aainstitutetext: School of Physics, Korea Institute for Advanced Study, Seoul
02455, Koreabbinstitutetext: Quantum Universe Center, Korea Institute for
Advanced Study, Seoul 02455, Korea

#  Twisted 6d (2, 0) SCFTs on a Circle

Zhihao Duan a Kimyeong Lee a June Nahmgoong b Xin Wang<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We study twisted circle compactification of 6d $(2,0)$ SCFTs to 5d
$\mathcal{N}=2$ supersymmetric gauge theories with non-simply-laced gauge
groups. We provide two complementary approaches towards the BPS partition
functions, reflecting the 5d and 6d point of view respectively. The first is
based on the blowup equations for the instanton partition function, from which
in particular we determine explicitly the one-instanton contribution for all
simple Lie groups. The second is based on the modular bootstrap program, and
we propose a novel modular ansatz for the twisted elliptic genera that
transform under the congruence subgroups $\Gamma_{0}(N)$ of
$\text{SL}(2,\mathbb{Z})$. We conjecture a vanishing bound for the refined
Gopakumar-Vafa invariants of the genus one fibered Calabi-Yau threefolds, upon
which one can determine the twisted elliptic genera recursively. We use our
results to obtain the 6d Cardy formulas and find universal behaviour for all
simple Lie groups. In addition, the Cardy formulas remain invariant under the
twist once the normalization of the compact circle is taken into account.

††preprint: KIAS-P21003

##  1 Introduction

Interacting conformal field theories in dimensions larger than four is one of
the most remarkable discoveries in string theory. Many of them do not have a
Lagrangian description, and to study their dynamics we have to make full use
of our knowledge on string theory accumulated in the past few decades. Six is
known to be the maximal dimension allowing for their presence Nahm:1977tg , so
one naturally starts from theories in six dimensions. In particular, 6d
$\mathcal{N}=(2,0)$ superconformal field theories (SCFTs) can arise from type
IIB string theory on the ALE spaces $\mathbb{C}^{2}/\Gamma_{ADE}$
Witten:1995zh ; Strominger:1995ac . Theories of $A$ and $D$ types can also be
engineered on parallel M5 branes without or with OM5 plane. These theories in
the tensor branch have 1/2 BPS self-dual strings and 1/2 BPS massless tensor
multiplets Haghighat:2013gba . Furthermore, these 6d $(2,0)$ SCFTs can be
compactified on a circle $S^{1}$ and the resulting low energy 5d theories are
5d $\mathcal{N}=2$ supersymmetric gauge theories with simply-laced $ADE$ type
gauge groups. The instantons of these 5d theories play the role of Kaluza-
Klein (KK) modes of the compactification Kim:2011mv . The massless tensor
multiplet becomes 1/2 BPS instantons and massless vector multiplet in the
Coulomb branch. The winding self-dual strings become 1/2 BPS W-bosons and The
unwound self-dual strings become magnetic monopole strings in the Coulomb
phase.

A further compactification of the 5d theory of $ADE$ type on a circle
$\tilde{S}^{1}$ with outer-automorphism twist leads to the 4d KK theories with
twisted affine algebra $G^{(n)}$ Tachikawa:2011ch . The exchange of ordering
of two compactifications is the 4d S-duality and provides the 5d
$\mathcal{N}=2$ supersymmetric gauge theory with non-simply-laced group. This
4d S-duality, or _Langlands duality_ , of the twisted affine Dynkin diagram is
the untwisted affine Dynkin diagram of non-simple-laced group. Thus 5d
$\mathcal{N}=2$ gauge theory can be obtained by the twisted compactification
of the 6d $(2,0)$ SCFTs of $A,D$ and $E_{6}$ types. This will be explained in
more detail in section 2. The low energy dynamics of these twisted theories
are the 5d $\mathcal{N}=2$ theories with the non-simply-laced group
$B_{r}=SO(2r+1),C_{r}=USp(2r)_{0},(C_{r})_{\pi}=USp(2r)_{\pi},G_{2}$ and
$F_{4}$ Tachikawa:2011ch . There are two kinds of 5d $USp(2r)$ gauge theories
with adjoint flavor, depending on the discrete theta parameter being $0$ or
$\pi$, which is sensitive to the homotopy group $\pi_{4}(G)$. The exact
correspondence between the 6d $(2,0)$ theories and the 5d $\mathcal{N}=2$
theories of non-simply-laced groups and the corresponding insertion of
orientifold planes are given in table 1.

In this work, we also want to explore the physics of these 5d theories
directly from twisting the 6d theory. A very partial list of this approach can
be found in Jefferson:2018irk ; Bhardwaj:2018yhy ; Bhardwaj:2018vuu ;
Bhardwaj:2019fzv ; Apruzzi:2019vpe ; Apruzzi:2019kgb ; Apruzzi:2019opn ;
Apruzzi:2019enx ; Bhardwaj:2020gyu  and references therein. For us, one of the
main interests is the elliptic genus of the BPS strings in the 6d $(2,0)$
theories with twisted circle compactification. The 6d partition function for
the counting of BPS states of BPS strings with KK momenta would be equivalent
to the 5d partition functions for dyonic instantons. From the 4d S-dual
perspective, magnetic monopoles strings in the twisted compactification of the
5d $ADE$ theory with KK momentum but without electric charge and instanton
number would correspond to these self-dual strings.

6d theory | twist | 5d theory | $\Gamma_{0}(n_{G})$ | orientifold
---|---|---|---|---
$A_{2r}$ | $\mathbb{Z}_{2}$ | $USp(2r)_{\pi}$ | $\Gamma_{0}(4)$ | $\widetilde{\text{O}4}{}^{+}$
$A_{2r-1}$ | $\mathbb{Z}_{2}$ | $SO(2r+1)$ | $\Gamma_{0}(2)$ | $\widetilde{\text{O}4}{}^{-}$
$D_{r+1}$ | $\mathbb{Z}_{2}$ | $USp(2r)_{0}$ | $\Gamma_{0}(2)$ | ${\text{O}4}^{+}$
$D_{4}$ | $\mathbb{Z}_{3}$ | $G_{2}$ | $\Gamma_{0}(3)$ |
$E_{6}$ | $\mathbb{Z}_{2}$ | $F_{4}$ | $\Gamma_{0}(2)$ |
Table 1: Twisted compactification of 6d $(2,0)$ theories

First of all, we analyze the BPS objects of these twisted theories from the 5d
point of view by studying the blowup equations of 5d instanton partition
function. The original blowup equations were derived by Nakajima and Yoshioka,
in order to prove that the Nekrasov’s instanton partition function gives a
deformation of the Seiberg-Witten prepotential for four dimensional
$\mathcal{N}=2$ supersymmetric gauge theory on $\mathbb{C}^{2}$. Later on,
they generalised them to five dimensional gauge theories and obtained
K-theoretic blowup equations Nakajima:2005fg , which count K-theoretic
Donaldson invariants as discussed in a paper with Göttsche Gottsche:2006bm .
Based on Sun:2016obh ; Grassi:2016nnt , in Huang:2017mis , a geometric
description was proposed, which makes it possible to define blowup equations
for non-Lagrangian theories like local $\mathbb{P}^{2}$. In Kim:2020hhh , the
geometric description was generalised to all KK theories of 6d SCFTs, and was
used to bootstrap the BPS invariants of the KK theories. The blowup equations
are functional equations, which turn out to be very powerful tool to bootstrap
the BPS invariants, instanton partition functions/elliptic genera for various
5d/6d SCFTs Keller:2012da ; Kim:2019uqw ; Huang:2017mis ; Gu:2018gmy ;
Gu:2019dan ; Gu:2019pqj ; Gu:2020fem ; Kim:2020hhh . In particular, the
elliptic blowup equations for 6d $(2,0)$ $ADE$ M-strings were proposed in
Gu:2019pqj . However, the number of the equations is not enough to solve the
elliptic genera recursively. Thanks to the 5d KK theory description of the 6d
theory, we develop novel methods to bootstrap the instanton partition
functions and the BPS invariants, which helps to recover the information of
the elliptic genera. Utilizing our method, we manage to compute the exact one-
instanton partition functions for all exceptional Lie groups with adjoint
matters, when turning off gauge fugacities. We also study the (elliptic)
blowup equations for the 6d $(2,0)$ KK theory with twisted circle
compactifications.

