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pH-responsive aldehyde-bearing cyclometalated iridium(III) complex for tracking intracellular pH fluctuations under external stimulation
Published for SISSA by
Springer
Received: November 8, 2021
Accepted: February 14, 2022
Published: March 9, 2022
Yuichi Enoki and Taizan Watari
Kavli Institute for the Physics and Mathematics of the Universe (WPI),
the University of Tokyo,
Kashiwa-no-ha 5-1-5, 277-8583, Japan
E-mail: yuichi.enoki@ipmu.jp, taizan.watari@ipmu.jp
Abstract: We compute the monodromy matrices on the special geometry of 4d N =
2 Heterotic-IIA dual vacua in some simple cases by numerical evaluation of the period
integrals, without assuming a geometric background. The integrality of the monodromy
matrices constrains some classification invariants of the string vacua. We also mention
some mathematical open problems on period polynomials for modular forms with poles.
Keywords: String Duality, Superstring Vacua, Superstrings and Heterotic Strings
ArXiv ePrint: 2111.01575
Open Access, c The Authors.
Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP03(2022)059
JHEP03(2022)059
Direct computation of monodromy matrices and
classification of 4d N = 2 heterotic-IIA dual vacua
Contents
1
2 Monodromy of projective symplectic sections
2.1 Preliminaries
2.2 Monodromy within the heterotic-perturbative region
2.3 Automorphic forms for the cases with rank(ΛS ) = 1 and ΛS = U
2
2
7
11
3 The case ΛS = h+2i
3.1 Mathematical facts related to the rank-1 cases in general
3.1.1 Lattice isometry group
3.1.2 The automorphic form
3.1.3 Period polynomials
3.2 Analysis on the ΛS = h+2i cases
3.2.1 A quick review
3.2.2 Monodromy matrix of g̃2 : the case n1/2 = 0
3.2.3 The cases n1/2 = 1, 2, 3, 4
13
13
13
15
15
17
17
19
23
4 The case ΛS = U
4.1 Analysis on the ΛS = U case
25
26
5 Discussion
30
A Brief notes on modular forms
A.1 Eisenstein series etc.
A.2 Explicit formulae of the modular forms {Φγ }
31
31
32
B Eichler cohomology and coarse/fine classification
33
1
Introduction
It is a hard task to classify string vacua, or to classify modular invariant superconformal field
theories (SCFT’s) on a worldsheet. Classification of Calabi-Yau manifolds will do a partial
job, but it is also a hard task to classify geometries; explicit construction of geometries one
by one in a certain method (e.g., complete intersections in toric varieties) does not tell us
how many other Calabi-Yau manifolds are overlooked by that method. Furthermore, not
all the SCFT’s may be associated with non-linear sigma models of Calabi-Yau manifolds,
in principle.
In this article, we focus on string vacua with the SO(3, 1) Lorentz symmetry and
N = 2 supersymmetry that have descriptions both by Heterotic string and Type IIA
string theory [1], and report a progress on the question above. The moduli space of such
–1–
JHEP03(2022)059
1 Introduction
2
Monodromy of projective symplectic sections
2.1
Preliminaries
The framework in the Heterotic string language. In a Heterotic string compactification with the SO(3, 1) Lorentz symmetry, N = 2 spacetime supersymmetry implies
special features [11, 12] in the worldsheet SCFT with central charge (c, c̃) = (22, 9). The
1
In the geometric phase of a Type IIA compactification, the Calabi-Yau threefold M has a K3-fibration
if it has a Heterotic dual. The datum ΛS is the lattice of the lattice polarization of the K3-fibration
π : M → P1 . See p. 6 for more information.
2
The study in this article was inspired by positive evidence in refs. [8, 9] that this requirement yields
non-trivial conditions on the integers for classification.
–2–
JHEP03(2022)059
4d N = 2 vacua forms a network of branches connected by the Coulomb-Higgs transitions,
and individual branches are assigned invariants for classification: a pair of lattices (ΛS , ΛT )
and a finite number of integers, to be explained in the main text. We will use facts that
are known since the 90’s, and obtain constraints on those integer parameters.
Here is a little more words on the idea. It has been known since the 90’s for a mirror
pair of a Type IIA compactification on a Calabi-Yau threefold M and a Type IIB compactification on a Calabi-Yau threefold W , that there are monodromies of a symplectic
projective section on the vector-multiplet moduli space; the monodromy matrices take values in the group of integer-valued symplectic transformations on H 3 (W ; Z). Monodromy
matrices have been computed explicitly for explicitly constructed and chosen mirror manifolds W . There should exist, however, the notion of monodromy and its appropriate matrix
representation, regardless of whether a given branch of 4d N = 2 compactification is given
by a non-linear sigma model of a Calabi-Yau threefold. In fact, with a careful reading of
references such as [2–5] and also [6, 7], one will find that, for certain classes of lattices
ΛS , the monodromy matrices can be computed from the abstract data1 of (ΛS , ΛT ) and
the integers, without knowing whether a mirror non-linear sigma model exists. We require
that the monodromy matrices should be integer-valued, and derive constraints on those
integers for classification.2
In this article, we work on two cases, ΛS = h+2i and ΛS = U , as a proof of concept. In
both cases, it turns out that each one of the branches of the moduli space has a region described by a non-linear sigma model; on the way to establish this claim, we have confirmed
that two independent ways to read out the second Chern class of the target manifold M
yield a consistent result. In the most non-trivial part of computing the monodromy matrices, we evaluated numerically the period polynomials of the automorphic forms of the
isometry groups associated with the lattices ΛS .
Sections 2.1–3.1 should be regarded as a process of reading the literatures; technical
materials in this part are entirely from references, except a few minor clarifications (footnotes 11 and 14). Technical analysis along the idea written in the middle of section 2.2
will start at the end of section 3.2.1. Some useful facts on modular forms are summarized
in the appendix A. In the appendix B, we will make a brief comment on how the coarse
and fine classifications in [10] are related to the Eichler cohomology.
right-moving sector contains a c̃ = 3 algebra of two free bosons and two free fermions forming a system with the N = 2 superconformal symmetry, and a c̃ = 6 N = 4 superconformal
algebra. In the left-moving sector, let ρ be the number of free chiral bosons; there is also
a Virasoro algebra of c = 22 − ρ. The total Hilbert space in the Ramond sector has the
structure of
(22−ρ,0)
(0,6)
tot
e
HR
= ⊕γ∈GS Hγ(ρ,3) ⊗ ⊕(h̃,I)
⊗H
˜ Hγ,(h̃,I)
˜
˜ ,
(h̃,I)
(2.1)
Φγ
η
(q) := Tr
22−ρ
h
i
L̃ − 6
F
q L0 − 22−ρ
24 q̄ 0 24 (−1) R
(22−ρ,0)
e(0,6)
⊕(h̃,I)
˜ H
˜ ⊗H
˜
γ,(h̃,I)
(2.2)
(h̃,I)
serve as classification invariants of individual branches of 4d N = 2 moduli space of Heterotic string compactifications. The set of generating functions {Φγ | γ ∈ GS } is a vectorvalued modular form of weight (11 − ρ/2) associated with the quadratic discriminant form
(GS , qS ) derived from the lattice ΛS . The modular nature implies that the whole {Φγ } is
completely fixed already, when all the first Fourier coefficients of Φγ ’s—Φγ = nγ q νγ + · · ·
where 0 ≤ νγ < 1 — are specified. For this reason, for a given ΛS , the vector-valued
modular form {Φγ } contains only a finite number of Z-valued classification invariants of
the branches of the moduli space.
It has been observed that a little more classification invariants are necessary in dise , Λ ) and {Φ }. Reference [10]
tinguishing branches of moduli space in addition to (Λ
γ
S
T
introduced yet another vector-valued modular form {Ψγ | γ ∈ GS } of weight-(13 − ρ/2) to
serve the purpose by following the idea in refs. [13, 14]. The modular form {Ψγ } is also
parametrized by a finite number of integers; more information on these integer parameters
will be provided later in this article, when it becomes necessary ((3.23) and (4.2)).
Prepotentials. The 4d field theory description of the Heterotic string compactification
e ∨ , so the magnetic
in question has (ρ + 2) abelian vector bosons. Electric charges are in Λ
S
3
An even unimodular (self-dual) lattice of signature (r+ , r− ) is denoted by IIr+ ,r− . The lattice II1,1 is
also denoted by U . For a lattice L, L[n] denotes the lattice that is isomorphic to L as a free abelian group,
and whose bilinear form (intersection form) is n times that of L.
–3–
JHEP03(2022)059
with each factor forming a representation space of the algebras of (c, c̃) = (ρ, 3), (22 − ρ, 0)
and (0, 6), respectively; irreducible Ramond-type representations of the c̃ = 6 N = 4
˜ The group GS and its elements γ are explained
superconformal algebra are labeled by (h̃, I).
shortly. The spectrum is assumed to yield a modular invariant partition function; spacetime
(R3,1 ) filling NS5-branes are assumed to be absent.
The (c, c̃) = (ρ, 3) sector is the N = (0, 1) supersymmetrization of a lattice CFT,
e — is even and of signature (2, ρ). In this article, we
where the lattice — denoted by Λ
S
e has a structure of Λ
e = U [−1] ⊕ Λ for an even
will consider the cases where the lattice Λ
S
S
S
e ,→ II
lattice3 ΛS of signature (1, ρ − 1), and also has a primitive embedding Λ
4,20 . The
S
e
orthogonal complement of ΛS within II4,20 is denoted by ΛT . The discriminant group and
the quadratic discriminant form of the lattice ΛS are denoted by GS and qS , respectively.
e , Λ ), the generating functions of indices (elliptic genus)
Besides the lattice pair (Λ
S
T
e . The mass m of BPS states of the 4d N = 2 spacetime supersymmetry
charges are in Λ
S
is given by
p
Z = (vI X I + mI FI )eK̂/2 ,
I
m = |Z|/ GN ,
K̂ = − ln i(X I F I − X FI )
(2.3)
(2.4)
Π = ΠF = (X I , FI )T
(2.7)
=: (1, (∂s F ), ta , (2 − t∂t − s∂s )F , −s, (∂ta F ))T
(2.8)
for a prepotential F of the given branch; the prepotential F (s, t) is of the form
X
s
dabc a b c
ζ(3) χ
1
F = (t, t) +
t tt −
+
2
3!
(2πi)3 2
(2πi)3 k∈N
(Im(hw,ti)≥0)
X
≥0
−
nw,k Li3 (e2πiks e2πihw,ti )
w∈Λ∨
S
aab a b ba a
t t − t
2
24
(2.9)
4
e S [resp. ΛS ] is denoted by CeIJ [resp. Cab ], where I, J = 0, ], 1, · · · , ρ [resp.
The intersection form of Λ
a, b = 1, · · · , ρ]. The lattice U [−1] has the intersection form (e0 , e0 ) = (e] , e] ) = 0 and (e0 , e] ) = −1.
5
Already the relative normalization between X I ’s and FI ’s has been fixed here, which means that the
normalization of F is also fixed. Now, there is no ambiguity in writing down the gauge kinetic term in a
4d N = 2 supergravity in the convention adopted in this article.
Let the covariant derivatives on purely electrically charged states be ∇ = d − iAI vI with I ∈
I
{0, ], 1, · · · , ρ}, and Fµν
:= ∂µ AIν − ∂ν AIµ . Then
L⊃−
1 Im(NIJ ) I Jµν
Fµν F
;
4
2π
(2.5)
the matrix NIJ is obtained by the electro-magnetic dual transformation (linear fractional transformation)
(A)
as in refs. [15, 16] from NIJ , where
(A)
N IJ = FIJ − 2i
Im(FIK )X
K
Im(FJL )X
P
Im(FP Q )X X
Q
using F (X) := (X 0 )2 F (ti =: X i /X 0 ), FI = ∂X I F and FIJ = ∂X I ∂X J F .
–4–
L
,
(2.6)
JHEP03(2022)059
for some projective symplectic section Π = (X I , FI )T , where ν = (v, m) = (vI , mI ) is the
electric and magnetic charges of a BPS state under the (ρ + 2) U(1) vector bosons. The
e ∨ [resp. magnetic charges
4d Newton constant is denoted by GN . Electric charges v ∈ Λ
S
e ] are described as v = eI v , v ∈ Z [resp. m = e mI , mI ∈ Z] by choosing a
m ∈ Λ
S
I
I
I
e ∨ [resp. {e
e
basis4 {eI=0,],1,··· ,ρ } of Λ
I=0,],1,··· ,ρ } of ΛS ]. So, relative normalization between
S
the electric part (X I ) and the magnetic part (FI ) of the section Π = (X I , FI )T is no longer
arbitrary.
For a given branch of the moduli space, the projective symplectic section Π = (X I , FI )T
is locally a function of flat coordinates (s, ta=1,··· ,ρ ) of the vector-multiplet moduli space;
we will provide more information on the coordinates ta=1,··· ,ρ in section 2.2, but we note
for the moment that t = ea ta can be regarded as an element of ΛS ⊗ C; the coordinate s
is normalized so that the gauge coupling constant of a 4d non-abelian gauge field from a
level-k current algebra in Heterotic string is given by (4π/gY2 M ) = kIm(s). The section Π
is of the form5
d0abc − dabc = [(δn)a Cbc + cyclic],
a0ab − aab = −Λab ,
b0a − ba = 24(δna ) − 24Λ00a (2.10)
for (δn)a , Λab , Λ00a ∈ Z.
