysis.
---

We wish to dedicate this paper to the memory of Costas Kounnas

###### Contents

  1. 1 Introduction
  2. 2 ${\cal N}=6$ supergravity in diverse dimensions
    1. 2.1 String Theory on (freely acting) asymmetric orbifolds
    2. 2.2 Character-valued partition functions and helicity super-traces
    3. 2.3 The asymmetric orbifold helicity supertraces
    4. 2.4 Preliminaries on U-duality
  3. 3 Low-energy effective action and threshold corrections
    1. 3.1 Supersymmetry constraints
    2. 3.2 (2,4) BPS $\bar{\mathcal{F}}^{2}\mathcal{R}^{2}$ type invariant
    3. 3.3 (2,2) BPS $\mathcal{R}^{4}$ type invariant
    4. 3.4 (1,1) BPS ${D}^{2}\mathcal{R}^{4}$ type invariants
    5. 3.5 (2,0) BPS $\mathcal{F}^{2k}\mathcal{R}^{4}$ and ${D}^{2}\mathcal{R}^{4}$ type invariants
    6. 3.6 BPS invariants in five dimensions
    7. 3.7 Consequences for string perturbation theory
  4. 4 Perturbative results in String Theory: tree level and one-loop
    1. 4.1 Vertex operators
    2. 4.2 ‘Chiral’ factorisation: closed-strings from open-strings at one-loop
    3. 4.3 ${\cal R}^{4}$ and $D^{2}{\cal R}^{4}$ couplings
    4. 4.4 $\bar{\cal F}^{2}{\cal R}^{2}$ at one-loop
    5. 4.5 Minimal theta series
    6. 4.6 Large radius expansions of ${\cal R}^{4}$ and $D^{2}{\cal R}^{4}$ couplings
  5. 5 BPS Branes in the asymmetric orbifold
    1. 5.1 Euclidean branes at the $D_{4}$ symmetric point
    2. 5.2 NS5 and KK(6,1)
  6. 6 Summary and conclusions
  7. A Hurwitz Quaternions and generalisations
  8. B Explicit helicity supertraces and BPS states indices
    1. B.1  $\mathds{Z}_{2}$ case
    2. B.2  $\mathds{Z}_{3}$ case
  9. C Elliptic functions and other useful formulae
  10. D Details on the 1-loop open string amplitude
  11. E Elliptic integrals
    1. E.1 Few simple integrals
    2. E.2 The integral of ${{{\mathcal{G}}}}(z){\mathcal{Y}}(z)$
  12. F Lattices and Eisenstein series
    1. F.1 $SL(2,\mathds{H})$ as $O(II_{1,1}\oplus D_{4})$
    2. F.2 $SL(2,\mathds{H})$ Eisenstein series
    3. F.3 $\Gamma^{\scalebox{0.6}{D}_{\scalebox{0.5}{4}}}_{0}(\alpha)\subset Sp(4,\mathds{H})$ as $O(II_{1,1}\oplus II_{1,1}[2]\oplus D_{4})$
    4. F.4 $Sp(4,\mathds{H})$ and $\Gamma^{\scalebox{0.6}{D}_{\scalebox{0.5}{4}}}_{0}(\alpha)$ Eisenstein series
  13. G Supergravity one-loop amplitude
  14. H The Dabholkar–Harvey $\mathds{Z}_{3}$ orbifold

##  1 Introduction

Supergravity theories with $\mathcal{N}$ extended supersymmetry admit
different ultra-violet completions in string theory. For $\mathcal{N}\geq 5$,
all the massless fields are in the graviton super-multiplet and there is no
lower-spin ‘matter’ super-multiplet. The two-derivative action is then
uniquely determined by supersymmetry and the scalar fields are moduli that
parametrise a symmetric space $G/K$ [1]. There is a unique string theory
ultra-violet completion of (ungauged) maximal $\mathcal{N}=8$ supergravity in
four dimensions.111The issue of its non-perturbative completion was addressed
in [2, 3]. This theory has been very useful in understanding dualities and
non-perturbative effects in string theory [4, 5], including D-brane and NS-
five-brane instantons. Supersymmetry and U-duality give very stringent
constraints on the low-energy effective action [6, 7, 8, 9, 10, 11, 12, 13,
14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], whose exact couplings are
known up to fourteen-derivative terms [26, 27, 28, 29, 30, 31, 32, 33, 34, 35,
36]. Such results offer an invaluable window into the non-perturbative regime
of string theory, like supersymmetric D-branes and their bound states. Similar
results have been obtained for theories with sixteen supercharges, i.e.
$\mathcal{N}=4$ in four dimensions [37, 38, 39, 40, 41, 42]. Another
interesting example is $\mathcal{N}=6$ supersymmetry in four dimensions. Such
theories provide the simplest examples of compactifications on T-folds [43],
and can therefore be used to study D-branes and NS-five-branes on non-
geometric backgrounds. One may wonder: how many independent string theories
give (ungauged) $\mathcal{N}=6$ supergravity at low energy? What is their
group of U-duality symmetries as an arithmetic subgroup of the non-compact
global $SO^{*}(12)$ symmetry group? Can one determine supersymmetry protected
couplings exactly using constraints from supersymmetry and U-duality? What is
the spectrum of supersymmetric D-branes in these theories?

The first construction of a string theory with $\mathcal{N}=6$ supersymmetry
was proposed by Ferrara and Kounnas in [44] as a model of free fermions. This
model can also be formulated as an asymmetric orbifold of $(T^{2}\times
T^{4})/\mathds{Z}_{2}$ in which $\mathds{Z}_{2}$ acts on the left moving
sector of $T^{4}$ as for the K3 orbifold conformal field theory and as a shift
on one additional circle in $T^{2}$. The D-branes and the four-graviton
amplitude have been analysed in [45, 46]. In this paper we refine this
analysis, with particular emphasis on the dependence on the NS-NS moduli
(internal metric and B-field)222The NS-NS moduli are the internal metric and
$B$ fields preserved by the orbifold that parametrise the Grassmannian
$SO(2,6)/(SO(2)\times SO(6))$ and the axio-dilaton that parametrise
$SL(2)/SO(2)$. We will sometimes call the Grassmannian coordinates ‘Narain
moduli’ [47]. The perturbative amplitudes only depend on the dilaton and the
Narain moduli, while the NS-NS axion and the R-R moduli do not contribute due
to shift symmetries resulting from their ‘axionic’ nature. of the couplings
and their automorphic symmetry under the T-duality group. We also demonstrate
that there is a similar string theory defined as an asymmetric orbifold of
$(T^{2}\times T^{4})/\mathds{Z}_{3}$. The obvious generalisation to
$\mathds{Z}_{4}$ and $\mathds{Z}_{6}$ orbifold theories does not work,
contrary to naive expectations. Another string theory with $\mathcal{N}=6$
supersymmetry was identified in [48] as an asymmetric orbifold in which
$\mathds{Z}_{3}$ acts on the left moving sector of $T^{6}$ as for the
corresponding Calabi–Yau orbifold conformal field theory. We find that this
theory has exactly the same (perturbative) spectrum as the $(T^{2}\times
T^{4})/\mathds{Z}_{3}$ and therefore conclude that they are dual to each
other. It seems therefore that there are only two inequivalent string theories
with $\mathcal{N}=6$ supersymmetry.

The bosonic sector of $\mathcal{N}=6$ supergravity is identical to the bosonic
sector of the $\mathcal{N}=2$ supergravity theory associated to the Jordan
algebra of Hermitian three by three matrices over the quaternions [49].333The
string version of the ‘magic’ $\mathcal{N}=2$ supergravity associated to the
octonionic algebra with $E_{7(-25)}$ global symmetry was constructed in [50].
In particular the scalar fields parametrise the special Kähler symmetric space
$SO^{*}(12)/U(6)$, whose coordinates can be defined as complexified ${3\times
3}$ Hermitian matrices $T$ over the quaternions. The duality symmetry acts on
this space by linear fractional transformations and the theory of modular
forms on $SO^{*}(12)/U(6)$ involves the choice of a discrete ring over the
quaternions admitting prime factorisation [51, 52]. We find that the relevant
ring for the $\mathds{Z}_{2}$ orbifold theory is the ring $\mathds{H}(2)$ of
Hurwitz quaternions isomorphic to the $D_{4}$ root lattice, while the relevant
ring for the $\mathds{Z}_{3}$ orbifold theory is the ring $\mathds{H}(3)$ of
Eisenstein quaternions (obtained through the Cayley–Dickson construction from
a pair of Eisenstein integers) isomorphic to the $A_{2}\oplus A_{2}$ root
lattice.

Following the discussion of [32] for the four-graviton amplitude in the
maximally supersymmetric theory, the type II superstring four-graviton
amplitude for the $\mathds{Z}_{K}$ asymmetric orbifolds with $K=2,3$ can be
decomposed at low energy as

$\mathcal{M}_{\scalebox{0.6}{type
II}}(T,s,t,u)=\mathcal{M}^{\scalebox{0.6}{sugra}}_{\scalebox{0.6}{type
II}}(T,s,t,u,\mu)+\mathcal{M}^{\scalebox{0.6}{Wilson}}_{\scalebox{0.6}{type
II}}(T,s,t,u,\mu)$ (1.1)

where $\mu$ is a regulating mass scale. As usual we factor out
$(2\pi)^{4}\delta(\sum_{a}k_{a})$ of the amplitude. The analytic term
$\mathcal{M}^{\scalebox{0.6}{Wilson}}_{\scalebox{0.6}{type II}}$ is
interpreted as defining the Wilsonian effective action obtained by integrating
out all the massive fields in string theory, whereas the first component
$\mathcal{M}^{\scalebox{0.6}{sugra}}_{\scalebox{0.6}{type II}}$ is not
analytic in the Mandelstam variables and corresponds to the supergravity
amplitude computed with the Wilsonian effective action. The former expands in
the Mandelstam variables as

$i\mathcal{M}^{\scalebox{0.6}{Wilson}}_{\scalebox{0.6}{type
II}}=\frac{\pi}{2^{10}}g_{\scalebox{0.6}{4}}^{8}\alpha^{\prime
4}\sum_{m,n,p\in\mathds{N}}\mathcal{E}_{\scriptscriptstyle(m,n,p)}(T)\bigl{(}\scalebox{1.0}{$\frac{g_{\scalebox{0.5}{4}}^{2}\alpha^{\prime}}{4}$}\bigr{)}^{\\!\scalebox{0.6}{$m{+}2n{+}3p$}}(s^{2}{+}t^{2}{+}u^{2})^{n}(s^{3}{+}t^{3}{+}u^{3})^{p}(s^{m}t^{s}_{8}+t^{m}t_{8}^{t}+u^{m}t_{8}^{u})t_{8}R^{4}$
(1.2)

where we included the Weyl rescaling to Einstein frame that involves the four-
dimensional effective string coupling constant

$g_{\scalebox{0.6}{4}}^{2}=\frac{K(2\pi)^{6}\alpha^{\prime 3}}{V(T^{6})}g_{\rm
s}^{\;2}$ (1.3)

with $g_{\rm s}$ the string coupling constant in ten dimensions and $V(T^{6})$
the volume of the six-torus, so much so that the volume of the orbifold is
$V(T^{6}/\mathds{Z}_{K})=V(T^{6})/K$. Denoting by $R_{\scalebox{0.6}{s}}$ and
$\tilde{R_{\scalebox{0.6}{s}}}$ the dimensionless radii of $T^{2}$ one finds
that

$g_{\scalebox{0.6}{4}}=\frac{2}{\sqrt{R_{\scalebox{0.6}{s}}\tilde{R_{\scalebox{0.6}{s}}}}}g_{\rm
s}\;.$ (1.4)

The gravitational coupling $\kappa$ is given in these conventions by

$\kappa^{2}=\pi\alpha^{\prime}g_{\scalebox{0.6}{4}}^{2}\;.$ (1.5)

The effective couplings $\mathcal{E}_{\scriptscriptstyle(m,n,p)}(T)$ are
U-duality invariant functions of the moduli. For maximal supersymmetry, only
the terms with $m=0$ appear in the low energy expansion and

$t_{8}=t^{s}_{8}+t_{8}^{t}+t_{8}^{u}$ (1.6)

is the Lorentz invariant tensor, ubiquitous in 4-point amplitudes [53], while
$t_{8}^{r}$ for $r=s$, $t$ and $u$ are the projections to the respective
channels defined in equation (3.6). The non-analytic component expands
similarly as

$\mathcal{M}^{\scalebox{0.6}{sugra}}_{\scalebox{0.6}{type
II}}(T,s,t,u,\mu)=-\frac{i}{2^{4}}\kappa^{2}t_{8}t_{8}R^{4}\frac{1}{stu}+\kappa^{4}\mathcal{M}^{\scalebox{0.6}{1-loop}}_{\scalebox{0.6}{sugra}}(s,t,u,\mu)+\kappa^{6}\mathcal{M}^{\scalebox{0.6}{2-loop}}_{\scalebox{0.6}{sugra}}(s,t,u,\mu)+\dots$
(1.7)

The onset of moduli dependence appears at order $\alpha^{\prime 5}$ with the
supergravity form-factor of the
$\mathcal{E}_{\scriptscriptstyle(0,0,0)}(T)R^{4}$ supersymmetric completion.
The fiducial scale $\mu$ is introduced to keep track of the ambiguity in the
split (1.1) associated to the presence of logarithmic divergences in the
supergravity amplitudes and form factors. Its counterpart in the Wilsonian
action gives an ambiguity in the definition of some couplings
$\mathcal{E}_{\scriptscriptstyle(m,n,p)}(T)$ in terms of lower order ones, in
which case we write $\mathcal{E}_{\scriptscriptstyle(m,n,p)}(T,\mu)$. There is
no ultra-violet divergence in $\mathcal{N}=6$ supergravity up to four-loop
included [54], and comparison with $\mathcal{N}=4$ and $5$ suggests that the
five-loop amplitude should be finite as well [55, 56, 57]. We will find
however that the $\mathcal{E}_{\scriptscriptstyle(0,0,0)}(T)R^{4}$ form factor
diverges at one-loop.

In $\mathcal{N}=6$ supergravity in four dimensions, there are two super-
multiplets of 4-point amplitudes, the $U(1)$ preserving super-multiplet of the
four-graviton amplitude and the $U(1)$ violating super-multiplet of the two-
graviphoton (of negative helicities) two-graviton (of positive helicities)
amplitude. There is a similar analytic component of this two-graviphoton two-
graviton amplitude and factorising out the helicity structure for simplicity
one has

$i\mathcal{A}^{\scalebox{0.6}{Wilson}}_{\scalebox{0.6}{type II}}=\pi
g_{\scalebox{0.6}{4}}^{6}\alpha^{\prime
3}\sum_{m,n\in\mathds{N}}\mathcal{F}_{\scriptscriptstyle(m,n)}(T)\bigl{(}\scalebox{1.1}{$\frac{g_{\scalebox{0.5}{4}}^{2}\alpha^{\prime}}{4}$}\bigr{)}^{\\!\scalebox{0.7}{$2m{+}3n$}}(s^{2}{+}t^{2}{+}u^{2})^{m}(s^{3}{+}t^{3}{+}u^{3})^{n}\;,$
(1.8)

where the permutation symmetry in the Mandelstam variables follows by
supersymmetry. The effective couplings
$\mathcal{F}_{\scriptscriptstyle(m,n)}(T)$ are $U(1)$ weight $2$ modular forms
of the U-duality group.

The perturbative expansions of
$\mathcal{M}^{\scalebox{0.6}{Wilson}}_{\scalebox{0.6}{type II}}$ and
$\mathcal{A}^{\scalebox{0.6}{Wilson}}_{\scalebox{0.6}{type II}}$ are obtained
from the expansion of the string amplitude in $\alpha^{\prime}$ before
integration over the genus $h$ moduli space of Riemann surfaces with four
punctures. Although the string theory amplitude is finite, these couplings
must then be regularised in order to tame the supergravity divergences due to
the massless fields.

In the maximally supersymmetric theory,
$\mathcal{E}_{\scriptscriptstyle(0,0,0)}(\phi)$,
$\mathcal{E}_{\scriptscriptstyle(0,1,0)}(\phi)$,
$\mathcal{E}_{\scriptscriptstyle(0,0,1)}(\phi,\mu)$ are protected by
supersymmetry and only receive perturbative corrections in string theory up to
one, two and three-loop respectively. They have been the subject of extensive
studies and have been determined as automorphic functions of
$E_{7}(\mathds{Z})$ in [12, 18, 19, 34, 36].

Similarly we demonstrate that $\mathcal{F}_{\scriptscriptstyle(0,0)}(T)$,
$\mathcal{E}_{\scriptscriptstyle(0,0,0)}(T)$,
$\mathcal{F}_{\scriptscriptstyle(1,0)}(T)$ and
$\mathcal{E}_{\scriptscriptstyle(1,0,0)}(T,\mu)$ are protected by
supersymmetry in $\mathcal{N}=6$ supergravity. The first two only receive
perturbative corrections in string theory up to one loop, and the last two up
to two loops. Moreover
$\mathcal{F}_{\scriptscriptstyle(1,0)}(T)=\det\\!\mathcal{D}\,\mathcal{E}_{\scriptscriptstyle(1,0,0)}(T,\mu)$
by supersymmetry, where $\det\mathcal{D}$ is an $SU(6)$ invariant third order
differential operator defined in (3.50).444In many respects
$\mathcal{F}_{\scriptscriptstyle(0,0)}(T)$,
$\mathcal{E}_{\scriptscriptstyle(0,0,0)}(T)$, and
$\mathcal{E}_{\scriptscriptstyle(1,0,0)}(T,\mu)$ are respectively analogues of
$\mathcal{E}_{\scriptscriptstyle(0,0,0)}(\phi)$,
$\mathcal{E}_{\scriptscriptstyle(0,1,0)}(\phi)$ and
$\mathcal{E}_{\scriptscriptstyle(0,0,1)}(\phi,\mu)$ in the maximally
supersymmetric theory.

The main purpose of this paper is to compute the one-loop contribution to
these four BPS protected couplings $\mathcal{F}_{\scriptscriptstyle(0,0)}(T)$,
$\mathcal{E}_{\scriptscriptstyle(0,0,0)}(T)$,
$\mathcal{F}_{\scriptscriptstyle(1,0)}(T)$ and
$\mathcal{E}_{\scriptscriptstyle(1,0,0)}(T)$ in the $\mathds{Z}_{K}$ orbifold
theory for $K=2$ and $3$. Using ‘chiral’ factorisation of the one-loop
intergrands and relying on the results of [58] for 4-point open-string
amplitudes of gauge bosons in supersymmetric D-brane worlds, we derive rather
compact expressions for the 4-graviton and 2-graviton 2-graviphoton
amplitudes. We then find that the one-loop four-point amplitudes can be
simplified by partly unfolding the genus one modular domain to the fundamental
domain of the congruent subgroup $\Gamma_{0}(K)\subset SL(2,\mathds{Z})$. In
this form they can be written as integrals of $\Gamma_{0}(K)$ modular forms
times the partition function for the Narain lattice $\mathds{F}_{2}(K)$
defined as

$\mathds{F}_{2}(2)=II_{1,1}\oplus II_{1,1}[2]\oplus
D_{4}\;,\qquad\mathds{F}_{2}(3)=II_{1,1}\oplus II_{1,1}[3]\oplus A_{2}\oplus
A_{2}\;,$ (1.9)

where $II_{1,1}[K]$ is the Lorentzian circle Narain lattice with a momentum
$m\in K\mathds{Z}$. Therefore the symmetries of the four-point one-loop
amplitude are the automorphisms ${\rm Aut}(\mathds{F}_{2}(K))$ of the lattice
$\mathds{F}_{2}(K)$. We also find that the character valued partition function
that determines the perturbative spectrum in string theory555Similar
techniques apply to theories with higher spin fields [59]. [60] can be written
in terms of the Narain partition function of $\mathds{F}_{2}(K)$ and its dual
lattice $\mathds{F}_{2}(K)^{*}$, such that the group ${\rm
Aut}(\mathds{F}_{2}(K))$ is a symmetry of the full perturbative spectrum of
the theory. In particular we find that for each charge of vanishing norm in
$\mathds{F}_{2}(K)$ there is a single massive spin two 1/2 BPS super-
multiplets and for each charge of vanishing norm in
$\mathds{F}_{2}(K)^{*}\smallsetminus\mathds{F}_{2}(K)$ there are respectively
two and one massive spin three-half 1/2 BPS super-multiplets for $K=2$ and
$3$. The true perturbative symmetry of these string theories may only be a
proper subgroup of the group of automorphisms ${\rm Aut}(\mathds{F}_{2}(K))$,
but for brevity we will nevertheless refer to ${\rm Aut}(\mathds{F}_{2}(K))$
as the T-duality group.

The perturbative T-duality group ${\rm Aut}(\mathds{F}_{2}(K))$ will be
identified as a group of ${4\times 4}$ symplectic matrices over the
quaternions $\mathds{H}(K)$ that we call
$\Gamma^{\scalebox{0.6}{D}_{\scalebox{0.5}{4}}}_{0*}(\alpha)$, where $\alpha$
is a prime in $\mathds{H}(K)$ of norm square $K$. In this notation $D_{4}$
refers to $SO(2,6)$ and not to the lattice $\mathds{H}(2)$. For brevity we
write $\Gamma^{\scalebox{0.6}{D}_{\scalebox{0.5}{4}}}_{0*}(\alpha)$ for both
$K=2$ and $3$ such that
$\Gamma^{\scalebox{0.6}{D}_{\scalebox{0.5}{4}}}_{0}(\alpha)\subset\mathds{H}(K)$,
is the subgroup with a lower left two by two matrix divisible by $\alpha$ in
$\mathds{H}(K)$. We will moreover conjecture that the U-duality group is
defined similarly as
$\Gamma^{\scalebox{0.6}{D}_{\scalebox{0.5}{6}}}_{0*}(\alpha)={\rm
Aut}(\mathds{F}_{3}(K))$ for the lattice of non-perturbative states

$\mathds{F}_{3}(K)=K\mathds{F}_{2}(K)^{*}\oplus[\mathds{H}(K)\oplus\mathds{H}(K)\oplus\alpha\mathds{H}(K)\oplus\alpha\mathds{H}(K)]\oplus\mathds{F}_{2}(K)$
(1.10)

where the middle component $\mathds{H}(K)^{2}\oplus(\alpha\mathds{H}(K))^{2}$
is the lattice of D-brane R-R charges and $K\mathds{F}_{2}(K)^{*}$ the lattice
of NS5-branes and KK(6,1)-branes. Here KK(6,1) refers to the Kaluza–Klein
monopole brane wrapping $T^{6}$ with a Taub-NUT fibration along one circle in
$T^{6}$ [61]. We will identify these lattices as the set of brane charges
invariant under the $\mathds{Z}_{K}$ orbifold action, while non-perturbative
states will generally be in

$\mathds{F}^{*}_{3}(K)=\mathds{F}_{2}(K)\oplus[\tfrac{1}{\alpha}\mathds{H}(K)\oplus\tfrac{1}{\alpha}\mathds{H}(K)\oplus\mathds{H}(K)\oplus\mathds{H}(K)]\oplus\mathds{F}^{*}_{2}(K)\;.$
(1.11)

Because of the asymmetric action of $\mathds{Z}_{K}$ on $T^{4}$, the D-branes
are necessarily bound states of branes wrapping a cycle and its Poincaré dual
in $T^{4}$ [45]. The D-brane states of the orbifold theory therefore preserve
at most one third of the supersymmetries, i.e. eight supercharges.