Secondly, as mentioned before, from the 6d point of view we should study the
elliptic genus of effective strings. The elliptic genus is known to transform
as a Jacobi modular form of weight zero, which imposes very strong constraints
on its structure. To make full use of it, we come up with a physically
motivated ansatz compatible with the modular property, and we are left with a
finite number of undetermined coefficients. If one can find sufficiently many
extra constraints either from gauge theory or geometry, one is able to fix all
the unknowns and hence the elliptic genus itself. This approach is often
dubbed modular bootstrap DelZotto:2016pvm ; Gu:2017ccq ; DelZotto:2017mee ;
Kim:2018gak ; DelZotto:2018tcj ; Duan:2018sqe ; Duan:2020cta ; Duan:2020imo ,
and was successfully applied to compute the elliptic genera of various $(1,0)$
and $(2,0)$ SCFTs in 6d. This method is also applicable to compact
elliptically fibered Calabi-Yau (CY) threefolds Huang:2015sta ; Huang:2020dbh
, leading to interesting results on Heterotic/Type II string duality
Cota:2019cjx ; Cota:2020zse , black hole physics Haghighat:2015ega  and
swampland conjectures Lee:2018urn ; Lee:2018spm ; Lee:2020gvu ; Cota:2020zse ,
to name a few.

In this paper, we find a lot of evidence that after twisted compactification
of $(2,0)$ SCFTs to 5d, there exists a twisted version of the elliptic genera
that are covariant under congruence subgroups $\Gamma_{0}(N)$ of
$\text{SL}(2,\mathbb{Z})$, which is shown in table 1. Geometrically, this
should correspond to the topological string partition function on genus one
fibered CY threefolds upon which M-theory compactifies Bhardwaj:2019fzv . We
develop novel modular ansatz for 6d twisted $(2,0)$ theories and determine the
elliptic genera at low base degrees using conjectural vanishing conditions for
the corresponding genus one fibered CY threefolds, extending the previous work
Duan:2020cta . We emphasize that although the twisted elliptic genera are not
related to the 6d elliptic genera in an apparent way, the choice of subgroups
indeed coincides nicely with one’s expectation from twisting the theory, which
is not at all obvious from the perspective of 5d instanton dynamics.

This paper is organized as follows. In section 2, we analyze in detail the
procedure of twisted circle compactification of $(2,0)$ SCFTs to 5d
$\mathcal{N}=2$ gauge theories. We also discuss the effect of twisting on the
self-dual strings. Then we move on to determining the partition functions. In
section 3, we tackle this problem from the perspective of instanton counting.
In particular, we show how to bootstrap the instanton partition function using
the blowup equations with adjoint matters. In section 4, our starting point
becomes twisted self-dual strings instead. We propose a novel ansatz for the
twisted elliptic genera, which, combined with conjectural vanishing conditions
of refined BPS invariants, can also bootstrap the twisted elliptic genera
recursively. In section 5, we consider the Cardy limit of the twisted elliptic
genera and find universal asymptotic behavior of the 6d free energy for all
simple Lie algebras. In section 6, we end this paper with concluding remarks
and possible future directions.

##  2 Twisted circle compactification of 6d $(2,0)$ SCFTs

In this section, we discuss some basic results about the twisted circle
compactification of $(2,0)$ theories. Section 2.1 is devoted to a field
theoretic derivation of the twisted compatification. It not only reviews
previous works in the literature, but also contains some new discussions
useful for later sections. In section 2.2, we analyze the effect of the
twisting on the $ADE$ M-strings. Finally in section 2.3, we re-examine the
results from the geometric engineering perspective.

###  2.1 Outer-automorphism twist

In this subsection, we discuss the compactification of 6d $(2,0)$ theories on
a circle with the outer-automorphism twist. From the classification 6d $(2,0)$
SCFTs only allow the simply-laced chain of M-strings, engineered from IIB
string theory on $ADE$ orbifold $\mathbb{C}^{2}/\Gamma_{ADE}$. After the
circle compactification, the 6d $(2,0)$ $ADE$ SCFTs become 5d $\mathcal{N}=2$
gauge with $ADE$ gauge group, whose instantons carry KK modes on the circle
Kim:2011mv . However, one can still consider a 5d $\mathcal{N}=2$ SYM with a
non-simply-laced gauge group. For those theories, their UV completions are
given by the 6d $(2,0)$ $ADE$ theories on a circle with outer-automorphism
twist Vafa:1997mh ; Tachikawa:2011ch .

Since the 6d theory is non-Lagrangian and also $G$ is not its gauge group, it
is hard to pin down the twisted circle compactification directly. To
circumvent this, we revisit and refine the argument given by Tachikawa
Tachikawa:2011ch . We compactify the 6d theory on a torus made of two distinct
circles $S^{1}$ and $\tilde{S}^{1}$. Now the order of compactification
matters, so we shall consider the following two compactification chains,

$\displaystyle\begin{array}[]{lcl}&\text{6d $(2,0)$
}G\text{-SCFT}&\\\\[2.84526pt] \hskip
42.67912pt{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\swarrow}\
\text{on}\ S^{1}\ \text{of}\
R_{6}&&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\searrow}\
\text{on}\ \tilde{S}^{1}\ \text{of}\ \tilde{R}_{6}\\\\[4.2679pt] \text{5d
}G-\text{MSYM}&&\ \text{5d }{G}^{\vee}-\text{MSYM}\\\\[2.84526pt] \hskip
28.45274pt{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\downarrow}\
\ \text{on}\ \tilde{S}^{1}\ \text{of}\ R_{5}&&\hskip
42.67912pt{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\downarrow}\
\ \text{on}\ S^{1}\ \text{of}\ \tilde{R}_{5}\\\\[4.2679pt] \text{4d\ KK}\
G^{(n)}-\text{MSYM}&\xleftrightarrow{\hskip 14.22636pt\text{S-dual}\hskip
14.22636pt}&\text{4d\ KK}\ {G}^{\vee(1)}-\text{MSYM}\end{array}$ (6)

where the blue arrow denotes the untwisted compactification on $S^{1}$, and
the red arrow denotes the twisted compactification on $\tilde{S}^{1}$. The
change of the compactification, which is the origin of the 4d S-duality,
provides the clues to the twisted compactification of the 6d (2,0) theories.
The direct compactification along $S^{1}$ without twist, there is no change of
the radius, $R_{6}=\tilde{R}_{5}$. We will see that in the convention where
the long roots of the Lie algebra of the gauge groups take the square length
two, the circle radius of the twisted circle $\tilde{S}^{1}$ changes such that
$n_{G}R_{5}=\tilde{R}_{6}$ with $n_{G}=1,2,3,4$, depending on the starting
theory and also its twisting $G^{(n)}$.