In the branch of the 4d N = 2 moduli space with classification invariants (ΛS , ΛT ),
{Φγ } and {Ψγ }, some of the parameters in the prepotential are determined by [13]
nw,k=0 = c[w] ((w, w)/2)
(w ∈ Λ∨
S , Im(hw, ti) > 0),
χ = −c0 (0),
(2.11)
where the Fourier expansion coefficients of {Φγ },
Fγ :=
X
Φγ
(τ ) =:
cγ (ν)q ν
24
η
ν
(2.12)
are used. This is done by computing Heterotic string genus-1 (1-loop) corrections
to the holomorphic R2 (gravitational) term and the probe gauge group kinetic function, and matching the results to the parameters in the effective theory on R3,1 (see
e.g., [13, 14, 17–19]). The same procedure also determines the parameters dabc in terms of
(ΛS , ΛT ), {Φγ } and {Ψγ } modulo the ambiguity in (2.10), but is able to constrain aab and
ba ’s only to the extent that
aab ,
ba ∈ R.
(2.13)
So, at this moment, we should think that (aab )+Z ∈ R/Z and (ba )+24Z ∈ R/24Z are also
invariants characterizing the branch in addition to (ΛS , ΛT ), {Φγ } and {Ψγ }.
For a choice of (ΛS , ΛT ), it is possible to work out a finite number of independent integer
parameters that specify the vector-valued modular forms {Φγ } and {Ψγ } completely. Those
independent integer parameters are further subject to some number of inequalities, as
discussed in ref. [10]. Starting from section 2.2, we derive additional consistency conditions
on those independent integer parameters, (aab )+Z and (ba )+24Z .
In the notation to be introduced in (2.30), Π0 = M (1, Λ)Π with Λ00 = Λ]] = Λ]0 = Λ0] = 0. Such
2(ρ + 2) × 2(ρ + 2) matrices form a subgroup of Sp(2(ρ + 2); Z). The integers δna and Λ00a in the main text
correspond to Λ]a and Λ0a + Λ]a , respectively.
0
Depending on whether we use the section ΠF or ΠF , monodromy matrices to be computed in this article
become either Mg̃ or M (1, Λ) · Mg̃ · M (1, −Λ).
6
–5–
JHEP03(2022)059
for some appropriately chosen parameters dabc , aab , ba , χ and nw,k where a, b, c = 1, · · · , ρ.
Rationale for the s and t-dependence in (2.8) is written, for example, in the appendix of
ref. [8] (reasonings in both Heterotic and Type IIA perspectives are used). Note that there
is no ambiguity left for the normalization of F , and hence of those parameters.
A prepotential itself is not a physical observable, although the spectrum of 4d N = 2
BPS states is. The spectrum remains unchanged under a symplectic transformation on
ΠF = (X I , FI )T and its conjugate action on the charge ν = (vI , mI ); when the symplectic
transformation is in a certain subgroup6 of Sp(2(ρ + 2); Z), the section Π0 after the trans0
formation may still be fitted by (2.8) for some prepotential F 0 of the form (2.9), Π0 = ΠF ,
but with parameters d0abc , a0ab , b0a different from the parameters before the transformation;
(Cab , dabc ) : H 2 (M ; C) 3 J := sD]∧ + ta Da 7−→
1
3!
Z
s
dabc a b c
J ∧ J ∧ J = Cab ta tb +
t tt,
2
3!
M
(2.14)
and the holomorphic R2 term in the 4d action ([28], [29], §8) involves the information [30]
2
(24, (c2 )a ) : H (M ; C) 3 J 7−→
Z
c2 (T M ) ∧ J = 24s + (c2 )a ta .
(2.15)
M
For a different choice of a basis {D]∧ , Da0 := Da + (δna )D]∧ }, the same element J ∈
H 2 (M ; C) is regarded as s0 D]∧ + ta Da0 with s0 = s − (δna )ta ; the topological numbers
R
R
0
0
0
0
0
0
M Da · Db · Dc =: dabc [resp.
M c2 (T M ) · Da =: (c2 )a ] are different from dabc [resp.
(c2 )a ] by +{(δn)a Cbc + cyclic} [resp. +24(δn)a ]. The symplectic transformations in (2.10)
parametrized by δna correspond to this change of basis. The symplectic transformations
in (2.10) parametrized by Λab and Λ00a just correspond to choosing different basis elements
of magnetically charged states.
It is known in a Type IIA vacuum given by an M -target non-linear sigma model
that [31, 32]
1
1
(aab )+Z = Da · Da · Db + Z = Da · Db · Da + Z
2
2
7
(2.16)
First, there must be a basis of Ramond-Ramond ground states that corresponds to a basis of the
spacetime U(1) vector bosons in which charges (vI , mI ) are integers. Second, within the ca-ring states with
the conformal weight (1/2, 1/2), one may also choose a basis that are tied under a spectral flow with some
(ρ + 1) elements in the basis of the Ramond-Ramond ground states. Let φ]∧ and φa=1,··· ,ρ be those ca-ring
states. Now, the data Cab (i.e., ΛS ) and the parameters dabc and nw,k are read out from the structure
constants of the ca-ring. The class of Type IIA compactifications to be considered in this article is those
with a state φ]∧ and the corresponding flat coordinate s so that the three point functions are of the form
hφ]∧ φ]∧ φ∗ i = O(e2πis ) and hφ]∧ φa φb i = Cab + O(e2πis ).
The vector-valued modular form {Φγ } encodes the helicity supertrace of purely electrically charged BPS
particles on R3,1 [24–26].
–6–
JHEP03(2022)059
Type IIA non-linear sigma model phases. Such data as ΛS ,→ II3,19 , {Φγ } and {Ψγ }
can also be characterized in the language of Type IIA string compactification. Certainly
the dictionary of the Heterotic-Type IIA duality has been studied and verified mostly in
cases the Type IIA description is given by a Calabi-Yau compactification, with a non-linear
sigma model on a worldsheet (cf [20–23]). It is not difficult to fix the dictionary with a guess
work,7 even in cases the Type IIA description is not necessarily associated with a CalabiYau-target non-linear sigma model. For this reason, Type IIA compactifications that are
not obviously related to a Calabi-Yau compactification are also within the framework of
study in this article.
If a branch of the moduli space contains a phase given by an M -target N = (2, 2) supersymmetric non-linear sigma model on a worldsheet, where M is a Calabi-Yau threefold,
then M has a ΛS -polarized regular K3-fibration π : M → P1 . One may choose a basis
{D]∧ , Da=1,··· ,ρ } of H 2 (M ; Z), with D]∧ represented by the K3-fiber class [27]. The lattice
ΛS in this context is SpanZ {Da=1,··· ,ρ } with its intersection form Cab = D]∧ · Da · Db . The
cubic part of the prepotential F is the trilinear intersection form on H 2 (M ; C),
and
(24Z)s + (ba )+24Z ta =
Z
M
c2 (T M ) + 24H 4 (M ; Z) ∧ J.
(2.17)
The latter property implies (ba )+24Z ≡ (c2 )a + 24Z. In this article, however, we do not
assume that (2.16), (2.17) are satisfied from the beginning, because there may be a branch
of the moduli space of Type IIA vacua that are not associated with any Calabi-Yau compactification. As a result of the monodromy analysis in this article, however, we will see
for some lattices ΛS that both (2.16), (2.17) are always satisfied.
Monodromy within the heterotic-perturbative region
The vector-multiplet moduli space of a given branch is known to have the following approximate form
e ) /[Γ ] = H
e
(Hs;Im(s) 1 /ZD) × D(Λ
S
S
s,Im(s) 1 × D(ΛS ) /ZD × [ΓS ]
(2.18)
in the region Im(s) 1 (the perturbative region in the Heterotic string language). Here,
Hs is the complex upper half-plane parametrized by s, and
n
o
(2.19)
t := ea ta ∈ ΛS ⊗ C,
(2.20)
e ) := P f ∈ Λ
e ⊗ C | (f, f) = 0, (f, f) > 0 ,
D(Λ
S
S
for which one can use the following parametrization:
f(t) = e0 + e]
(t, t)
+ e a ta ,
2
e . In the region Im(s) 1, we have chosen in eq. (2.8)
using the basis {e0 , e] , ea=1,··· ,ρ } of Λ
S
the electric part of the projective symplectic section Π as
(X I ) = (fI (t)) + O(e2πis ),
(fI (t)) = 1,
(t, t) a
,t
2
T
.
(2.21)
Within the quotient group, a generator D acts only on Hs , and it does as s → s + 1.
e ) containing8
The group [ΓS ] is a subgroup of the lattice isometry group Isom(Λ
S
h
i
e ) → Isom(G , q ) .
ΓS := Ker Isom(Λ
S
S S
(2.22)
e ⊂ II
The subgroup [ΓS ] may be as large as the set of lattice isometries of Λ
4,20 that can
S
be lifted to an isometry of II4,20 (cf [9]). In the cases to be worked out explicitly in this
article, where ΛS = h+2i and ΛS = U , this ambiguity does not matter because all those
e ).
subgroups agree with the entire Isom(Λ
S
Within the moduli space (2.18), there are complex codimension-1 loci where 4d effective
field theory has extra massless states charged under the (ρ + 2) U(1) vector fields. States
e ∨ satisfying −2 ≤ v 2 < 0 become massless [3] at
with an electric charge v ∈ Λ
S
n
o
e ) | (v, f) = 0 .
X(v) := [f] ∈ D(Λ
S
8
(2.23)
Isom(GS , qS ) is the group of automorphisms of the abelian group GS preserving qS . An isometry
eS → Λ
e S induces an automorphism g : GS 3 x + Λ
e S 7−→ g · x + Λ
e S ∈ GS . Here, qS (g · x) = qS (x).
g:Λ
–7–
JHEP03(2022)059
2.2
Their multiplicty is governed by the classification invariant nγ=[v] . Those light states give
rise to logarithmic singularity in the gauge coupling constants of the (ρ + 2) U(1) vector
fields around [3]
e
Xsingl. := ∪v∈Λ
e∨ ;−2≤v2 <0 X(v) ⊂ D(ΛS );
(2.24)
S
e )\X
Hs;Im(s) 1 × (D(Λ
S
singl. )
(2.25)
and its image [(s0 , t0 )] in the moduli space (2.18). For a loop γg̃ in
e )\X
(Hs,Im(s) 1 /ZD) × (D(Λ
S
singl. ) /[ΓS ]
(2.26)
with the base point [(s0 , t0 )], one may think of analytically continuing the section ΠF (s, t)
of a fixed prepotential F along the path γ̃g̃ in (2.25), the lift of γg̃ . Then there must be a
matrix Mg̃ ∈ Sp(2(ρ + 2); Z)H.el so that9
contin.along γg̃
(ΠF
=proj. Mg̃ · ΠF
@(s,t)∼(s0 ,t0 ) )
@(s,t)∼(s0 ,t0 ) ,
(2.27)
In a simplified notation, we may drop reference to a fixed F , and write
Π(sg̃ , tg̃ ) =proj. Mg̃ · Π(s, t).
(2.28)
The matrix Mg̃ has the form of M (g, Λg̃ ) ∈ Sp(2(ρ + 2); Z)H.el , where [15, 16]
(
!
:=
U Z
W V
!
M (U, Λ) :=
U 0
V ·Λ V
Sp(2(ρ + 2); Z)
H.el
)
∈ Sp(2(ρ + 2); Z)
,
U = (V T )−1 ,
Z=0 ,
ΛT = Λ.
(2.29)
(2.30)
The element g in Mg̃ = M (g, Λg̃ ) is the lattice isometry that maps the starting point [f(t0 )]
e ) to the endpoint [f(tg̃ )] ∈ D(Λ
e ).
of the path γ̃g̃ in the covering space D(Λ
S
S
0
10
For the assignment γg̃ 7−→ Mg̃ to be a homomorphism,
Π(s(g̃a ·g̃b ) , t(g̃a ·g̃b ) ) =proj Mg̃a · Mg̃b · Π(s, t),
9
(2.31)
Alternatively, one may assign a symplectic matrix Mg̃ for a loop γg̃ in (2.26) by using analytic continuation of Π in the reverse direction of γg̃ , or placing the matrix Mg̃ in the left hand side of (2.27). In this
article, we follow the assignment that looks popular in the literatures. In order for this assignment to be a
homomorphism, we set the composition law of the loops as in (2.32). See also the appendix B.
10
Because the condition (2.27) leaves the freedom of changing the matrix Mg̃ by multiplying
−12(ρ+2)×2(ρ+2) , we should expect that the assignment γg̃ 7−→ Mg̃ ∈ Sp(2(ρ + 2); Z)H.el can be a homomorphism only projectively, with the fudge factor −12(ρ+2)×2(ρ+2) . That should be kept in mind when
we talk of multiplying matrices Mg̃a · Mg̃b , although we will use little space to mention this issue in the
main text of this article.
–8–
JHEP03(2022)059
note that X(v) and X(v 0 ) may conincide when v and v 0 are parallel. So, the projective symplectic section Π = (X I , FI )T has non-trivial monodromies around the complex
e ).
codimension-1 locus Xsingl. in D(Λ
S
Fix a base point (s0 , t0 ) in
e ) has to end at g · (g · [f(t )]) in D(Λ
e ). So,
the path γ̃(g̃a ·g̃b ) starting from [f(t0 )] ∈ D(Λ
a
0
S
S
b
the composition law of the paths in (2.25) starting from (s0 , t0 ) [resp. the loops in (2.26)
with the base point [(s0 , t0 )]] should be
γ̃(g̃a ·g̃b ) := (γ̃g̃b )ga ◦ γ̃g̃a ,
h
γ(g̃a ·g̃b ) := γg̃b ◦ γg̃a
resp.
i
(2.32)
e
MD = M (1, C).
(2.33)
e ) in the
For a loop γ̃g̃(v) that maintains constant s = s0 and goes around X(v) ⊂ D(Λ
S
11
counter-clockwise direction (gaining phase +2π), the matrix representation is [3]
Mg̃(X(v)) = M (1, Λg̃(X(v)) ),
1
Λg̃(X(v)) = −
2
(−2≤(v 0 ,v 0 )2 <0)
n[v0 ] (v 0 ⊗ v 0 )IJ ;
X
(2.34)
v 0 ∈(Qv∩Λ∨
S)
e
the matrix Λg̃(X(v)) is Z-valued, because contributions always come in pair from v 0 and
−v 0 , and n[v0 ] = n[−v0 ] .