We shall argue in particular that NS5-branes instantons have integer charges
as a winding number over $T^{6}/\mathds{Z}_{K}$. Assuming T-duality and some
minimal assumptions about the non-perturbative symmetry we shall identify the
non-perturbative coupling $\mathcal{F}_{\scriptscriptstyle(0,0)}$ as a the
theta series

$\mathcal{F}_{\scriptscriptstyle(0,0)}=-\frac{\pi}{3}\det T_{2}\sum_{{\bf
q}\in\mathds{H}(K)^{2}\oplus\frac{1}{\alpha}\mathds{H}(K)}e^{2\pi i{\bf
q}^{\dagger}T{\bf q}}\;.$ (1.12)

This is the unique rank one theta series invariant under the conjectured
U-duality group $\Gamma^{\scalebox{0.6}{D}_{\scalebox{0.5}{6}}}_{0*}(\alpha)$.
This 1/2 BPS protected coupling does not receive corrections from
D-instantons, consistently with the property that there is no 1/2 BPS
(Euclidean) D-brane.

The organisation of the paper is as follows. In Section 2 we review asymmetric
orbifolds that allow for string embeddings of ${\cal N}=6$ supergravity. We
prove that only $K=2,3$ give rise to perturbatively consistent models in
$D=4,5$ while $K=4,6$ don’t, contrary to what one would have naively expected.
We study the perturbative BPS states and emphasise the role of Hurwitz
quarternions in the definition of the BPS charges and the T-duality group. We
then turn our attention on supersymmetry constraints on threshold corrections
to the low-energy effective action in Section 3 and prove non-renormalisation
theorems in string theory for the threshold corrections to the
$\mathcal{R}^{4},D^{2}\mathcal{R}^{4}$ and
$\bar{\mathcal{F}}^{2}\mathcal{R}^{2}$ couplings. In Section 4, we compute the
one-loop threshold corrections to these couplings. In order to achieve our
goals, we exploit chiral factorisation of the integrands and rely on the
results of [58] for 4-point open-string amplitudes of vector bosons at one-
loop in vacuum configurations with (unoriented) intersecting and magnetised
D-branes preserving ${\cal N}\leq 4$ supersymmetry in $D=4$. In Section 4.5 we
rewrite the perturbative $\bar{\mathcal{F}}^{2}\mathcal{R}^{2}$ coupling as a
rank one theta series and conjecture its nonpertubative completion (1.12). We
discuss in more detail the D-brane and NS5-brane instantons in Section 5. We
conclude with a discussion of our results and an anticipation of forthcoming
papers on the non-perturbative $\mathcal{R}^{4},D^{2}\mathcal{R}^{4}$
couplings.

Several appendices contains technical details on Hurwitz quaternions, helicity
super-traces, elliptic functions, Eisenstein series, one-loop computations in
string theory, matching of normalisations with supergravity and equivalence
with another orbifold theory.

##  2 ${\cal N}=6$ supergravity in diverse dimensions

Maximal supergravity with 32 supercharges (i.e. ${\cal N}=8$ in $D=4$) can be
defined in any dimension $D\leq 11$. It is the low energy effective theory of
either M-theory on $\mathds{R}^{1,D-1}\times T^{11-D}$ or type IIA/B string
theory on $\mathds{R}^{1,D-1}\times T^{10-D}$. In $D=10$ type IIA and IIB are
respectively the non-chiral $({\cal N}_{L},{\cal N}_{R})=(1,1)$ and chiral
$({\cal N}_{L},{\cal N}_{R})=(2,0)$ supergravity theory. The consistency of
$({\cal N}_{L},{\cal N}_{R})=(2,0)$ supergravity inherited from type IIB
string theory relies on gravitational anomaly cancellation [62].

The maximal number of supercharges below 32 is 24 (i.e. ${\cal N}=6$ in
$D=4$). The classical theory can be defined in $D=6$ dimensions with $({\cal
N}_{L},{\cal N}_{R})=(1,2)$ as the consistent truncation of maximal
supergravity corresponding to the subgroup
$SU(2)_{L}\times_{\mathds{Z}_{2}}SU(2)_{R}\times SO(1,5)\subset SO(5,5)$,
where one sets to zero all the fields with a non-trivial $SU(2)_{L}$ weight.
The $SU(2)_{L}$ singlet sector can be understood as the untwisted sector of
the orbifold by a discrete subgroup $\mathds{Z}_{K}\subset SU(2)_{L}$ that can
be realised as an asymmetric $\mathds{Z}_{K}$ orbifold of $T^{4}$ in type II
string theory. The $({\cal N}_{L},{\cal N}_{R})=(1,2)$ supergravity theory is
plagued by gravitational anomalies and is inconsistent at the quantum level in
six dimensions. String theory ‘cures’ the problem thanks to the presence of a
gravitino multiplet in the twisted sector of the asymmetric $\mathds{Z}_{K}$
orbifold leading to supersymmetry enhancement and to maximal non-chiral
$({\cal N}_{L},{\cal N}_{R})=(2,2)$ supergravity. Only in dimension $D\leq 5$,
one can define a freely acting orbifold with the common asymmetric action of
$\mathds{Z}_{K}$ on $T^{4}$ and a shift on an additional circle $S^{1}$.

The theory in five dimensions is ${\cal N}=3$ supergravity666Here as above, in
$D=6$ and $D=10$, we count the number of super-symmetries in terms of the
relevant spinorial representation in $D=5$ with 8 real (4 complex) components.
and includes: one graviton, 6 gravitini, 15 vectors (after dualizing all
massless 2-forms), 10 dilatini and 14 scalars. The latter parameterize the
non-compact symmetric space ${\cal
M}^{D=5}_{\mathcal{N}=6}=SU^{*}(6)/Sp(3)$.777We write $Sp(3)=USp(6)$ the
subgroup of $SU(6)$ preserving a symplectic form.

The $\mathcal{N}=6$ theory in four dimensions can be obtained by dimensional
reduction from $D=5$, the massless content is: one graviton, 6 gravitini, 16
vectors, 26 dilatini and 30 scalars. The latter parameterize the non-compact
symmetric space ${\cal M}^{D=4}_{\mathcal{N}=6}=SO^{*}(12)/U(6)$ [63]. Quite
remarkably ${\cal M}^{D=4}_{\mathcal{N}=6}$ is a special Kähler manifold,
which allows $\mathcal{N}=6$ supergravity to have the same bosonic sector as a
peculiar $\mathcal{N}=2$ supergravity coupled to 15 vector multiplets [49].

Further reduction to $D=3$ produces a (non-propagating) graviton, 6 (non-
propagating) gravitini, 64 dilatini and 64 scalars (after dualizing all 17
vectors). The latter parameterize the non-compact quaternionic-Kähler
symmetric space ${\cal
M}^{D=3}_{\mathcal{N}=6}=E_{7(-5)}/(SU(2)\times_{\mathds{Z}_{2}}Spin(12))$.

###  2.1 String Theory on (freely acting) asymmetric orbifolds

Due to the large amount of supersymmetry, only Type II strings admit
perturbative vacuum configurations with $\mathcal{N}=6$ supersymmetry [44].
The easiest way to achieve the breaking of $\mathcal{N}=8$ to $\mathcal{N}=6$
is to rely on asymmetric orbifolds [64, 65] that preserve all supercharges in
one sector (say R-moving) and break half of them in the other sector (say
L-moving). One can accomplish the task with a combination of ‘asymmetric’
twists $\tau_{L,R}$ and shifts $\sigma_{L,R}$ [66, 67]. In the simplest
${\mathds{Z}}_{2}$ case, $\tau_{L}$ reflects 4 internal Left-moving
coordinates $\tau_{L}X_{L}^{i}=-X^{i}_{L}$ as well as their (world-sheet)
fermionic partners $\tau_{L}\Psi_{L}^{i}=-\Psi^{i}_{L}$, while $\sigma$ acts
‘geometrically’ by an order 2 shift on another (compact circle) direction:
$\sigma X=X+\pi\sqrt{\alpha^{\prime}}\tilde{R_{\scalebox{0.6}{s}}}$ where
$2\pi\sqrt{\alpha^{\prime}}\tilde{R_{\scalebox{0.6}{s}}}$ is the
periodicity.888In general $\sigma$ can act in a non L-R symmetric fashion,
compatibly with modular invariance. All these cases are however related by the
T-duality of the original theory and can be obtained by field redefinitions.
It will be convenient to consider a geometric shift for which the Narain
moduli are all in the untwisted sector. In this way the twisted sector does
not contribute new massless states and supersymmetry is broken from
$\mathcal{N}=4=2_{L}+2_{R}$ (maximal) to $\mathcal{N}=3=1_{L}+2_{R}$ in $D=5$.
The resulting construction is the simplest possible (supersymmetric) T-fold
[43]. This construction straightforwardly generalises to $D=4$ or $D=3$ by
compactification on additional circles.

For later use, let us describe the construction in more details, relying on
the characters of $SO(2n)$ current algebra at level one

$O_{2n}={\vartheta_{3}^{n}+\vartheta_{4}^{n}\over 2\eta^{n}}\,,\quad
V_{2n}={\vartheta_{3}^{n}-\vartheta_{4}^{n}\over 2\eta^{n}}\,,\quad
S_{2n}={\vartheta_{2}^{n}+i^{n}\vartheta_{1}^{n}\over 2\eta^{n}}\,,\quad
C_{2n}={\vartheta_{2}^{n}-i^{n}\vartheta_{1}^{n}\over 2\eta^{n}}\;,$ (2.1)

where $\vartheta_{\alpha}$ are Jacobi (elliptic) theta functions and $\eta$ is
the Dedekind function.

One defines the weight $(d/2,d/2)$ modular form $\mathcal{Z}$ associated to a
partition function $Z$ via an integral

$Z={\rm
Tr}[(-1)^{F}]=\int_{-\frac{1}{2}}^{\frac{1}{2}}d\tau_{1}\;\mathcal{Z}\;,$
(2.2)

that implements the level-matching condition.999The vacuum energy would be
$V_{{\rm
eff}}\propto\int_{\mathcal{F}}{d^{2}\tau}\tau_{2}^{\frac{d}{2}-2}\mathcal{Z}$.

The Type II one-loop partition functions on $T^{d}$ then compactly read
101010Here we do not include the divergent $D$-dimensional zero mode
contribution.

$\mathcal{Z}_{{\rm
IIA}}=\frac{\overline{V}_{8}-\overline{C}_{8}}{\overline{\eta}^{8}}\frac{V_{8}-S_{8}}{\eta^{8}}\Lambda_{II_{d,d}}\quad,\quad\mathcal{Z}_{{\rm
IIB}}=\frac{\overline{V}_{8}-\overline{S}_{8}}{\overline{\eta}^{8}}\frac{V_{8}-S_{8}}{\eta^{8}}\Lambda_{II_{d,d}}\;,$
(2.3)

where one identifies the contribution from the fermions ${\cal Q}=V_{8}-S_{8}$
and ${\cal Q}^{\prime}=V_{8}-C_{8}$ as the supersymmetric characters in
$D=10$, whereby $SO(8)$ acts as the little group. Here $\Lambda_{II_{d,d}}$ is
the Narain partition function for the Lorentzian lattice $II_{d,d}$.

The $\mathds{Z}_{2}$ breaking of supersymmetry in $D=6$ is achieved on the
fermionic component via [68, 69]

${\cal Q}=V_{8}-S_{8}=V_{4}O_{4}+O_{4}V_{4}-S_{4}S_{4}-C_{4}C_{4}\rightarrow
V_{4}O_{4}-O_{4}V_{4}-S_{4}S_{4}+C_{4}C_{4}={\cal Q}_{o}-{\cal Q}_{v}\;,$
(2.4)

with the ‘untwisted’ supersymmetric characters given by

${\cal Q}_{o}=V_{4}O_{4}-S_{4}S_{4}\quad,\quad{\cal
Q}_{v}=O_{4}V_{4}-C_{4}C_{4}\;.$ (2.5)

Performing an $S$-modular transformation ($\tau\rightarrow-1/\tau$) on ${\cal
Q}_{o}$ produces two ‘twisted’ supersymmetric characters

${\cal Q}_{s}=O_{4}S_{4}-C_{4}O_{4}\quad,\quad{\cal
Q}_{c}=V_{4}C_{4}-S_{4}V_{4}\;,$ (2.6)

while ${\cal Q}_{s}$ accommodates massless states, ${\cal Q}_{c}$ doesn’t.

As already mentioned, in $D=6$ the twisted sector produces two gravitini
multiplets that enhance supersymmetry back to 32 supercharges, i.e.
$(\mathcal{N}_{L},\mathcal{N}_{R})=(2,2)$. In $D\leq 5$ one can prevent the
appearance of twisted massless states by introducing the order 2 ‘shift’
$\sigma$ defined above. In particular starting with $T^{4+d}=T^{4}\times
T^{d}$ with $d=6-D$ and $T^{4}$ the maximal torus of $SO(8)=D_{4}$, one can
immediately determine the invariant lattice and the perturbative charge
lattice of the model.

Other possibilities may be envisaged to break $\mathcal{N}=8$ to
$\mathcal{N}=6$ in $D<6$ that rely on other asymmetric $\mathds{Z}_{K}$
orbifolds with $K=3,4,6$, that act on the supersymmetric character ${\cal Q}$
according to

${\cal Q}\rightarrow\sum_{k=0}^{K-1}\omega^{k}{\cal Q}_{k}\quad{\rm
with}\quad\omega=e^{2\pi i\over K}\;.$ (2.7)

For this purpose one further decomposes the second $SO(4)$ current algebra
character into $SU(2)\times SU(2)^{\prime}$ characters as

$\mathcal{Q}=V_{4}\chi_{0}\chi_{0}^{\prime}+O_{4}\chi_{\frac{1}{2}}\chi_{\frac{1}{2}}^{\prime}-S_{4}\chi_{\frac{1}{2}}\chi_{0}^{\prime}-C_{4}\chi_{0}\chi_{\frac{1}{2}}^{\prime}=\mathcal{V}\chi_{0}^{\prime}+{\cal
H}\chi_{\frac{1}{2}}^{\prime}\;,$ (2.8)

where one identifies the vector multiplet super-character ${\cal
V}=V_{4}\chi_{0}-S_{4}\chi_{1/2}$, the (half) hypermultiplet super-character
${\cal H}=O_{4}\chi_{1/2}-C_{4}\chi_{0}$, and $SU(2)$ as the R-symmetry group.
The orbifold acts on the $SU(2)^{\prime}$ characters $\chi_{s}^{\prime}$ only.
For instance for $K=3$ the $\mathds{Z}_{3}$ orbifold acts on the
$SU(2)^{\prime}$ characters as

$\chi_{0}^{\prime}\rightarrow\xi_{0}+\bar{\omega}\xi_{+4}+\omega\xi_{-4},\quad\chi_{1/2}^{\prime}\rightarrow\omega\xi_{+2}+\bar{\omega}\xi_{-2}+\xi_{6}\quad,$
(2.9)

where $\xi_{p}$ are the characters of a free boson with radius
$R=\sqrt{6\alpha^{\prime}}$ and ${\cal Q}_{0}={\cal V}\xi_{0}+{\cal
H}\xi_{6}$, ${\cal Q}_{1}={\cal H}\xi_{2}+{\cal V}\xi_{-4}$ and ${\cal
Q}_{2}={\cal H}\xi_{-2}+{\cal V}\xi_{+4}$. Starting from the boson partition
function on $T^{4}$ at the $SU(3)\times SU(3)$ symmetric point

$\frac{\Lambda_{II_{4,4}}}{\eta^{2}\bar{\eta}^{2}}=\bigl{(}|\chi_{[00]}|^{2}+|\chi_{[10]}|^{2}+|\chi_{[01]}|^{2}\bigr{)}^{2}\;,$
(2.10)

and acting with $\mathds{Z}_{3}$ only on the Left-movers one finds

$\frac{\Lambda_{II_{4,4}}}{\eta^{2}\bar{\eta}^{2}}\rightarrow\bigl{(}|\chi_{[00]}|^{2}+\omega|\chi_{[10]}|^{2}+\bar{\omega}|\chi_{[01]}|^{2}\bigr{)}^{2}\;.$
(2.11)

In general one may consider a subgroup $\mathds{Z}_{K}\subset SU(2)$ acting on
$T^{4}$. Consistency of the asymmetric orbifold requires that the number of
‘chiral’ fixed points $N_{\rm f}=4\sin(\pi/K)^{2}$ be an integer, which is the
case only for $K=2,3,4,6$. For $K=4$ one starts with the maximal torus of
$SU(2)^{4}$. For $K=6$ from $SU(3)_{1}\times SU(3)_{2}$, with the second
$SU(3)$ current algebra at level $2$. The lattices for $K=2,3,4,6$ are
therefore $D_{4}$, $A_{2}\oplus A_{2}$, $A_{1}\oplus A_{1}\oplus A_{1}\oplus
A_{1}$ and $A_{2}\oplus A_{2}[2]$, respectively. Quite remarkably, these
lattices can be realised as integral quaternions and admit in this way a non-
abelian ring structure. We will therefore denote these lattices as
$\mathds{H}(K)$, following the definition (A.18). The ring $\mathds{H}(2)$,
that we will often abreviate as $\mathds{H}$, is the ring of Hurwitz
quaternions, $\mathds{H}(3)$ is a generalisation of the Eisenstein integers
that we call the Eisenstein quaternions, while $\mathds{H}(4)$ and
$\mathds{H}(6)$ are subrings of the Hurwitz quaternions. This is consistent
with the property that the duality group in $\mathcal{N}=6$ is naturally
defined over the quaternions, and the T-duality group is a theta congruent
subgroup of $SL(2,\mathds{H}(K))$ and $Sp(4,\mathds{H}(K))$ in five and four
dimensions dimensions.

Yet the condition that $N_{\rm f}$ be an integer is not enough for the
asymmetric orbifold to be consistent, and one needs to check that the number
of bosonic ground states in the twisted sectors be an integer too [64, 65].
Indeed, the twisted sectors of the asymmetric orbifold are only defined
abstractly by modular invariance, and one must verify that this construction
can be realised on a Hilbert space. The number of bosonic ground states is
determined by $N_{\rm f}=\sqrt{\det(1-g)}$ (where $g$ is the L-R symmetric
extension of the twist $\tau_{L}$) divided by the square root of the index
$|\Lambda^{*}/\Lambda|$ of the $\tau_{L}$ invariant lattice $\Lambda\subset
II_{4,4}$ [64, 65]. For the $\mathds{Z}_{K}$ asymmetric orbifold discussed
above, $\Lambda=\mathds{H}(K)$ and $|\mathds{H}(K)^{*}/\mathds{H}(K)|=K^{2}$.
One obtains

$\sqrt{\frac{\det(1-g)}{|\mathds{H}(K)^{*}/\mathds{H}(K)|}}=\frac{4\sin(\pi/K)^{2}}{K}\underset{K=2,3,4,6}{=}\\{2,1,\tfrac{1}{2},\tfrac{1}{6}\\}$
(2.12)

and the asymmetric orbifold is only consistent for $K=2$ and $3$. We will find
more explicitly below that $4\sin^{2}(\pi/K)/K$ is the number of massive 1/2
BPS gravitini multiplets for a given charge of vanishing norm in the Narain
lattice in the first twisted sector. To this end we will now turn our
attention onto the identification and counting of (perturbative) BPS states
via character-valued partition functions and helicity super-traces [27, 28,
60].

###  2.2 Character-valued partition functions and helicity super-traces

Thanks to Jacobi aequatio satis abstrusa, the 10-D super-characters ${\cal Q}$
and ${\cal Q}^{\prime}$ vanish, which implies equal number of bosons and
fermions at each mass level and ensures the absence of tachyons [53]. As a
consequence, the one-loop contribution to the vacuum energy is zero, which
guarantees quantum stability of flat ten-dimensional space-time and of
toroidal compactifications thereof. Actually supersymmetry is more powerful in
that also 1-, 2- and 3-point amplitudes are zero for massless external states
both in Type IIA and B as well as in Type I and Heterotic strings. The first
non-trivial amplitude with only massless external states at one-loop is a
4-point amplitude. In type II, the prototypical case is the 4-graviton
amplitude. After partial supersymmetry breaking, lower-point amplitudes with
(non-conserved) complex momenta are non-zero but some are still protected. The
easiest way to identify the special amplitudes and derive the associated (on-
shell) effective action is to introduce the concept of ‘character valued
partition functions’ and helicity super-traces [27, 28, 60].