Let us first consider the left chain. To begin with, we compactify these 6d
theories on a circle of radius $R_{6}$ with periodic boundary condition
$x_{6}\sim x_{6}+2\pi R_{6}$, to get 5d $\mathcal{N}=2$ gauge theories of
$ADE$ type. As instantons represent the KK modes of mass $1/R_{6}$, the 5d
gauge coupling constant $g_{5}$ is fixed to satisfy
$8\pi^{2}/g_{5}^{2}=1/R_{6}$. We further compactify this 5d theory on a circle
of radius $R_{5}$ so that $x_{5}\sim x_{5}+2\pi R_{5}$ with twisted boundary
condition

$\displaystyle\phi(x_{5}+2\pi R_{5})=\sigma(\phi(x_{5}))\,,$ (7)

where $\sigma$ is an automorphism of order $n_{G}$. The twisted 5d theory has
the effective 4d coupling constant
$4\pi/g_{4}^{2}=8\pi^{2}R_{5}/g_{5}^{2}=R_{5}/R_{6}$ at the low energy. In the
presence of twisting, the generators of the corresponding Lie algebra get
split to the eigenstates under the corresponding automorphism $\sigma$. Each
mode can have KK momentum of $(\mathbb{Z}+s/n_{G})/R_{5}$ with $n_{G}$ being
the order of the corresponding automorphism and $s=0,1,\cdots n_{G}-1$. The
adjoint field of the original 5d theory gets split into various fractional KK
modes. The case for $A_{2r}$ needs an additional consideration with
automorphism including the $\mathbb{Z}_{2}$ outer-automorphism kac1990infinite
; Tachikawa:2011ch . With a matrix $R=(1,i\sigma_{2}\otimes{\bf 1}_{r})$, the
automorphism for $A_{2r}=SU(2r+1)$ is the map $\sigma(\phi)=-R\phi^{T}R^{-1}$.
As $R^{2}=\text{diag}(1,-{\bf 1}_{2r})$, this twisting is actually of order
four. See the table 1 for the list of $n_{G}$.

After carefully analyzing the decomposition of the adjoint representation into
the representations of its invariant subalgebra as well as the fractional
momentum dependence $e^{(k+s/n_{G})x_{5}/R_{5}}$, we obtain the well-known
result below:

$\displaystyle A^{(2)}_{2r}$ $\displaystyle:\ {\bf adj}\ {\rm of}\ A_{2r}$
$\displaystyle\longrightarrow$ $\displaystyle{\bf adj}_{0}\oplus{\bf
fund}_{\frac{1}{4}}\oplus{\boldsymbol{\Lambda}}^{2}_{\frac{1}{2}}\oplus{\bf
1}_{\frac{1}{2}}\oplus{\bf fund}_{\frac{3}{4}}\ \ {\rm of}\ C_{r}\,,$
$\displaystyle A^{(2)}_{2r-1}$ $\displaystyle:\ {\bf adj}\ \ {\rm of}\
A_{2r-1}$ $\displaystyle\longrightarrow$ $\displaystyle{\bf
adj}_{0}\oplus{\boldsymbol{\Lambda}}^{2}_{\frac{1}{2}}\ \ {\rm of}\ C_{r}\,,$
$\displaystyle D^{(2)}_{r+1}$ $\displaystyle:\ {\bf adj}\ {\rm of}\ D_{r+1}$
$\displaystyle\longrightarrow$ $\displaystyle{\bf adj}_{0}\oplus{\bf
fund}_{\frac{1}{2}}\ \ {\rm of}\ B_{r}\,,$ $\displaystyle E^{(2)}_{6}$
$\displaystyle:\ {\bf adj}\ {\rm of}\ E_{6}$ $\displaystyle\longrightarrow$
$\displaystyle{\bf adj}_{0}\oplus{\bf fund}_{\frac{1}{2}}\ {\rm of}\ F_{4}\,,$
$\displaystyle D^{(3)}_{4}$ $\displaystyle:\ {\bf adj}\ {\rm of}\ D_{4}$
$\displaystyle\longrightarrow$ $\displaystyle{\bf adj}_{0}\oplus{\bf
fund}_{\frac{1}{3}}\oplus{\bf fund}_{\frac{2}{3}}\ \ \text{of}\ G_{2}\,,$ (8)

where ${\bf adj}_{s}$, ${\bf fund}_{s}$, and ${\boldsymbol{\Lambda}}^{2}_{s}$
denote the adjoint, fundamental and rank two anti-symmetric representations.
It turns out that the extended weights on the right hand side of figure 8,
including also the KK momentum, give rise to the twisted affine algebra
$G^{(n)}$ kac1990infinite . The identification of the simple roots of the
original Lie algebra $G$ under the twist is shown on the leftmost of table 2.
Under this identification, the invariant and non-invariant roots under the
twist become long and short roots of $G^{(n)}$, respectively.

The second column in table 2 gives the Dynkin diagrams for the twisted affine
Lie algebras. They encode the simple roots of the 4d theory plus the affine
root with minimally positive momentum. Ignoring the momentum dependence, the
root vectors for the twisted affine algebra are given below:

$\displaystyle\begin{array}[]{|c|c|l|l|l|}\hline\cr
G^{(n)}&\text{4d}&\qquad\text{Long}&\qquad\text{Short}&\ \text{Special}\\\
\hline\cr A^{(2)}_{2r}&C^{\prime}_{r}&\quad\ \pm\sqrt{2}e_{a}&\
\frac{1}{\sqrt{2}}(\pm e_{a}\pm e_{b})&\pm\frac{1}{\sqrt{2}}e_{a}\\\
A^{(2)}_{2r-1}&C_{r}&\quad\ \pm\sqrt{2}e_{a}&\ \frac{1}{\sqrt{2}}(\pm e_{a}\pm
e_{b})&\quad$---$\\\ D^{(2)}_{r+1}&B_{r}&\quad\pm e_{a}\pm e_{b}&\qquad\pm
e_{a}&\quad$---$\\\ E^{(2)}_{6}&F_{4}&\quad\pm e_{a}\pm e_{b}&\frac{1}{2}(\pm
e_{1}\pm...e_{4}),\ \pm e_{a}&\quad$---$\\\
D^{(3)}_{4}&G_{2}&\scriptstyle\sqrt{2}(\pm 1,0),\frac{1}{\sqrt{2}}(\pm
1,\pm\sqrt{3})&\scriptstyle\sqrt{\frac{2}{3}}(0,\pm 1),\
\frac{1}{\sqrt{2}}(\pm 1,\pm\frac{1}{\sqrt{3}})&\quad$---$\\\
\hline\cr\end{array}$ (15)

In the above table, $\\{e_{1},e_{2},\cdots e_{r}\\}$ are $r$-dim orthonormal
vectors and $a\neq b$ . While there is no difference between $C_{r}^{\prime}$
and $C_{r}$ at the Lie algebra level, there is a difference in the 4d physics
due to the range of the continuous 4d $\theta$ parameter. We will discuss the
detail later on in this subsection. The upshot is that we can represent the
equation (8) more succinctly, highlighting the length and the fractional
momentum $e^{ip_{5}x_{5}/R_{5}}$:

$\displaystyle A^{(2)}_{2r}$ $\displaystyle:\ {\bf adj}\ {\rm of}\ A_{2r}$
$\displaystyle\longrightarrow$
$\displaystyle\text{long}_{k}\oplus\text{short}_{\frac{k}{2}}\oplus\text{special}_{k\pm\frac{1}{4}}\oplus
1_{k+\frac{1}{2}}\ \ {\rm of}\ C^{\prime}_{r}$ $\displaystyle A^{(2)}_{2r-1}$
$\displaystyle:\ {\bf adj}\ \ {\rm of}\ A_{2r-1}$
$\displaystyle\longrightarrow$
$\displaystyle\text{long}_{k}\oplus\text{short}_{\frac{k}{2}}\ \ {\rm of}\ \
C_{r}$ $\displaystyle D^{(2)}_{r+1}$ $\displaystyle:\ {\bf adj}\ {\rm of}\
D_{r+1}$ $\displaystyle\longrightarrow$
$\displaystyle\text{long}_{k}\oplus\text{short}_{\frac{k}{2}}\ \ {\rm of}\
B_{r}$ $\displaystyle E^{(2)}_{6}$ $\displaystyle:\ {\bf adj}\ {\rm of}\
E_{6}$ $\displaystyle\longrightarrow$
$\displaystyle\text{long}_{k}\oplus\text{short}_{\frac{k}{2}}\ \ {\rm of}\
F_{4}$ $\displaystyle D^{(3)}_{4}$ $\displaystyle:\ {\bf adj}\ {\rm of}\
D_{4}$ $\displaystyle\longrightarrow$
$\displaystyle\text{long}_{k}\oplus\text{short}_{\frac{k}{3}}\ \ \text{of}\
G_{2}\ ,$ (16)

where the subscript indicates $p_{5}$ with $k\in\mathbb{Z}$. For
$A^{(2)}_{2r}$, there are additional special roots with half size of long
roots, which carries non-vanishing KK momenta $k\pm\frac{1}{4},\
k\in\mathbb{Z}$. There are only $r$ independent Cartan elements at $p_{5}=0$
in the twisted theory $G^{(n)}$. The higher momentum modes for the Cartan
elements $\alpha_{\text{long}}\cdot H$ and $\alpha_{\text{short}}\cdot H$ for
long and short simple roots are given exactly by $E_{\alpha_{\text{long}}}$
and $E_{\alpha_{\text{short}}}$ as shown in the equation (16).

The $A^{(2)}_{2r}$ theory is the intriguing case. Even for this case, the
$SU(2)$ generator elements for the long simple root carry integer momentum
$k\in\mathbb{Z}$ while the short simple roots can carry integer or half-
integer momenta, namely, $p_{5}=\frac{k}{2},k\in\mathbb{Z}$. In addition there
is an identity element with only half-integer momentum,
$p_{5}=k+\frac{1}{2},k\in\mathbb{Z}$. Finally, the elements for special roots
can carry only nontrivial momenta $k\pm\frac{1}{4},\ k\in\mathbb{Z}$. The
identity element of half integer momentum is generated, for example, by the
commutation relations of special generators with momentum $k+\frac{1}{4}$.

Up to now we have looked at the $x_{5}$ dependence of all the modes in the 5d
$\mathcal{N}=2$ gauge multiplet with twisting. If we have not turned on any
scalar expectation value $\langle\phi\rangle$ or gauge holonomy $\langle
A_{5}\rangle$, then one can quantize each mode in the vanishing coupling limit
with gauge fixing, and get the massless and massive modes with KK momentum
$p_{5}/R_{5}$. The value of the KK momentum for adjoint vector multiplet for
each root is identical to that given in the equation (16), and so the energy
of the mode would be
$E\cong\sqrt{p_{1}^{2}+p_{2}^{2}+p_{3}^{2}+p_{5}^{2}/R_{5}^{2}}$ for the given
3d spatial momentum $(p_{1},p_{2},p_{3})$.

Let us now consider the BPS massive spectrum in the Coulomb phase. From the 6d
perspective we have turned on the VEV of single scalar field
$\langle\Phi\rangle$ in the 6d (2,0) theory, which has the mass dimension two
and decides the tension of self-dual strings. Reduced on $S^{1}$, the 5d
scalar vev $\langle\phi\rangle=R_{6}\langle\Phi\rangle$ has the mass dimension
one and decides the W-boson mass, which is a self-dual string wrapping along
the $x_{6}$ circle. Meanwhile the 5d gauge coupling constant is related to the
instantons which are KK momentum states via $8\pi^{2}/g_{5}^{2}=1/R_{6}$. A
further reduction on $\tilde{S}^{1}$ gives 4d $\mathcal{N}=4$ KK theory with
4d gauge coupling constant which is related to the 5d one via
$4\pi/g_{4}^{2}=8\pi^{2}R_{5}/g_{5}^{2}=R_{5}/R_{6}$. The scalar vev and the
$A_{5}$ holonomy get combined into a complex Coulomb moduli
$\mathring{\phi}=\langle\phi+iA_{5}\rangle$. In this 4d KK theory, there are
1/2 BPS W-bosons, magnetic monopoles, Kaluza-Klein modes for
$x_{5}$-compactification and instantons for $x_{6}$-compactification. Both
instantons and KK modes get fractionalized in the general Coulomb phase. Let
us just consider the W-bosons and magnetic monopoles for the SU(2) of a root
in $\textbf{adj}_{0}$ of the untwisted or twisted $G^{(n)}$ 4d KK theory.
Their masses are respectively,

$\displaystyle
m_{\text{W}}=|\alpha\cdot\langle\phi\rangle|=R_{6}|\alpha\cdot\langle\Phi\rangle|,\
\ m_{\text{M
}}=\frac{4\pi}{g_{4}^{2}}|\alpha^{\vee}\cdot\langle\phi\rangle|=\frac{2}{\alpha^{2}}R_{5}|\alpha\cdot\langle\Phi\rangle|\,.$
(17)

The 4d magnetic monopoles arise from the 5d magnetic monopole strings wrapping
the twisted circle $\tilde{S}^{1}$. Since the 5d monopole strings are the 6d
self-dual strings in the 5d theory, the physics of magnetic monopoles with KK
momentum along $\tilde{S}^{1}$ captures the twisted 5d theory on a circle. The
magnetic monopoles for roots provide the massive dual W-bosons for the dual
gauge theory in the Coulomb branch. The 4d dual magnetic group has the
magnetic dual gauge group $G^{\vee}$ with the dual roots
$\alpha^{\vee}=2\alpha/\alpha^{2}$ made of the roots $\alpha$ of the zero
momentum sector of the twisted affine algebra $G^{(n)}$.

Now, let us consider the second compactification chain in the equation (6).
First, we compactify the original 6d theory on the twisted circle
$\tilde{S}^{1}$. The resulting 5d theory is also $\mathcal{N}=2$ gauge theory
with some gauge group ${G}^{\vee}$, which is determined by self-dual strings
with twisted compactification. Then, we further compactify the 5d theory on
the untwisted circle $S^{1}$ to obtain the 4d $\mathcal{N}=4$ KK theory with
gauge group ${G}^{\vee(1)}$. Now let us look at the bottom line of the
equation (6). Exchanging the order of compactification amounts to the
S-duality transformation of the 4d $\mathcal{N}=4$ KK theory with gauge group
$G^{(n)}$, which exchanges the W-bosons and magnetic monopoles and replaces
$G^{(n)}$ with the _Langlands dual_ gauge group ${G}^{\vee(1)}$, constructed
from the affine coroots. Surely, for simply-laced $ADE$ cases (without actual
twisting), their Langlands dual ${G}^{\vee(1)}$ are identical to the original
groups $G^{(n)}$. Then the S-duality in 4d is a self-duality
$4\pi/g_{4}^{2}=R_{5}/R_{6}\cong\tilde{g}_{4}^{2}/4\pi=\tilde{R}_{6}/\tilde{R}_{5}$
and we set $R_{5}=\tilde{R}_{6},R_{6}=\tilde{R}_{5}$.