To determine all the monodromy representation matrices, the remaining task is to
choose a set of generators {gi } of [ΓS ], find a path γ̃g̃i that starts from (s0 , [f(t0 )]) and
ends at (sg̃0i , gi · [f(t0 )]) for each element gi , and to work out its monodromy matrices.
All the paths in (2.25) from (s0 , t0 ) that become loops in the moduli space (2.26) are
obtained as compositions of those paths γ̃g̃i along with the path γ̃D and the loops γ̃g̃(X(v)) .
Such generators may be subject to some relations, but monodromy representation matrices
should automatically satisfy the relations by construction.
Here is the program we start to carry out in this article. We start off by assuming that
there is a branch of moduli space whose classification invariants are (ΛS , ΛT ), {Φγ }, {Ψγ },
(aab )+Z and (ba )+24Z . By exploiting known facts, we choose a set of generators of [ΓS ],
find lifts γ̃g̃ of those generators, and compute their monodromy matrices in terms of those
invariants. When some of the monodromy matrices turn out not to be integer valued for
those invariants, that is a contradiction. There should be no such branch of moduli space.
This is how we obtain additional consistency conditions on the classification invariants.
There are ρ elements g∞(d) ∈ ΓS ⊂ [ΓS ] (where d = 1, · · · , ρ) whose monodromy
matrices can be computed immediately. To be more precise, let γ̃g̃∞(d) be the path in (2.25)
where s remains constant at s0 , and ta varies from ta0 to ta0 + δ ad in a straight line; we choose
11
The dilaton superfield S and the prepotential F (X) of [3] are 2s and 2F in this article. The matrix c̃
in ([3], (4.17)) is also 2Λg̃(X(v)) here. This systematic difference by a factor 2 is not a matter of convention;
the normalization of F has been fixed unambiguously by (2.3), (2.8) and/or (2.5), (2.6), using an integral
basis of charges. We demand that the matrices Λg̃ are integer valued.
–9–
JHEP03(2022)059
in the standard notation of the composition rule in fundamental groups.
The monodromy matrices are completely understood for loops γg̃ in (2.26) that are still
e )\X
regarded as loops when lifted to (Hs,Im(s) 1 /ZD) × (D(Λ
S
singl. ). All those monodromy
matrices are of the form M (1, Λg̃ ). For the loop γD in the cylindrical fiber Hs /ZD, the
monodromy matrix is
t0 so that Im(ta0 ) 1, and t0 is far away from the loci X(v) of extra 4d massless particles.
Then
d
ddd
3
bd
− add − 12
(Λg̃∞(d) )IJ =
0
dadd
2 − aad
0 dddb
2 − adb
0
0
0
ddab
,
(2.35)
IJ
where I ∈ {0, ], a} and J ∈ {0, ], b}. So, the following conditions are obtained:
ddda
− aad ∈ Z,
2
dddd
bd
− add −
∈ Z,
3
12
∀
a, b, d ∈ {1, · · · , ρ}.
(2.36)
Without loss of information, the conditions (2.36) can also be stated as
ddab ∈ Z,
dadd + daad ∈ 2Z,
2dddd + bd ∈ 12Z
(2.37)
and
aab ∈
dabb
+ Z.
2
(2.38)
Two remarks are in order here. First, if we allow ourselves to replace the parameters
ba mod 12Z in the condition (2.37) by (c2 )a , then the condition (2.37) is precisely equal
to Wall’s condition [33] for existence of a diffeomorphism class of manifolds [M ] with
the topological trilinear intersection (2.14) and the second Chern class (2.15) given by
(Cab , dabc ) and (24, (c2 )a ). When we start from a hypothetical branch of Het-Type IIA
dual moduli space characterized by (ΛS , ΛT ), {Φγ }, {Ψγ }, (aab )+Z and (ba )+24Z , however,
there is a priori no guarantee that (c2 )a mod 24 is equal to (ba )+24Z . So, it is one of the
tasks in this article to examine whether (ba )+24Z are equal to (c2 )a + 24Z; we wait until
this check is done to conclude12 that such a branch of moduli space has a region described
by a non-linear sigma model of the target space M .
Another thing to note at this moment is that only the parameters dabc are determined
by the classification invariants {Φγ } and {Ψγ }, so only the two conditions out of (2.37)
set constraints on the integer parameters of {Φγ } and {Ψγ }. Those integer parameters
do not determine the real parts of aab and ba as stated in (2.13), so the condition (2.38)
and the last one in (2.37) determine the values of (aab )+Z and of (ba )+24Z + 12Z. We will
see in the case ΛS = h+2i in section 3 that similar conditions from other generators of
[ΓS ] are combined with (2.37) to yield further constraints on the integer parameters of
{Φγ } and {Ψγ }.
12
Alternatively, one may reason already at this moment (cf. footnote 26) that a branch of moduli space
satisfying (2.37) must be realized by a Type IIA compactification on a manifold M , where [M ] is a diffeomorphism class whose existence is guaranteed by Wall’s theorem; here, the data (ba )+24Z do not specify one
diffeomorphism class uniquely due to the +24Λ0]a ambiguity in (2.10), but one may still think that one of
those diffeomorphism classes will realize the branch in question. The remaining question then is to find out
how the relation (ba )+24Z = (c2 )a + 24Z in geometric phases comes about theoretically; the study in the
main text may therefore be regarded as addressing this question, besides the program of narrowing down
the range of theoretically consistent classification invariants.
– 10 –
JHEP03(2022)059
dabd ∈ Z,
2.3
Automorphic forms for the cases with rank(ΛS ) = 1 and ΛS = U
To determine the matrices Λg̃ in Mg̃ (g, Λg̃ ) for generators {g} of the lattice isometry group
[ΓS ], the following observation is useful. In the region with Im(s) 1 in the vector
multiplet moduli space, various terms in the prepotential (2.9) can be grouped into
s
F = (t, t) + F (1) (t) + O(e2πis )
2
(2.39)
1
J 2 (1) g̃
(g I=0
(t ) = F (1) (t) + (Λg̃ )IJ f(t)I f(t)J
J f(t) ) F
2
(2.40)
for a duality transformation g̃ = M (g, Λg̃ ). Readers are referred to [2] for a detailed proof.13
We just note here that F (1) (tg̃ ) on the left-hand side means the analytic continuation of
g
e )\X
F (1) (t) along the path γ̃g̃ in Hs × (D(Λ
S
singl ) from t to t , seen as a function of
the original coordinates t. The analytic continuation of the single component function
e )\X
F (1) along a path γ̃g̃ — its projection to D(Λ
S
singl in fact — almost determines the
matrix Λg̃ through (2.40). The remaining ambiguity in the matrix Λg̃ [resp. Mg̃ ] is +nCe
[resp. ×(MD )n ] with n ∈ Z, because (f, f) = 0. This ambiguity corresponds to the lost
e ) to D(Λ
e ).
topological information when the path γ̃g̃ is projected from (Hs /ZD) × D(Λ
S
S
The easiest case where we can do this is for g̃∞(d) .
F
(1)
(t
g̃∞(d)
)−F
(1)
ddbc b c
(t) =
tt +
2
dddb
− adb tb +
2
dddd add
bd
−
−
,
6
2
24
(2.41)
from which the matrix (2.35) follows. We will see another example of easy analytic continuation in section 4.
For some other generators g of [ΓS ], it is not always easy to work out the analytic
continuation of F (1) along a path γ̃g̃ . A combination of two observations in the literature
comes to rescue, however, at least for some classes of lattices ΛS .
e ) is almost an automorphic form
The first observation is that the function F (1) on D(Λ
S
e
of weight −2 under the lattice isometry group Isom(ΛS ); it has singularity along Xsingl , and
is not precisely automorphic because of the second term in (2.40). One may even hope that
one of appropriate derivatives of F (1) may turn into a genuine meromorphic automorphic
form, because the second term in (2.40) is an at most quartic polynomial in ta ’s.
In the cases of rank(ΛS ) = 1 (where ΛS = h+2ki with k ∈ {1, 2, · · · } = N) and in the
case ΛS = U , the fifth derivative and the third derivative of F (1) , respectively, is indeed
an automorphic form, denoted by f∗ . The function F (1) is an iterated integral of the
automorphic form f∗ then. The violation of the automorphic transformation law of F (1) ,
the term involving the matrix Λg̃ in (2.40), is determined by the property of f∗ . More
review is provided in sections 3 and 4.
13
Use X I FI = (X 0 )2 (2F (1) + O(e2πis )) and X I = fI (t) + O(e2πis ).
– 11 –
JHEP03(2022)059
by the s-dependence. The second term F (1) is the 1-loop correction in Heterotic string,
which consists of the cubic polynomial in t as well as e2πit terms. In the discussion
around (4.26) of [2], the relation (2.27) is exploited to derive the relation
The other observation is that those automorphic forms f∗ are determined by the classification invariants {Φγ }. An idea is that there is a function G(1) (t) that is related to F (1)
and f∗ on one hand, and also to {Φγ } on the other. The function G(1) is in the 1-loop
correction term of the Kähler potential of the vector-multiplet scalar fields,
K̂ =: − ln
s − s̄
+ K̂ (0) (t) + K̂ (1) (s, t) + O((Im(s))−2 ) + O(e−2πIm(s) ),
i
(2.42)
where
"
#
K̂ (1) (s, t) =:
2i
G(1) (t),
s − s̄
(2.43)
and
G(1) (t) =
h
i
i
VGS
(1)
2(F − F (1) ) + (t − t̄)d (∂d F (1) + ∂d F (1) ) =: −
.
4π
(t − t̄, t − t̄)
(2.44)
(1)
Thus, Gab̄ can be expressed in terms of F (1) . Precise formulae for the automorphic form
(1)
f∗ in terms of G(1) — just the trace of its derivative matrix Gab̄ is enough in fact — will
be quoted as (3.11), (3.12) and (4.20), (4.21) for the ρ = 1 cases and the ΛS = U case,
respectively.
(1)
The matrix-valued14 function Gab̄ is related, on the other hand, to the index {Φγ } as
follows [3, 5, 6]:
(1)
K̂ (0)ab̄ Gab̄ =
2i
(4π)2
dτRe dτIm X
Z
F nd (τIm )
1+ ρ2
ρ
Fγ ∂τ [(τIm ) 2 Zγ ].
(2.45)
γ
(1)
Although we just need the fact that the combination K (0)ab̄ Gab̄ is determined uniquely by
{Fγ } = {Φγ /η 24 } in this article, let us also explain the notations in (2.45). The integral
on the right-hand side is over the g = 1 worldsheet complex structure τ = τRe + iτIm in
the Heterotic string language, within one fundamental region F nd of PSL2 Z in the upper
P
2
2
half plane H. In the integrand, Zγ := v∈γ q pL (v) /2 q̄ pR (v) /2 , where q = e2πiτ , q̄ = e−2πiτ̄ ,
e ∨ → Rρ,2 is the momenta associated with the (ρ, 2) chiral bosons on the
and (pL , pR ) : Λ
S
Heterotic string worldsheet. The expression on the right-hand side of (2.45) can also be
rewritten in a form where the PSL2 Z-invariance is a little more manifest:
1 ρ+2˜
I[F ] − 8I[∂ S F ] .
32π
3
(2.45) =
(2.46)
˜ ] and I[F 0 ] in the expression are
Here, I[F
˜ ] :=
I[F
Z
dτRe dτIm X
dτRe dτIm X
Zγ Ê2 Fγ − c̃γ (0) , I[F 0 ] :=
Zγ Fγ0 − c0γ (0)
τIm
τIm
F nd
F nd
γ
γ
Z
(2.47)
14
(1)
(0)
The matrix Gab̄ is not always in a form of K̂ab̄ I for some scalar-valued integral function I of moduli
(1)
(0)
t. For example, in the case of ΛS = U , Gρū 6= 0 but K̂ρū = 0.
We have confirmed by closely following the derivations in [6], however, that the scalar-valued integral
(1)
in (2.45) determines the trace of the matrix Gab̄ for a general lattice ΛS .
– 12 –
JHEP03(2022)059
(t − t̄, t − t̄)
K̂ (0) (t) := − ln −
,
2
The case ΛS = h+2i
3
3.1
Mathematical facts related to the rank-1 cases in general
To implement the general idea described in section 2.3 for cases with various lattices ΛS ,
there are more results we can exploit from the literatures. In section 3.1, we summarize
those things for cases with ρ = rank(ΛS ) = 1.
A rank-1 positive definite even lattice15 has to be of the form h+2ki for k = 1, 2, · · · .
Any primitive embeddings ΛS = h+2ki ,→ II3,19 are identical modulo isometries of II3,19 ,
and ΛT := [(ΛS )⊥ ⊂ II3,19 ] is determined modulo lattice isometry:
e = U [−1] ⊕ h+2ki ,
Λ
S
ΛT = h−2ki ⊕ E8⊕2 ⊕ U ⊕2 .
(3.1)
The following review is for a general k ∈ {1, 2, · · · } = Z>0 , but we will pick up the
ΛS = h+2i case in section 3.2 and carry out in detail the program in sections 2.2–2.3.
3.1.1
Lattice isometry group
e (e.g., [5]). We begin
Here is what is known about the group of isometries of the lattice Λ
S
with describing a group Γ0 (k)+ . Here, k is a positive integer.
The group Γ0 (k)+ is a group of transformations on the complex upper half plane. It
contains Γ0 (k)/{±1} as a subgroup, where
(
Γ0 (k) :=
15
g=
a b
c d
!
)
∈ SL2 Z
k|c ,
g : H 3 t 7−→
at + b
∈ H.
ct + d
(3.2)
The rank-1 lattice ΛS = h+2ki has a basis {ea=1 } with the intersection form given by (e1 , e1 ) = 2k.