Recalling that the (Left-moving) Lorentz generators are given by

$J^{MN}=\Psi^{M}\Psi^{N}+X^{M}\partial X^{N}-X^{N}\partial X^{M}$ (2.13)

one can consider including a coupling to the zero-modes of $J^{MN}$ in the
Cartan subalgebra, i.e. replace $Z={\rm Tr}[(-)^{F}]$ with $Z={\rm
Tr}[(-)^{F}\prod_{a}e^{2\pi iv_{a}J_{a}}]$. For the theory in four dimensions,
we only need to include the zero mode $J_{3}=J^{12}$ in the four-dimensional
little group and that does not act on the compact bosons, so we consider
$Z={\rm Tr}[(-)^{F}e^{2\pi ivJ_{3}}]$. In practice, this amounts to including
a fugacity in the partition function. Writing the maximally supersymmetric
Left-mover partition function as a sum over spin structures

$\frac{V_{8}-S_{8}}{\eta^{8}}=\frac{1}{2}\Bigl{[}\frac{\vartheta_{3}(0)^{4}}{\eta^{12}}-\frac{\vartheta_{4}(0)^{4}}{\eta^{12}}-\frac{\vartheta_{2}(0)^{4}}{\eta^{12}}-\frac{\vartheta_{1}(0)^{4}}{\eta^{12}}\Bigr{]}\;.$
(2.14)

and including the fugacity produces for each spin structure $\alpha=1,2,3,4$
111111In general one would have
${\vartheta_{\alpha}(0)^{4}\over\eta^{12}}\rightarrow\prod_{a}2\sin\pi
v_{a}{\vartheta_{\alpha}(v_{a})\over\vartheta_{1}(v_{a})}=\prod_{a}{\vartheta_{\alpha}(v_{a})\over\eta^{3}}{\sin\pi
v_{a}\vartheta_{1}^{\prime}(0)\over\pi\vartheta_{1}(v_{a})}=\prod_{a}{\xi(v_{a})\vartheta_{\alpha}(v_{a})\over\eta^{3}}$
where $\xi(v_{a})={\sin\pi
v_{a}\vartheta_{1}^{\prime}(0)/\pi\vartheta_{1}(v_{a})}\;.$

${\vartheta_{\alpha}(0)^{4}\over\eta^{12}}\rightarrow 2\sin\pi
v{\vartheta_{\alpha}(v)\over\vartheta_{1}(v)}{\vartheta_{\alpha}(0)^{3}\over\eta^{9}}={\sin\pi
v\vartheta_{1}^{\prime}(0)\over\pi\vartheta_{1}(v)}{\vartheta_{\alpha}(v)\over\eta^{3}}{\vartheta_{\alpha}(0)^{3}\over\eta^{9}}=\xi(v){\vartheta_{\alpha}(v)\vartheta_{\alpha}(0)^{3}\over\eta^{12}}$
(2.15)

where

$\xi(v)={\sin\pi v\,\vartheta_{1}^{\prime}(0)\over\pi\vartheta_{1}(v)}\;.$
(2.16)

Summing over the spin structures with the help of Riemann identities yields

$\displaystyle\frac{V_{8}-S_{8}}{\eta^{8}}$ $\displaystyle\rightarrow$
$\displaystyle\frac{\xi(v)}{2}\Bigl{(}\frac{\vartheta_{3}(v)\vartheta_{3}(0)^{3}}{\eta^{12}}-\frac{\vartheta_{4}(v)\vartheta_{4}(0)^{3}}{\eta^{12}}-\frac{\vartheta_{2}(v)\vartheta_{2}(0)^{3}}{\eta^{12}}-\frac{\vartheta_{1}(v)\vartheta_{1}(0)^{3}}{\eta^{12}}\Bigr{)}$
(2.17) $\displaystyle=$
$\displaystyle\frac{\xi(v)\vartheta_{1}(\frac{v}{2})^{4}}{\eta^{12}}\;,$

that correctly reproduces the (vanishing) partition function in the limit
$v\rightarrow 0$, using $\vartheta_{1}^{\prime}(0)=2\pi\eta^{3}$.

Taking derivatives with respect to $v$ and then setting it to zero one brings
down powers of $J_{3}$ and obtains what are known as helicity super-traces,
denoted by $B_{n}$. Indeed recall that the four Cartan generators of $SO(8)$
represent the ‘helicities’ in the four planes transverse to the light-cone in
ten dimensions, and $J_{3}$ determines the four-dimensional helicity. For
massive states in four dimensions $J_{3}$ is the Cartan generator of the
$SU(2)$ little group. In five dimensions one must distinguish the Left- and
Right-mover $J_{3L,R}$ that define the Cartan generators of the
$SU(2)_{L}\times SU(2)_{R}$ little group of massive states. One defines
accordingly the ‘character-valued’ partition function $\mathcal{Z}(v,\bar{v})$
and introduces the differential operators

${{\cal H}}={1\over 2\pi i}{\partial\over\partial v}\;,\quad\bar{{{\cal
H}}}=-{1\over 2\pi i}{\partial\over\partial\bar{v}}\;,$ (2.18)

such that

${\cal B}_{2n}=({{\cal H}}+\bar{{{\cal H}}})^{2n}{\cal
Z}(v,\bar{v})|_{v=\bar{v}=0}\;.$ (2.19)

The helicity supertrace is then defined by implementing the level matching
condition as

$B_{2n}={\rm
Tr}[(-1)^{2J_{3}}J_{3}^{\,2n}]=\int_{-\frac{1}{2}}^{\frac{1}{2}}d\tau_{1}\;{\cal
B}_{2n}\;.$ (2.20)

$B_{2n+1}=0$ by CPT symmetry. By construction $B_{0}=0$ in all supersymmetric
theories with a supersymmetric ground state, since BPS multiplets have the
same number of bosonic and fermionic degrees of freedom. For $\mathcal{N}$
extended supersymmetry with $\mathcal{N}$ even in four dimensions, one finds
that $B_{2n}=0$ for $2n<\mathcal{N}$, and $B_{\mathcal{N}}$ only receives
contributions from 1/2 BPS states in the theory. The non-BPS long
supermultiplets only contribute to $B_{2n}$ for $n\geq\mathcal{N}$, and the
helicity supertraces for $\mathcal{N}/2\leq n<\mathcal{N}$ are BPS protected
observables of the theory. The corresponding ‘modular forms’
$\mathcal{B}_{2n}$ are associated to BPS protected threshold corrections.

In particular the leading one-loop $\mathcal{R}^{4}$ type supersymmetry
invariant in the low-energy effective action of Type II strings in $D=10$ and
in toroidal compactifications is associated to the helicity super-trace ${\cal
B}_{8}$, the ${D}^{4}\mathcal{R}^{4}$ to ${\cal B}_{12}$ and the
${D}^{6}\mathcal{R}^{4}$ to ${\cal B}_{14}$.121212Similarly the one-loop
$F^{4}$ supersymmetry invariant in Type I and Heterotic strings is related to
${\cal B}_{4}$ and ${D}^{2}F^{4}$ to ${\cal B}_{6}$ [38, 40, 41, 42].

###  2.3 The asymmetric orbifold helicity supertraces

The ‘character-valued’ partition function can be obtained similarly for the
Type II asymmetric $\mathds{Z}_{K}$ orbifolds. We will concentrate on the
relevant theories with $K=2,3$, although the first part of this section
applies to the ‘inconsistent’ cases of $K=4,6$, too.

The contribution from the bosonic zero modes in the untwisted sector with the
insertion of $(\tau_{L},\sigma)^{r}\in\mathds{Z}_{K}$ for $r=1$ to $K-1$ gives
the Narain lattice partition function

$\Lambda_{II_{2,2}\oplus\mathds{H}(K)}[^{0}_{r}]=\Lambda_{II_{2,2}\oplus\mathds{H}(K)}[e^{\frac{2\pi
ir\tilde{m}}{K}}]=\sum_{m,n,\tilde{m},\tilde{n}\in\mathds{Z}}\sum_{q\in\mathds{H}(K)}e^{\frac{2\pi
ir\tilde{m}}{K}}e^{\pi i\tau p_{L}(m,n,\tilde{m},\tilde{n},q)^{2}-\pi
i\bar{\tau}p_{R}(m,n,\tilde{m},\tilde{n},q)^{2}}$ (2.21)

where $\tilde{m}$ is the momentum along the circle on which $\sigma$ acts as a
geometric shift $\sigma
X=X+\frac{2\pi}{K}\sqrt{\alpha^{\prime}}\tilde{R_{\scalebox{0.6}{s}}}$. There
is no loss of generality in choosing the geometric shift along a particular
circle in $T^{2}$ since the other cases could be obtained by T-duality.

Since the current $J_{3}$ does not act on the fields affected by the orbifold,
the contribution from the Left-mover oscillators in each spin structure
$\alpha$ is modified as in (2.15) as

$4\sin^{2}(\tfrac{\pi
r}{K})\frac{\vartheta_{\alpha}(0)^{2}\vartheta_{\alpha}(\frac{r}{K})^{2}}{\vartheta_{1}(\frac{r}{K})^{2}\eta^{6}}\rightarrow
4\sin^{2}(\tfrac{\pi
r}{K})\frac{\xi(v)\vartheta_{\alpha}(\frac{v}{2})\vartheta_{\alpha}(0)\vartheta_{\alpha}(\frac{r}{K})^{2}}{\vartheta_{1}(\frac{r}{K})^{2}\eta^{6}}$
(2.22)

and they recombine accordingly as in (2.17)

$\displaystyle\frac{\xi(v)}{2}\Bigl{(}\frac{\vartheta_{3}(\frac{v}{2})\vartheta_{3}(0)\vartheta_{3}(\frac{r}{K})^{2}}{\vartheta_{1}(\frac{r}{K})^{2}\eta^{6}}-\frac{\vartheta_{4}(\frac{v}{2})\vartheta_{4}(0)\vartheta_{4}(\frac{r}{K})^{2}}{\vartheta_{1}(\frac{r}{K})^{2}\eta^{6}}-\frac{\vartheta_{2}(\frac{v}{2})\vartheta_{2}(0)\vartheta_{2}(\frac{r}{K})^{2}}{\vartheta_{1}(\frac{r}{K})^{2}\eta^{6}}-\frac{\vartheta_{1}(\frac{v}{2})\vartheta_{1}(0)\vartheta_{1}(\frac{r}{K})^{2}}{\vartheta_{1}(\frac{r}{K})^{2}\eta^{6}}\Bigr{)}$
(2.23) $\displaystyle=$
$\displaystyle\frac{\xi(v)\vartheta_{1}(\frac{v}{2})^{2}\vartheta_{1}(\frac{v}{2}+\frac{r}{K})\vartheta_{1}(\frac{v}{2}-\frac{r}{K})}{\vartheta_{1}(\frac{r}{K})^{2}\eta^{6}}\;,$

in such a way that the total contribution from the untwisted sector with
insertion of $(\tau_{L},\sigma)^{r}\in\mathds{Z}_{K}$ is

$4\sin^{2}(\tfrac{\pi
r}{K})\frac{\xi(v)\vartheta_{1}(\frac{v}{2})^{2}\vartheta_{1}(\frac{v}{2}+\frac{r}{K})\vartheta_{1}(\frac{v}{2}-\frac{r}{K})}{\vartheta_{1}(\frac{r}{K})^{2}\eta^{6}}\overline{\frac{\xi(v)\vartheta_{1}(\frac{v}{2})^{4}}{\eta^{12}}}\Lambda_{II_{2,2}\oplus\mathds{H}(K)}[^{0}_{r}]\;.$
(2.24)

As usual the twisted sector is determined by modular invariance. We define the
contribution from the bosonic zero modes for the twisted sector $s\neq 0$ mod
$K$ dividing $r$ as 131313For $K=2,3$ this includes all $s\neq 0$ mod $K$,
while this excludes $(r,s)=(1,2)$ and $(3,2)$ for $K=4$ and
$(r,s)=(1,2),\,(3,2),\,(5,2),\,(1,3),\,(2,3),\,(4,3),\,(5,3),\,(1,4),\,(3,4),\,(5,4)$
for $K=6$.

$\Lambda_{II_{2,2}\oplus\mathds{H}(K)}[^{s}_{r}]=-\frac{1}{K}\sum_{\begin{subarray}{c}m,n,\tilde{m}\in\mathds{Z}\\\
\tilde{n}\in\mathds{Z}+\frac{s}{K}\end{subarray}}\sum_{q\in\mathds{H}(K)^{*}}e^{\frac{2\pi
ir\tilde{m}}{K}-2\pi ir|q|^{2}}e^{\pi i\tau
p_{L}(m,n,\tilde{m},\tilde{n},q)^{2}-\pi
i\bar{\tau}p_{R}(m,n,\tilde{m},\tilde{n},q)^{2}}$ (2.25)

where we included the factor of $-\frac{1}{K}$ appearing in the Poisson
summation over $\mathds{H}(K)$ in the definition in such a way that

$\Lambda_{II_{2,2}\oplus\mathds{H}(K)}[^{0}_{s}](-\tfrac{1}{\tau},-\tfrac{1}{\bar{\tau}})=\tau\bar{\tau}^{3}\Lambda_{II_{2,2}\oplus\mathds{H}(K)}[^{\scalebox{0.6}{-}s}_{0}](\tau,\bar{\tau})\;.$
(2.26)

The complete ‘character-valued’ partition function can finally be written with
these definitions as

${\cal
Z}^{\mathds{Z}_{K}}_{D=4}(v,\bar{v})=\frac{\xi(v)}{K}\overline{\frac{\xi({v})\vartheta_{1}({v}/2)^{4}}{\eta(\tau)^{12}}}\Biggl{[}\frac{{\vartheta_{1}({v\over
2})^{4}}}{\eta(\tau)^{12}}\Lambda_{II_{6,6}}\\\
+\sum_{\begin{subarray}{c}r,s\,{\rm mod}K\\\
(r,s)\neq(0,0)\end{subarray}}4\sin^{2}\Bigl{(}\frac{\pi{\rm
gcd}(r,s)}{K}\Bigr{)}\frac{{\vartheta_{1}({v\over 2})^{2}\vartheta_{1}({v\over
2}+\frac{r+s\tau}{K})\vartheta_{1}({v\over
2}-\frac{r+s\tau}{K})}}{\vartheta_{1}(\frac{r+s\tau}{K})^{2}\eta(\tau)^{6}}\Lambda_{II_{2,2}\oplus\mathds{H}(K)}[^{s}_{r}]\Biggr{]}$
(2.27)

with the sum over $r$ and $s$ from $0$ to $K-1$ excluding $r=s=0$. Recall that
we have fixed the external four-dimensional momenta to zero.

Setting $i\pi v=i\pi\bar{v}=\log(y)$ with $y$ the fugacity for the angular
momentum $J_{3}$, one can define the partition function as a trace over the
Hilbert space of perturbative string states as

$Z^{\mathds{Z}_{K}}_{D=4}(y)={\rm Tr}[e^{2\pi i\tau L_{0}-2\pi
i\bar{\tau}\bar{L}_{0}}(-y)^{2J_{3}}]=\int_{-\frac{1}{2}}^{\frac{1}{2}}d\tau_{1}{\cal
Z}^{\mathds{Z}_{K}}_{D=4}(v,\bar{v})\;,$ (2.28)

where the integral over $\tau_{1}$ implements level matching, as above. The
partition function $Z^{\mathds{Z}_{K}}_{D=4}(y)$ expands as a sum over
$\frac{k}{6}$-BPS supermultiplets $\psi\in\mathcal{E}_{j}^{k/6}$ of highest
spin $j+3-\frac{k}{2}$ with $k=1,2,3$ and $k=0$ fo non-BPS states as

$\displaystyle Z^{\mathds{Z}_{K}}_{D=4}(y)$ $\displaystyle=$
$\displaystyle\sum_{j\in\mathds{N}/2}(-1)^{2j}\frac{y^{2j+1}-y^{-2j-1}}{y-y^{-1}}\Biggl{[}-(y^{\frac{1}{2}}-y^{-\frac{1}{2}})^{6}\sum_{\psi\in\mathcal{E}_{j}^{\scalebox{0.5}{$1/2$}}}e^{-\pi\tau_{2}\,\alpha^{\prime}\\!M_{\psi}^{2}}$
$\displaystyle+(y^{\frac{1}{2}}-y^{-\frac{1}{2}})^{8}\sum_{\psi\in\mathcal{E}_{j}^{\scalebox{0.5}{$1/3$}}}e^{-\pi\tau_{2}\,\alpha^{\prime}\\!M_{\psi}^{2}}-(y^{\frac{1}{2}}-y^{-\frac{1}{2}})^{10}\sum_{\psi\in\mathcal{E}_{j}^{\scalebox{0.5}{$1/6$}}}e^{-\pi\tau_{2}\,\alpha^{\prime}\\!M_{\psi}^{2}}+(y^{\frac{1}{2}}-y^{-\frac{1}{2}})^{12}\sum_{\psi\in\mathcal{E}_{j}^{\scalebox{0.5}{$0$}}}e^{-\pi\tau_{2}\,\alpha^{\prime}\\!M_{\psi}^{2}}\Biggr{]}\;,$

where $M_{\psi}$ is the mass of the supermultiplet $\psi$. The large
$\tau_{2}$ expansion of (2.27) (not expanding the lattice partition functions)
is determined by

$\displaystyle\overline{\frac{\xi({v})\vartheta_{1}(\frac{v}{2})^{4}}{\eta(\tau)^{12}}}\frac{\xi(v){\vartheta_{1}(\frac{v}{2})^{2}\vartheta_{1}(\frac{v}{2}+u)\vartheta_{1}(\frac{v}{2}-u)}}{\vartheta_{1}(u)^{2}\eta(\tau)^{6}}$
(2.30) $\displaystyle=$ $\displaystyle\frac{(1-\zeta y)(1-\zeta
y^{-1})}{(1-\zeta)^{2}}(y^{\frac{1}{2}}-y^{-\frac{1}{2}})^{6}\Bigl{(}1+(y^{\frac{1}{2}}-y^{-\frac{1}{2}})^{4}e^{-2\pi
i\bar{\tau}}+\dots\Bigr{)}$ $\displaystyle\hskip
113.81102pt\times\Bigl{(}1-\frac{(1-\zeta y)(1-\zeta
y^{-1})}{\zeta}(y^{\frac{1}{2}}-y^{-\frac{1}{2}})^{2}e^{2\pi
i\tau}+\dots\Bigr{)}$

where $\zeta=e^{2\pi iu}$ for $u=\frac{r+s\tau}{K}$. This exhibits that the
corresponding constant term determines the spectrum of 1/2 BPS
supermultiplets, the holomorphic part the spectrum of 1/3 BPS supermultiplets,
the anti-holomorphic part the spectrum of 1/6 BPS supermultiplets while the
generic terms determine the spectrum of non-BPS supermultiplets. This is
consistent with the BPS black hole solitons in supergravity [70]. The 1/2 BPS
black holes have a rank one electromagnetic charge that is a vector ${Q}$ with
${Q}^{2}=0$ in the perturbative Narain lattice. The 1/3 BPS black holes have a
rank two electromagnetic charge that is a vector ${Q}$ with ${Q}^{2}<0$ in the
perturbative Narain lattice, consistently with the level matching condition in
the Left-moving sector. The 1/6 BPS black holes with an electromagnetic charge
in the perturbative Narain lattice can only be rank two, with a vector ${Q}$
with ${Q}^{2}>0$, consistently with the level matching condition in the Right-
moving sector.

Using (2.27) and (2.30) one obtains the partition function for 1/2 BPS states

$Z^{\scalebox{0.6}{$\frac{1}{2}$-BPS}}_{D=4}(y)=(y^{\frac{1}{2}}-y^{-\frac{1}{2}})^{6}\int_{-\frac{1}{2}}^{\frac{1}{2}}d\tau_{1}\Biggl{[}(y+y^{-1})\sum_{m,n,\tilde{m},\tilde{n}\in\mathds{Z}}\sum_{q\in\mathds{H}(K)}e^{\pi
i\tau p_{L}({Q})^{2}-\pi i\bar{\tau}p_{R}({Q})^{2}}\\\
-\Biggl{(}\sum_{\begin{subarray}{c}m,n,\tilde{n}\in\mathds{Z}\\\ \tilde{m}\in
K\mathds{Z}+1\end{subarray}}+\sum_{\begin{subarray}{c}m,n,\tilde{n}\in\mathds{Z}\\\
\tilde{m}\in
K\mathds{Z}-1\end{subarray}}\Biggr{)}\sum_{q\in\mathds{H}(K)}e^{\pi i\tau
p_{L}({Q})^{2}-\pi i\bar{\tau}p_{R}({Q})^{2}}\\\
-\frac{4\sin^{2}(\frac{\pi}{K})}{K}\sum_{\begin{subarray}{c}m,n,\tilde{m}\in\mathds{Z}\\\
\tilde{n}\in\mathds{Z}+\frac{1}{K}\end{subarray}}\sum_{q\in\mathds{H}(K)}e^{\pi
i\tau p_{L}({Q})^{2}-\pi i\bar{\tau}p_{R}({Q})^{2}}\\\
+\sum_{s=2}^{K-1}\sum_{r=0}^{K}\frac{4\sin^{2}\bigl{(}\frac{\pi{\rm
gcd}(r,s)}{K}\bigr{)}}{K}\Lambda_{II_{2,2}\oplus\mathds{H}(K)}[^{s}_{r}]\Biggr{]}\;.$
(2.31)

The first two lines correspond to the untwisted sector, with one spin two 1/2
BPS supermultiplet for each null charge ${Q}\in II_{2,2}\oplus\mathds{H}(K)$
with $\tilde{m}=0$ mod $K$, and one spin three half 1/2 BPS supermultiplet for
each null charge ${Q}\in II_{2,2}\oplus\mathds{H}(K)$ with $\tilde{m}=\pm 1$
mod $K$ (two supermultiplets for $K=2$). For the first twisted sector, one
gets for each charge of vanishing norm with
$\tilde{n}\in\mathds{Z}+\frac{1}{K}$ and $q\in\mathds{H}(K)^{*}$,
${4\sin(\pi/K)^{2}}/{K}$ massive spin three half 1/2 BPS supermultiplets. This
confirms that (2.12) must be an integer and therefore that the asymmetric
orbifold theory is only consistent for $K=2,3$.