For cases with twisting as in the equation (6), the situation becomes more
interesting. Note that we use the convention where the longest roots have the
square length two, so we need to scale down the length of all affine coroots
by $\sqrt{n_{G}}$, i.e., $\alpha_{{G}^{\vee(1)}}=\alpha^{\vee}/\sqrt{n_{G}}$.
From (17) for the magnetic monopole mass
$m_{\text{M}}=n_{G}R_{5}|\langle\alpha_{G^{\vee(1)}}\cdot\Phi/\sqrt{n_{G}}|$,
we find that the S-duality in these cases exchanges the coupling constant in
the following way,

$\displaystyle\frac{1}{n_{G}}\frac{g_{4}^{2}}{4\pi}=\frac{R_{6}}{n_{G}R_{5}}\
\cong\ \frac{4\pi}{\tilde{g}_{4}^{2}}=\frac{\tilde{R}_{5}}{\tilde{R}_{6}}\,.\
$ (18)

which arises from the identification of the compactification radius as
follows:

$\displaystyle\tilde{R}_{5}=R_{6},\ \ \tilde{R}_{6}=n_{G}R_{5}.$ (19)

Here there is a scaling of the tensor scalar VEV for W-bosons and magnetic
monopoles by $\sqrt{n_{G}}$ for the invariance of the mass. For the earlier
works on the S-duality for 4d theories with non-simple-laced group, see, for
example, Dorey:1996hx ; Argyres:2006qr ; Kim:2004xx .

The Dynkin diagram of Langlands dual algebra $G^{(1)}_{r}$ is shown in the
third column of the table 2. Except for the $A^{(2)}_{2r}$ case, the dual
roots are simply the Dynkin diagram of the dual roots
$\alpha^{\vee}/\sqrt{n}_{G}$ of simple roots $\alpha$ of $G^{(n))}$ algebra in
the second column of the table 2.

For $A^{(2)}_{2r}$, its dual group turns out to be $(C^{(1)}_{r})_{\pi}$ which
arise from the 5d $(C_{r})_{\pi}=USp(2r)_{\pi}$ theory with discrete theta
parameter $\pi$. We will provide several evidences for this later in this
subsection. The list of explicit expressions for roots of the Lie algebra
$G^{\vee(1)}_{r}$ with the zero $\tilde{S}^{1}$ momentum are as follows:

$\displaystyle\begin{array}[]{|c|c|c|c|}\hline\cr
G^{(n)}&G^{\vee}_{r}&\text{Long}&\text{Short}\\\ \hline\cr
A^{(2)}_{2r}&(C_{r})_{\pi}&\pm\sqrt{2}e_{r}&\ \frac{1}{\sqrt{2}}(\pm e_{a}\pm
e_{b})\\\ A^{(2)}_{2r-1}&B_{r}&\ \pm e_{a}\pm e_{b}&\pm e_{a}\\\
D^{(2)}_{2r-1}&(C_{r})_{0}&\pm\sqrt{2}e_{a}&\frac{1}{\sqrt{2}}(\pm e_{a}\pm
e_{b})\\\ E^{(2)}_{6}&F_{4}&\pm e_{a}\pm e_{b}&\frac{1}{2}(\pm
e_{1}\pm...e_{4}),\ \pm e_{a}\\\ D^{(3)}_{4}&G_{2}&\scriptstyle\sqrt{2}(\pm
1,0),\frac{1}{\sqrt{2}}(\pm
1,\pm\sqrt{3})&\scriptstyle\sqrt{\frac{2}{3}}(0,\pm 1),\
\frac{1}{\sqrt{2}}(\pm 1,\pm\frac{1}{\sqrt{3}})\\\ \hline\cr\end{array}$ (26)

For all cases, the Langlands dual gauge group ${G}^{\vee(1)}$ in 4d actually
becomes untwisted, meaning that all affine roots have the same KK mode. This
can be seen from (16). Moreover, since descending from 5d to 4d involves no
twisting, it is straightforward to find out what is ${G}^{\vee}$. The final
result is listed in table 2. Note that ${G}^{\vee}$ is not a subalgebra of $G$
if ${G}^{\vee}\in\\{B_{r},\ C_{r}\\}$, which is only possible if 6d (2,0)
theory with$G$ is not a gauge theory.

$G\ /\ \text{Out}(G)$ |  $G^{(n)}$ (4d $G^{\prime}$ ) | ${G}^{\vee(1)}$ | 5d ${G}^{\vee}$
---|---|---|---
$A_{2r}\ /\ \mathbb{Z}_{2}$ |  $A_{2r}^{(2)}$ (4d $C_{r}^{\prime}$ ) | $(C_{r}^{(1)})_{\pi}$ | $(C_{r})_{\pi}$
[edge length=.75cm, involution/.style=blue,stealth-stealth, labels*=α_1,α_r-1,α_r,α_r+1,α_r+2,α_2r , involutions=16;25;34]A*.****.* |  A[2]even |  C[1] |  [edge length=.75cm, labels*=α_1,α_2,α_r-1,α_r]C**.**
|  $\widetilde{\text{O3}}{}^{-}$ $\widetilde{\text{O3}}{}^{+}$ |  $\text{O3}^{+}$ $\widetilde{\text{O3}}{}^{+}$ | $\widetilde{\text{O4}}{}^{+}$
$A_{2r-1}\ /\ \mathbb{Z}_{2}$ |  $A_{2r-1}^{(2)}$ (4d $C_{r}$ ) | $B_{r}^{(1)}$ | $B_{r}$
[edge length=.75cm, involution/.style=blue,stealth-stealth, labels*=α_1,α_r-1,α_r,α_r+1,α_2r-1 , involutions=15;24]A*.***.* |  A[2]odd |  B[1] |  [edge length=.75cm, labels*=α_1,α_2,α_r-1,α_r]B**.**
|  $\text{O3}^{-}$ $\text{O3}^{+}$ |  $\text{O3}^{-}$ $\widetilde{\text{O3}}{}^{-}$ | $\widetilde{\text{O4}}{}^{-}$
$D_{r+1}\ /\ \mathbb{Z}_{2}$ |  $D_{r+1}^{(2)}$ (4d $B_{r}$ ) | $(C_{r}^{(1)})_{0}$ | $(C_{r})_{0}$
[edge length=.75cm, involution/.style=blue,stealth-stealth, labels=α_1,α_2,α_r-1,α_r,α_r+1 , involutions=54]D**.*** |  D[2] |  C[1] |  [edge length=.75cm, labels*=α_1,α_2,α_r-1,α_r]C**.**
|  $\widetilde{\text{O3}}{}^{-}$ $\widetilde{\text{O3}}{}^{-}$ |  $\text{O3}^{+}$ $\text{O3}^{+}$ | $\text{O4}^{+}$
$D_{4}\ /\ \mathbb{Z}_{3}$ |  $D_{4}^{(3)}$ (4d $G_{2}$ ) | $G_{2}^{(1)}$ | $G_{2}$
[edge length = .75cm, involution/.style=blue,stealth-stealth, labels=α_1,α_2,α_3,α_4, involutions = 31;43;14]D4 |  D[3]4 |  G[1]2 |  [labels*=α_1,α_2, edge length=.75cm]G2
$E_{6}\ /\ \mathbb{Z}_{2}$ |  $E_{6}^{(2)}$ (4d $F_{4}$ ) | $F_{4}^{(1)}$ | $F_{4}$
[edge length = .75cm, involution/.style=blue,stealth-stealth, labels*=α_1,α_6,α_2,α_3,α_4,α_5, involutions = 16;35]E6 |  E[2]6 |  F[1]4 |  [edge length=.75cm, labels*=α_1,α_2,α_3,α_4]F4
Table 2: Dynkin diagrams for (affine) Lie algebras along the compactification
chain (6). Brane interpretation is given when possible. Note that $G^{(n)}$
and ${G}^{\vee(1)}$ are related by the S-duality between 4d theories. The
$A^{(2)}_{2r}$ case is subtler and its S-duality is explained in the text.