– 13 –
JHEP03(2022)059
for weight −1 − ρ/2 and 1 − ρ/2 vector valued modular forms F and F 0 , respectively;
P
P
c̃γ (0) and c0γ (0) are Fourier coefficients in Fγ E2 =: ν c̃γ (ν)q ν and Fγ0 (τ ) =: ν c0γ (ν)q ν ,
˜ ] and I[F 0 ] do not diverge. When we use
respectively, inserted so that the integrals I[F
S
the Ramanujan-Serre derivative ∂ Fγ for F 0 , the constant c00 (0) for F 0 = ∂ S F is equal to
[−(1 + ρ/2)/12] times c̃0 (0) for F , so those insertions do not have a net effect. Both Ê2 Fγ
and ∂ S Fγ transform under the group PSL2 Z with the weight (1 − ρ/2, 0), and Zγ with the
weight (ρ/2, 1), so they are combined to have the weight (1,1). If one wants to write down
the expressions (2.46), (2.47) in terms of the coefficients cγ (ν) of Fγ , although that is not
necessary in this article, information in [13, 17] is useful (see also [10] and its references).
Now, it is not necessary to construct a Heterotic string (c, c̃) = (22, 9) SCFT, or a
Type IIA (c, c̃) = (9, 9) SCFT. We may just assume that there exists a branch of moduli
space with a set of classification invariants (ΛS , ΛT ) and {Φγ }, to get started. Compute the
automorphic forms abstractly from the data ΛS and {Φγ }, and then determine the residual
terms in the automorphic transformation of F (1) in (2.40), for generators g of [ΓS ]. By
demanding that the matrices Λg̃ should be Z-valued, we will find out that string vacua
cannot exist for certain choices of the classification invariants. That is what we do in the
following sections.
Furthermore, there is an exact sequence16
1 → Γ0 (k)/{±1} → Γ0 (k)+ → (Z2 )s → 1,
(3.3)
where s is the number of distinct primes appearing in the prime decomposition of k,
Q
k = si=1 pki i . For an element of (Z2 )s , say = (1 , 2 , · · · , s ) with i ∈ {0, 1}, the coset
Q
(Γ0 (k)/{±1}) · is described as follows. To prepare notations, let k1 := i pki i i and
Q k (1−i )
k2 := i pi i
, so k1 k2 = k and (k1 , k2 ) = 1. The coset (Γ0 (k)/{±1}) · is
(
!
)
a, b, c, d ∈ Z, k2 |c k1 |d, (ad − bc) = 1 ;
g : t 7−→ tg :=
k1 at + b
.
k1 ct + d
(3.4)
Now, all the information is here to verify the exact sequence (3.3).
e ). To see this, note that one can
The group Γ0 (k)+ forms a subgroup of Isom(Λ
S
e : for
assign to an element g of Γ0 (k)+ in (3.4) the following linear transformation on Λ
S
n ∈ U [−1] ⊕ h+2ki described by e0 n0 + e] n] + ea=1 na=1 =: (n0 , n] , n1 )T ∈ U [−1] ⊕ h+2ki,
g : n 7→ g · n,
2
d /k1 c2 /k2
g = k 2 b2 k 1 a 2
2cd
2kab .
bd/k1 ac/k2 ad + bc
(3.5)
e ) is an
This is a lattice isometry.17 The assignment from g ∈ Γ0 (k)+ to g ∈ Isom(Λ
S
injective homomorphism, so our abuse of notation will be tolerated.
Besides the image of Γ0 (k)+ , one will be aware of two more isometries; one is to
e ,
multiply (−1) to ΛS = h+2ki, denoted by (−1)2k , and the other is to multiply (−1) to Λ
S
denoted by −id. The group of all those isometries has a structure
(Γ0 (k)+ o Z2 h(−1)2k i) × Z2 h−idi ,
(3.6)
e ) in fact.18 Furthermore, any one of the isometries of Λ
e can
and this is the group Isom(Λ
S
S
be combined with an appropriate isometry on ΛT so that the pair of isometries defines an
isometry of II4,20 .
e ) is parametrized as
The space D(Λ
S
f(t) = (1, kt2 , t)T
(3.7)
with t = ea=1 ta=1 ∈ ΛS ⊗ C so that (f, f) = 0; to be a little more precise, the space
e ) is {t ∈ C = Λ ⊗ C | Im(t) 6= 0}. The isometry (−id) acts trivially on D(Λ
e ), and
D(Λ
S
S
S
Non-trivial elements in this (Z2 )s correspond to Atkin-Lehner involutions.
17
e S then g · n ∈ Λ
e S ; furthermore, for any n, n0 ∈ Λ
e S , (g · n0 , g · n) = (n0 , n).
That is, if n ∈ Λ
18
e S ). Using the fact that
Sketch of the proof: conversely, suppose that there is an isometry g ∈ Isom(Λ
e
g(e0 ) and g(e] ) form a basis for some sublattice of ΛS isometric to U , one can show that their components
can be represented as the first and second columns of (52) for some k1 , k2 , a, b, c and d which satisfy the
conditions in (51), after modifying g by Z2 h(−1)2k i and Z2 h−idi if necessary. The last basis vector g(ea=1 )
is uniquely determined up to (−1)2k multiplication.
16
– 14 –
JHEP03(2022)059
g=±
k1 a b
k1 c d
(−1)2k : t 7→ −t maps one connected component to the other. So,19 it is fine to focus on
the subgroup Γ0 (k)+ in studying the monodromy representations.
3.1.2
The automorphic form
For g ∈ Γ0 (k)+ of the form (3.5), (3.4), the relation (2.40) reads
(ck1 t + d)4 (1) g̃
1
F (t ) = F (1) (t) + (Λg̃ )IJ fI (t)fJ (t).
2
2
k1
(3.8)
dtg
k1
=
,
dt
(ck1 t + d)2
(3.9)
k13 (∂ 5 F (1) )(tg̃ ) = (ck1 t + d)6 (∂ 5 F (1) )(t).
(3.10)
to arrive at [4, 5] (Bol’s identity)
This means that
f∗ (t) :=
1
(∂ 5 F (1) )(t)
(2πi)5 t
(3.11)
is a meromorphic automorphic form of weight-(+6) of the group Γ0 (k)+ ; the logarithmic
singularities of (∂ 2 F ) at Xsingl turn into poles of order 3.
The automorphic form f∗ is determined uniquely by the index {Φγ }. This is because,
historically, there is a relation ([7] and [5], (3.3))
(1)
4 −2
2
(0)tt̄
[t−4
Gtt̄
2 ∂t t2 ][t2 ∂t t2 ]∂t K̂
= ··· = −
i 5 (1)
i
(∂t F ) = − (2πi)5 f∗ (t).
4k
4k
(3.12)
So, one can just combine this with (2.45). An alternative, and more practical perspective
is to note that f∗ ∝ ∂t5 F (1) contains only the coefficients nw,0 , which are all determined by
{Φγ } through (2.11).
The function F (1) (t) is the five-fold iterated integral of the automorphic form f∗ (t)
with respect to the coordinate (2πit). Although the integral F (1) has a remnant of the
automorphic transformation property of f∗ , there is also a violation term in (3.8). Let us
now turn to the question how the violation term is determined from f∗ .
3.1.3
Period polynomials
Let f (σ) be a cusp meromorphic modular form of weight-w for a group Γ, and the group
Γ be either one of Γ0 (k)+ for some k, or its finite index subgroup that at least contains
the σ → σ + 1 transformation. The automorphic form f∗ in (3.11) for the ρ = 1 cases and
f∗ in (4.20) for the ΛS = U case satisfy this property; the weight is (w = +6) and the
19
e S ) should actually be replaced by [ΓS ]\D(Λ
e S )/Z2 , where the Z2 action
The moduli space [ΓS ]\D(Λ
C
C
e
exchanges the role of f and f (and pR and pR ). So, we may restrict D(ΛS ) = {t ∈ C | Im(t) 6= 0} to
e S ) gauge generated by (−1)2k is also fixed.
{t ∈ C | Im(t) > 0} to fix this Z2 gauge, when the Z2 ⊂ Isom(Λ
– 15 –
JHEP03(2022)059
After taking derivatives with respect to t five times, the 2nd term on the right hand side
drops. On the left-hand side, one may use (3.4) and
argument σ ∈ H is t in the ρ = 1 cases, while w = +4 and σ = u in the ΛS = U case. In
the following, we will summarize necessary facts from the Eichler-Shimura theory of the
modular transformation law of the (w − 1)-fold iterated integral of f (σ) [3, 5]. For more
information, see e.g., [34].
Let us begin with the following observation. The (w − 1)-fold iterated integral along
an arbitrary path γ̃ starting from a base point σ0 ∈ H,
I[σ; f, γ̃] := (2πi)w−1
Z σ
Z σ1
dσ2 · · ·
dσ1
σ0
σ0
Z σw−2
dσw−1 f (σw−1 ),
(3.13)
σ0
(2πi)w−1
FEich (σ; f, γ̃) :=
(w − 2)!
Z σ
dσ 0 f (σ 0 )(σ − σ 0 )w−2 .
(3.14)
σ0
A general form of the (w − 1)-fold indefinite integral is
F (σ; f, γ̃) := I[σ; f, γ̃] + Q(σ) = FEich (σ; f, γ̃) + Q(σ),
(3.15)
where Q(σ) is a polynomial of σ of degree at most (w − 2). The polynomial Q(σ) cannot
be determined from the modular form f , so it is a kind of integration constants.
The Eichler integral has the following modular transformation property. For g ∈
Γ0 (k)+ and a path γ̃g̃ from σ0 to σ0g ,
FEich (σ g̃ ; f, (γ̃)g ◦ γ̃g̃ ) =
(k1 cσ + d)2−w
1−w/2
k1
(FEich (σ; f, γ̃) + Pg̃ (σ; f )) ,
(3.16)
where
(2πi)w−1
Pg̃ (σ, f ) :=
(w − 2)!
Z
(γ̃g̃ )g−1
dσ f (σ 0 )(σ − σ 0 )w−2 .
(3.17)
So, the (w − 1)-fold iterated integral of a weight-w modular form almost has a modular
transformation property, with a weight w + (w − 1)(−2) = (2 − w). How much the modular
transformation property is violated is computed by the period polynomials Pg̃ (σ, f ), which
is a polynomial of σ of degree (w − 2).
In most of math textbooks and literatures, Eichler integrals and period polynomials
are introduced for modular forms that do not have a pole in the interior of the upper half
plane. In the context of this article, however, we have no choice but to deal with modular
forms f that have a pole in the interior. So, we have to define Eichler integrals and period
polynomials by specifying homotopy classes of integration contours that stay away from
the poles of f (σ).
Let us now return to the original context in this article. Once the period polynomial
is worked out for g̃, then the polynomial (cf. the appendix B)
(k1 cσ + d)w−2
w
−1
2
F (σ g , f, (γ̃)g ◦ γ̃g̃ ) − F (σ, f, γ̃) = Pg̃ (σ, f ) +
k1
(k1 cσ + d)w−2
w
−1
2
Q(σ g ) − Q(σ)
k1
(3.18)
determines the matrix Λg̃ through (3.8) and (4.19).
– 16 –
JHEP03(2022)059
has an alternative expression called the Eichler integral:
For the group Γ, we need to choose a set of generators {gi }, their lifts g̃i and the
e )\X
paths γ̃g̃i in D(Λ
S
singl . Those generators, in general, are not independent, but are
subject to relations that follow from the composition law of the paths (2.32) and homotopy
equivalence of the paths. Once the monodromy representation matrices Mg̃i (gi , Λg̃i ) are
found for those generators, however, those matrices automatically satisfy the relations. To
see this, suppose that g̃a , g̃b are some lifts of ga and gb , respectively, and let g̃c := g̃a · g̃b
and gc := ga · gb , so γ̃g̃c = (γg̃b )ga ◦ γg̃a , as a reminder. Now, note that
Pg̃c (σ, f ) =
ϕgb (σ)w−2
Pg̃a (σ gb , f ) + Pg̃b (σ, f ),
(3.19)
and that
ϕgc (σ)w−2
Q(σ gc ) − Q(σ) =
(k1,b k1,a )w/2−1
ϕ (σ gb )w−2
ga
w/2−1
k1,a
(3.20)
ϕg (σ)w−2
ϕg (σ)w−2
Q((σ gb )ga ) − Q(σ gb ) bw/2−1 + bw/2−1 Q(σ gb ) − Q(σ) ,
k1,b
k1,b
where ϕgb (σ) = cb k1,b σ + db using c, k1 , d of g = gb in (3.4), (3.5) for cb , k1,b and db .
Combining them together (with f = f∗ ), we automatically have
Λg̃c = gbT · Λg̃a · gb + Λg̃b ,
3.2
Mg̃a · Mg̃b = Mg̃c .
(3.21)
Analysis on the ΛS = h+2i cases
In this article, we pick up just one case ΛS = h+2i from the series of ρ = 1 cases, and carry
out the program outlined earlier.
3.2.1
A quick review
Classification invariants. In the case of ΛS = h+2i, it is known that GS = Z2 , the
vector valued modular form {Φγ } is parametrized by two free low-energy BPS indices
n0 , n1/2 that are allowed to take value in
n0 = −2,
n1/2 ∈ {0, 1, 2, 3, 4}.
(3.22)
Concrete expressions of {Φγ } for those (n0 , n1/2 ) are found in section A.2 and references
there; for notations and the range of (n0 , n1/2 ), see [10]. It is also known [10] (cf. also [35])
that one more classification invariant parameter is necessary besides the data (n0 , n1/2 ), in
order to distinguish known distinct branches of moduli space with the lattice ΛS = h+2i.
It is
bR ∈ 2−1 Z≥0 .