From now on we will therefore consider $K=2,3$ only. In these cases (2.27)
simplifies because $\sin^{2}(\pi{\rm gcd}(r,s)/K)=\sin^{2}(\pi/K)$ for all
non-zero $r$ and $s$. In order to make the physical lattice of charges
manifest, it is convenient to write the Lorentzian lattice partition function
as a sum over orbits

$\Lambda_{II_{6,6}}=4\sin^{2}(\tfrac{\pi}{K})\Biggl{(}\Lambda_{II_{2,2}\oplus\mathds{H}(K)}\frac{\wp(\frac{1}{K})}{4\pi^{2}}-\frac{1}{K}\sum_{r=0}^{K-1}\Lambda_{II_{2,2}\oplus\mathds{H}(K)^{*}}[e^{2\pi
ir|q|^{2}}]\frac{\wp(\frac{r-\tau}{K})}{4\pi^{2}}\Biggr{)}\;.$ (2.32)

Using identity (C.18) one gets

$\frac{\xi(v){\vartheta_{1}({v\over 2})^{2}\vartheta_{1}({v\over
2}+\frac{r+s\tau}{K})\vartheta_{1}({v\over
2}-\frac{r+s\tau}{K})}}{\vartheta_{1}(\frac{r+s\tau}{K})^{2}\eta(\tau)^{6}}=\frac{\xi(v)\vartheta_{1}({v\over
2})^{4}}{\eta(\tau)^{12}}\frac{\wp(\frac{r+s\tau}{K})-\wp(\frac{v}{2})}{4\pi^{2}}\;.$
(2.33)

The symmetries of the Weierstrass function imply that the distinct values of
$\wp(\frac{r+s\tau}{K})$ reduce to $\wp(\frac{1}{K})$ and
$\wp(\frac{\tau+r}{K})$ for $r=0$ to $K-1$. Moreover using

$\Lambda_{II_{2,2}\oplus\mathds{H}(K)}-\frac{1}{K}\sum_{r=0}^{K-1}\Lambda_{II_{2,2}\oplus\mathds{H}(K)^{*}}[e^{2\pi
ir|q|^{2}}]=0\;,$ (2.34)

the ‘character-valued’ partition function (2.27) can be rewritten as

$\displaystyle{\cal Z}^{\mathds{Z}_{K}}_{D=4}(v,\bar{v})$ $\displaystyle=$
$\displaystyle
4\sin^{2}(\tfrac{\pi}{K})\Biggl{|}\frac{\xi(v)\vartheta_{1}({v}/2)^{4}}{\eta(\tau)^{12}}\Biggr{|}^{2}\Biggl{[}\frac{\wp(\frac{1}{K})-\wp(\frac{v}{2})}{4\pi^{2}}\frac{1}{K}\sum_{r=0}^{K-1}\Lambda_{II_{2,2}\oplus\mathds{H}(K)}[e^{\frac{2\pi
ir\tilde{m}}{K}}]$ (2.35) $\displaystyle\hskip
42.67912pt-\frac{1}{K^{2}}\sum_{r=0}^{K-1}\frac{\wp(\frac{\tau+r}{K})-\wp(\frac{v}{2})}{4\pi^{2}}\Lambda_{II_{1,1}\oplus
II_{1,1}[\frac{1}{K}]\oplus\mathds{H}(K)^{*}}[e^{\pi
ir{Q}^{2}}]\Biggr{]}\;,\;$

where

$\Lambda_{II_{1,1}\oplus II_{1,1}[\frac{1}{K}]\oplus\mathds{H}(K)^{*}}[e^{\pi
ir{Q}^{2}}]=\sum_{s=0}^{K-1}\sum_{\begin{subarray}{c}m,n,\tilde{m}\in\mathds{Z}\\\
\tilde{n}\in\mathds{Z}+\frac{s}{K}\end{subarray}}\sum_{q\in\mathds{H}(K)^{*}}e^{\pi
i(\tau+r)p_{L}(m,n,\tilde{m},\tilde{n},q)^{2}-\pi
i(\bar{\tau}+r)p_{R}(m,n,\tilde{m},\tilde{n},q)^{2}}\;.$ (2.36)

One finally arrives to the very compact expression for the character-valued
partition function

$\displaystyle Z^{\mathds{Z}_{K}}_{D=4}(y)$ $\displaystyle=$ $\displaystyle
4\sin^{2}(\tfrac{\pi}{K})\int_{-\frac{1}{2}}^{\frac{1}{2}}d\tau_{1}\overline{\frac{\xi({v})\vartheta_{1}(\frac{v}{2})^{4}}{\eta(\tau)^{12}}}\frac{\xi(v){\vartheta_{1}(\frac{v}{2})^{2}\vartheta_{1}(\frac{v}{2}{+}\frac{1}{K})\vartheta_{1}(\frac{v}{2}{-}\frac{1}{K})}}{\vartheta_{1}(\frac{1}{K})^{2}\eta(\tau)^{6}}\Lambda_{II_{1,1}\oplus
II_{1,1}[K]\oplus\mathds{H}(K)}$
$\displaystyle-\frac{4\sin^{2}(\tfrac{\pi}{K})}{K^{2}}\int_{-\frac{K}{2}}^{\frac{K}{2}}d\tau_{1}\overline{\frac{\xi({v})\vartheta_{1}(\frac{v}{2})^{4}}{\eta(\tau)^{12}}}\frac{\xi(v){\vartheta_{1}(\frac{v}{2})^{2}\vartheta_{1}(\frac{v}{2}{+}\frac{\tau}{K})\vartheta_{1}(\frac{v}{2}{-}\frac{\tau}{K})}}{\vartheta_{1}(\frac{\tau}{K})^{2}\eta(\tau)^{6}}\Lambda_{II_{1,1}\oplus
II_{1,1}[\frac{1}{K}]\oplus\mathds{H}(K)^{*}}\;,$

where we have traded the sum over $r$ from $0$ to $K-1$ with the extension of
the integration domain of $\tau_{1}$ and used (2.33) reversely. The main
advantage of this formula is that it makes manifest that the physical lattice
of bosonic zero modes is the lattice $\mathds{F}_{2}(K)^{*}$ dual to

$\mathds{F}_{2}(K)=II_{1,1}\oplus II_{1,1}[K]\oplus\mathds{H}(K)\;.$ (2.38)

Therefore one finds that the spectrum of supermultiplets with a given charge
${Q}\in\mathds{F}_{2}(K)^{*}$ depends only of its norm square ${Q}^{2}$ and
the property that it lies or not in the sublattice $\mathds{F}_{2}(K)$. All
the charges in $\mathds{F}_{2}(K)$ correspond to states in the untwisted
sector, but charges
${Q}\in\mathds{F}_{2}(K)^{*}\smallsetminus\mathds{F}_{2}(K)$ are in the
twisted sector if and only if $\tilde{n}$ is not an integer. However, (2.3)
exhibits that the physical distinction between states relies on the condition
of being or not in $\mathds{F}_{2}(K)$ and we will therefore sometimes refer
by abuse of language to all the states with charge
${Q}\in\mathds{F}_{2}(K)^{*}\smallsetminus\mathds{F}_{2}(K)$ as twisted
states, including the ones with $\tilde{n}$ an integer.

In particular the constant term in the large $\tau_{2}$ expansion (not
expanding the lattice partition function) gives the partition function of 1/2
BPS supermultiplets

$Z^{\scalebox{0.6}{$\frac{1}{2}$-BPS}}_{D=4}(y)=\bigl{(}y^{\frac{1}{2}}-y^{-\frac{1}{2}}\bigr{)}^{6}\bigl{(}y+y^{-1}\bigr{)}\sum_{\begin{subarray}{c}{Q}\in\mathds{F}_{2}(K)\\\
{Q}^{2}=0\end{subarray}}e^{-2\pi\tau_{2}p_{R}({Q})^{2}}\\\
-\frac{4\sin^{2}(\frac{\pi}{K})}{K}\bigl{(}y^{\frac{1}{2}}-y^{-\frac{1}{2}}\bigr{)}^{6}\sum_{\begin{subarray}{c}{Q}\in\mathds{F}_{2}(K)^{*}\smallsetminus\mathds{F}_{2}(K)\\\
{Q}^{2}=0\end{subarray}}e^{-2\pi\tau_{2}p_{R}({Q})^{2}}\;.$ (2.39)

One has therefore a single spin two 1/2 BPS supermultiplet for each untwisted
charge ${Q}\in\mathds{F}_{2}(K)$ with ${Q}^{2}=0$, including the massless
gravity multiplet for ${Q}=0$, and respectively two (for $\mathds{Z}_{2}$) and
one (for $\mathds{Z}_{3}$) spin three half 1/2 BPS supermultiplets for each
‘twisted’ charge ${Q}\in\mathds{F}_{2}(K)^{*}\smallsetminus\mathds{F}_{2}(K)$
with ${Q}^{2}=0$. Note that an $\mathcal{N}=8$ spin two 1/2 BPS supermultiplet
decomposes into one $\mathcal{N}=6$ spin two 1/2 BPS multiplet and two spin
three half 1/2 BPS multiplets, so we understand that the two spin three half
1/2 BPS supermultiplets are in the single ‘twisted sector’ for $K=2$, and each
spin three half 1/2 BPS multiplet is in one of the two (conjugate) ‘twisted
sectors’ for $K=3$.

The character valued partition function (2.3) is not protected in that it
receives contribution from all kinds of supermultiplets. Yet the helicity
supertraces

$B^{\mathds{Z}_{K}}_{2n}=\int_{-\frac{1}{2}}^{\frac{1}{2}}d\tau_{1}\;{\cal
B}^{\mathds{Z}_{K}}_{2n}=\int_{-\frac{1}{2}}^{\frac{1}{2}}d\tau_{1}\;({{\cal
H}}+\bar{{{\cal H}}})^{2n}{\cal
Z}^{\mathds{Z}_{K}}_{D=4}(v,\bar{v})|_{v=\bar{v}=0}\;,$ (2.40)

with $n\leq 5$ are protected in that they only receive contribution from short
BPS multiplets.

So far as the perturbative threshold contributions are concerned, due to the
splitting between Left- and Right-movers on the word-sheet, we are interested
in the decomposition of BPS representations of (space-time) ${\cal N}=6$
supersymmetry into $({\cal N}_{L},\mathcal{N}_{R})=(2,4)$. To this end we
introduce the notations

$\displaystyle\langle({{\cal H}}+\bar{{{\cal H}}})^{2n}\rangle_{u}$
$\displaystyle=$ $\displaystyle({{\cal H}}+\bar{{{\cal
H}}})^{2n}\Biggl{(}4\sin^{2}(\tfrac{\pi}{K})\frac{\xi(v){\vartheta_{1}(\frac{v}{2})^{2}\vartheta_{1}(\frac{v}{2}{+}u)\vartheta_{1}(\frac{v}{2}{-}u)}}{\vartheta_{1}(u)^{2}\eta(\tau)^{6}}\overline{\frac{\xi({v})\vartheta_{1}(\frac{v}{2})^{4}}{\eta(\tau)^{12}}}\Biggr{)}\Bigg{|}_{v=\bar{v}=0}\;,$
$\displaystyle\langle{{\cal H}}^{2n}\rangle^{u}_{L}$ $\displaystyle=$
$\displaystyle{{\cal
H}}^{2n}\Biggl{(}4\sin^{2}(\tfrac{\pi}{K})\frac{\xi(v){\vartheta_{1}(\frac{v}{2})^{2}\vartheta_{1}(\frac{v}{2}{+}u)\vartheta_{1}(\frac{v}{2}{-}u)}}{\vartheta_{1}(u)^{2}\eta(\tau)^{6}}\Biggr{)}\Bigg{|}_{v=0}\;,$
$\displaystyle\langle\bar{{{\cal H}}}^{2n}\rangle^{u}_{R}$ $\displaystyle=$
$\displaystyle\bar{{{\cal
H}}}^{2n}\Biggl{(}\overline{\frac{\xi({v})\vartheta_{1}(\frac{v}{2})^{4}}{\eta(\tau)^{12}}}\Biggr{)}\Bigg{|}_{\bar{v}=0}\;,$
(2.41)

where $u=(r-s\tau)/K$ accounts for the sector and

${\cal B}^{\mathds{Z}_{K}}_{2n}=\langle({{\cal H}}+\bar{{{\cal
H}}})^{2n}\rangle_{\\!\frac{1}{K}}\Lambda_{\mathds{F}_{2}(K)}-\frac{1}{K^{2}}\sum_{r=0}^{K-1}\langle({{\cal
H}}+\bar{{{\cal
H}}})^{2n}\rangle_{\\!\frac{\tau+r}{K}}\Lambda_{\mathds{F}_{2}(K)^{*}}[e^{\pi
ir{Q}^{2}}]\;.$ (2.42)

In particular, $\langle{{\cal H}}^{0}\rangle^{u}_{L}=0$ and
$\langle\bar{{{\cal H}}}^{0}\rangle^{u}_{R}=\langle\bar{{{\cal
H}}}^{2}\rangle^{u}_{R}=0$ and one obtains that

${\cal B}^{\mathds{Z}_{K}}_{0}={\cal B}^{\mathds{Z}_{K}}_{2}={\cal
B}^{\mathds{Z}_{K}}_{4}=0\;.$ (2.43)

The first non-vanishing contribution is 1/2 BPS protected with

$\langle({{\cal H}}+\bar{{{\cal H}}})^{6}\rangle_{u}=15\langle{{\cal
H}}^{2}\rangle^{u}_{L}\langle\bar{{{\cal H}}}^{4}\rangle^{u}_{R}={45\over
4}\times 4\sin^{2}(\tfrac{\pi}{K})\;,$ (2.44)

and preserves $({\cal N}_{L},\mathcal{N}_{R})=(1,2)$ supersymmetry. The 1/3
BPS protected contribution

$\displaystyle\langle({{\cal H}}+\bar{{{\cal H}}})^{8}\rangle_{u}$
$\displaystyle=$ $\displaystyle 28\langle{{\cal
H}}^{2}\rangle_{L}^{u}\langle\bar{{{\cal
H}}}^{6}\rangle_{R}^{u}+70\langle{{\cal
H}}^{4}\rangle_{L}^{u}\langle\bar{{{\cal H}}}^{4}\rangle^{u}_{R}$ (2.45)
$\displaystyle=$ $\displaystyle 28\times
4\sin^{2}(\tfrac{\pi}{K})\frac{1}{2}\times\frac{15}{8}+70\times
4\sin^{2}(\tfrac{\pi}{K})\Bigl{(}\frac{3\wp(u)}{8\pi^{2}}+\frac{1}{4}\Bigr{)}\times\frac{3}{2}$
$\displaystyle=$ $\displaystyle{315\over 2}\times
4\sin^{2}(\tfrac{\pi}{K})\Bigl{(}\frac{\wp(u)}{4\pi^{2}}+\frac{1}{3}\Bigr{)}$

preserves $({\cal N}_{L},\mathcal{N}_{R})=(0,2)$ supersymmetry. The insertion
of four Left-moving currents and four Right-moving currents gives the 1/3 BPS
${\cal R}^{4}$ coupling, and indeed we will find that the holomorphic modular
form $\wp(\frac{1}{K})$ determines this coupling in the low-energy effective
action.

The 1/6 BPS protected contribution is

$\displaystyle\langle({{\cal H}}+\bar{{{\cal H}}})^{10}\rangle_{u}$
$\displaystyle=$ $\displaystyle 45\langle{{\cal
H}}^{2}\rangle^{u}_{L}\langle\bar{{{\cal
H}}}^{8}\rangle^{u}_{R}+210\langle{{\cal
H}}^{4}\rangle_{L}^{u}\langle\bar{{{\cal
H}}}^{6}\rangle^{u}_{R}+210\langle{{\cal
H}}^{6}\rangle_{L}^{u}\langle\bar{{{\cal H}}}^{4}\rangle^{u}_{R}$ (2.46)
$\displaystyle=$ $\displaystyle 45\times 4\sin^{2}(\tfrac{\pi}{K}){1\over
2}\times\frac{21}{32}(\overline{E_{4}}+2)+210\times
4\sin^{2}(\tfrac{\pi}{K})\Bigl{(}\frac{3\wp(u)}{8\pi^{2}}+\frac{1}{4}\Bigr{)}\times\frac{15}{8}$
$\displaystyle\hskip 28.45274pt+210\times
4\sin^{2}(\tfrac{\pi}{K})\frac{3}{32}\Bigl{(}E_{4}+\frac{5\wp(u)}{\pi^{2}}+1\Bigr{)}\times\frac{3}{2}$
$\displaystyle=$ $\displaystyle\frac{945}{64}\times
4\sin^{2}(\tfrac{\pi}{K})\Bigl{(}\overline{E_{4}}+2E_{4}+\frac{20\wp(u)}{\pi^{2}}+\frac{32}{3}\Bigr{)}$

where the first (anti-holomorphic) term preserves $({\cal
N}_{L},\mathcal{N}_{R})=(1,0)$ supersymmetry, while the two other
(holomorphic) terms preserve $({\cal N}_{L},\mathcal{N}_{R})=(0,2)$
supersymmetry.

Substituting these expressions in (2.42) one obtains the helicity supertraces

$\displaystyle B^{\mathds{Z}_{K}}_{6}$ $\displaystyle=$
$\displaystyle\frac{45}{4}\times
4\sin^{2}(\tfrac{\pi}{K})\Biggl{(}\int_{-\frac{1}{2}}^{\frac{1}{2}}d\tau_{1}\,\Lambda_{\mathds{F}_{2}(K)}-\frac{1}{K^{2}}\int_{-\frac{K}{2}}^{\frac{K}{2}}d\tau_{1}\,\Lambda_{\mathds{F}_{2}(K)^{*}}\Biggr{)}$
(2.47) $\displaystyle B^{\mathds{Z}_{K}}_{8}$ $\displaystyle=$
$\displaystyle\frac{315}{8}\times
4\sin^{2}(\tfrac{\pi}{K})\Biggl{(}\int_{-\frac{1}{2}}^{\frac{1}{2}}d\tau_{1}\,\Lambda_{\mathds{F}_{2}(K)}\Bigl{(}\frac{4}{3}+\frac{\wp(\frac{1}{K})}{\pi^{2}}\Bigr{)}$
$\displaystyle\hskip
113.81102pt-\frac{1}{K^{2}}\int_{-\frac{K}{2}}^{\frac{K}{2}}d\tau_{1}\,\Lambda_{\mathds{F}_{2}(K)^{*}}\Bigl{(}\frac{4}{3}+\frac{\wp(\frac{\tau}{K})}{\pi^{2}}\Bigr{)}\Biggr{)}$
$\displaystyle B^{\mathds{Z}_{K}}_{10}$ $\displaystyle=$
$\displaystyle\frac{945}{64}\times
4\sin^{2}(\tfrac{\pi}{K})\Biggl{[}\int_{-\frac{1}{2}}^{\frac{1}{2}}d\tau_{1}\,\Lambda_{\mathds{F}_{2}(K)}\Biggl{(}\frac{32}{3}\\!+\\!E_{4}(-\bar{\tau})\\!+\\!2E_{4}(\tau)\\!+\\!\frac{20\wp(\frac{1}{K})}{\pi^{2}}\Biggr{)}$
$\displaystyle\hskip
56.9055pt-\frac{1}{K^{2}}\int_{-\frac{K}{2}}^{\frac{K}{2}}d\tau_{1}\,\Lambda_{\mathds{F}_{2}(K)^{*}}\Biggl{(}\frac{32}{3}\\!+\\!E_{4}(-\bar{\tau})\\!+\\!2E_{4}(\tau)\\!+\\!\frac{20\wp(\frac{\tau}{K})}{\pi^{2}}\Biggr{)}\Biggr{]}\;.$

To interpret these helicity supertraces as counting BPS supermultiplets up to
sign, it is useful to decompose them as a sum over supermultiplets using (2.3)

$\displaystyle B^{\mathds{Z}_{K}}_{6}$ $\displaystyle=$
$\displaystyle-\frac{45}{4}\sum_{j\in\mathds{N}/2}(-1)^{2j}(2j+1)\sum_{\psi\in\mathcal{E}_{j}^{\scalebox{0.5}{$1/2$}}}e^{-\pi\tau_{2}\,\alpha^{\prime}\\!M_{\psi}^{2}}\;,$
(2.48) $\displaystyle B^{\mathds{Z}_{K}}_{8}$ $\displaystyle=$
$\displaystyle\frac{315}{8}\sum_{j\in\mathds{N}/2}(-1)^{2j}(2j+1)\Biggl{(}-\frac{2(2j+1)^{2}+1}{3}\sum_{\psi\in\mathcal{E}_{j}^{\scalebox{0.5}{$1/2$}}}e^{-\pi\tau_{2}\,\alpha^{\prime}\\!M_{\psi}^{2}}+4\sum_{\psi\in\mathcal{E}_{j}^{\scalebox{0.5}{$1/3$}}}e^{-\pi\tau_{2}\,\alpha^{\prime}\\!M_{\psi}^{2}}\Biggr{)}\;,$
$\displaystyle B^{\mathds{Z}_{K}}_{10}$ $\displaystyle=$
$\displaystyle\frac{945}{64}\sum_{j\in\mathds{N}/2}(-1)^{2j}(2j+1)\Biggl{(}-\frac{6(2j+1)^{4}+10(2j+1)^{2}+5}{3}\sum_{\psi\in\mathcal{E}_{j}^{\scalebox{0.5}{$1/2$}}}e^{-\pi\tau_{2}\,\alpha^{\prime}\\!M_{\psi}^{2}}$
$\displaystyle\hskip
128.0374pt+40\bigl{(}(2j+1)^{2}+1\bigr{)}\sum_{\psi\in\mathcal{E}_{j}^{\scalebox{0.5}{$1/3$}}}e^{-\pi\tau_{2}\,\alpha^{\prime}\\!M_{\psi}^{2}}-240\sum_{\psi\in\mathcal{E}_{j}^{\scalebox{0.5}{$1/6$}}}e^{-\pi\tau_{2}\,\alpha^{\prime}\\!M_{\psi}^{2}}\Biggr{)}\;.$

The explicit formulae one obtains from these equations are displayed in
Appendix B. We will find that $\mathcal{B}^{\mathds{Z}_{K}}_{6}$ is associated
to the 1/2 BPS protected $\mathcal{F}^{2}\mathcal{R}^{2}$ coupling,
$\mathcal{B}^{\mathds{Z}_{K}}_{8}$ to the 1/3 BPS protected $\mathcal{R}^{4}$
coupling, and $\mathcal{B}^{\mathds{Z}_{K}}_{10}$ to the 1/6 BPS protected
${D}^{2}\mathcal{R}^{4}$ coupling.