It is also productive to understand the above results from the brane systems.
Recall that the 5d $\mathcal{N}=2$ theories with
$B_{r},(C_{r})_{0},(C_{r})_{\pi}$ and $D_{r}$ gauge groups arise from multiple
D4-branes on $\widetilde{\text{O4}}{}^{-},$ $\text{O4}^{+},$
$\widetilde{\text{O4}}{}^{+}$, and $\text{O4}^{-}$ planes, respectively
Hori:1998iv ; Tachikawa:2011ch . Note that after a circle compactification and
T-duality, we get the interpretation in terms of O3-planes Gimon:1998be ;
Hanany:2000fq . The S-dual of these O3-planes for the 5d theory on a circle
without twist is exactly the O3-planes for the twisted $G^{(n)}$ shown in
table 2. One important point is that in 4d, there are two kinds of
$C_{r}=USp(2r)$ theories depending on the range of the 4d continuous
$\theta$-parameter. The standard 4d $C_{r}=USp(2r)$ theory on D3-branes with
$\text{O3}^{+}$is S-dual to 4d $B_{r}=SO(2r+1)$ theory on D3-branes with
$\widetilde{\text{O3}}{}^{-}$.

Another 4d $C_{r}^{\prime}=USp(2r)$ theory on D3-branes with
$\widetilde{\text{O3}}{}^{+}$ is self-dual. This $C_{r}^{\prime}$ theory has
the continuous theta $\theta=2\pi$, such that the magnetic monopole for the
long simple root has nontrivial electric charge due to additional Witten
effect Witten:1979ey . On the other hand, the dyonic bound state of two such
monopoles, a.k.a. Sen’s state Sen:1994yi  with unit electric charge, has no
net electric charge when one includes also the Witten effect. Thus there
exists a 1/2 BPS massive vector multiplet of zero electric charge and twice
magnetic charge for the long simple root, making it self-dual under the
S-transformations. 1/2 BPS single magnetic monopole states for short simple
roots have zero electric charge sector when one considers both the dyonic BPS
states and the Witten effect together. Thus this theory is S-duality
invariant. Note that $\text{O3}^{+}$ and $\widetilde{\text{O3}}{}^{+}$ are
related by the T-transformation of the $\text{SL}(2,\mathbb{Z})$. Some studies
of the twisted theories and magnetic monopole spectrum were done before
Kim:2004xx ; Tachikawa:2011ch .

The S-duality of $A^{(2)}_{2r}$ is a bit complicated as its affine Dynkin
diagram is self-dual under the
$\alpha\leftrightarrow\alpha^{\vee}=2\alpha/\alpha^{2}$. Its S-dual theory is
the 4d KK theory with affine $(C_{r}^{(1)})_{\pi}$. From the brane
perspective, the S-dual of $A^{(2)}_{2r}$ theory arising on D3 branes between
$\widetilde{\rm O3}{}^{-}$ and $\widetilde{\rm O3}{}^{+}$ is the
$(C^{(1)}_{r})_{\pi}$ theory arising from on D3 branes between
$\text{O3}{}^{+}$ and $\widetilde{\rm O3}^{+}$. From it we learn that the
$A_{2r}^{(2)}$ 4d KK theory cannot be self-dual. The Montonen-Olive duality is
a quantum symmetry and so is sensitive to full quantum spectrum.

There is an additional subtlety in the S-duality. In the left-side line of the
compactification (6) to 4d, we have 1/2 BPS W-bosons, instantons, magnetic
monopoles and $x_{5}$ KK momentum. We are interested in the magnetic monopoles
with $x_{5}$ KK momentum, which would correspond to the 5d W-bosons and
instantons of the dual theory with non-simple-laced group, obtained by the
right-side line of the compactification (6). These 5d W-bosons and instantons
are the wrapped self-dual strings with KK momentum of the twisted
compactification of 6d (2,0) theories, which are of our primary interest.

In 4d $\mathcal{N}=4$ theories, in the Coulomb branch with only a single
scalar field vev turned on $\langle\phi\rangle$, the simple roots of the Lie
algebra appear naturally. While W-bosons of mass
$\alpha\cdot\langle\phi\rangle$ exist as 1/2 BPS elementary particle for all
positive roots, classically only for each simple root there exists an
_elementary_ 1/2 BPS magnetic monopole with four zero modes, three for the 3d
position and one for the internal phase. Once we quantize the moduli space of
these elementary monopoles, the threshold bound states without electric charge
appear for each positive roots, forming the W-boson of the dual gauge
theoryLee:1996if ; Gauntlett:1996cw . Along the left-side of the
compactification to 4d, we want to consider these elementary BPS magnetic
monopole for each simple roots with zero $x^{5}$ momentum and mass in (17). We
want to add the $x^{5}$ momentum by simply combining the KK momentum carried
by the Cartan for each simple roots as given in (16). Namely, one is imagining
chargeless KK momentum combined with magnetic monopoles for each simple root
$\alpha$ of $G^{\vee}$. This would generate a simple prescription for
elementary monopoles with arbitrary $x_{5}$ KK momentum and without any
electric and instanton charge. The full dynamics of these objects is at least
as complicated as self-dual strings with KK momentum on the twisted circle.

In the right-side of the compactification chain, they correspond to 5d
W-bosons with instantons for the twisted compactification, with the radius
$\tilde{R}_{6}=n_{G}R_{5}$. This makes the $x^{6}$ momentum $n_{G}p_{5}$
integer-valued. On the other hand, a single $SU(2)$ instanton embedded in each
simple root $\alpha$ carries the instanton number $2/\alpha^{2}$. Both the 4d
monopole and 5d instanton description give the same result for the allowed
instanton number of a single W-boson or the KK momentum of an elementary
string, corresponding to each simple root in 5d $G^{\vee}$ as follows:

$\displaystyle\begin{array}[]{|c|c|c|c|c|c|}\hline\cr\text{Twist}&\text{5d}\
G^{\vee}&\text{Long}&\text{Short}&\text{Special}\\\\[2.84526pt] \hline\cr
A^{(2)}_{2r}&(C_{r})_{\pi}&[\alpha_{r}]_{2k,4k}&[\alpha_{a}]_{2k}&[\frac{1}{2}\alpha_{r}]_{4k\pm
1}\\\\[2.84526pt]
A^{(2)}_{2r-1}&B_{r}&[\alpha_{a}]_{k}&[\alpha_{r}]_{2k}&$---$\\\\[2.84526pt]
D^{(2)}_{r+1}&(C_{r})_{0}&[\alpha_{r}]_{k}&[\alpha_{a}]_{2k}&$---$\\\\[2.84526pt]
E^{(2)}_{6}&F_{4}&[\alpha_{1}]_{k},[\alpha_{2}]_{k}&[\alpha_{3}]_{2k},[\alpha_{4}]_{2k}&$---$\\\\[2.84526pt]
D^{(3)}_{4}&G_{2}&[\alpha_{1}]_{k}&[\alpha_{2}]_{3k}&$---$\\\\[2.84526pt]
\hline\cr\end{array}$ (33)

where $a=1,2,\cdots r-1$. Before we move on to $A^{(2)}_{2r}$ case, one would
ask if any role is played by monopoles related to the affine simple root of
$G^{(n)}$. One could imagine an $SU(2)$ generator $(\alpha\cdot
H,E_{\alpha}e^{ip_{5}/R_{5}},E_{-\alpha}e^{-ip_{5}/R_{5}})$ and building up a
magnetic monopole solution. It is static and has $x_{5}$ dependence, so it
naturally has nontrivial $F_{i5}$ component with an instanton number along the
left-side compactification in (6). Thus they are not of our current interest.