(3.23)
For more information, see [10].20
20
It is not that one branch of moduli space has a unique value of bR . Reference [10] assigned a value of
bR to a branch of moduli space by finding a branch of enhanced gauge symmetry (probe gauge group), and
reading out the 1-loop beta function bR ; when there are multiple branches of enhanced gauge symmetry,
there are multiple values of bR assigned to the original branch. It is the set of bR ’s that is assigned to the
original branch of moduli space, to be precise. The set of bR ’s should be such that difference among those
bR ’s are in 6Z so that those bR ’s result in a consistent determination of the effective theory parameters
in (3.24), (3.25).
– 17 –
JHEP03(2022)059
w/2−1
k1,b
Once one set of the classification invariants n0 , n1/2 , bR is given, then [10]21
d111 = 4 − bR − n1/2 + 6δna=1 ,
(3.24)
(c2 )1 = 52 − 4bR − 10n1/2 + 24δna=1 ;
(3.25)
e )
Isometry group and loci of extra massless fields. The group Γ0 (k)+ ⊂ Isom(Λ
S
e
is now PSL2 Z. Because Isom(GS , qS ) is trivial for ΛS = h+2ki, all the isometries of ΛS lift
to isometries of the lattice II4,20 . So, we demand that all the elements g of PSL2 Z have
corresponding duality transformations g̃ and matrices Mg̃ in Sp(2(ρ + 2); Z)H.el .
As argued already, it is enough to find a set of generators {gi } of PSL2 Z ⊂ [ΓS ],
and construct their lifts, γ̃g̃i and Mg̃i ; the matrices Mg̃i automatically satisfy appropriate
±1 , g ±1 } as a set
relations that follow from the relations of {gi }’s and γ̃g̃i ’s. We choose {g∞
2
e ) 3 t 7→ tg∞ = t + 1 ∈ D(Λ
e ), and g : t 7→ tg2 = −1/t. The map
of generators; g∞ : D(Λ
2
S
S
g3 : t 7→ tg3 = −1/(t + 1) is obtained as g3 = g2−1 · g∞ = g2 · g∞ .
The loci of extra massless fields Xsingl consist of the PSL2 Z orbits of
[0]
[0]
[0]
X(v∗ ) = {t = t∗ := i},
v∗ = (1, 1, 0),
(3.26)
[1]
[1]
X(v∗ ) = {t = t∗ := e2πi/3 },
[1]
v∗ = (1, 1, 1),
(3.27)
[0]
[1]
e ∨ to express v and v in Λ
e ∨ . For
where we have used an integral basis (e0 , e] , ea=1 )T of Λ
∗
∗
S
S
[0]
[1]
e ) that goes around X(v ) [resp. X(v )] by phase +2π, the monodromy
a loop γ̃ in D(Λ
∗
∗
S
matrix is given by (2.34), with the data n0 [resp. n1/2 ] determining the matrices Λg̃(X(v[0] ))
∗
[resp. Λg̃(X(v[1] )) ].
∗
e ) to be the limit t = lim
We choose the base point t0 in D(Λ
0
t0,Im →+∞ (it0,Im ) along
S
the pure imaginary axis; this choice is expressed as t0 ≃ +i∞ in the rest of this article.
To choose a lift duality transformation g̃i for gi ∈ [ΓS ], a path γ̃g̃i from t0 is specified as
e );
follows: the path γ̃g̃∞ is a straight line from t0 to t0 + 1 in the large Im(t) region of D(Λ
S
g2
the path γ̃g̃2 is a path from t0 ≃ +i∞ to t0 ≃ +i almost straight down the imaginary
[0]
axis in the complex t-plane that avoids t = t∗ = +i by detouring into the 2nd quadrant
(see figure 1 (a)). We choose the path γ̃(g̃2 )−1 almost the same as γ̃g̃2 , but it detours around
[0]
the point t = t∗ = +i by stepping into the 1st quadrant of the t-plane. One can verify
by using the path composition rule (2.32) that g̃2−1 introduced in this way is indeed the
inverse element of g̃2 .
21
Reference [10] argued that δna are integers with a language that is valid when the Type IIA description
has a phase given by a Calabi-Yau-target non-linear sigma model in the branch of enhanced gauge symmetry.
In fact, we can argue that δna ∈ Z whether the enhanced symmetry branch has a phase of non-linear sigma
model description or not; we can just apply the argument leading to (2.36), (2.37), (2.38) to the branch of
enhanced gauge symmetry.
– 18 –
JHEP03(2022)059
here, d111 is the effective theory parameter dabc in (2.9), and (c2 )1 is a coefficient appearing
in the holomorphic R2 term in the 4d effective theory (mentioned briefly at (2.15)). The two
effective theory parameters d111 and (c2 )1 are determined modulo δn1 ∈ Z; this ambiguity
corresponds to the symplectic transformations M (1, Λ) with Λ]1 = −Λ01 = δn1 ∈ Z,
changing the flat coordinate s by Zta=1 .
We will see by (3.44), however, that not all bR ∈ 2−1 Z are theoretically possible.
The monodromy matrix Mg̃ for g̃ = g̃∞ . We have already discussed in (2.35)–(2.37)
what the monodromy matrix Mg̃∞ should be. In the present context (ρ = 1), the matrix
Mg̃∞ is integer valued if and only if
d111 ∈ Z,
a11 ∈
d111
+ Z,
2
2d111 + b1 ∈ 12Z.
(3.28)
It follows immediately from (3.28) that a11 ∈ Z/2 and b1 ∈ Z. Furthermore, the classification invariants should be subject to
(3.29)
so we should have the parameter bR in Z not in 2−1 Z. The condition (3.28) determines
(b1 )+24Z only mod +12Z, so there are still two possible values of (b1 )+24Z . To compare
(b1 )+24Z and (c2 )1 modulo +12Z,
(c2 )1 ≡ 4 − 4bR + 2n1/2 ,
(b1 )+24Z ≡ (−2d111 ) ≡ −8 + 2bR + 2n1/2 .
(3.30)
So, (c2 )1 and (b1 )+24Z are different when compared mod 12Z, if bR is odd. They are equal
mod 12Z when bR is even.
3.2.2
Monodromy matrix of g̃2 : the case n1/2 = 0
Let us now determine the monodromy matrix Mg̃2 for the other generator of PSL2 Z. To find
out the matrix Λg̃2 by analytic continuation, we use the method described in sections 2.3
and 3.1. It is enough to evaluate the period polynomials in (3.18).
To start, let us work on the case n1/2 = 0. The automorphic form f∗ in (3.11) is of
weight (w = 6), under the group PSL2 Z. It is determined uniquely by the combination
of (3.12) and (2.45) in terms of the indices {Φγ }(n0 ,n1/2 )=(−2,0) . It is known that [4] (more
explanation available around (3.46))
1 (18E43 − 5E62 )(E43 − E62 )
(2πi)3
9E63
1
5
2
≃
2496
Li
(q)
+
2
·
223752
Li
(q
)
+
·
·
·
−2
−2
(2πi)3
f∗ (t) :=
(3.31)
(3.32)
is the right choice. One way to argue for (3.31) is to note that f∗ satisfies all the properties
[0]
expected from physics, including appropriate singularity22 at t = t∗ = i and non-singular
[1]
behavior at t = t∗ = e2πi/3 . A more practical way is to see that the coefficients 2496,
223752, etc. of w5 Li−2 (q w ) (for w = ea=1 w ∈ Λ∨
S ) agree with nw,0 = c[w] ((w, w)/2) (see
secgtion A.2).
22
The Laurent series expansion of f∗ (t) in (3.31) is
f∗ (t) ≃
1
−16
+ O((t − i)−2 ),
(2πi)6 (t − i)3
(3.33)
from which F0 ≃ 8i(t − i) ln[t − i]/(2πi) + O((t − i)2 ) and F1 ≃ −8(t − i) ln[t − i]/(2πi) + O((t − i)2 )
follow. This singularity reflects the non-zero beta function of the massless SU(2) gauge theory in the 4d
field theory (the W-boson is from nγ=0 = −2); we should expect n1/2 = 0 massless charged hypermultiplets
at t ≃ e2πi/3 , so f∗ is expected to be non-singular there when n1/2 = 0.
– 19 –
JHEP03(2022)059
bR ∈ −n1/2 + Z,
(a)
(b)
(c)
Now, we wish to compute the period polynomials23 P(g̃2 )±1 (t, f∗ ). As we have chosen
t0 ≃ +i∞ and the path γ̃g̃2 [resp. γ̃g̃−1 ] to be in the 2nd [resp. 1st] quadrant of the t-plane,
2
−1
the integration contour (γ̃g̃2 )g2 [resp. (γ̃g̃−1 )g2 ] is in the 1st [resp. 2nd] quadrant, from +i
2
to +i∞ (figure 1 (b)). The numerical integration of (3.17) along this contour can be split
into two segments by exploiting the modular transformation property of f∗ and change of
variables (figure 1 (c)). It is not hard to find out numerically by using Mathematica that
P(g̃2 )±1 (t, f∗ ) ≃ −0.610599i(t4 − 1) ± 0.5(t4 + 2t2 + 1) − 1.5(t3 + t),
≃
ζ(3) (−252) 4
(t − 1) ±
(2πi)3
2
t4 + 2t2 + 1
2
(3.34)
3
− (t3 + t).
2
(3.35)
Now, let us add the integration constants in the (w − 1) = 5-fold iterated integral, as
in (3.15). The polynomial Q(t) should be at most degree-4, but we set it to24
Q(t) =
0 4 d111 3 a11 2 b1
ζ(3) χ
t +
t −
t − t−
,
4!
3!
2
24
(2πi)3 2
(3.36)
because we know for a Het-IIA dual vacuum that there must be a symplectic frame where
the quartic term is absent in the prepotential (2.9). The relation (3.8), (3.18) determines
the matrix Λg̃±1 as follows:
2
ζ(3)
±1 + (2πi)
3 (χ + 252)
±1
Λ(g̃2 )±1 =
23
b1
d111
3
24 − 6 − 2
b1
3
− d111
6 − 2
24
ζ(3)
b1
d111
3
e
±1 − (2πi)
+ CC.
3 (χ + 252)
24 − 6 − 2
b1
d111
3
0
24 − 6 − 2
±1
(3.37)
The indefinite integrals I[σ; f∗ , γ̃] and FEich (σ; f∗ , γ̃) and also the period polynomials Pg̃±1 (t, f∗ ) con2
verge in the limit of the base point t0 ≃ +i∞ (t0,Im → +∞), because of the cusp property of f∗ .
24
As we have chosen t0 ≃ +i∞, the (w − 1) = 5-fold iterated integral of f∗ only contains power series of
2πit
e
. The integration constant terms Q(t) in (3.15) should be precisely the non-e2πit terms of F (1) .
– 20 –
JHEP03(2022)059
Figure 1. Paths γ̃g̃∞ , γ̃g̃2 and γ̃g̃−1 in the upper half plane t ∈ H are shown by dot-dashed, solid
2
and dashed lines in (a) along with the boundaries of fundamental regions in thinner solid lines. The
[0]
[1]
PSL2 Z orbit of t∗ and t∗ are indicated by •’s and N’s, respectively. The contours of integration for
the period polynomials Pg̃2 (t, f∗ ) and Pg̃−1 (t, f∗ ) are drawn by solid and dashed lines, respectively,
2
in (b). The contour for Pg̃2 (t, f∗ ) may be split into two segments (solid and dashed) in (c), and the
integral over the dashed part of the contour may be further rewritten as an integral over the other
solid line contour in (c).
The +CCe ambiguity should be reduced to +ZCe (or dropped) because the ambiguity beyond
+ZCe does not help making this Λg̃2 matrix integer valued.
As a second check, remember that we have chosen the two paths γ̃g̃2 and γ̃g̃−1 both as
2
lifts of
1
e )
∈ Isom(Λ
S
g2 = 1
(3.38)
−1
in a way they are for the inverse duality transformation of each other under the path
composition law (2.32). Indeed, one can verify that both M (g2 , Λg̃2 ) · M (g2 , Λg̃−1 ) and
2
M (g2 , Λg̃−1 ) · M (g2 , Λg̃2 ) are equal to the identity matrix by using (3.37), (3.38); we do not
2
need to use a specific value for χ, d111 and b1 . One will also find that
(M (g2 , Λg̃2 ))2 = M (1, Λg̃(X(v[0] )) ),
∗
M (g2 , Λg̃−1 )
2
2
= M (1, −Λg̃(X(v[0] )) ),
∗
(3.39)
without substituting any value into χ, d111 and b1 .
Finally, one may define two lifts of the map g3 : t 7→ −1/(t + 1). One is g̃3 := g̃2 · g̃∞
e ). The
and the other g̃30 := g̃2−1 · g̃∞ ; note that g2 · g3 = g∞ and g22 = 1 in Isom(Λ
S
corresponding paths γ̃g̃3 and γ̃g̃30 are determined from (2.32) and are shown in figure 2 (a).
A straightforward matrix computation confirms that
e
(Mg̃3 )3 = (Mg̃2 · Mg̃∞ )3 = M (1, (d111 /2 − 2)C),
3
0
3
(Mg̃30 ) = (Mg̃−1 · Mg̃∞ ) = M (1, Λ ),
(3.40)
(3.41)
2
e
Λ0 = −Λg̃(X(v[0] )) − Λg̃(X(v[0] )) − Λg̃(X(v[0] )) + (d111 /2 − 2)C;
∗
[0]
∗∗
∗∗∗
(3.42)
[0]
e ∨ and v
here, v∗∗ = (1, 2, 2) ∈ Λ
∗∗∗ = (2, 1, 2), with the corresponding states becoming
S
[0]
[0]
massless at t∗∗ = (−1 + i)/2 and t∗∗∗ = (−1 + i), respectively. It is appropriate that the
relations (3.40), (3.42) hold without including monodromy contributions from the PSL2 Z
[1]
orbit of t∗ on the right-hand sides (see figure 2 (b)), because the computation (3.35) is for
f∗ in (3.31), which is for n1/2 = 0.