###  2.4 Preliminaries on U-duality

As suggested by the coset structure $G/K$ of the scalar manifolds, ${\cal
N}=6$ supergravity admits global non-compact symmetries $G$ acting linearly on
the vector (field-strengths) and non-linearly on the scalars. As usual the
fermions transform under R-symmetry (isotropy group $K$) that acts by local
transformations on the coset representative $\mathcal{V}\in G$. More
specifically for $D=4$, one has the coset representative
$(\mathcal{V}_{A},\mathcal{V}_{ijA},\bar{\mathcal{V}}^{kl}{}_{A},\bar{\mathcal{V}}_{A})$
in the Majorana–Weyl representation of $Spin^{*}(12)$ that transforms as

$\mathcal{V}_{A}(x)\rightarrow\det\\!k(x)\,\mathcal{V}_{B}(x)g^{B}{}_{A}\;,\qquad\mathcal{V}_{ijA}(x)\rightarrow
k_{i}{}^{k}(x)k_{j}{}^{l}(x)\mathcal{V}_{klB}(x)g^{B}{}_{A}\;,$ (2.49)

with $g^{B}{}_{A}\in Spin^{*}(12)$ a constant $32\times 32$ real matrix and
$k_{i}{}^{j}(x)\in U(6)$ a $6\times 6$ complex matrix. The sixteen vector
fields field strengths and their dual define thirty-two field strengths
transforming as

$F_{\mu\nu}^{A}(x)\rightarrow g^{-1A}{}_{B}F_{\mu\nu}^{B}(x)\;,$ (2.50)

while the Weyl fermions transform under $U(6)$ as

$\displaystyle\psi_{\mu\alpha i}(x)$ $\displaystyle\rightarrow$ $\displaystyle
k_{i}{}^{j}(x)\psi_{\mu\alpha j}(x)\;,$ $\displaystyle\chi_{\alpha ijk}(x)$
$\displaystyle\rightarrow$ $\displaystyle
k_{i}{}^{l}(x)k_{j}{}^{p}(x)k_{k}{}^{q}(x)\chi_{\alpha
lpq}(x)\;,\quad\lambda_{\alpha
i}(x)\rightarrow\det\\!k(x)^{-1}k_{i}{}^{j}(x)\lambda_{\alpha j}(x)\;.$ (2.51)

The $Spin^{*}(12)$ invariant symplectic form $\omega_{AB}$ is normalised such
that

$\displaystyle\omega_{AB}$ $\displaystyle=$ $\displaystyle
i\bar{\mathcal{V}}_{A}\mathcal{V}_{B}+\frac{i}{2}\mathcal{V}_{ijA}\bar{\mathcal{V}}^{ij}{}_{B}-\frac{i}{2}\bar{\mathcal{V}}^{ij}{}_{A}\mathcal{V}_{ijB}-i\mathcal{V}_{A}\bar{\mathcal{V}}_{B}\;,$
(2.52) $\displaystyle\omega^{AB}\mathcal{V}_{A}\bar{\mathcal{V}}_{B}$
$\displaystyle=$ $\displaystyle
i\;,\quad\omega^{AB}\mathcal{V}_{A}\bar{\mathcal{V}}^{ij}{}_{B}=0\;,\quad\omega^{AB}\mathcal{V}_{ijA}\bar{\mathcal{V}}^{kl}{}_{B}=-2i\delta_{ij}^{kl}\;,\quad\omega^{AB}\mathcal{V}_{ijA}{\mathcal{V}}_{klB}=0\;.$

One decomposes as usual the Maurer-Cartan form into $Q_{H}\in\mathfrak{u}(6)$
and $P$ in the coset component such that

$\displaystyle d\mathcal{V}_{A}$ $\displaystyle=$
$\displaystyle(Q_{H})^{k}{}_{k}\mathcal{V}+\frac{1}{2}P^{ij}\mathcal{V}_{ijA}$
$\displaystyle d\mathcal{V}_{ijA}$ $\displaystyle=$
$\displaystyle-2(Q_{H})^{k}{}_{[i}\mathcal{V}_{j]kA}+\frac{1}{4}\varepsilon_{ijklpq}P^{kl}\bar{\mathcal{V}}^{pq}{}_{A}+\bar{P}_{ij}\bar{\mathcal{V}}_{A}$
(2.53)

and $P^{ij}$ is holomorphic in the fifteen complex scalar fields $T^{M}$, i.e.

$P^{ij}=P_{M}{}^{ij}dT^{M}\;,\qquad\bar{P}_{ij}=P_{\bar{M}ij}d\bar{T}^{\bar{M}}\;.$
(2.54)

Due to charge quantisation and to the absence of continuous symmetries in
(super)gravity, only a discrete U-duality subgroup of $SO^{*}(12)$ may survive
as an exact symmetry of the quantum theory, such as string theory. The
U-duality group is defined as the non-perturbative duality symmetry of string
theory and the T-duality group as the perturbative symmetry. In this paper we
determine the T-duality group as the symmetry of the four-point one-loop
amplitudes and of the character valued partition function that counts states
for a given charge in the Narain lattice $\mathds{F}_{2}(K)^{*}$. The
T-duality group defined in this way does not preserve the untwisted sector,
and mixes twisted states with untwisted states with charge
$\mathcal{Q}\in\mathds{F}_{2}(K)^{*}\smallsetminus\mathds{F}_{2}(K)$. It is
possible that the true T-duality group is restricted to the automorphisms of
the perturbative lattice preserving the untwisted sector, if for example the
scattering of massive twisted states differs from the scattering of unwtisted
states with a charge in
$\mathds{F}_{2}(K)^{*}\smallsetminus\mathds{F}_{2}(K)$. We may nevertheless
conjecture that this is indeed the T-duality group, i.e. the symmetry of the
complete perturbative theory, and for simplicity we shall refer to the group
of automorphisms of $\mathds{F}_{2}(K)$ as the T-duality group in the
following.

We shall start with the five-dimensional theory, with global non-compact
symmetry $SU^{*}(6)$. The computation of (2.35) applies directly to five
dimensions with $\mathds{F}_{2}(K)$ replaced by

$\mathds{M}_{2}(K)=II_{1,1}[K]\oplus\mathds{H}(K)\;.$ (2.55)

The perturbative states with bosonic zero modes $Q\in\mathds{M}_{2}(K)$ are
untwisted and the other states with
$Q\in\mathds{M}_{2}(K)^{*}\smallsetminus\mathds{M}_{2}(K)$ contribute in the
same way to the the four-point one-loop amplitudes and of the character valued
partition function. The T-duality group in five dimensions as defined above is
therefore the group of automorphisms of the lattice $\mathds{M}_{2}(K)$. To
determine this group of automorphisms it is convenient to define
$\mathds{M}_{2}(K)$ as the set of Hermitian two by two matrices over the
integral quaternions $\mathds{H}(K)$ with a top left component $\tilde{m}=0$
mod $K$, i.e.

$Q=\left(\begin{array}[]{cc}\tilde{m}&\,q\\\
q^{*}&\tilde{n}\end{array}\right)\in\mathds{M}_{2}(K)\qquad\mbox{with}\;\tilde{n}\in\mathds{Z}\,,\;\tilde{m}\in
K\mathds{Z}\,,\;q\in\mathds{H}(K)\;.$ (2.56)

Then $Q^{2}=2\det Q$ and the action of $SO(1,5)$ on this charge is defined as

$Q\rightarrow\gamma^{\dagger}Q\gamma\;,$ (2.57)

where $\gamma=(^{a}_{c}{}^{b}_{d})$ is a two by two matrix over the
quaternions of unit modulus determinant $\bigl{|}ad-aca^{-1}b\bigr{|}=1$. The
condition that the lattice $\mathds{M}_{2}(K)$ is preserved implies that
$a,d\in\mathds{H}(K)$, $b\in\frac{1}{\alpha}\mathds{H}(K)$ and
$c\in\alpha\mathds{H}(K)$ where $\alpha$ is a quaternion of norm square
$|\alpha|^{2}=K$ in $\mathds{H}(K)$. For both $K=2,3$, $\alpha$ is defined
uniquely up to left multiplication by a unit $u\in\mathds{H}(K)$ (with
$|u|^{2}=1$). $\alpha\mathds{H}(K)$ is therefore a two-sided ideal in
$\mathds{H}(K)$ and $\mathds{H}(K)^{*}=\frac{1}{\alpha}\mathds{H}(K)$. We
denote this arithmetic group
$\Gamma^{\scalebox{0.6}{A}_{\scalebox{0.5}{3}}}_{0*}(\alpha)$ and will refer
to it as the T-duality group of the $K=2$, $3$ asymmetric orbifold theories in
five dimensions.141414It is related to $SL(2,\mathds{H}(K))$ by a similarity
transformation, but the notation indicates that it is generated from
$\Gamma^{\scalebox{0.6}{A}_{\scalebox{0.5}{3}}}_{0}(\alpha)\subset
SL(2,\mathds{H}(K))$ with $c=0$ mod $\alpha$ and $\gamma_{\rm
F}=(^{0}_{\alpha}{}^{1\\!/\\!\alpha}_{\;0})$.

In four dimensions the group of automorphisms of $\mathds{F}_{2}(K)$ is
defined similarly (up to a similarity transformation) as the subgroup
$\Gamma^{\scalebox{0.6}{D}_{\scalebox{0.5}{4}}}_{0*}(\alpha)$ of symplectic
matrices $(^{A}_{C}{}^{B}_{D})$ with $A,B,C,D$ two by two rational matrices
over $\mathds{H}(K)$ such that

$D^{\dagger}A-B^{\dagger}C=\mathds{1}\;,\qquad
D^{\dagger}B=B^{\dagger}D\;,\qquad A^{\dagger}C=C^{\dagger}A\;,$ (2.58)

that is generated by the group
$\Gamma^{\scalebox{0.6}{D}_{\scalebox{0.5}{4}}}_{0}(\alpha)\subset
Sp(4,\mathds{H}(K))$ of integral matrices $(^{A}_{C}{}^{B}_{D})$ with $C=0$
mod $\alpha$ and the symplectic group element

$\gamma_{\rm F}=\left(\begin{array}[]{cccc}0&\,0&-\frac{1}{\alpha^{*}}&\,0\\\
0&\,0&\,0&-\frac{1}{\alpha^{*}}\\\ \;\alpha&\,0&\,0&\,0\\\
0&\,\alpha&\,0&\,0\end{array}\right)\;.$ (2.59)

This additional generator $\gamma_{\rm F}$ is not an element of the original
T-duality group $O(6,6,\mathds{Z})\supset Sp(4,\mathds{H}(K))$ prior to the
orbifold and is similar to the Fricke duality introduced in [71]. We shall
refer accordingly to
$\Gamma^{\scalebox{0.6}{D}_{\scalebox{0.5}{4}}}_{0*}(\alpha)$ as the T-duality
group of the $K=2$, $3$ asymmetric orbifold theories in four dimensions.

We will find strong evidence in [72] that the U-duality group in five
dimensions is defined similarly as
$\Gamma^{\scalebox{0.6}{A}_{\scalebox{0.5}{5}}}_{0*}(\alpha)$: the group
preserving $\alpha\mathds{H}(K)\oplus\alpha\mathds{H}(K)\oplus\mathds{H}(K)$
by left action, and which is related to $SL(3,\mathds{H}(K))$ by similarity
transformation. The U-duality group in four dimensions will be conjectured to
be $\Gamma^{\scalebox{0.6}{D}_{\scalebox{0.5}{6}}}_{0*}(\alpha)$, the group
generated by
$\Gamma^{\scalebox{0.6}{D}_{\scalebox{0.5}{6}}}_{0}(\alpha)\subset
Sp(6,\mathds{H}(K))$ and $\gamma_{\rm
F}\in\Gamma^{\scalebox{0.6}{D}_{\scalebox{0.5}{4}}}_{0*}(\alpha)\subset\Gamma^{\scalebox{0.6}{D}_{\scalebox{0.5}{6}}}_{0*}(\alpha)$,
where
$(^{A}_{C}{}^{B}_{D})\in\Gamma^{\scalebox{0.6}{D}_{\scalebox{0.5}{6}}}_{0}(\alpha)$
is such that $C=0$ mod $\alpha$ as a three by three matrix. This is the group
of automorphisms of the 32-dimensional lattice

$\mathds{F}_{3}(K)=K\mathds{F}_{2}(K)^{*}\oplus[\mathds{H}(K)\oplus\mathds{H}(K)\oplus\alpha\mathds{H}(K)\oplus\alpha\mathds{H}(K)]\oplus\mathds{F}_{2}(K)$
(2.60)

in the Majorana–Weyl spinor representation of $Spin^{*}(12)$ of ‘untwisted
states’ in the sense explained below (2.38). We will find strong evidence that
one-half BPS states are associated to rank one charges $\Gamma$ in

$\mathds{F}^{*}_{3}(K)=\mathds{F}_{2}(K)\oplus[\tfrac{1}{\alpha}\mathds{H}(K)\oplus\tfrac{1}{\alpha}\mathds{H}(K)\oplus\mathds{H}(K)\oplus\mathds{H}(K)]\oplus\mathds{F}^{*}_{2}(K)$
(2.61)

for which the projection $\Gamma\times\Gamma$ to the adjoint representation of
$\mathfrak{so}^{*}(12)$ vanishes. For each $\Gamma\in\mathds{F}_{3}(K)$ there
is one spin two supermultiplet and for each
$\Gamma\in\mathds{F}^{*}_{3}(K)\smallsetminus\mathds{F}_{3}(K)$ there are
$\frac{2}{K-1}$ spin three half supermultiplets.

One may then consider the decomposition of the spectrum under T-duality and
U-duality orbits. Perturbative one-half BPS states have a charge
$\mathcal{Q}\in\mathds{F}_{2}^{*}$ of vanishing norm. We will show in Appendix
F.3 that such charges are either in the
$\Gamma^{\scalebox{0.6}{D}_{\scalebox{0.5}{4}}}_{0*}(\alpha)$ orbit of a
charge with only non-zero component the momentum $\tilde{m}={\rm
gcd}_{\mathds{F}^{*}_{2}}(\mathcal{Q})$ along the twisted circle or with only
non-zero component the momentum $m={\rm gcd}_{\mathds{F}_{2}}(\mathcal{Q})$
along the additional circle. A charge associated to a spin 3/2 supermultiplet
in $\mathds{F}_{2}^{*}\smallsetminus\mathds{F}_{2}$ is therefore in the
$\Gamma^{\scalebox{0.6}{D}_{\scalebox{0.5}{4}}}_{0*}(\alpha)$ orbit of a
Kaluza–Klein mumentum charge along the twisted circle, with mass

$\sqrt{\alpha^{\prime}}M(\mathcal{Q})=\frac{|\tilde{m}|}{\tilde{R_{\scalebox{0.6}{s}}}}\;.$
(2.62)

For $\tilde{R_{\scalebox{0.6}{s}}}>>1$ there are $\frac{4}{K-1}$ gravitino
supermultiplets with the minimal mass ${1}/{\tilde{R_{\scalebox{0.6}{s}}}}$
for $\tilde{m}=\pm 1$. The two derivative low energy effective theory
truncated to the massless graviton supermultiplet and two such gravitino
supermultiplets should be the corresponding Cremmer–Scherk–Schwarz gauging of
$\mathcal{N}=8$ supergravity [73]. This gauged supergravity theory may be a
consistent truncation, but it is not a good effective theory since there are
also massive spin two supermuliplets for each Kaluza–Klein momentum of mass
$\frac{|\tilde{m}|}{\tilde{R_{\scalebox{0.6}{s}}}}$ with $\tilde{m}=0$ mod $K$
and massive spin three-half supermuliplets for each Kaluza–Klein momentum of
mass $\frac{|\tilde{m}|}{\tilde{R_{\scalebox{0.6}{s}}}}$ with $\tilde{m}\neq
0$ mod $K$ that are of the same order. For $K=2$ there are moreover four
gravitino supermultiplets with the same minimal mass. The analysis for the
non-perturbative one-half BPS states is very similar, and all the rank one
charges $\Gamma\in\mathds{F}^{*}_{3}(K)\smallsetminus\mathds{F}_{3}(K)$ are in
the $\Gamma^{\scalebox{0.6}{D}_{\scalebox{0.5}{6}}}_{0*}(\alpha)$ orbit of a
perturbative Kaluza–Klein charge along the twisted circle.

##  3 Low-energy effective action and threshold corrections

Different $\mathcal{N}=6$ realisations such as the $K=2$ and $K=3$ asymmetric
orbifold theories considered in this paper give rise to different low-energy
effective actions, notwithstanding the uniqueness of the 2-derivative action.
Comparison is easier for special couplings that are BPS-saturated and whose
dependence on the scalar fields is tightly constrained by differential
equations and U-duality. Once again the prototypical example is the
$\mathcal{R}^{4}$ term in Type IIB, whose threshold function is the real
analytic Eisenstein series $E_{\frac{3}{2},0}(S)$ of the $SL(2,\mathds{Z})$
S-duality group [7]. Supersymmetry constrains this threshold function to be an
eigen-function of the Laplace operator on the upper-complex half plane with
eigenvalue $\frac{3}{4}$ [11], which together with $SL(2,\mathds{Z})$
invariance determines the function uniquely up to normalisation [10]. The
maximal supergravity $\mathcal{R}^{4}$ threshold function is more generally
understood in dimension $D\geq 3$ to be associated to the minimal automorphic
representation of the (split real form) exceptional group $E_{11-D}$ [15, 34],
which is unique for $D\leq 6$. This analysis extends to higher derivative
couplings and the ${D}^{4}\mathcal{R}^{4}$ threshold function is associated to
the next to minimal automorphic representation [19]. The fact that these
threshold functions are attached to small automorphic representations follows
from differential equations implied by supersymmetry [20, 21, 24], which can
be derived in four dimensions by classifying all the linearised supersymmetry
invariants using the techniques developed in [74, 75]. In this section we
extend this analysis to the BPS protected couplings of $\mathcal{N}=6$
supergravity.

###  3.1 Supersymmetry constraints

The field content of $D=4$ $\mathcal{N}=6$ supergravity includes fifteen
complex scalar fields $T^{M}$ parametrising the special Kähler symmetric space
$SO^{*}(12)/U(6)$, where $U(6)$ is the R-symmetry group, the metric field
$g_{\mu\nu}$ of Weyl tensor $C_{\alpha\beta\gamma\delta}$, six
Rarita–Schwinger fields $\psi_{\mu\alpha i}$ of field strength
$\rho_{\alpha\beta\gamma i}$ in the fundamental of $U(6)$, sixteen vector
fields and their Gaillard–Zumino duals transforming together as a
32-dimensional Majorana–Weyl spinor $A_{\mu}^{A}$ of $Spin^{*}(12)$, and
twenty plus six Weyl spinors $\chi_{\alpha ijk}$ and $\lambda_{\alpha}^{i}$.
One defines as usual the scalar dressed Gaillard–Zumino field strengths

$F_{\alpha\beta
ij}=\frac{1}{2}\sigma^{ab}{}_{\alpha\beta}e_{a}{}^{\mu}e_{b}{}^{\nu}\mathcal{V}_{ijA}F_{\mu\nu}^{A}\;,\qquad
F_{\alpha\beta}=\frac{1}{2}\sigma^{ab}{}_{\alpha\beta}e_{a}{}^{\mu}e_{b}{}^{\nu}\bar{\mathcal{V}}_{A}F_{\mu\nu}^{A}\;,$
(3.1)

where
$\mathcal{V}(T,\bar{T})=(\bar{\mathcal{V}}_{A},\bar{\mathcal{V}}^{ij}{}_{A},\mathcal{V}_{ijA},\mathcal{V}_{A})$
define the coset representative in the $Spin^{*}(12)$ Majorana-Weyl spinor
representation. The $\mathcal{N}=6$ supermultiplet decomposes according to the
increasing $U(1)$ weight 151515Here we call ‘weight’ the charge with respect
to $U(1)\subset U(6)$ R-symmetry / isotropy group. from
$w=0,\frac{1}{6},\frac{1}{3},\frac{1}{2},\frac{2}{3},\frac{5}{6},1$ as

$C_{\alpha\beta\gamma\delta}\;,\quad\rho_{\alpha\beta\gamma i}\;,\quad
F_{\alpha\beta ij}\;,\quad\chi_{\alpha ijk}\;,\quad
P_{M}{}^{ij}dT^{M}\;,\quad\bar{\lambda}_{\dot{\alpha}}^{i}\;,\quad\bar{F}_{\dot{\alpha}\dot{\beta}}\;,$
(3.2)

and its CPT conjugate. In the linearised approximation, one can identify the
linearised complex superfield $W^{ij}$

$P_{M}{}^{ij}dT^{M}\approx dW^{ij}\;,$ (3.3)

which defines the ultra-short supersymmetry representation determined by the
action of the superspace derivative as [76]

$\begin{split}D_{\alpha}^{i}W^{jk}&=\frac{1}{2}\varepsilon^{ijkpqr}\chi_{\alpha
pqr}\;,\\\ D_{\alpha}^{i}\chi_{\beta jkl}&=3\delta^{i}_{[j}F_{\alpha\beta
kl]}\;,\\\ D_{\alpha}^{i}F_{\beta\gamma
jk}&=2\delta^{i}_{[j}\rho_{\alpha\beta\gamma k]}\;,\\\
D_{\alpha}^{i}\rho_{\beta\gamma\delta
j}&=\delta^{i}_{j}C_{\alpha\beta\gamma\delta}\;.\end{split}\hskip
28.45274pt\begin{split}\bar{D}_{\dot{\alpha}i}W^{jk}&=\delta_{i}^{[j}\bar{\lambda}_{\dot{\alpha}}^{k]}\;,\\\
\bar{D}_{\dot{\alpha}i}\bar{\lambda}_{\dot{\beta}}^{j}&=\delta_{i}^{j}\bar{F}_{\dot{\alpha}\dot{\beta}}\;,\\\
&\\\ &\end{split}$ (3.4)

We will need to define differential operators on the symmetric space
$SO^{*}(12)/U(6)$. The moduli space metric is by construction

$2G_{M\bar{N}}dT^{M}d\bar{T}^{\bar{N}}={\rm tr}_{{\scriptscriptstyle\bf
12}}P\bar{P}=\frac{1}{4}{\rm tr}_{{\scriptscriptstyle\bf 32}}P\bar{P}\;,$
(3.5)

with components

$G_{M\bar{N}}=P_{M}{}^{ij}\bar{P}_{\bar{N}ij}$ (3.6)

and we define the covariant derivative

$\mathcal{D}_{ij}=\bar{P}_{\bar{N}ij}G^{M\bar{N}}(\partial_{M}+Q_{M})\
,\qquad\bar{\mathcal{D}}^{ij}=P_{M}{}^{ij}G^{M\bar{N}}(\bar{\partial}_{\bar{M}}+Q_{\bar{M}})\
,$ (3.7)

where $Q_{M}{}^{i}{}_{j}$ defined in (2.53) acts on $U(6)$ tensors in the
corresponding representation. These covariant derivatives acting on a $U(6)$
scalar function $f(T,\bar{T})$ satisfy the commutation relation

$[\mathcal{D}_{ij},\bar{\mathcal{D}}^{kl}]\mathcal{D}_{pq}\,f(T,\bar{T})=2\delta_{j][p}^{kl}\mathcal{D}_{q][i}\,f(T,\bar{T})\
.$ (3.8)

The superspace derivative of a function of the scalars gives

$D_{\alpha}^{i}\mathcal{E}(T,\bar{T})=\Bigl{(}{\frac{1}{2}\varepsilon^{ijkpqr}\chi_{\alpha
pqr}\mathcal{D}_{jk}+\lambda_{\alpha\,j}\bar{\mathcal{D}}^{ij}}\Bigr{)}\mathcal{E}(T,\bar{T})\;.$
(3.9)

One classifies the BPS protected couplings using superconformal
representations of $SU(2,2|6)$ according to [74]. The superfield $W^{ij}$
defines an ultra-short representation of the superconformal algebra
$\mathfrak{su}(2,2|6)$ with the dilatation weight $L[W^{ij}]=1$ and the
R-symmetry weight $R[W^{ij}]=-\frac{1}{3}$, with the convention that
$R[D_{\alpha}^{i}]=-\frac{1}{6}$ and $L[D_{\alpha}^{i}]=\frac{1}{2}$
[77].161616The $SU(2,2|6)$ R-charge is not the $U(1)$ R-charge of the super-
Poincaré algebra, but we have instead the $U(1)$ weight $w=-2R-h$ where $h$ is
the helicity of the field. The BPS protected couplings are the top components
of short multiplets, which must be Lorentz invariant but not necessarily
$U(6)$ invariants. The polynomials in $W^{ij}$ defining short multiplets with
a Lorentz invariant top component are represented in table 1.