Let us finally consider the 5d $(C_{r})_{\pi}$ case arising from twisting
$A^{(2)}_{2r}$. After the twisted compactification from 5d to 4d, the equation
(16) shows that the possible KK momentum states are $k,k/2,k\pm 1/4,k+1/2$
with $k\in\mathbb{Z}$, and the possible magnetic charges are long, short and
special. When one combines them together and takes the S-dual picture with
momentum factor $n_{G}=4$, The possible BPS states one gets in 5d turn out to
be given in the equation (33) . Without KK momentum, there are states for each
root of $C_{r}$. Here we take the view from the 6d and so they appear as a
composite of elementary strings for simple roots of $C_{r}$. These elementary
strings for simple roots can carry KK momentum $2k,k\in\mathbb{Z}$. In
addition, there exist strings of half electric charge $\frac{1}{2}\alpha_{r}$
of the long simple root. This string has to carry nontrivial KK momentum
$4k\pm 1,k\in\mathbb{Z}$ so that it does not appear in the gauge multiplet
spectrum of 5d theory. This is consistent with the known instanton dynamics.
Single instanton for $(C_{r})_{\pi}$ case carrying nontrivial electric charge
belongs to fundamental representation of $USp(2r)=C_{r}$. For the Cartan
elements, the $x_{6}$ KK momentum is just 4 times of what is given in (16):
the $\alpha_{r}\cdot H$ mode carries the KK momentum $4k$, the
$\alpha_{a}\cdot H,a=1,...r-1$ modes carry the KK momentum $2k$, and finally
the identity element carries the KK momentum $4k+2$ with integer $k$.

###  2.2 Elliptic genera of twisted $ADE$ M-strings

Now, let us consider the partition function of the twisted 6d $(2,0)$
theories. More precisely, we will study the partition function of 6d $(2,0)$
theory on $\mathbb{R}^{4}\times T^{2}$ where a spatial circle in $T^{2}$ gives
the outer-automorphism twist. As we have discussed so far, it is equivalent to
the partition function of 5d $\mathcal{N}=2$ gauge theory on
$\mathbb{R}^{4}\times S^{1}$ with a non-simply-laced gauge group. The full
$\mathbb{R}^{4}\times T^{2}$ partition function is defined schematically as
follows,

$\displaystyle
Z_{\sigma}(\tau,\epsilon_{1,2},m,\mathbf{v})=\text{Tr}\Bigr{[}(-1)^{F}e^{2\pi
i\tau
P}e^{-\epsilon_{1}J_{1}-\epsilon_{2}J_{2}}e^{-(\epsilon_{+}+m)R_{1}-(\epsilon_{+}-m)R_{2}}e^{-\mathfrak{n}\cdot\mathbf{v}}\Bigr{]}.$
(34)

Here, $\sigma$ resembles the twisting operation, $P$ is the KK momentum along
the spatial circle in $T^{2}$, $J_{1,2}$ are two angular momenta on
$\mathbb{R}^{4}$, and $R_{1,2}$ are two charges of $SO(5)$ R-symmetry. $m$ can
be regarded as a mass deformation, known as the M-string mass. Lastly,
$\mathfrak{n}$ is the charge of the twisted M-strings in 6d, which becomes the
electric charge of W-bosons in 5d. Henceforth, for simplicity we will omit
$\sigma$ in the subscript of $Z$, with the correct twisting understood from
the context.

The 6d BPS partition function (34) admits two different descriptions: the
instanton partition function and the elliptic genus. First, one can study a 6d
$(2,0)$ theory from the 5d $\mathcal{N}=2$ gauge theory with KK instantons.
Then, the 6d partition function admits the following expansion,

$\displaystyle
Z(\tau,\epsilon_{1,2},m,\mathbf{v})=Z_{\text{pert}}(\epsilon_{1,2},m,\mathbf{v})\Bigr{(}1+\sum_{k=1}^{\infty}Z_{k}^{\text{inst}}(\epsilon_{1,2},m,\mathbf{v})q^{k}\Bigr{)},\quad(q=e^{2\pi
i\tau}).$ (35)

Here, $Z_{\text{pert}}$ is the perturbative partition function which captures
the perturbative W-bosons in 5d, and its form is given as follows,

$\displaystyle Z_{\text{pert}}=\text{PE}\Bigr{[}{\sinh{m\pm\epsilon_{+}\over
2}\over\sinh{\epsilon_{1,2}\over
2}}\sum_{\alpha\in\Delta^{+}(G)}e^{-\alpha\cdot\mathbf{v}}\Bigr{]},$ (36)

where $\Delta^{+}(G)$ is the positive root system of 5d gauge algebra
$\mathfrak{g}$. Also, $Z_{k}$ is the $k$-instanton partition function, which
captures the non-perturbative degrees in 5d with $k$-unit of KK momentum. For
the classical gauge groups, the instanton partition functions are well-studied
with ADHM construction Nekrasov:2004vw ; Hwang:2014uwa ; Hwang:2016gfw . On
the other hand, the exceptional gauge groups do not have ADHM construction,
and their instanton partition functions are much less understood.

Instead of the 5d description, one can study the 6d partition function from
the BPS spectra on the twisted M-strings on $T^{2}$. Then, the 6d partition
function admits the following expansion,

$\displaystyle
Z=Z_{0}\times\Bigr{(}1+\sum_{\mathfrak{n}}\mathbb{E}_{\mathfrak{n}}(\tau,\epsilon_{1,2},m)e^{-\mathfrak{n}\cdot\mathbf{v}}\Bigr{)}.$
(37)

In (37), the expansion parameter is the string fugacity
$e^{-\mathfrak{n}\cdot\mathbf{v}}$, and the coefficient $Z_{\mathfrak{n}}$ is
called the elliptic genus with charge $\mathfrak{n}=(n_{1},n_{2},...,n_{r})$.
Here, $Z_{0}$ is call the Abelian contribution which is the neutral part of
the partition function independent of the tensor VEV $\mathbf{v}$. The Abelian
contribution of general gauge group takes the following form,

$\displaystyle U(1):\quad Z_{0}$
$\displaystyle=\text{PE}\Bigr{[}{\sinh{m\pm\epsilon_{-}\over
2}\over\sinh{\epsilon_{1,2}\over 2}}{q\over 1-q}\Bigr{]},$ $\displaystyle
A_{r},\ D_{r},\ E_{r}:\quad Z_{0}$
$\displaystyle=\text{PE}\Bigr{[}{\sinh{m\pm\epsilon_{-}\over
2}\over\sinh{\epsilon_{1,2}\over 2}}{rq\over 1-q}\Bigr{]},$ $\displaystyle
B_{r}:\quad Z_{0}$
$\displaystyle=\text{PE}\Bigr{[}{\sinh{m\pm\epsilon_{-}\over
2}\over\sinh{\epsilon_{1,2}\over 2}}\Bigr{(}(r-1){q\over 1-q}+{q^{2}\over
1-q^{2}}\Bigr{)}\Bigr{]},$ $\displaystyle(C_{r})_{0}:\quad Z_{0}$
$\displaystyle=\text{PE}\Bigr{[}{\sinh{m\pm\epsilon_{-}\over
2}\over\sinh{\epsilon_{1,2}\over 2}}\Bigr{(}(r-1){q^{2}\over 1-q^{2}}+{q\over
1-q}\Bigr{)}\Bigr{]},$ $\displaystyle(C_{r})_{\pi}:\quad Z_{0}$
$\displaystyle=\text{PE}\Bigr{[}{\sinh{m\pm\epsilon_{-}\over
2}\over\sinh{\epsilon_{1,2}\over 2}}\Bigr{(}(r-1){q^{2}\over
1-q^{2}}+{q^{4}\over 1-q^{4}}+{q^{2}\over 1-q^{4}}\Bigr{]},$ $\displaystyle
G_{2}:\quad Z_{0}$
$\displaystyle=\text{PE}\Bigr{[}{\sinh{m\pm\epsilon_{-}\over
2}\over\sinh{\epsilon_{1,2}\over 2}}\Bigr{(}{q\over 1-q}+{q^{3}\over
1-q^{3}}\Bigr{)}\Bigr{]},$ $\displaystyle F_{4}:\quad Z_{0}$
$\displaystyle=\text{PE}\Bigr{[}{\sinh{m\pm\epsilon_{-}\over
2}\over\sinh{\epsilon_{1,2}\over 2}}\Bigr{(}{2q\over 1-q}+{2q^{2}\over
1-q^{2}}\Bigr{)}\Bigr{]}.$ (38)

Note that $Z_{0}$ of non-simply-laced gauge group is made of $U(1)$ partition
functions with different momentum fugacities. In general, it can be read off
from the decomposition rule (16). In this paper, we will mostly focus on the
non-Abelian part $Z/Z_{0}$ and the elliptic genera
$\mathbb{E}_{\mathfrak{n}}$.