Note that we have tools to compute the monodromy matrices Mg̃ from first principle,
using numerical evaluation of the period polynomials. It is not that the relations such
– 21 –
JHEP03(2022)059
Before proceeding further, let us run a few checks with known results, to validate this
method of computing the monodromy matrices. There is a branch with ΛS = h+2i that
has been studied extensively in the literature. That is the Heterotic construction known as
the ST -model, whose Type IIA dual is for a Calabi-Yau threefold M = (12) ⊂ P4[1:1:2:2:6] .
This geometry indicates that d111 = 4, b1 = 52, and χ = −252. When those values
are substituted into (3.37), the monodromy matrix Mg̃−1 = M (g2−1 , Λg̃−1 ) reproduces the
2
2
monodromy matrix (−S1 ) in [36], which was determined by using computations of the
mirror manifold of M .
(b)
Figure 2. The paths γ̃g̃3 = (γ̃g̃∞ )g2 ◦ γ̃g̃2 for g̃3 = g̃2 · g̃∞ and γ̃g̃30 = (γ̃g̃∞ )g2 ◦ γ̃g̃−1 for g̃30 = g̃2−1 · g̃∞
2
are drawn in the complex t plane in (a) by the solid and dashed oriented lines, respectively. In
the panel (b), the solid and dashed oriented lines are the paths (loops in fact) for (g̃3 )3 and (g̃30 )3 ,
respectively. For more information, see the caption of figure 1.
as (3.39), (3.40), (3.42) are imposed to constrain Λg̃ that is otherwise intractable;25 we
computed the matrices Λg̃ from first principle and expressed them in terms of the (implicit
n1/2 = 0 and) integration constants d111 , b1 and χ; the relations (3.39), (3.40), (3.42) are
satisfied automatically, as explained in (3.19)–(3.21).
Let us now go back to the program of imposing the integrality of the monodromy
matrices to narrow down theoretically possible choices of the classification invariants. Demanding that all the matrix entries in (3.37) are integers, we obtain conditions that are
independent from (3.28). A common solution to those conditions is parametrized as
χ = −252,
d111 = 2D,
b1 = 8D + 36 + 24B,
a11 = A,
(3.43)
for D, A, B ∈ Z. The first two conditions impose extra conditions on the classification
invariants (n0 , n1/2 , bR ) = (−2, 0, bR ). The required value χ = −252 here is the same
as the value [−c0 (0)]n1/2 =0 = −252 determined in an independent reasoning (2.11), so
the extra condition is satisfied (see also section 5). The condition that d111 is even (for
n1/2 = 0) implies that only
bR ∈ 2Z
(3.44)
are for branches of theoretically consistent Het-IIA dual moduli space.
25
Although refs. [3, 5] observed that the period polynomials Pg̃ are relevant to the Λg̃ part of the monodromy matrices Mg̃ (as reviewed in section 3.1.3), the period polynomials Pg̃ of paths γ̃g̃ were not exploited
to compute Mg̃ there. When it comes to the discussion on monodromy matrices, refs. [3, 5] used the monodromy matrices M (1, Λg̃(X(v)) ) in (2.34), which follow from 4d 1-loop beta functions, for loops γ̃g̃(X(v)) to
impose conditions such as (3.39), (3.40), (3.42), (4.17), and presented an example of Λg̃i ’s satisfying those
conditions.
– 22 –
JHEP03(2022)059
(a)
We are also ready to compare the two parameters of the 4d effective theory, (b1 )+24Z
and (c2 )1 modulo 24Z (not just mod 12Z as in (3.30)).
(b1 )+24Z − (c2 )1 ≡ (36 + 16 − 4bR ) − (52 − 4bR ) ≡ 0.
(3.45)
3.2.3
The cases n1/2 = 1, 2, 3, 4
It is straightforward to employ the same method to determine the monodromy matrix Mg̃2
for the cases with ΛS = h+2i and n1/2 > 0. The weight-6 automorphic form f∗ in (3.11)
under PSL2 Z must be determined uniquely for individual n1/2 ∈ {0, 1, 2, 3, 4}, and it should
be of the form
f∗ (t) =
1 aE49 + bE46 E62 + cE43 E64 + dE66
(2πi)3
E43 E63
(3.46)
for some coefficients a, b, c, d because ∂t2 F (1) (and hence the three-fold integral of f∗ ) should
[0]
[1]
have logarithmic singularity at t ≃ t∗ = i (where E6 (t) ≃ 0) and also at t ≃ t∗ =
e2πi/3 (where E4 (t) ≃ 0). The coefficients a, b, c, d may, in principle, be determined by
exploiting (3.12), (2.45); we demand instead that the e2πit series expansion of f∗ agrees
26
To be precise, what we have confirmed is existence and unique determination of a diffeomorphism class
of real 6-dimensional manifolds [M ] that reproduces (Cab , dabc ) and (24, (c2 )a ) of a branch with (ΛS , ΛT ),
{Φγ } and {Ψγ }, along with a few consistency conditions (2.38), (3.45) for the branch to be realized by a nonlinear sigma model with the target space M . We have not explored any observable from 4d hypermultiplets
for consistency check on the geometric phase interpretation.
27
On the subtle possibility that something other than BPS dyon spectra might be used for an even finer
classification, see [10, 35].
– 23 –
JHEP03(2022)059
So, for any branch of Het-IIA dual vacua with integral monodromy matrices, we have seen
that the two parameters (b1 )+24Z and (c2 )1 yield one common value that is interpreted as
R
M c2 (T M )Da=1 if the branch is given by a non-linear sigma model with the target space
M in the Type IIA description.
Now, the conditions (2.37) is read precisely as Wall’s condition for (necessary and)
sufficient condition for a diffeomorphism class [M ] of real 6-manifolds M to exist, with the
trilinear intersection form on H 2 (M ; Z) and the 2nd Chern class in H 4 (M ; Z) characterized
by (C11 , d111 ) and (24, (c2 )1 ). The conditions (2.38), (3.45) are equal to the additional
conditions (2.16), (2.17) for the central charge of 4d N = 2 supersymmetry from D-branes
appropriate in a phase of non-linear sigma model description. So, we have seen that all
the branches of theoretically consistent Het-IIA dual moduli space with ΛS = h+2i and
n1/2 = 0 contain phases described by a non-linear sigma model26 in the Type IIA language.
Now, three (or possibly more)27 distinct branches of moduli space remain (among
those with ΛS = h+2i and n1/2 = 0). The integer parameters A and B differ by +Z
do not lead to distinct spectra of electrically/magnetically charged 4d N = 2 BPS states
(note that ∆a11 ∈ Z and ∆b1 ∈ 24Z and remember (2.10)). ∆D ∈ 3Z also fall into
the ambiguity in (2.10). So, D+3Z ∈ {2, 1, 0}+3Z (or equivalently (bR )+6Z ∈ {0, 2, 4}+6Z )
label the three branches. Calabi-Yau threefolds for those three branches are known in the
literature [35, 37].
with the e2πit series expansion of (2πi)−5 ∂t5 F (1) , where the coefficients of the latter are
determined by nw,k=0 = cγ ((w, w)/2)’s (see section A.2). It turns out that
a = 2,
b=−
23
,
9
c=
5 + 2n1/2
,
9
d=−
2n1/2
.
9
(3.47)
The period polynomial Pg̃2 (t; f∗ ) can be evaluated numerically for individual n1/2 ∈
{1, 2, 3, 4}, just like we have done for the case n1/2 = 0. The results are fitted very well by
the formula
ζ(3) (−252 + 56n1/2 ) 4
t4 +2t2 +1
3 n1/2
Pg̃2 (t; f∗ ) ≃
(t
−
1)
+
+ − +
(t3 + t). (3.48)
3
(2πi)
2
2
2
4
Λg̃2 =
ζ(3)
1 + (2πi)
3 X0
1
ζ(3)
1 − (2πi)
3 X0
1
n1/2
b1
d111
3
24 − 6 − 2 + 4
n1/2
b1
d111
3
24 − 6 − 2 + 4
n1/2
b1
3
− d111
6 − 2 + 4
24
n1/2
b1
3
e
− d111
+ ZC,
6 − 2 + 4
24
0
(3.49)
where
X0 := χ + 252 − 56n1/2 .
(3.50)
When we use the value of χ determined by the reasoning reviewed in (2.11) (see also
section A.2), the combination X0 vanishes for all n1/2 .
The value of (b1 )+24Z is determined when we demand that all the entries of the matrix
Λg̃2 be integers. This condition on b1 mod 24Z and the conditions (3.28) combined imposes
one condition d111 − n1/2 ∈ 2Z, or equivalently
bR ∈ 2Z≥0
(3.51)
on the classification invariant, besides determining (b1 )+24Z and (a11 )+Z . One also finds
that
(b1 )+24Z = 52 − 4bR − 10n1/2 + 24Z = (c2 )1 + 24Z
(3.52)
by using (3.24), (3.25).
We have therefore seen that both (b1 )+24Z and (c2 )1 yield one common thing that can
be interpreted as the 2nd Chern class of the target space in the Type IIA language. Now,
the conditions (2.37) is read as the sufficient conditions for existence of a diffeomorphism
class [M ] of real 6-dimensional manifolds whose trilinear intersection form and the 2nd
Chern class on H 2 (M ; Z) agree with what we compute from the data ΛS , {Φ} and bR .
It is reasonable to conclude (cf. footnote 26) that all those branches with ΛS = h+2i,
n1/2 ∈ {0, 1, 2, 3, 4} and Z-valued monodromy matrices have a region described by the
non-linear sigma model in the Type IIA language, with M the target space. Table 1
– 24 –
JHEP03(2022)059
This is combined with the contributions from the integration constant terms (3.36) to
determine (3.18), and hence the matrix Λg̃2 .
of ref. [37] shows that there is a known construction for Calabi-Yau threefolds M whose
diffeomorphism classes [M ] correspond to n1/2 = 1, bR + 6Z = {0, 2}+6Z , n1/2 = 2, bR +
6Z = {0, 2}+6Z , n1/2 = 3, bR + 6Z = {0}+6Z . At this moment, it is not clear whether a
geometry with (n1/2 , bR + 6Z) = (1, 4+6Z ), (2, 4+6Z ), (3, 2+6Z ), (3, 4+6Z ), (4, {0, 2, 4}+6Z ) is
simply not within the range of parameters of the toric ambient space scanned in [37], or
not within the category of complete intersection of a toric variety, or such a geometry does
not exist for a reason we do not understand yet.
The case ΛS = U
4
(ρ)
(u)
PSL2 Z × PSL2 Z o (Z2 hσi × Z2 (−1)ρu ) × Z2 (−id),
(ρ)
(4.1)
(u)
e ) = {(ρ, u) ∈ C | Im(ρ)Im(u) > 0} as linear
where PSL2 Z and PSL2 Z act on D(Λ
S
fractional transformations on ρ and u, respectively, and (−1)ρu : (ρ, u) 7→ (−ρ, −u). The
map σ brings (ρ, u) to (ρσ , uσ ) = (u, ρ). The discriminant group GS is trivial for ΛS = U ,
e ) lifts to isometries of
and so is the group Isom(GS , qS ). So, all the elements in Isom(Λ
S
e = II
Isom(II4,20 ) for the embedding Λ
,→
II
⊕
II
=
II
2,2
2,2
2,18
4,20 . For this reason, we
S
e
will demand in this article that all the elements g ∈ Isom(ΛS ) should have a lift duality
transformation g̃ whose monodromy matrix is Z-valued.28
Before starting to work out the monodromy matrices, let us also quote the results on
classification invariants for the case ΛS = U . The invariant {Φ} is the unique scalar valued
weight (11 − ρ/2 = 10) modular form starting with nγ=0 = −2, which is Φ = −2E4 E6 .
One more classification invariant is
bR ∈ 2−1 Z≥0 .
(4.2)
With this invariant, the parameters of the effective theory in the Im(u) > Im(ρ) phase are
given by [10, 13, 14]
dρρu = n0 − 2 + 2δnρ ,
dρuu = n0 + 2δnu ,
0
(c2 )ρ = 12(2 + n ) − 4 + 24δnρ ,
0
dρρρ = 2,
(c2 )u = 12(2 + n ) + 24δnu ;
duuu = 0,
(4.3)
(4.4)
Here, n0 := 2 − bR /6.
28
(ρ)
(u)
It is fine to focus on (PSL2 Z × PSL2 Z) o Z2 hσ̃i for the reasons explained already in footnotes 10
and 19.
– 25 –
JHEP03(2022)059
As another example, let us work on the case ΛS = U . Historically, the relevance of an
automorphic form f∗ to the monodromy matrices has been observed in this ΛS = U case for
the first time [2, 3], so the following presentation is inevitably very close to what is written
in the literature. What we do here is to be faithful to the first principle calculation of
the monodromy matrices to narrow down the theoretically possible choices of classification
invariants, instead of finding relevance of f∗ in a string vacuum with a known construction.
Let us start off by reviewing known things. On the lattice ΛS = U , let us choose a
e ) is
basis {eρ , eu } so that (eρ , eρ ) = (eu , eu ) = 0 and (eρ , eu ) = 1. The moduli space D(Λ
S
parametrized by f = (1, (t, t)/2, t)T = (1, ρu, ρ, u)T , where t = eρ ρ + eu u ∈ ΛS ⊗ C. The
e ) is (e.g., [38, 39])
lattice isometry group Isom(Λ
S
Im(u)
Im(ρ)
(a)
(b)
Figure 3. The paths γ̃σ̃ and γ̃σ̃−1 are shown by the solid and dashed curves, respectively, in the
Im(u) = 0 slice of the (ρ, u) ∈ C2 space in (a); the u = ρ singularity is shown by the dot-dashed
(u)
line. The panel (b) shows the path γ̃g̃2(u) in the ρ = ρ0 slice of the (ρ, u) parameter space. The
(u)
singularity at u ∈ PSL2 Z · ρ0
4.1
are marked by the open circles.