BPS degree | Polynomial | $SU(6)$ | $L$ | $R$
---|---|---|---|---
$(2,4)$ |  $W^{n}$, $n\geq 4$ | $[0,n,0,0,0]$ | $n$ | $-\frac{n}{3}$
$(2,2)$ |  $W^{n}\bar{W}^{m}$, $n\geq 2$, $m\geq 2$ | $[0,n,0,m,0]$ | $n+m$ | $\frac{m-n}{3}$
$(1,1)$ |  $W^{n+k}\bar{W}^{m+k}$, $k\geq 2$, | $[k,n,0,m,k]$ | $n+m+2k$ | $\frac{m-n}{3}$
$(2,0)$ |  $W^{n}\bar{W}^{m+2k}$, $n\geq 2$, $k\geq 2$ | $[0,n+k,0,m,0]$ | $n+m+2k$ | $\frac{m+2k-n}{3}$
Table 1: Short multiplets from polynomials in the superfield $W$.

A BPS multiplet has $(p,q)$ degree if the chiral primary is annihilated by
$D_{\alpha}^{i}$ for $i=1$ to $p$ and $\bar{D}_{\alpha j}$ for $j=7-q$ to $6$,
preserving therefore a fraction $\frac{p+q}{12}$ of the supersymmetries. The
$(p,q)$ BPS short multiplets are conveniently represented in harmonic
superspace using the harmonic variables in $U(6)/(U(p)\times U(6-p-q)\times
U(q))$ [78]. The $(2,2)$ BPS primary with $n=m=2$ defines the
$\mathcal{R}^{4}$ linearised invariant while the $(1,1)$ primary with $n=m=0$
and $k=2$ defines the ${D}^{2}\mathcal{R}^{4}$ linearised invariant [54]. The
representations with $m$, $n$, $k$ generic allow to determine the differential
equations satisfied by the threshold function as in [24].

We will see that each of these $(p,q)$ BPS degrees corresponds to an
automorphic representation of $Spin^{*}(12)$. Functions and tensors on the
symmetric space that are eigen functions of all Casimir operators, i.e. the
three independent invariant operators

$\displaystyle\Delta$ $\displaystyle=$ $\displaystyle
2\bar{\mathcal{D}}^{ij}\mathcal{D}_{ij}\;,\quad\Delta^{\\!\times}=\frac{1}{32}\varepsilon^{ijklpq}\varepsilon_{rstupq}\bar{\mathcal{D}}^{rs}\bar{\mathcal{D}}^{tu}\mathcal{D}_{ij}\mathcal{D}_{kl}\;,$
$\displaystyle\Delta^{\\!\rm det}$ $\displaystyle=$
$\displaystyle\frac{1}{48^{2}}\varepsilon^{ijklpq}\varepsilon_{rstuvw}\bar{\mathcal{D}}^{rs}\bar{\mathcal{D}}^{tu}\bar{\mathcal{D}}^{vw}\mathcal{D}_{ij}\mathcal{D}_{kl}\mathcal{D}_{pq}\;,$
(3.10)

for $Spin^{*}(12)$, are associated to nilpotent orbits of $Spin^{*}(12)$. To
be more precise, the variety associated to their annihilator in the universal
enveloping algebra (realised as differential operators) is the closure of a
union of nilpotent orbits [79]. In particular the differential equations
implied by supersymmetry correspond to small nilpotent orbits. In general, a
nilpotent orbit of the complex simple Lie algebra ${\mathfrak{g}}$ is
determined by its normal ${\mathfrak{sl}}_{2}$ triple such than the semi-
simple element $h\in{\mathfrak{sl}}_{2}$ is an integral weight that defines a
$\mathds{Z}$ grading of the Lie algebra:
${\mathfrak{g}}=\bigoplus_{n=-n_{\scalebox{0.5}{max}}}^{n_{\scalebox{0.5}{max}}}{\mathfrak{g}}^{\scriptscriptstyle(n)}$,
and the nilpotent element $e\in{\mathfrak{sl}}_{2}$ is generic in
${\mathfrak{g}}^{\scriptscriptstyle(2)}$, i.e. its $G^{\scriptscriptstyle(0)}$
orbit is open in ${\mathfrak{g}}^{\scriptscriptstyle(2)}$ for the Zariski
topology.171717Zariski topology is defined in such a way that the closed sets
are solutions to algebraic equations. The closure of a nilpotent orbit is the
set of Lie algebra elements that satisfy the same algebraic equations, e.g.
they have the same nilpotency degree in all irreducible representation. The
dimension of the nilpotent orbit $G\cdot e$ is ${\rm
dim}({\mathfrak{g}}^{\scriptscriptstyle(1)})+2\sum_{n=2}^{n_{\scalebox{0.5}{max}}}{\rm
dim}({\mathfrak{g}}^{\scriptscriptstyle(n)})$ and is always even, because
$G\cdot e$ is a symplectic variety. The space of functions / tensors
associated to the nilpotent orbit corresponds to a quantification of the
Zariski closure $\overline{G\cdot e}$ of $G\cdot e$, and can be represented in
principle by functions supported on the Lagrangian subspace of dimension
$d_{\rm GK}=\frac{1}{2}{\rm
dim}({\mathfrak{g}}^{\scriptscriptstyle(1)})+\sum_{n=2}^{n_{\scalebox{0.5}{max}}}{\rm
dim}({\mathfrak{g}}^{\scriptscriptstyle(n)})$. The tensors we are interested
in for the string theory low-energy effective action are automorphic forms
invariant under the U-duality group, and the associated representations are
called automorphic representations. One defines the Gelfand–Kirillov dimension
$d_{\rm GK}$ of the associated automorphic representation, which defines the
dimension of its Fourier support. The relation between BPS couplings and
nilpotent orbits of $SO^{*}(12)$ is summarised in Figure 1, where the white
dotes are nilpotent orbits of $SO(12,\mathds{C})$ that do not intersect with
${\mathfrak{so}}^{*}(12)$ and are therefore irrelevant. The lines between two
nilpotent orbits mean that the smaller orbit is in the Zariski closure of the
bigger one [79]. The Dynkin label is related to the BPS degree so that
$U(p)\times U(6-p-q)\times U(q)$ is the stabiliser of the Cartan
representative of the normal triple in $U(6)$ and the $U(1)$ weight
coefficient is such that the G-analytic superfields are of weight $2$.

•••$\circ$••$\circ$$\mathcal{R}$$0$$\bar{\mathcal{F}}^{2}\mathcal{R}^{2}$$\Lambda_{2}+w$$\mathcal{R}^{4}$$\Lambda_{2}+\Lambda_{4}$${D}^{2}\mathcal{R}^{4},\,\bar{\mathcal{F}}^{2k}\mathcal{R}^{4}$$2\Lambda_{2}-w$${D}^{2}\mathcal{R}^{4}$$2\Lambda_{1}+2\Lambda_{5}$$0$-------$18$$20$$28$$30$$32$$34$dim
Figure 1: Closure Hasse diagram of the small nilpotent orbits of
$SO(12,\mathds{C})$. The normal triple Dynkin label is given in Bourbaki
convention for the fundamental weights $\Lambda_{i}=A^{-1ij}\alpha_{j}$ of
$SU(6)$ and $w$ the $U(1)$ weight.

###  3.2 (2,4) BPS $\bar{\mathcal{F}}^{2}\mathcal{R}^{2}$ type invariant

To describe the class of supersymmetry invariants obtained from the $(2,4)$
chiral primaries in Table 1, it is convenient to introduce the associated
harmonic superspace, with harmonic variables $u^{\hat{r}}{}_{i}$ for
$\hat{r}=1,2$ and $u^{r}{}_{i}$ for $r=3,4,5,6$ parametrising
$U(6)/(U(2)\times U(4))$. The details of this construction can be found in
[78]. In practice the harmonic variables simply allow to restrict the set of
indices to define the corresponding short multiplets. The G-analytic
superfield $W^{12}=u^{1}{}_{i}u^{2}{}_{j}W^{ij}$ satisfies

$D_{\alpha}^{\hat{r}}W^{12}=0\ ,\qquad\bar{D}_{\dot{\alpha}r}W^{12}=0\ .$
(3.11)

The chiral primaries are therefore all the harmonics of $(W^{12})^{n}$, that
combine into the irreducible representation $[0,n,0,0,0]$ of $SU(6)$. The top-
component can be shown to include the graviphoton–graviton coupling

$[D_{\alpha}^{r}]^{8}[\bar{D}_{\dot{\alpha}\hat{r}}]^{4}(W^{12})^{4+n}\sim(W^{12})^{n}\bar{F}_{\dot{\alpha}\dot{\beta}}\bar{F}^{\dot{\alpha}\dot{\beta}}C_{\alpha\beta\gamma\delta}C^{\alpha\beta\gamma\delta}+\dots$
(3.12)

which is the only term with two Weyl tensors. This coupling is therefore
determined by the amplitude with two gravitons of polarisation $+2$ and two
graviphotons of polarisation $-1$ (in the convention in which all particles
are incoming). Following the same analysis as in [20, 21, 24], one deduces
that the non-linear supersymmetric density ${\cal
L}_{\scriptscriptstyle(2,4)}[\Theta]$ must be determined by a form
$\Theta(T,\bar{T})$ of $U(1)$ weight $2$ such that

${\cal L}_{\scriptscriptstyle(2,4)}[\Theta]={\cal
L}_{\scriptscriptstyle(2,4)}^{\scriptscriptstyle(0)}+\mathcal{D}_{ij}\Theta(T){\cal
L}_{\scriptscriptstyle(2,4)}^{{\scriptscriptstyle(1)}ij}+\mathcal{D}_{ij}\mathcal{D}_{kl}\Theta(T){\cal
L}_{\scriptscriptstyle(2,4)}^{{\scriptscriptstyle(2)}ij,kl}+\dots+\mathcal{D}_{ij}\mathcal{D}_{kl}\cdots\mathcal{D}_{pq}\Theta(T){\cal
L}_{\scriptscriptstyle(2,4)}^{{\scriptscriptstyle(8)}ij,kl,\dots,pq}\;,$
(3.13)

where ${\cal L}_{\scriptscriptstyle(2,4)}^{{\scriptscriptstyle(n)}}$ are
$Spin^{*}(12)$ invariant densities that do not depend explicitly on the scalar
fields $T^{M}$ and transform in the $[0,n,0,0,0]$ irreducible representation
of $SU(6)$. More precisely, ${\cal
L}_{\scriptscriptstyle(2,4)}^{{\scriptscriptstyle(n)}}$ are functions of
$\hat{C},\hat{\rho},\hat{F},\hat{P},\chi,\lambda$ and their super-covariant
derivatives, as well as the naked gravitino fields that only appear in wedge
products. They are determined in the linearised approximation by the
linearised invariant as

$[D_{\alpha}^{r}]^{8}[\bar{D}_{\dot{\alpha}\hat{r}}]^{4}(W^{12})^{4+n}=(W^{12})^{n}{\cal
L}_{{\scriptscriptstyle(2,4)}\scalebox{0.6}{lin}}^{\scriptscriptstyle(0)}+n(W^{12})^{n-1}{\cal
L}_{{\scriptscriptstyle(2,4)}\scalebox{0.6}{lin}}^{{\scriptscriptstyle(1)}12}+n(n-1)(W^{12})^{n-2}{\cal
L}_{{\scriptscriptstyle(2,4)}\scalebox{0.6}{lin}}^{{\scriptscriptstyle(2)}12,12}+\dots$
(3.14)

For example

${\cal
L}_{\scriptscriptstyle(2,4)}^{\scriptscriptstyle(0)}\sim\hat{\bar{F}}^{2}\hat{C}^{2}+\dots\;,\qquad{\cal
L}_{\scriptscriptstyle(2,4)}^{{\scriptscriptstyle(8)}[0,8,0,0,0]}\sim\bar{\lambda}^{4}_{[0,2,0,0,0]}\chi^{8}_{[0,6,0,0,0]}\;.$
(3.15)

More schematically, we shall write (3.13) as

${\cal
L}_{\scriptscriptstyle(2,4)}[\Theta]=\sum_{n=0}^{8}\bigl{(}\mathcal{D}^{n}\Theta(T)\bigr{)}|_{[0,0,0,n,0]}\,{\cal
L}_{\scriptscriptstyle(2,4)}^{[0,n,0,0,0]}\ .$ (3.16)

The fact that the supersymmetry invariant expands in this way implies that the
derivative expansion of the form $\Theta(T)$ must only involve these specific
irreducible representations, i.e.

$\bar{\mathcal{D}}^{ij}\Theta(T)=0\
,\qquad\varepsilon^{ijklpq}\mathcal{D}_{kl}\mathcal{D}_{pq}\Theta(T)=0\ .$
(3.17)

Note that $\Theta(T)$ is a form of $U(1)$ weight $2$, so its covariant
derivative involves the Kähler connexion, and
$\frac{\Theta(T)}{\det(T-\bar{T})}$ is a holomorphic function of $T$. For a
form $\Theta_{w}$ of generic weight $w$ one has

$[\mathcal{D}_{ij},\bar{\mathcal{D}}^{kl}]\Theta_{w}=\frac{w}{2}\delta_{ij}^{kl}\Theta_{w}\
,\qquad[\mathcal{D}_{ij},\bar{\mathcal{D}}^{kl}]\mathcal{D}_{pq}\Theta_{w}=2\delta_{j][p}^{kl}\mathcal{D}_{q][i}\Theta_{w}+\frac{w}{2}\delta_{ij}^{kl}\mathcal{D}_{pq}\Theta_{w}\
,$ (3.18)

therefore

$\bar{\mathcal{D}}^{kl}\mathcal{D}_{[ij}\mathcal{D}_{pq]}\Theta_{w}-\mathcal{D}_{[ij}\mathcal{D}_{pq]}\bar{\mathcal{D}}^{kl}\Theta_{w}=(2-w)\delta^{kl}_{[ij}\mathcal{D}_{pq]}\Theta_{w}\
,$ (3.19)

and equation (3.17) is only consistent for $w=2$. This is indeed the $U(1)$
weight of $\Theta(T)$, since the field strength $\bar{F}$ has $U(1)$ weight
$w=-1$.

The $Spin^{*}(12)$ representation associated to the differential equation
(3.17) is the unique representation attached to the minimal nilpotent orbit of
${\mathfrak{so}}^{*}(12)$. The minimal nilpotent orbit is represented by an
element of degree $2$ in the Heisenberg parabolic decomposition

${\mathfrak{so}}^{*}(12)={\bf 1}^{\scriptscriptstyle(-2)}\oplus({\bf 2},{\bf
8})^{\scriptscriptstyle(-1)}\oplus\bigl{(}{\mathfrak{gl}}_{1}\oplus{\mathfrak{su}}(2)\oplus{\mathfrak{so}}^{*}(8)\bigr{)}^{\scriptscriptstyle(0)}\oplus({\bf
2},{\bf 8})^{\scriptscriptstyle(1)}\oplus{\bf 1}^{\scriptscriptstyle(2)}\;,$
(3.20)

and is of dimension $18$. One can understand this from the property that a Lie
algebra element in the coset component
${\mathfrak{so}}(12,\mathds{C})\ominus{\mathfrak{gl}}(6,\mathds{C})$
parametrised by a complex element $(\bar{W}_{ij},W^{ij})$ with only
$W^{12}\neq 0$ is in the complex minimal nilpotent orbit of
${\mathfrak{so}}(12,\mathds{C})$. The Cartan element of
${\mathfrak{u}}(6,\mathds{C})$ defining the normal triple parametrising this
orbit, i.e. $\Lambda_{2}+w$ in Bourbaki convention,181818We define the
fundamental weights of $SU(6)$ such that $\sum_{i}n^{i}\Lambda_{i}$ is the
highest weight for the representation of Dynkin label
$[n^{1},n^{2},n^{3},n^{4},n^{5}]$, with $[1,0,0,0,0]={\bf 6}$,
$[0,1,0,0,0]={\bf 15}$, $[0,0,1,0,0]={\bf 20}$, etc. We define the fundamental
weights of $SO^{*}(12)$ such that $\Lambda_{1}$ is the highest weight for the
vector representation, $\Lambda_{2}$ the adjoint, $\Lambda_{5}$ the pseudo-
real Weyl spinor and $\Lambda_{6}$ the real Weyl spinor. defines the harmonic
coset space as its semi-simple $U(6)$ orbit.

The Gelfand–Kirillov dimension of the minimal automorphic representation is
$9=18/2$, and corresponds to holomorphic forms of $T$ with Fourier expansion

$\Theta_{2}=-\tfrac{i}{8}\det(T-\bar{T})\sum_{\begin{subarray}{c}Q\in
L\subset\mathds{R}^{15}\\\ Q\times Q=0\end{subarray}}\tilde{c}(Q)e^{2\pi
iQ_{M}T^{M}}$ (3.21)

where $Q$ is in the minimal $SU^{*}(6)$ obit inside the real antisymmetric
rank two tensor representation of dimension $9$, and $\tilde{c}(Q)$ are
constants. Supersymmetry allows for the sum to be replaced by an integral, but
the axion shift symmetry of string theory implies that $Q$ only takes value in
a lattice $L$ inside $\mathds{R}^{15}$.

###  3.3 (2,2) BPS $\mathcal{R}^{4}$ type invariant

The $\mathcal{R}^{4}$ type invariant can be defined in the linearised
approximation from the (2,2) chiral primaries in Table 1. It is convenient to
introduce the associated harmonic superspace, with harmonic variables
$u^{\hat{r}}{}_{i}$ for $\hat{r}=1,2$, $u^{\check{r}}{}_{i}$ for
$\check{r}=3,4$ and $u^{r}{}_{i}$ for $r=5,6$ parametrising $U(6)/(U(2)\times
U(2)\times U(2))$. The associated G-analytic superfields
$W^{12}=u^{1}{}_{i}u^{2}{}_{j}W^{ij}$ and
$\bar{W}_{56}=u^{i}{}_{5}u^{j}{}_{6}\bar{W}_{ij}$ satisfy

$D_{\alpha}^{\hat{r}}W^{12}=0\ ,\qquad\bar{D}_{\dot{\alpha}r}W^{12}=0\;,\quad
D_{\alpha}^{\hat{r}}\bar{W}_{56}=0\
,\qquad\bar{D}_{\dot{\alpha}r}\bar{W}_{56}=0\ .$ (3.22)

The chiral primaries are therefore all the harmonics of
$(W^{12})^{n}(\bar{W}_{56})^{m}$, that combine into the irreducible
representation $[0,n,0,m,0]$ of $SU(6)$. The top-component can be shown to
include the $\mathcal{R}^{4}$ Bell–Robinson square term

$[D_{\alpha}^{\check{r}}]^{4}[D_{\alpha}^{{r}}]^{4}[\bar{D}_{\dot{\alpha}\hat{r}}]^{4}[\bar{D}_{\dot{\alpha}\check{r}}]^{4}(W^{12})^{2+n}(\bar{W}_{56})^{2+m}\sim(W^{12})^{n}(\bar{W}_{56})^{m}C^{2}\bar{C}^{2}+\dots$
(3.23)

As in the previous section, one therefore concludes that the non-linear
invariant is determined by a function $\mathcal{E}(T,\bar{T})$, and expands
schematically as

${\cal
L}_{\scriptscriptstyle(2,2)}[\mathcal{E}]=\sum_{\begin{subarray}{c}n,m=0\\\
0\leq n+m\leq
12\end{subarray}}^{10}\bigl{(}\mathcal{D}^{n}\bar{\mathcal{D}}^{m}\mathcal{E}(T,\bar{T})\bigr{)}|_{[0,m,0,n,0]}{\cal
L}_{\scriptscriptstyle(2,2)}^{[0,n,0,m,0]}$ (3.24)

where for example

$\displaystyle{\cal L}_{\scriptscriptstyle(2,2)}^{[0,0,0,0,0]}$
$\displaystyle\sim$ $\displaystyle C^{2}\bar{C}^{2}+\dots\;,$ (3.25)
$\displaystyle\quad{\cal L}_{\scriptscriptstyle(2,2)}^{[0,6,0,6,0]}$
$\displaystyle\sim$
$\displaystyle\chi^{8\,[0,0,0,6,0]}\bar{\chi}^{8\,[0,6,0,0,0]}\;,\qquad{\cal
L}_{\scriptscriptstyle(2,2)}^{[0,2,0,10,0]}\sim\bar{\lambda}^{4\,[0,0,0,2,0]}\chi^{8\,[0,0,0,6,0]}\bar{\chi}^{4\,[0,2,0,2,0]}\;.$

The restriction to the irreducible $U(6)$ representations appearing in the
supersymmetry invariant implies the differential equations

$\varepsilon^{ijklpq}\mathcal{D}_{kl}\mathcal{D}_{pq}\mathcal{E}=0\
,\qquad\mathcal{D}_{ik}\bar{\mathcal{D}}^{jk}\mathcal{E}=\frac{1}{6}\delta_{i}^{j}\mathcal{D}_{kl}\bar{\mathcal{D}}^{kl}\mathcal{E}\
,\qquad\varepsilon_{ijklpq}\bar{\mathcal{D}}^{kl}\bar{\mathcal{D}}^{pq}\mathcal{E}=0\
.$ (3.26)

Using the commutation relations (3.8) one finds that

$\mathcal{D}_{[ij}\bar{\mathcal{D}}^{lp}\mathcal{D}_{k]p}\mathcal{E}=\bar{\mathcal{D}}^{lp}\mathcal{D}_{[ij}\mathcal{D}_{kp]}\mathcal{E}-\frac{3}{2}\delta^{l}_{[k}\mathcal{D}_{ij]}\mathcal{E}\
.$ (3.27)

Using then (3.26) one gets

$\Delta\mathcal{E}=2\mathcal{D}_{ij}\bar{\mathcal{D}}^{ij}\mathcal{E}=-18\mathcal{E}\
.$ (3.28)

The function $\mathcal{E}_{\scriptscriptstyle(2,2)}$ defining the
$\mathcal{R}^{4}$ type invariant therefore satisfies the equations

$\varepsilon^{ijklpq}\mathcal{D}_{kl}\mathcal{D}_{pq}\mathcal{E}_{\scriptscriptstyle(2,2)}=0\
,\qquad\mathcal{D}_{ik}\bar{\mathcal{D}}^{jk}\mathcal{E}_{\scriptscriptstyle(2,2)}=-\frac{3}{2}\delta_{i}^{j}\mathcal{E}_{\scriptscriptstyle(2,2)}\
,\qquad\varepsilon_{ijklpq}\bar{\mathcal{D}}^{kl}\bar{\mathcal{D}}^{pq}\mathcal{E}_{\scriptscriptstyle(2,2)}=0\
.$ (3.29)

This implies in particular

$\Delta^{\\!\times}\mathcal{E}_{\scriptscriptstyle(2,2)}=0\;,\qquad\Delta^{\\!\rm
det}\mathcal{E}_{\scriptscriptstyle(2,2)}=0\;.$ (3.30)

where $\Delta^{\\!\times}$ and $\Delta^{\\!\rm det}$ were defined in (3.10).
The associated representation is attached to the next to minimal nilpotent
orbit of dimension 28 that is represented by an element of degree $2$ in the
parabolic decomposition

${\mathfrak{so}}^{*}(12)=({\bf 6},{\bf
1})^{\scriptscriptstyle(-2)}\oplus(\overline{\bf 4},{\bf
4})^{\scriptscriptstyle(-1)}\oplus\bigl{(}{\mathfrak{gl}}_{1}\oplus{\mathfrak{su}}^{*}(4)\oplus{\mathfrak{so}}^{*}(4)\bigr{)}^{\scriptscriptstyle(0)}\oplus({\bf
4},{\bf 4})^{\scriptscriptstyle(1)}\oplus({\bf 6},{\bf
1})^{\scriptscriptstyle(2)}\;.$ (3.31)

The Gelfand–Kirillov dimension of the next to minimal automorphic
representation is therefore $14$. The complex element $(\bar{W}_{ij},W^{ij})$
with only $\bar{W}_{56}$ and $W^{12}$ non-zero parametrises a generic complex
vector in the $({\bf 6},{\bf 1})^{\scriptscriptstyle(2)}$.

###  3.4 (1,1) BPS ${D}^{2}\mathcal{R}^{4}$ type invariants

A ${D}^{2}\mathcal{R}^{4}$ type invariant can be defined in the linearised
approximation from the (1,1) chiral primaries in Table 1. It is convenient to
introduce the associated harmonic superspace, with harmonic variables
$u^{1}{}_{i}$, $u^{r}{}_{i}$ for ${r}=2,3,4,5$ and $u^{6}{}_{i}$ parametrising
$U(6)/(U(1)\times U(4)\times U(1))$. The associated G-analytic superfields
$W^{1r}=u^{1}{}_{i}u^{r}{}_{j}W^{ij}$ and
$\bar{W}_{r6}=u^{i}{}_{r}u^{j}{}_{6}\bar{W}_{ij}$ satisfy

$D_{\alpha}^{1}W^{1r}=0\ ,\qquad\bar{D}_{\dot{\alpha}6}W^{1r}=0\;,\quad
D_{\alpha}^{1}\bar{W}_{r6}=0\ ,\qquad\bar{D}_{\dot{\alpha}6}\bar{W}_{r6}=0\ .$
(3.32)

In this case the measure can be defined in the non-linear theory [57], and one
finds that the class of supersymmetry invariants defined in this section can
be identify with the set of non-linear G-analytic integrands. It will
nonetheless be more convenient to use the linearised analysis. The chiral
primaries are all the harmonics of
$(W^{1r})^{n}(\bar{W}_{6s})^{m}(W^{1t}\bar{W}_{6t})^{k}$ with $k\geq 2$, that
combine into the irreducible representation $[k,n,0,m,k]$ of $SU(6)$. The top-
component can be shown to include the ${D}^{2}\mathcal{R}^{4}$ term 191919the
only terms that contribute to the four graviton in the linearised
approximation are where
$[D_{\alpha}^{r}]^{8}[D_{\alpha}^{6}]^{2}[\bar{D}_{\dot{\alpha}1}]^{2}[\bar{D}_{\dot{\alpha}r}]^{8}(W^{1r}\bar{W}_{6r})^{2}\sim[D_{\alpha}^{r_{1}}]^{4}D_{\alpha}^{6}\bar{D}_{\dot{\alpha}s_{1}}W^{1r}[D_{\alpha}^{r_{2}}]^{4}D_{\alpha}^{6}\bar{D}_{\dot{\alpha}s_{2}}W^{1s}[\bar{D}_{\dot{\alpha}1}][\bar{D}_{\dot{\alpha}s_{1}}]^{3}\bar{W}_{r6}[\bar{D}_{\dot{\alpha}1}][\bar{D}_{\dot{\alpha}s_{2}}]^{3}\bar{W}_{s6}\sim\partial
C\partial C\bar{C}\bar{C}$.

$[D_{\alpha}^{r}]^{8}[D_{\alpha}^{6}]^{2}[\bar{D}_{\dot{\alpha}1}]^{2}[\bar{D}_{\dot{\alpha}r}]^{8}(W^{1r})^{n}(\bar{W}_{s6})^{m}(W^{1t}\bar{W}_{6t})^{2+k}\sim(W^{1r})^{n}(\bar{W}_{s6})^{m}(W^{1t}\bar{W}_{6t})^{k}\nabla
C^{2}\nabla\bar{C}^{2}+\dots$ (3.33)

One therefore concludes as in the previous section that the non-linear
invariant is determined by a function $\mathcal{E}(T,\bar{T})$, and expands
schematically as

${\cal
L}_{\scriptscriptstyle(1,1)}[\mathcal{E}]=\sum_{\begin{subarray}{c}n,m,k\\\
0\leq n+m+2k\leq
16\end{subarray}}\bigl{(}\mathcal{D}^{n+k}\bar{\mathcal{D}}^{m+k}\mathcal{E}(T,\bar{T})\bigr{)}|_{[k,m,0,n,k]}{\cal
L}_{\scriptscriptstyle(1,1)}^{[k,n,0,m,k]}$ (3.34)

where for example

${\cal L}_{\scriptscriptstyle(1,1)}^{[0,0,0,0,0]}\sim\nabla
C^{2}\nabla\bar{C}^{2}+\dots\;,\quad{\cal
L}_{\scriptscriptstyle(1,1)}^{[2,4,0,8,2]}\sim\lambda^{2}_{[0,0,0,0,2]}\chi^{8}_{[0,0,0,6,0]}\bar{\chi}^{10}_{[2,4,0,2,0]}\;.$
(3.35)

The restriction of the $U(6)$ irreducible representations appearing in the
supersymmetry invariant implies the differential equations

$\varepsilon^{ijklpq}\mathcal{D}_{kl}\mathcal{D}_{pq}\mathcal{E}=0\
,\qquad\varepsilon_{ijklpq}\bar{\mathcal{D}}^{kl}\bar{\mathcal{D}}^{pq}\mathcal{E}=0\
.$ (3.36)

These constraints fix the eigenvalues of $\Delta^{\\!\times}$ and
$\Delta^{\\!\rm det}$ to be zero. The eigenvalue of the Laplace operator is
determined by supersymmetry, but does not follow directly from representation
theory via (3.36). We shall find indirectly that $\mathcal{E}$ satisfies the
equation

$\Delta\mathcal{E}=2\mathcal{D}_{ij}\bar{\mathcal{D}}^{ij}\mathcal{E}=-18\mathcal{E}\
.$ (3.37)

In principle this condition could be worked out explicitly using the same
method as in [11, 20].

The associated representation is attached to the nilpotent orbit of dimension
34 and is represented by a generic element of degree $2$ in the Heisenberg
parabolic decomposition

${\mathfrak{so}}^{*}(12)={\bf 1}^{\scriptscriptstyle(-4)}\oplus({\bf 2},{\bf
8})^{\scriptscriptstyle(-2)}\oplus\bigl{(}{\mathfrak{gl}}_{1}\oplus{\mathfrak{su}}(2)\oplus{\mathfrak{so}}^{*}(8)\bigr{)}^{\scriptscriptstyle(0)}\oplus({\bf
2},{\bf 8})^{\scriptscriptstyle(2)}\oplus{\bf 1}^{\scriptscriptstyle(4)}\;.$
(3.38)

The Gelfand–Kirillov dimension of the associated automorphic representations
is therefore $17$. There is a one-parameter family of such representations
determined by the eigen-value of the $SO^{*}(12)/U(6)$ Laplacian that
corresponds to the maximal parabolic character of the $GL(1)$ factor, i.e.
$g_{\scalebox{0.6}{4}}^{-2s}$ with $g_{\scalebox{0.6}{4}}$ the effective
string coupling in our parametrisation.

As for the ${D}^{6}\mathcal{R}^{4}$ correction in type IIB theory [17], one
expects that ${D}^{2}\mathcal{R}^{4}$ will get a contribution depending
quadratically in the $\bar{F}^{2}C^{2}$ threshold function $\Theta$ in the
$\mathcal{N}=6$ supergravity effective action. This is consistent with power
counting, as $\bar{F}^{2}C^{2}$ has dimension $2+4$ and
${D}^{2}\mathcal{R}^{4}$ dimension $2+8$. Moreover one can check that this is
consistent with the automorphic representation of Gelfand–Kirillov dimension
$17$ as (3.17) implies

$\varepsilon^{ijklpq}\mathcal{D}_{kl}\mathcal{D}_{pq}\bigl{(}\Theta(T)\bar{\Theta}(\bar{T})\bigr{)}=0\
,\qquad\varepsilon_{ijklpq}\bar{\mathcal{D}}^{kl}\bar{\mathcal{D}}^{pq}\bigl{(}\Theta(T)\bar{\Theta}(\bar{T})\bigr{)}=0\;,$
(3.39)

consistently with (3.36). We conjecture therefore that the Wilsonian effective
action at this order includes a term in
$\mathcal{E}_{\scriptscriptstyle(1,1)}(T,\bar{T}){D}^{2}\mathcal{R}^{4}$ with
the function $\mathcal{E}_{\scriptscriptstyle(1,1)}(T,\bar{T})$ satisfying to

$\varepsilon^{ijklpq}\mathcal{D}_{kl}\mathcal{D}_{pq}\mathcal{E}_{\scriptscriptstyle(1,1)}=0\
,\qquad\varepsilon_{ijklpq}\bar{\mathcal{D}}^{kl}\bar{\mathcal{D}}^{pq}\mathcal{E}_{\scriptscriptstyle(1,1)}=0\
,\quad
2\mathcal{D}_{ij}\bar{\mathcal{D}}^{ij}\mathcal{E}_{\scriptscriptstyle(1,1)}=-18\mathcal{E}_{\scriptscriptstyle(1,1)}-\Theta(T)\bar{\Theta}(\bar{T})\;.$
(3.40)

We shall further argue in [72] that such a source term is indeed present in
perturbative string theory. It originates from the singularity of the two-loop
amplitude integrand at the separating degeneration locus, as for the
${D}^{6}\mathcal{R}^{4}$ threshold function in $\mathcal{N}=8$ [80] and
${D}^{2}\mathcal{F}^{4}$ in $\mathcal{N}=4$ [42].

###  3.5 (2,0) BPS $\mathcal{F}^{2k}\mathcal{R}^{4}$ and
${D}^{2}\mathcal{R}^{4}$ type invariants

The class of supersymmetry invariants one can construct from the $(2,0)$
chiral primaries in Table 1 is conveniently described using the associated
harmonic superspace, with harmonic variables $u^{\hat{r}}{}_{i}$ for
$\hat{r}=1,2$ and $u^{r}{}_{i}$ for $r=3,4,5,6$ parametrising
$U(6)/(U(2)\times U(4))$. The G-analytic superfields
$W^{12}=u^{1}{}_{i}u^{2}{}_{j}W^{ij}$ and
$\bar{W}_{rs}=u^{i}{}_{r}u^{j}{}_{s}\bar{W}_{ij}$ satisfy

$D_{\alpha}^{\hat{r}}W^{12}=0\ ,\qquad D_{\alpha}^{\hat{r}}\bar{W}_{rs}=0\ .$
(3.41)

The chiral primaries are therefore all the harmonics of
$(W^{12})^{n}(\tfrac{1}{2}\varepsilon^{rstu}\bar{W}_{rs}\bar{W}_{tu})^{k}(\bar{W}_{rs})^{m}$
with $n\geq 2$ and $k\geq 2$, that combine into the irreducible representation
$[0,n+k,0,m,0]$ of $SU(6)$. The top-component includes the
graviphoton–graviton coupling

$\displaystyle[D_{\alpha}^{r}]^{8}[\bar{D}_{\dot{\alpha}i}]^{12}(W^{12})^{2+n}(\tfrac{1}{2}\varepsilon^{rstu}\bar{W}_{rs}\bar{W}_{tu})^{k+2}(\bar{W}_{rs})^{m}$
(3.42) $\displaystyle\sim$
$\displaystyle(W^{12})^{n}(\tfrac{1}{2}\varepsilon^{rstu}\bar{W}_{rs}\bar{W}_{tu})^{k}(\bar{W}_{rs})^{m}(\bar{F}^{12})^{2}C^{2}\bar{C}^{2}+\dots$

where the indices of $(\bar{W}_{rs})^{m}$ are all meant to be uncontracted and
projected to the traceless component. The linearised analysis therefore
suggests that the non-linear supersymmetry invariant takes the form

${\cal
L}_{\scriptscriptstyle(2,0)}[\mathcal{E}_{ij,kl}]=\sum_{n,m,k}\bigl{(}\mathcal{D}^{n}\bar{\mathcal{D}}^{m+2k}\mathcal{E}_{ij,kl}(T,\bar{T})\bigr{)}|_{[0,m,0,2+n+2k,0]}{\cal
L}_{\scriptscriptstyle(2,0)}^{[0,2+n+2k,0,m,0]}$ (3.43)

where $\mathcal{E}_{ij,kl}(T,\bar{T})$ is a form of $U(1)$ weight
$\frac{2}{3}$ in the $[0,0,0,2,0]$ of $SU(6)$ satisfying to

$\varepsilon^{ijrstu}\mathcal{D}_{rs}\mathcal{D}_{tu}\mathcal{E}_{kl,pq}=0\
,\qquad\mathcal{D}_{ir}\bar{\mathcal{D}}^{jr}\mathcal{E}_{kl,pq}=\frac{1}{6}\delta_{i}^{j}\mathcal{D}_{rs}\bar{\mathcal{D}}^{rs}\mathcal{E}_{ij,kl}\
.$ (3.44)

However, this linearised invariant only contributes to six and higher point
amplitudes whereas we expect the non-linear supersymmetry invariant to
contribute to four-point amplitude. Although it is always possible to
linearised a non-linear supersymmetry invariant, one cannot neglect the
possibility that a given non-linear invariant produce different types of
linearised invariant in the linear approximation. This is precisely the
situation encountered in $\mathcal{N}=8$ supergravity for the $(2,0)$-BPS
${D}^{6}\mathcal{R}^{4}$ type invariant [24].

The weight of $\mathcal{E}_{ij,kl}(T,\bar{T})$ is compatible with the
requirement that

$\mathcal{E}_{ij,kl}(T,\bar{T})=\bigl{(}\varepsilon_{ijpqrs}\mathcal{D}_{kl}+\varepsilon_{klpqrs}\mathcal{D}_{ij}-2\varepsilon_{pqrs[ij}\mathcal{D}_{kl]}\bigr{)}\bar{\mathcal{D}}^{pq}\bar{\mathcal{D}}^{rs}\mathcal{E}(T,\bar{T})\;,$
(3.45)

and we cannot neglect the possibility that there be other linearised
invariants in the expansion of the non-linear invariant in derivatives of
$\mathcal{E}(T,\bar{T})$. To construct these linearised invariants one can use
the $(2,2)$ harmonic superspace introduced in Section 3.3, which is by
construction consistent with the $(2,0)$-BPS condition. The first natural
candidate is

$[D_{\alpha}^{\check{r}}]^{4}[D_{\alpha}^{{r}}]^{4}[\bar{D}_{\dot{\alpha}\hat{r}}]^{4}[\bar{D}_{\dot{\alpha}\check{r}}]^{4}(W^{12})^{2+n}(\bar{W}_{56})^{m}(\partial\bar{W}_{56}\cdot\partial\bar{W}_{56})\sim(W^{12})^{n}(\bar{W}_{56})^{m}C^{2}(\nabla\bar{C})^{2}+\dots$
(3.46)

which is compatible with the gradient expansion

$\sum_{n,m}\bigl{(}\mathcal{D}^{n}\bar{\mathcal{D}}^{m}\mathcal{E}(T,\bar{T})\bigr{)}|_{[0,m,0,n,0]}{\cal
L}_{\scriptscriptstyle(1,1)}^{[0,n,0,m,0]}$ (3.47)

such that ${\cal L}_{\scriptscriptstyle(1,1)}^{[0,0,0,0,0]}\sim
C^{2}(\nabla\bar{C})^{2}$ plus supersymmetric completion. Up to integration by
part, these densities are the same as in 3.33 for $k=0$. Another natural
candidate is

$[D_{\alpha}^{\check{r}}]^{4}[D_{\alpha}^{{r}}]^{4}[\bar{D}_{\dot{\alpha}\hat{r}}]^{4}[\bar{D}_{\dot{\alpha}\check{r}}]^{4}(W^{12})^{2+n}(\bar{W}_{56})^{m}(F_{56}F_{56})\sim(W^{12})^{n}(\bar{W}_{56})^{m}C^{2}(\nabla\bar{F})^{2}+\dots$
(3.48)

which is of $U(1)$ weight $2$. It is therefore compatible with the gradient
expansion

$\sum_{n,m}\bigl{(}\mathcal{D}^{n}\bar{\mathcal{D}}^{m}\det\\!\mathcal{D}\,\mathcal{E}(T,\bar{T})\bigr{)}|_{[0,m,0,n,0]}{\cal
L}_{{\scriptscriptstyle(1,1)}+2}^{[0,n,0,m,0]}$ (3.49)

where

$\det\\!\mathcal{D}\,\mathcal{E}(T,\bar{T})=\frac{1}{48}\varepsilon^{ijklpq}\mathcal{D}_{ij}\mathcal{D}_{kl}\mathcal{D}_{pq}\,\mathcal{E}(T,\bar{T})$
(3.50)

and ${\cal L}_{{\scriptscriptstyle(1,1)}+2}^{[0,0,0,0,0]}\sim
C^{2}({\nabla}\bar{\mathcal{F}})^{2}$ plus supersymmetric completion. There is
no other available candidate $(2,2)$ harmonic superspace integral. We expect
therefore the complete invariant to expand in all these structures with

$\displaystyle{\cal
L}_{{\scriptscriptstyle(2,0)}+{\scriptscriptstyle(0,2)}}[\mathcal{E}]$
$\displaystyle=$
$\displaystyle\sum_{n,m,k}\bigl{(}\mathcal{D}^{n+1}\bar{\mathcal{D}}^{m+2k+2}\mathcal{E}(T,\bar{T})\bigr{)}|_{[0,m,0,2+n+2k+2,0]}{\cal
L}_{\scriptscriptstyle(2,0)}^{[0,2+n+2k,0,m,0]}$ (3.51)
$\displaystyle+\sum_{n,m}\bigl{(}\mathcal{D}^{n}\bar{\mathcal{D}}^{m}\mathcal{E}(T,\bar{T})\bigr{)}|_{[0,m,0,n,0]}{\cal
L}_{\scriptscriptstyle(1,1)}^{[0,n,0,m,0]}$
$\displaystyle+\sum_{n,m}\bigl{(}\mathcal{D}^{n}\bar{\mathcal{D}}^{m}\det\\!\mathcal{D}\,\mathcal{E}(T,\bar{T})\bigr{)}|_{[0,m,0,n,0]}{\cal
L}_{{\scriptscriptstyle(1,1)}+2}^{[0,n,0,m,0]}$
$\displaystyle+\sum_{n,m}\bigl{(}\mathcal{D}^{n}\bar{\mathcal{D}}^{m}\det\\!\bar{\mathcal{D}}\,\mathcal{E}(T,\bar{T})\bigr{)}|_{[0,m,0,n,0]}{\cal
L}_{{\scriptscriptstyle(1,1)}-2}^{[0,n,0,m,0]}$
$\displaystyle+\sum_{n,m,k}\bigl{(}\bar{\mathcal{D}}^{n+1}\mathcal{D}^{m+2k+2}\mathcal{E}(T,\bar{T})\bigr{)}|_{[0,2+n+2k+2,0,m,0]}{\cal
L}_{\scriptscriptstyle(0,2)}^{[0,m,0,2+n+2k,0]}\;,$

where ${\cal L}_{\scriptscriptstyle(0,2)}^{[0,m,0,2+n+2k,0]}$ is the complex
conjugate of ${\cal L}_{\scriptscriptstyle(2,0)}^{[0,2+n+2k,0,m,0]}$ and
${\cal L}_{{\scriptscriptstyle(1,1)}-2}^{[0,n,0,m,0]}$ the complex conjugate
of ${\cal L}_{{\scriptscriptstyle(1,1)}+2}^{[0,m,0,n,0]}$.