Analysis on the ΛS = U case
Now, let us choose a base point of paths in a connected part Hρ × Hu of the covering
e ),
space D(Λ
S
(u)
(u)
(u)
t0 = (ρ0 , u0 ) =
lim
u0,Im →+∞
(iρ0,Im , iu0,Im );
(4.5)
ρ0,Im is chosen to be a finite positive value much larger than 1. All the loci of extra massless
(u)
fields are [3] of the form of ρ = ug for some g ∈ PSL2 Z, and form a single orbit under
e ). Monodromy matrices for loops in the covering space is completely understood,
Isom(Λ
S
(ρ)
(u)
so we are left to choose a set of generators {gi } of [PSL2 Z × PSL2 Z] o Z2 hσi, find one
lift g̃i (and γ̃g̃i ) for each gi and compute the matrix Mg̃i . As a set of generators, we choose
±1
±1
±1
{g∞(u)
, g∞(ρ)
, σ, g2(u)
}, where g∞(u) and g2(u) keep ρ invariant and act on u as u 7→ u + 1
and u 7→ −1/u, respectively. Similarly, g∞(ρ) : ρ 7→ ρ + 1.
To specify one lift g̃i for each gi , we describe the corresponding path γ̃g̃i , as follows.
(u)
(u)
The paths γ̃g̃∞(ρ) and γ̃g̃∞(u) both start from t0 , and have ∆ρ = +1 for fixed u = u0 and
(u)
(u)
(u)
∆u = +1 for fixed ρ = ρ0 , respectively, to reach the endpoint (t0 )g∞(ρ) and (t0 )g∞(u) ,
(u)
respectively. The path γ̃2(u) starts from t0 , and moves in the u-plane down the imaginary
(u)
(u)
(u)
axis (and fixed ρ = ρ0 ) to u = −1/(u0 ), while avoiding u = ρ0 by detouring into
(u)
the 2nd quadrant in the u-plane. Finally, the path γ̃σ̃ [resp. γ̃σ̃−1 ] starts from t0 and
(u) σ
(u) (u)
ends at (t0 ) = (u0 , ρ0 ), avoiding the u = ρ singularity on the way by temporarily
setting Re(ρ) to be positive [resp. negative]; see figure 3 (a). The corresponding duality
transformation (analytic continuation of the section Π = (X, F )T along those paths) are
denoted by g̃∞,ρ , g̃∞,u , g̃2,u , and σ̃ ±1 , respectively.
– 26 –
JHEP03(2022)059
Re(ρ)
The monodromy matrices for g̃∞(ρ) and g̃∞(u) are given by g∞(ρ) , g∞(u) and (2.35). The
effective theory parameters are subject to the conditions (2.37), (2.38), or more explicitly,
dρρρ ,
dρρu ,
duuu ∈ Z,
dρuu ,
(aρρ )+Z = (dρρρ /2) + Z,
(4.6)
(auu )+Z = (duuu /2) + Z,
(4.7)
(aρu )+Z = dρρu /2 + Z,
(4.8)
dρρu ∈ dρuu + 2Z,
(4.9)
2duuu + bu ∈ 12Z,
2dρρρ + bρ ∈ 12Z.
(4.10)
bR ∈ 6Z≥0
(4.11)
in a branch of theoretically consistent Het-IIA dual moduli space, not 2−1 Z≥0 . The values
of (aρρ )+Z and (auu )+Z are also fixed, when combined with the parametrization (4.3).
The values of (bρ )+24Z and (bu )+24Z are determined only modulo +12Z. At this level of
precision,
(bρ )+12Z = (−2dρρρ )+12Z = −4 + 12Z,
(c2 )ρ ≡ −4
(mod 12Z),
(4.12)
(bu )+12Z = (−2duuu )+12Z = 12Z,
(c2 )u ≡ 0
(mod 12Z).
(4.13)
So, we are left with (ba )+24Z ≡ (c2 )a mod 24, or (c2 )a + 12 mod 24, chosen individually for
a = ρ and a = u.
The monodromy representation matrices Mσ̃±1 = M (σ, Λσ̃±1 ) are given by com(u) (u)
(u) (u)
puting the analytic continuation of F (1) from t0 = (ρ0 , u0 ) to tσ0 = (u0 , ρ0 )
along the paths γ̃σ̃±1 . In addition to the polynomial terms of F (1) , one more term
±1
c(−1)Li3 (e2πi(u−ρ) )/(2πi)3 in F (1) contributes to F (1) (tσ̃ ) − F (1) (t) in (2.40). Using29
h
Li3 (e2πi(u−ρ) )
i
along γ̃σ̃±1
(2πi)3
Li3 (e2πi(u−ρ) )
1
3
1
=
(u − ρ)3 ∓ (u − ρ)2 + (u − ρ)
3
(2πi)
3!
2
2
−
(4.14)
and c(−1) = −2 (see section A.2), one finds that
±1
F (1) (tσ̃ ) − F (1) (t) =
4 + b ρ − bu
±1 − aρρ + auu 2 ±1 + aρρ − auu 2
u +
ρ ∓ ρu +
(ρ − u);
2
2
24
(4.15)
cubic terms cancel when we use (4.3). The monodromy matrices Mσ̃ = M (σ, Λσ̃ ) and
Mσ̃−1 = M (σ, Λσ̃−1 ) are now determined by (2.40).
σ=
1
1
1
,
1
Λσ̃±1 =
00
0 0
4+bρ −bu
24
0
−
4+bρ −bu
24
0
e
+ ZC;
±1 + aρρ − auu
∓1
∓1
±1 − aρρ + auu
0
0
(4.16)
The numerator in the 1st term is the function of (ρ, u) whose value is Li3 (e2πi(u −ρ ) )(ρ0 ,u0 )=(ρσ ,uσ ) ; the
0
0
value is determined by analytically continuing Li3 (e2πi(u −ρ ) ) for (ρ0 , u0 ) along (γ̃)σ ◦ γ̃σ̃±1 ◦ (γ̃)−1 .
29
– 27 –
JHEP03(2022)059
Those conditions imply that the classification invariant bR can take values only in
the lower-left corner of the symmetric matrices Λσ̃±1 are omitted just to save space. The
two matrices automatically satisfy the relations (cf. footnotes 11), (25 and 30)
(Mσ̃±1 )2 ∈ M (1, ±Λg̃(X(vρu )) ) · (MD )n ,
n ∈ Z,
(4.17)
e ∨ are the charges of the states that become massless at
where ±vρu = ±(0, 0, −1, 1) ∈ Λ
S
u = ρ.
By demanding that the matrices Λσ̃±1 are Z-valued, we obtain one more condition.
(bρ )+24Z + 4 = (bu )+24Z ;
(4.18)
(u)
(u)
(u)
(u)
find out how F (1) (ρ0 , −1/u0 ) is related to F (1) (ρ0 , u0 ), we use numerical evaluation
of the period polynomial.
(u)
For g(u) ∈ PSL2 Z mapping u 7→ (au + b)/(cu + d), the relation (2.40) reads
1
(cu + d)2 F (1) (ρ, ug̃(u) ) = F (1) (ρ, u) + f(ρ, u)T Λg̃(u) f(ρ, u).
2
(4.19)
(u)
The 3rd derivative of F (1) with respect to u transforms as a modular form under PSL2 Z
of weight (w = +4) [2], so we set
f∗ (u; ρ) :=
1
∂ 3 F (1) (ρ, u).
(2πi)3 u
(4.20)
This modular form f∗ (u; ρ) is determined uniquely by the unique Φ (independent of the
classification invariant bR ∈ 6Z), because ([3], (2.15)) (note also (2.45))
(1)
2
(0)ab̄
¯
−ρ22 [u−2
Gab̄
2 ∂u u2 ]∂u ∂ρ̄ K̂
i
i
= · · · = ∂u3 F (1) (ρ, u) = (2πi)3 f∗ (u; ρ).
2
2
(4.21)
It is known [2, 3] that this uniquely determined f∗ (u; ρ) for all the ΛS = U cases is of the
form (cf [40])
1 −i ju (u)
j(ρ) ju (u) j(ρ) − j(i)
,
(2πi)3 π j(u) − j(ρ) j(u) jρ (ρ) j(u) − j(i)
1
2
E 4 E6
=
E4 (u) 24 (ρ).
3
(2πi) j(ρ) − j(u)
η
f∗ (u; ρ) =
(4.22)
(4.23)
Note that f∗ (u, ρ) for a fixed value of ρ has the behavior f∗ (u) ≃ e2πiu ×(const.)+O(e2πi2u )
at large Im(u), so this is a cusp form of weight (w = 4) with poles (at u = ρ and its PSL2 Z
images) in the upper half plane of u.
The part of the prepotential F (1) (ρ, u) should be reproduced from f∗ (u; ρ) by a (w −
1) = 3-fold integral with respect to the coordinate 2πiu. The indefinite integral (the Eichler
(u)
integral) I[u; f∗ , γ̃, ρ] = FEich (u; f∗ , γ̃, ρ) starting from u = u0 ≃ i∞ converges (see the
review in section 3.1.3), and makes sense uniquely when the path γ̃ stops at Im(u) Im(ρ);
for this reason, the (w − 1)-fold integral (3.15) is ready for an easy interpretation in the
– 28 –
JHEP03(2022)059
two choices are left, bρ + 4 ≡ bu ≡ 0 mod 24 and ≡ 12 mod 24.
Finally, let us determine the monodromy matrix of the duality transformation g̃2(u) . To
Im(u) > Im(ρ) phase. The integration constant terms Q(u; ρ) in (3.15) is a polynomial of
u of at most degree (w − 2) = 2, which should be of the form
dρuu
auu
Q(u; ρ) =
ρ−
u2 +
2
2
+
dρρu 2
bu
ρ − aρu ρ −
u
2
24
(4.24)
dρρρ 3 aρρ 2 bρ
ζ(3) χ X n(n,0),0
ρ −
ρ −
−
+
Li3 (e2πiρn ) .
6
2
24 (2πi)3 2 n≥1 (2πi)3
[Pg̃2(u) (u; f∗ , ρ)]u,(u2 +1) ≃ −2(ρ2 + 1)u +
ρ2 + 1 2
(u + 1),
2
(4.25)
and the (u2 − 1) term satisfies
ζ(3) 480
480 X
[Pg̃2(u) (u; f∗ , ρ)](u2 −1) + (u2 − 1)
+
Li3 (e2πiρn )
3
(2πi) 2
(2πi)3 n≥1
≃ (u2 − 1)
5
1
ρ − ρ3 ,
6
3
(4.26)
where we have used χ = −c(0) = −480 and n(n,0),0 = c(n · 0) = c(0) = 480 from (2.11)
(see section A.2). So, the deviation (3.18) from the modular transformation property for
F (1) (ρ, u) is now purely polynomial in both ρ and u. Moreover, the (u2 − 1)ρ3 terms cancel
because dρρρ = 2 in (4.3). So, we obtain
1 − auu
Λg̃2(u) =
b +12d
−20
ρ
ρuu
0
−2 + b12u
24
bρ +12dρuu −20
1 − aρρ −2 − dρρu −
e
24
+ ZC.
1 + aρρ
2aρu
1 + auu
(4.27)
Almost all the entries of the matrix Λg̃2(u) are automatically integers based on the conditions that have been derived. One new condition is obtained, however, which determines
(bρ )+24Z uniquely, and consequently also determines (bu )+24Z because of (4.18).
(bρ )+24Z = 20 + 12n0 + 24Z,
(c2 )ρ ≡ 20 + 12n0
(mod 24Z),
(4.28)
(bu )+24Z = (bρ )+24Z ,
(c2 )u ≡ (c2 )ρ + 4
(mod 24Z).
(4.29)
So the uniquely determined (ba )+24Z ’s agree with (c2 )a mod 24Z.
– 29 –
JHEP03(2022)059
Neither the Eichler integral FEich (u; f∗ , γ̃, ρ) nor the integration constant terms give rise to
the u3 term in the Im(u) > Im(ρ) phase, but that is consistent with the known parametrization duuu = 0 (for Im(u) > Im(ρ)) by the classification invariants (see (4.3)).
Now let us determine Λg̃2(u) in the monodromy matrix Mg̃2(u) = M (g2(u) , Λg̃2(u) ), by
using (4.19), (3.18). We evaluated the period polynomial Pg̃2(u) (u; f∗ , ρ) for several values
of ρ by carrying out numerical integrals just like in the ΛS = h+2i cases. It turns out that
there is a nice fitting formula for the numerical integrals. The u and (u2 + 1) terms in the
polynomial Pg̃2(u) (u; f∗ , ρ) are
5
Discussion
In this article, we have given a proof of concept of computing the monodromy matrices
directly to narrow down possible choices of classification invariants of Heterotic-IIA dual
vacua; sometimes the computation involves numerical evaluation of period polynomials.
Besides the obvious directions going beyond a proof of concept, there are a few questions
of theoretical (mathematical) nature about period polynomials, which we note here.
One is about the ζ(3)χ/[2(2πi)3 ] term in the prepotential. The parameter χ in the
effective theory prepotential (2.9) is determined, through (2.11) on one hand (which is based
˜ ] in (2.47)), and also through the imaginary coefficients
on computation of G(1) through I[F
in the period polynomial (3.35), (3.48) and (4.26) on the other. In both reasonings, the
parameter χ is determined from the data {Φγ }. If a given {Φγ } is for a theoretically
consistent branch of the Heterotic-IIA dual moduli space, then the two procedures should
result in the same value of χ. At this moment, however, the authors do not have an idea how
to prove mathematically that the two procedures yield the same value of χ for any {Φγ }.