Using the constraints (3.44) on (3.45) one concludes that the real function
$\mathcal{E}(T,\bar{T})$ must satisfy the constraint

$\varepsilon^{ijrstu}\varepsilon_{klpqvw}\mathcal{D}_{rs}\mathcal{D}_{tu}\bar{\mathcal{D}}^{pq}\bar{\mathcal{D}}^{vw}\mathcal{E}(T,\bar{T})=0\,,\qquad\mathcal{D}_{ik}\bar{\mathcal{D}}^{jk}\mathcal{E}(T,\bar{T})=-\frac{3}{2}\delta_{i}^{j}\mathcal{E}(T,\bar{T})\
,$ (3.52)

and in particular the three Casimir operators give (3.10)

$\Delta\mathcal{E}(T,\bar{T})=-18\mathcal{E}(T,\bar{T})\;,\qquad\Delta^{\\!\times}\mathcal{E}(T,\bar{T})=0\;,\qquad\Delta^{\\!\rm
det}\mathcal{E}(T,\bar{T})=0\;.$ (3.53)

Note that the structure in (3.48) must necessarily be present, because one
does not find any solution $\mathcal{E}(T,\bar{T})$ to these equations such
that $\det\\!\mathcal{D}\,\mathcal{E}(T,\bar{T})=0$ without satisfying
moreover (3.26) as for the $(2,2)$-BPS invariant.

The existence of this invariant will be confirmed by the one-loop amplitude
computation in string theory. The associated representation is attached to the
nilpotent orbit of dimension 30 that is realised by an element of degree $2$
in the abelian parabolic decomposition

${\mathfrak{so}}^{*}(12)=\overline{\bf
15}^{\scriptscriptstyle(-2)}\oplus\bigl{(}{\mathfrak{gl}}_{1}\oplus{\mathfrak{su}}^{*}(6)\bigr{)}^{\scriptscriptstyle(0)}\oplus{\bf
15}^{\scriptscriptstyle(2)}\;.$ (3.54)

The Gelfand–Kirillov dimension of the associated automorphic representations
is therefore $15$.

There is a one-parameter family of such representations fixed by the
eigenvalue of the $SO^{*}(12)/U(6)$ Laplace operator that is represented by
the maximal parabolic character associated to the $GL(1)$ factor, i.e.
$R^{3s}$ in our conventions for $R$ the circle radius.

According to this structure we find that $\mathcal{E}(T,\bar{T})$ appears in
the four-graviton amplitude $\mathcal{E}_{\scriptscriptstyle(1,0,0)}$ at order
$s$, while $\det\\!\mathcal{D}\,\mathcal{E}(T,\bar{T})$ appears in the two-
graviphoton two-graviton amplitude at order $s^{2}$.

This class of supersymmetry invariants generalises to higher derivative
couplings by including the additional G-analytic superfields
$\bar{F}_{\dot{\alpha}\dot{\beta}}^{12}=u^{1}{}_{i}u^{2}{}_{j}\bar{F}_{\dot{\alpha}\dot{\beta}}^{ij}$
satisfying to

$D_{\alpha}^{\hat{r}}\bar{F}_{\dot{\alpha}\dot{\beta}}^{12}=0\ .$ (3.55)

The chiral primaries are in one-to-one correspondance with the harmonics of
202020The conformal primaries can be obtained as
$[S_{\dot{\alpha}}^{r}]^{8}(W^{12})^{n}(\tfrac{1}{2}\varepsilon^{rstu}\bar{W}_{rs}\bar{W}_{tu})^{k-2}(\bar{W}_{rs})^{m}(\bar{F}^{12})^{2\ell+4}$
with the special supersymmetry
$S_{\dot{\alpha}}^{r}=u^{r}{}_{i}S_{\dot{\alpha}}^{i}$. They only differ from
(3.56), by a total superspace derivative and therefore define the same
supersymmetry invariants.

$(W^{12})^{n}(\tfrac{1}{2}\varepsilon^{rstu}\bar{W}_{rs}\bar{W}_{tu})^{k}(\bar{W}_{rs})^{m}(\bar{F}^{12})^{2\ell}\;,$
(3.56)

that combine into the irreducible representation $[0,n+k+2\ell,0,m,0]$ of
$SU(6)$. The top-component includes the graviphoton–graviton coupling

$\displaystyle[D_{\alpha}^{r}]^{8}[\bar{D}_{\dot{\alpha}i}]^{12}(W^{12})^{2+n}(\tfrac{1}{2}\varepsilon^{rstu}\bar{W}_{rs}\bar{W}_{tu})^{k+2}(\bar{W}_{rs})^{m}(\bar{F}^{12})^{2\ell-2}$
(3.57) $\displaystyle\sim$
$\displaystyle(W^{12})^{n}(\tfrac{1}{2}\varepsilon^{rstu}\bar{W}_{rs}\bar{W}_{tu})^{k}(\bar{W}_{rs})^{m}(\bar{F}^{12})^{2\ell}C^{2}\bar{C}^{2}+\dots$

The linearised analysis therefore suggests that the non-linear supersymmetry
invariant takes the form

${\cal
L}_{\scriptscriptstyle(2,0)}^{\scriptscriptstyle(\ell)}[\mathcal{E}_{12^{2\ell}}]=\sum_{n,m,k}\bigl{(}\mathcal{D}^{n}\bar{\mathcal{D}}^{m+2k}\mathcal{E}_{12^{2\ell}}(T,\bar{T})\bigr{)}|_{[0,m,0,2\ell+n+2k,0]}{\cal
L}_{\scriptscriptstyle(2,0)}^{[0,2\ell+n+2k,0,m,0]}$ (3.58)

where $\mathcal{E}_{12^{2\ell}}(T,\bar{T})$ is a form of $U(1)$ weight
$w=2\ell/3$ in the $[0,0,0,2\ell,0]$ of $SU(6)$ satisfying to

$\varepsilon^{ijrstu}\mathcal{D}_{rs}\mathcal{D}_{tu}\mathcal{E}_{12^{2\ell}}=0\
,\qquad\mathcal{D}_{ir}\bar{\mathcal{D}}^{jr}\mathcal{E}_{12^{2\ell}}=\frac{1}{6}\delta_{i}^{j}\mathcal{D}_{rs}\bar{\mathcal{D}}^{rs}\mathcal{E}_{12^{2\ell}}\
.$ (3.59)

It is associated to a discrete series representation attached to the same
nilpotent orbit corresponding the the maximal parabolic character
$R^{6+3\ell}$.

###  3.6 BPS invariants in five dimensions

In five dimensions the linearised superfields do not define superconformal
multiplets and linearised supersymmetry invariants cannot all be written as
superspace integrals of appropriate G-analytic integrands. To classify five-
dimensional supersymmetry invariants one must instead rely on the four-
dimensional ones, and check if they consistently lift to five dimensions. In
particular one finds that there is no 1/2 BPS $\mathcal{F}^{2}\mathcal{R}^{2}$
invariant in five dimensions, whereas the $\mathcal{R}^{4}$ and
${D}^{2}\mathcal{R}^{4}$ type invariants consistently lift. In this subsection
we introduce the linearised superfield in five dimensions to determine the
structure of the four-graviton amplitude for generic polarisations.

The fourteen real scalar fields $\phi^{M}$ parametrise the symmetric space
$SU^{*}(6)/Sp(3)$ and one defines the coset Maurer–Cartan form

$P_{M}{}^{ij}d\phi^{M}\,,$ (3.60)

in the real antisymmetric symplectic traceless representation of $Sp(3)$, with
$i=1$ to $6$ and for the symplectic form $\Omega_{ij}$ satisfying
$\Omega^{ik}\Omega_{jk}=\delta^{i}_{j}$

$P^{*}_{Mij}=\Omega_{ik}\Omega_{jl}P_{M}{}^{kl}\;,\qquad\Omega_{ij}P_{M}{}^{ij}=0\;\;.$
(3.61)

The $\mathcal{N}=3$ supermultiplet ($\mathcal{N}=6$ in $D=4$) involves the
supercovariant fields

$P_{M}{}^{ij}d\phi^{M}\;,\quad\chi_{\alpha}^{ijk}\;,\;\lambda_{\alpha}^{i}\;,\quad
F_{\alpha\beta}^{ij}\;,\;F_{\alpha\beta}\;,\quad\rho_{\alpha\beta\gamma}^{i}\;,\quad
C_{\alpha\beta\gamma\delta}\;,$ (3.62)

where $\alpha=1$ to $4$ are $Sp(1,1)=Spin(1,4)$ indices that are symmetrised,
while all the $Sp(3)$ indices are antisymmetrised and symplectic-traceless.
One defines the fifteen field strengths and the Weyl tensor using $Spin(1,4)$
gamma matrices as

$\displaystyle F_{\alpha\beta}^{ij}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\gamma^{ab}{}_{\alpha\beta}e_{a}{}^{\mu}e_{b}{}^{\nu}\mathcal{V}^{ij}{}_{IJ}F_{\mu\nu}^{IJ}\;,\qquad
F_{\alpha\beta}=\frac{1}{2}\gamma^{ab}{}_{\alpha\beta}e_{a}{}^{\mu}e_{b}{}^{\nu}\mathcal{V}_{IJ}F_{\mu\nu}^{IJ}\;,$
$\displaystyle C_{\alpha\beta\gamma\delta}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\gamma^{ab}{}_{(\alpha\beta}\gamma^{cd}{}_{\gamma\delta)}e_{a}{}^{\mu}e_{b}{}^{\nu}e_{c}{}^{\sigma}e_{c}{}^{\rho}R_{\mu\nu\sigma\rho}\;.$
(3.63)

In the linearised approximation, one can identify the real and symplectic-
traceless linearised superfield $W^{ij}$

$P_{M}{}^{ij}d\phi^{M}\approx dW^{ij}\;,$ (3.64)

which defines the ultra-short supersymmetry representation that is acted on by
the superspace derivative according to

$\displaystyle D_{\alpha}^{i}W^{jk}$ $\displaystyle=$
$\displaystyle\chi_{\alpha}^{ijk}+2\Omega^{i[j}\lambda_{\alpha}^{k]}+\frac{1}{3}\Omega^{jk}\lambda_{\alpha}^{i}\;.$
$\displaystyle D_{\alpha}^{i}\chi_{\beta}^{jkl}$ $\displaystyle=$
$\displaystyle
3\Omega^{i[j}(F_{\alpha\beta}^{kl]}-i\partial_{\alpha\beta}W^{kl]})-\frac{3}{2}\Omega^{[kl}(F_{\alpha\beta}^{j]i}-i\partial_{\alpha\beta}W^{j]i})\;,$
$\displaystyle D_{\alpha}^{i}\lambda_{\beta}^{j}$ $\displaystyle=$
$\displaystyle\Omega^{ij}F_{\alpha\beta}+\frac{1}{2}F^{ij}_{\alpha\beta}+\frac{3i}{2}\partial_{\alpha\beta}W^{ij}\;,$
(3.65)

where we define
$\partial_{\alpha\beta}=\frac{1}{2}\gamma^{a}{}_{\alpha\beta}\partial_{a}$. To
describe the linearised invariants relevant to the four-graviton amplitude it
is convenient to introduce the 1/3 BPS harmonic variables
$(u^{r}{}_{i},u^{\hat{r}}{}_{i},u_{ri})\in Sp(3)/(U(2)\times Sp(1))$ with
$r=1,2$ and $\hat{r}=3,4$ that satisfy

$\Omega^{ij}u^{r}{}_{i}u^{s}{}_{j}=0\;,\quad\Omega^{ij}u^{r}{}_{i}u_{sj}=2\delta^{r}_{s}\;,\quad\Omega^{ij}u^{r}{}_{i}u^{\hat{s}}{}_{j}=0\;,\quad\Omega^{ij}u^{\hat{r}}{}_{i}u^{\hat{s}}{}_{j}=\varepsilon^{\hat{r}\hat{s}}\;.$
(3.66)

One then has the two G-analytic superfields
$W^{12}=u^{1}{}_{i}u^{2}{}_{j}W^{ij}$ and
$F_{\alpha\beta}^{12}=u^{1}{}_{i}u^{2}{}_{j}F_{\alpha\beta}^{ij}$, satisfying
$D_{\alpha}^{r}W^{12}=D_{\alpha}F_{\beta\gamma}^{12}=0$ and one can define the
measure

$[D^{16}]=[D^{\hat{r}}_{\alpha}]^{8}[D_{{r\alpha}}]^{8}\;.$ (3.67)

From the four-dimensional analysis we know that the four-point
$\mathcal{R}^{4}$ invariant is unique, and it can be realised in five
dimensions as

$[D^{16}](W^{12})^{4}\sim\prod_{a=1}^{4}\varepsilon^{\alpha_{a}\beta_{a}\gamma_{a}\delta_{a}}C_{\alpha_{1}\alpha_{2}\alpha_{3}\alpha_{4}}C_{\beta_{1}\beta_{2}\beta_{3}\beta_{4}}C_{\gamma_{1}\gamma_{2}\gamma_{3}\gamma_{4}}C_{\delta_{1}\delta_{2}\delta_{3}\delta_{4}}+\dots=\frac{1}{256}t_{8}t_{8}\mathcal{R}^{4}+\dots$
(3.68)

where we introduce the standard $t_{8}$ tensor

$t_{8}F^{4}=24{\rm tr}_{\bf 5}F^{4}-6({\rm tr}_{\bf 5}F^{2})^{2}=-96{\rm
tr}_{\bf 4}F^{4}+48({\rm tr}_{\bf 4}F^{2})^{2}\;,$ (3.69)

with the notation 212121One raises $Sp(1,1)$ indices with the symplectic form
as $F_{\alpha}{}^{\beta}=F_{\alpha\gamma}\Omega^{\gamma\beta}$ and
$\varepsilon^{\alpha\beta\gamma\delta}=3\Omega^{[\alpha\beta}\Omega^{\gamma\delta]}$.

${\rm tr}_{\bf 5}F^{2}=F_{ab}F^{ba}\;,\quad{\rm tr}_{\bf
5}F^{4}=F_{ab}F^{bc}F_{cd}F^{da}\;,\quad{\rm tr}_{\bf
4}F^{2}=F_{\alpha}{}^{\beta}F_{\beta}{}^{\alpha}\;,\quad{\rm tr}_{\bf
4}F^{4}=F_{\alpha\beta}F^{\beta\gamma}F_{\gamma\delta}F^{\delta\alpha}\;.$
(3.70)

We also know from the four-dimensional analysis that there is a unique
linearised ${D}^{2}\mathcal{R}^{4}$ invariant, that can be written in five
dimensions as

$\displaystyle[D^{16}](W^{12})^{2}F_{\alpha\beta}^{12}F^{\alpha\beta 12}$
$\displaystyle\sim$
$\displaystyle\prod_{a=1}^{4}\varepsilon^{\alpha_{a}\beta_{a}\gamma_{a}\delta_{a}}\partial_{\epsilon_{1}(\alpha_{1}}C_{\alpha_{2}\alpha_{3}\alpha_{4}\epsilon_{2}}\partial^{(\epsilon_{1}}{}_{(\beta_{1}}C_{\beta_{2}\beta_{3}\beta_{4})}{}^{\epsilon_{2})}C_{\gamma_{1}\gamma_{2}\gamma_{3}\gamma_{4}}C_{\delta_{1}\delta_{2}\delta_{3}\delta_{4}}+\dots$
(3.71) $\displaystyle=$
$\displaystyle\frac{1}{3072}t_{8}^{(s)}t_{8}\partial_{a}R\partial^{a}RR^{2}+\dots$

where one uses integration by part in the last step, and the $t_{8}^{(s)}$
tensor is defined as
222222$(t_{8}^{(s)}){}^{a_{1}a_{2}b_{1}b_{2}c_{1}c_{2}d_{1}d_{2}}$ is
antisymmetric under the permutations of $a_{1}a_{2}$, $b_{1}b_{2}$,
$c_{1}c_{2}$ and $d_{1}d_{2}$ and symmetric under the permutations of the
pairs $(a_{1}a_{2})$ and $(b_{1}b_{2})$ and the pairs $(c_{1}c_{2})$ and
$(d_{1}d_{2})$. Therefore it is determined by its contraction with two
independent rank two tensors $F_{1ab}$ and $F_{2ab}$.

$t_{8}^{(s)}F_{1}^{\,2}F_{2}^{\;2}=8{\rm tr}_{\bf
5}F_{1}F_{2}F_{1}F_{2}-4({\rm tr}_{\bf 5}F_{1}F_{2})^{2}+2{\rm tr}_{\bf
5}F_{1}^{\;2}{\rm tr}_{\bf 5}F_{2}^{\;2}=-32{\rm tr}_{\bf
4}F_{1}^{\;2}F_{2}^{\;2}+16{\rm tr}_{\bf 4}F_{1}^{\;2}{\rm tr}_{\bf
4}F_{2}^{\;2}\,.$ (3.72)

For the amplitude it is convenient to introduce the three permutations of the
tensor $t_{8}^{(s)}$

$\displaystyle
t_{8}^{(s)}(F_{1}F_{2}F_{3}F_{4})=t_{8}^{(t)}(F_{3}F_{1}F_{2}F_{4})=t_{8}^{(u)}(F_{2}F_{3}F_{1}F_{4})$
$\displaystyle=$ $\displaystyle 4{\rm tr}_{\bf 5}F_{1}F_{3}F_{2}F_{4}+4{\rm
tr}_{\bf 5}F_{2}F_{3}F_{1}F_{4}-2{\rm tr}_{\bf 5}F_{1}F_{3}{\rm tr}_{\bf
5}F_{2}F_{4}-2{\rm tr}_{\bf 5}F_{2}F_{3}{\rm tr}_{\bf 5}F_{1}F_{4}+2{\rm
tr}_{\bf 5}F_{1}F_{2}{\rm tr}_{\bf 5}F_{3}F_{4}$

that satisfy $t_{8}=t_{8}^{(s)}+t_{8}^{(t)}+t_{8}^{(u)}$. The notation is
justified because for four-dimensional polarisations,
$t_{8}^{(s)}F_{1}F_{2}F_{3}F_{4}$ vanishes if $F_{1}$ and $F_{2}$ do not have
the same helicity. This structure of the supersymmetry invariant agrees with
the six-dimensional four-graviton amplitude [81] reduced to five dimensions.

###  3.7 Consequences for string perturbation theory

The solutions to the differential equations implied by supersymmetry derived
in this section severely constrain possible perturbative corrections. We shall
find in particular that the BPS saturated couplings can only receive
corrections at specific loop orders in string perturbation theory. In this
subsection we analyse these equations for the perturbative threshold functions
that only depend on the 13 NS-NS moduli, excluding the axion dual to the
external Kalb-Ramond two-form. These include the dilaton defining the
effective string coupling $g_{\scalebox{0.6}{4}}$, and the Narain lattice
moduli $t^{m},\bar{t}^{\bar{m}}$ that parametrise $SO(2,6)/(SO(2)\times
SO(6))$, corresponding to the internal components of the metric and B-field
that survive the orbifold projection. We define the tangent frame differential
operator on the Narain lattice moduli space

$\mathcal{D}_{ab}=\bar{P}_{\bar{n}\,ab}G^{m\bar{n}}(\partial_{m}+Q_{m}),\qquad\bar{\mathcal{D}}^{ab}=P_{m}{}^{ab}G^{m\bar{n}}(\bar{\partial}_{\bar{n}}+Q_{\bar{n}})\
,$ (3.74)

as for $SO^{*}(12)/U(6)$ in (3.7), where $a=1,2,3,4$ of $SU(4)$. The Laplace
operator decomposes as

$2\mathcal{D}_{ij}\bar{\mathcal{D}}^{ij}\mathcal{E}^{\scalebox{0.6}{per}}(g_{\scalebox{0.6}{4}},t,\bar{t})=\biggl{(}\frac{1}{4}g_{\scalebox{0.6}{4}}\frac{\partial\;}{\partial
g_{\scalebox{0.5}{4}}}\Bigl{(}g_{\scalebox{0.6}{4}}\frac{\partial\;}{\partial
g_{\scalebox{0.5}{4}}}+18\Bigr{)}+2\mathcal{D}_{ab}\bar{\mathcal{D}}^{ab}\biggr{)}\mathcal{E}^{\scalebox{0.6}{per}}(g_{\scalebox{0.6}{4}},t,\bar{t})\;.$
(3.75)

By gauge invariance, the non-perturbative couplings only depend on the R-R
moduli $C$ and the axio-dilaton axion $b$ through periodic functions.
Expanding in Fourier series, the Laplace equation on a function of type
$\mathcal{E}_{Q}(g_{\scalebox{0.6}{4}},t,\bar{t})e^{2\pi iQC}$ is then
exponentially suppressed in $\frac{|V(Q)|}{g_{\scalebox{0.5}{4}}}$ at weak
coupling, and equivalently
$\mathcal{E}_{k}(g_{\scalebox{0.6}{4}},t,\bar{t},C)e^{2\pi ikb}$ is
exponentially suppressed in $\frac{|k|}{g_{\scalebox{0.5}{4}}^{2}}$. One
interprets physically that the former correspond to Euclidean Dp-brane
instantons [7, 37, 82, 83] and the latter to Euclidean NS5 brane instantons.
It follows in particular that the perturbative couplings do not depend on the
R-R and the external Kalb-Ramond two-form axions.

The relevant components of (3.36) are