The other is about the overall transcendental factors of period polynomials. Even
for a Hecke-eigen cusp form f with integral Fourier coefficients that is without a pole
in the interior of the complex upper half plane, the period polynomial Pg2 (σ, f ) do not
30
(ρ)
(u)
A set of monodromy matrices of generators of (PSL2 Z×PSL2 Z)oZ2 (hσi is presented in ([3], (4.16)).
e in this article, but they are not equal no
The matrix c̃/2 for σ in ([3], (4.16)) is the closest to Λσ̃+1 + C/2
matter how we set the value of aρρ , auu , bρ and bu in (4.16). Either the choices of the symplectic frame
at the base point t0 are different (not just for ∆aab ∈ Z and ∆ba ∈ 24Z) between [3] and here, or the
monodromy matrix for σ in ([3], (4.16)) is not precisely for the path γ̃σ̃ but for some other path in the set
e S )\Xsingl ).
of (loops) ◦ γ̃σ̃ ◦ (loops), where “loops” refer to the loops in the covering space (Hs /ZD) × (D(Λ
We did not try to find out whether the matrices presented in ([3], (4.16)) are for the bR ≡ 0 (12) branches,
or for the bR ≡ 6 (12) branch.
– 30 –
JHEP03(2022)059
Now that we have confirmed that (ba )+24Z and (c2 )a ’s allow a common thing interpreted
as the 2nd Chern class, we can regard the conditions (4.6), (4.9), (4.10) as Wall’s necessary
and sufficient condition for existence of a diffeomorphism class [M ] of real 6-dimensional
manifolds, with the trilinear intersection of H 2 (M ; Z) and the 2nd Chern class given by dabc
and (c2 )a , respectively. In any one of the branches of Het-IIA moduli space with ΛS = U ,
therefore, there is a region described by a non-linear sigma model with the target space M
in the Type IIA language (cf. footnote 26). The properties (4.7), (4.8) and (ba ) ≡ (c2 )a
mod 24 guarantee that necessary conditions (2.16), (2.17) for D-brane central charges in a
geometric phase are satisfied.
Those branches of moduli space with ΛS = U come with the unique Φ and the invariant
bR ∈ 6Z≥0 . Two bR ’s that differ by 12Z (i.e., two n0 ’s that differ by 2Z) result in an identical
spectrum of 4d N = 2 BPS dyons. The two branches with distinct dyon spectra,30 one
with bR ≡ 0 (12) and the other with bR ≡ 6 (12) (i.e., even n0 and odd n0 ) have known
descriptions both in the Type IIA and Heterotic languages. In Type IIA, the corresponding
Calabi-Yau threefold is the elliptic fibration over the Hirzebruch surface F0 (or F2 ) and F1 ,
respectively. In Heterotic language, the internal space is K3 ×T 2 and the 24 instantons on
K3 are distributed by 12 + 12 (or 10 + 14) and 11 + 13, respectively (cf. footnote 27).
Acknowledgments
We thank I. Antoniadis for kindly explaining derivations in ref. [6] to one of the authors. We
also thank Y. Sato for discussions. The study in this article was supported in part by JSPS
Fellowship for Young Scientists (YE), Leading Graduate School FMSP program (YE), the
brain circulation program (TW), a Grant-in-Aid for Scientific Research on Innovative Areas
no. 6003 (TW), and by the WPI program (YE, TW), all from MEXT, Japan.
A
Brief notes on modular forms
In this appendix, we collect some conventions and facts from the literatures for convenience
of readers.
A.1
Eisenstein series etc.
Eisenstein series.
∞
X
n=1
∞
X
E2 = 1 − 24 q + 3q 2 + · · · = 1 − 24
2
q n σ1 (n) = 1 − 24
E4 = 1 + 240 q + 9q + · · · = 1 + 240
m=1
X
n=1
1 − qm
,
∞
X
m3 q m
n
q σ3 (n), = 1 + 240
n=1
E6 = 1 − 504 q + 33q 2 + · · · = 1 − 504
∞
X
mq m
m=1
q n σ5 (n) = 1 − 504
1 − qm
(A.1)
,
X m5 q m
m=1
1 − qm
(A.2)
,
(A.3)
where q = e2πiσ , with the argument σ taking value in the complex upper half plane H.
The Eisenstein series E4 (σ) and E6 (σ) are modular forms of weight 4 and 6, respectively,
– 31 –
JHEP03(2022)059
always have coefficients in Q; the polynomial Pg2 (σ, f ) may be split into even-power terms
and odd-power terms, and then the ratio of coefficients among the even-power terms and
also the ratio of those among the odd-power terms are known to be in Q, but there is a
common factor for the even-degree terms and another common factor for the odd-degree
terms, which are not even necessarily algebraic (they are given by special values of Lfunctions, more general than the special values of the zeta function) [34, 41, 42]. In the
applications in this article, f = f∗ is chosen from a more general class, in that f∗ has
poles in the upper half plane, and is not guaranteed to be a Hecke eigenform. In light of
these general expectations for period polynomials, the fact that the coefficients of the odd
degree terms in (3.48), (4.25), (4.26) are in Q — not just their ratios are — hints that there
are still things to be understood. For {Φγ } to be for a theoretically consistent branch of
the Heterotic-IIA dual moduli space, the overall transcendental factors of the odd part of
the period polynomial Pg̃ (t, f∗ ) cannot be transcendental, or even algebraic outside of Q.
Either there are more math to be understood, or only finite number of {Φγ }’s have the
overall transcendental factor in Q in the odd part and are for the theoretically consistent
dual moduli space, we do not speculate in this article.
for SL(2; Z), but E2 (τ ) is not modular (it is Mock modular).31 The space of scalar-valued
modular forms can be identified with the polynomial ring C[E4 , E6 ].
In the main text, we will also use the Dedekind Eta function η and the j-invariant
η := q 1/24
∞
Y
(1 − q n ),
j :=
n=1
E43
,
η 24
(A.4)
which are of weight +1/2 and 0, respectively. There is a relation E43 − E62 = 1728η 24 .
S
∂ F :=
For a modular form F of weight k, the Ramanujan-Serre
1 ∂
k
− E2 F = η 2k q∂q
2πi ∂σ 12
F
η 2k
.
(A.5)
∂ S F is a modular form of weight k + 2. It is useful to know that ∂ S E4 = (−1/3)E6 and
∂ S E6 = (−1/2)E42 .
A.2
Explicit formulae of the modular forms {Φγ }
We will use some of the Fourier coefficients of {Φγ }γ∈GS in the main text, so here is the
data. For more information, readers are referred to the literatures cited in [10].
The case ΛS = U . In this case, the group GS is trivial, and {Φγ } consists of single
component Φ, which is equal to n0 E4 E6 = −2E4 E6 , without any free parameters.
F =
Φ
−2
=
+ 480 + 282888q + 17058560q 2 + · · · ;
24
η
q
(A.6)
the Fourier coefficients c(−1) = −2 and c(0) = 480 are used in (4.15), (4.26) along
with (2.11); c(1) = 282888 and c(2) = 17, 058, 560 can also be used to verify that f∗
in (4.23) satisfies (4.20), (2.11).
The case ΛS = h+2i. In this case, GS ∼
= Z2 , so {Φγ } consists of two components, Φ0
and Φ1/2 . It is known that [37, 43, 44]
Φ0 = −2 + (300 − 56n1/2 )q + (217200 − 13680n1/2 )q 2 + · · ·
5
1
9
Φ1/2 = n1/2 q 4 + (2496 + 360n1/2 )q 4 + (665600 + 30969n1/2 )q 4 + · · · ;
(A.7)
we have chosen a parametrization so that the coefficients of the leading terms of Φ0 and
Φ1/2 are n0 = −2 and n1/2 , respectively. It is straightforward to compute Fγ = Φγ /η 24 .
Let us just note that
c0 (0) = 252 − 56n1/2 , c1/2 (1/4) = 2496 + 384n1/2 , c0 (1) = 223, 752 − 15, 042n1/2 ,
(A.8)
c1/2 (9/4) = 38, 637, 504 + 1, 129, 856n1/2 ,
c0 (4) = 9, 100, 224, 984 − 115, 446, 576n1/2 .
Those facts are used in (3.50), (2.11) and also in determining f∗ in (3.31), (3.46), (3.47).
31
We will also use Ê2 (σ) := E2 (σ) − 3/(πIm(σ)) in (2.47).
– 32 –
JHEP03(2022)059
Ramanujan-Serre derivative.
derivative ∂ S is defined by
B
Eichler cohomology and coarse/fine classification
Pf : G 3 g 7−→ Pg (t, f ) ∈ C[t]deg=w−2 .
(B.1)
On the abelian group C[t]deg=w−2 of the C-coefficient polynomials of degree (w − 2), the
group G acts as
C[t]deg=w−2 3 χ(t) 7−→ g · (χ(t))) =
(ϕg (t))w−2
w
−1
2
k1
χ(tg ) ∈ C[t]deg=w−2 .
(B.2)
The assignment Pf for f in (B.1) is regarded as a 1-cocycle
Pf ∈ Z 1 (G, C[t]deg=w−2 )
(B.3)
in the sense of group cohomology. The integration constant terms Q(t) to be added to the
Eichler integral (cf (3.15)) gives rise to the ambiguity to be added to Pg (t, f ) as in (3.18).
This additional ambiguity is interpreted as the coboundary from 0-cochains in group cohomology, so a cusp form f of weight w determines an element in the group cohomology
[Pf ] ∈ H 1 (G, C[t]deg=w−2 ).
(B.4)
In this article, we had to deal with modular forms f∗ that may have a pole in the
interior of the complex upper half plane. Now, the group G = [ΓS ] such as PSL2 Z (for
ΛS = h+2i) and a finite index subgroup of Γ0 (k)+ (for h+2ki) is replaced by
e )\X
π1 (D(Λ
S
singl )/[ΓS ], [t0 ] ,
(B.5)
and the period polynomials allow us to think of an assignment
Pf∗ : π1 3 γg̃ 7−→ Pg̃ (t, f∗ ) ∈ C[t]deg=w−2 ,
(B.6)
where we use an abbreviated notation π1 for the group in (B.5). The condition (3.19)
implies that this assignment is a 1-cocycle of the group π1 (be aware that we use the
standard loop composition law in the group π1 here, whereas the composition of the duality
transformation g̃a · g̃b corresponds to (2.32)).
It has been explained in this article that the modular form f∗ is determined uniquely
by the indices (elliptic genus) {Φγ }. This fact means that the classification of branches of
– 33 –
JHEP03(2022)059
Here we leave a brief note on how the Eichler cohomology is related to the coarse versus fine
classifications of branches of the Heterotic-IIA dual moduli space discussed in [10]. The
following discussion is written for the cases with rank(ΛS ) = 1; some kinds of generalization
for higher-rank cases may be possible, but we will not try to cover the higher rank cases
here.
The Eichler cohomology is often formulated in the literatures as follows. For a group
G such as PSL2 Z, Γ0 (k)+ and their finite index subgroups that acts on the complex upper
half plane H, and for a cusp form f of weight w under the group G, the period polynomials
can be used to think of an assignment
moduli space by (implicit ΛS and) {Φγ } — referred to as the coarse classification in [10]
— is equivalent to classification by the Eichler cohomology
Pf∗ + B 1 (π1 , C[t]deg=4 ) ∈ H 1 (π1 , C[t]deg=4 ).
(B.7)
e
Λ : π1 3 γg̃ 7−→ Λg̃ + ZCe ∈ Sym2 (Z⊕(2+ρ) )/ZC.
(B.8)
The group [ΓS ] acts on the abelian group Sym2 (Z⊕(2+ρ) )/ZCe as
e 7−→ g T · (Λ + ZC)
e · g.
(Λ + ZC)
(B.9)
The condition (3.21) implies that the assignment Λ above is a 1-cocycle of the group π1
(be aware of the composition law, once again).
There is no unique choice of a symplectic frame at the base point t0 . The ambiguity
in the choice of a frame discussed in footnote 6 is parametrized by just one matrix
Λ0 ∈ Sym2 (Z⊕(2+ρ) ),
(Λ0 )00 = (Λ0 )0] = (Λ0 )]] = 0.
(B.10)
This can be seen as a 0-cochain, and the change in the matrix Λg̃ under the change in the
choice of a symplectic frame at the base point is regarded as the coboundary of a 0-cochain
of the form above. So, one may say that the main text of this article classified branches of
the Heterotic-IIA moduli space by using
e
Λ + B 1 (π1 , Sym2 (Z⊕(2+ρ) )condt0 n(B.10) /ZC).
(B.11)
Now, we are ready to describe a role played by the invariants {Ψγ } for finer classification in [10], using the language of group cohomology. For branches of the moduli space
that share the same {Φγ }, and hence the same f∗ , there are still distinct dyon spectra
and their monodromy characterized by (B.11). Such branches distinguished by (B.11) are
one-to-one with the set
h
i
Pf∗ + B 1 (π1 , C[t]deg=4 ) ∩ Z 1 (π1 , Z[t]deg=4 ) /B 1 (π1 , Z[t]deg=4, condt0 n (B.10) ); (B.12)
e have been converted to polynomials through fT (t) · Λ ·
symmetric matrices Λ (mod +ZC)
f(t) as we have done in the main text. The set (B.12) is not necessarily of single element
(e.g. [35, 45]); the classification invariant {Ψγ } was introduced in [10] so that elements in
this set are distinguished. Presumably the authors of [3, 5] anticipated classification by
something like (B.12); to get it done, we had to deal with integers (rather than Q, R or C)
by paying enough attention to normalizations in this article; direct computation of period
polynomials made a systematic study of the set (B.12) possible.
– 34 –
JHEP03(2022)059
On the other hand, we also studied the monodromy representations that are formulated
by using the BPS dyon spectrum of branches of moduli space. The diagonal part of the
monodromy matrices is in [ΓS ], and is determined by the lattice ΛS ; the off-diagonal part
(the matrices Λg̃ ) carry more detailed information of the branches of the moduli space. We
have seen in this article that one can think of the assignment
Open Access. This article is distributed under the terms of the Creative Commons
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