For 6d $(2,0)$ $A$-type theories, the elliptic genus can be computed from 2d
quiver gauge theories on M-strings Haghighat:2013gba . For other types of
theories, such 2d gauge theory description has been yet unknown. Instead,
$\text{SL}(2,\mathbb{Z})$ modular property of the elliptic genera can be used
to determine them. Such procedure is called the ‘modular bootstrap’, and it
has been studied in various 6d theories DelZotto:2016pvm ; Gu:2017ccq ;
DelZotto:2017mee ; Kim:2018gak ; DelZotto:2018tcj ; Duan:2018sqe ;
Duan:2020cta ; Duan:2020imo , including $(2,0)$ $D$, $E$-type theories
Gu:2017ccq ; Duan:2020cta . In this paper, we will focus on the twisted circle
compactification of $(2,0)$ theories whose elliptic genera are unknown.
Specifically, we will extend the modular bootstrap program to be applicable
for non-simply-laced theories also.

As will be explained later, for the twisted M-strings of type ${G}^{\vee}$
given by the $\mathbb{Z}_{n_{G}}$ twist of $G$, the elliptic genus is not the
Jacobi form of $\text{SL}(2,\mathbb{Z})$, but of the particular congruence
subgroup of $\text{SL}(2,\mathbb{Z})$. Such subgroup
$\Gamma_{0}(n_{G})\subset\text{SL}(2,\mathbb{Z})$111We use both letters
$n_{G}$ and $N$ for congruence subgroups in this paper, hoping no confusion
will occur. is given as follows,

$\displaystyle\Gamma_{0}(n_{G})=\Bigr{\\{}\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in\text{SL}(2,\mathbb{Z}):c\equiv 0\ (\text{mod}\
n_{G})\Bigr{\\}}\,.$ (39)

Under the $\Gamma_{0}(n_{G})$ action, as a Jacobi modular form the elliptic
genus transforms as follows,

$\displaystyle\mathbb{E}_{\mathfrak{n}}\Bigr{(}{a\tau+b\over
c\tau+d}\Bigr{|}{z\over c\tau+d}\Bigr{)}=\exp\Bigr{[}-{\pi ic\over
c\tau+d}\mathfrak{i}_{\mathfrak{n}}(z)\Bigr{]}\cdot\mathbb{E}_{\mathfrak{n}}(\tau|z),\quad\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in\Gamma_{0}(n_{G}),$ (40)

where $z$ collectively denotes the elliptic parameters $\epsilon_{\pm}$ and
$m$. The elliptic genus is a modular form with index
$\mathfrak{i}_{\mathfrak{n}}$ and weight 0. For the untwisted case with
$n_{G}=1$, the index can be obtained from the 6d anomaly polynomial, through
the anomaly inflow from 6d to 2d Kim:2016foj . Then, the index of the
untwisted M-strings are given as follows,

$\displaystyle\mathfrak{i}_{\mathfrak{n}}(z)={\epsilon_{1}\epsilon_{2}\over
2}\mathfrak{n}^{T}\Omega\mathfrak{n}+(m^{2}-\epsilon_{+}^{2})\sum_{a=1}^{r}n_{a},\quad(n_{G}=1),$
(41)

where $\Omega$ is the Cartan matrix of the simply-laced Lie algebra
$\mathfrak{g}$.

For the elliptic genus of twisted M-strings, its index can be read off from
the index of the untwisted M-strings. In the last subsection, we explained how
the roots transform under the twisted compactification. From table 2, one can
obtain the following mapping between the twisted and the untwisted string,

$\displaystyle B_{r}:\ \hat{\mathfrak{n}}=(n_{1},n_{2},...,n_{r})$
$\displaystyle\leftarrow$ $\displaystyle A_{2r-1}:\
\mathfrak{n}=(n_{1},n_{2},...,n_{r},n_{r-1},...,n_{1})$
$\displaystyle(C_{r})_{0}:\ \hat{\mathfrak{n}}=(n_{1},n_{2},...,n_{r})$
$\displaystyle\leftarrow$ $\displaystyle D_{r+1}:\
\mathfrak{n}=(n_{1},n_{2},...,n_{r-1},n_{r},n_{r})$ $\displaystyle G_{2}:\
\hat{\mathfrak{n}}=(n_{1},n_{2})$ $\displaystyle\leftarrow$ $\displaystyle
D_{4}:\ \mathfrak{n}=(n_{2},n_{1},n_{2},n_{2})$ $\displaystyle F_{4}:\
\hat{\mathfrak{n}}=(n_{1},n_{2},n_{3},n_{4})$ $\displaystyle\leftarrow$
$\displaystyle E_{6}:\ \mathfrak{n}=(n_{1},n_{2},n_{3},n_{2},n_{1},n_{4})$
$\displaystyle(C_{r})_{\pi}:\ \hat{\mathfrak{n}}=(n_{1},n_{2},...,n_{r})$
$\displaystyle\leftarrow$ $\displaystyle A_{2r}:\
\mathfrak{n}=(n_{1},...,n_{r},n_{r},...,n_{1}).$ (42)

See table 2 for our convention of the node enumeration. We claim that the
index of the twisted M-strings of type ${G}^{\vee}$ and charge
$\hat{\mathfrak{n}}$ can be obtained from the index of the untwisted M-strings
of type $G$ and charge $\mathfrak{n}$ as follows,

$\displaystyle\mathfrak{i}_{\hat{\mathfrak{n}}}(z)\text{ of
}{G}^{\vee}=\frac{1}{n_{G}}\Bigr{(}\mathfrak{i}_{\mathfrak{n}}(z)\text{ of
}G\Bigr{)}\,.$ (43)

Using (41) and (43), one can universally write down the index of the M-strings
in any Lie algebra $\mathfrak{g}$ in the following expression,

$\displaystyle\mathfrak{i}_{\mathfrak{n}}$
$\displaystyle={\epsilon_{1}\epsilon_{2}\over 2}\mathfrak{n}(\Omega
D)\mathfrak{n}^{T}+({m^{2}-\epsilon_{+}^{2}})\sum_{a=1}^{r}D_{aa}\mathfrak{n}_{a},$
$\displaystyle\mathfrak{n}$
$\displaystyle=\begin{cases}(n_{1},n_{2},...,{n_{r}\over 2})\text{ if
}G=(C_{r})_{\pi}\\\ (n_{1},n_{2},...,{n_{r}})\text{ otherwise}\end{cases}.$
(44)

Here, except for $(C_{r})_{\pi}$ theories, $\Omega$ is the Cartan matrix of
$\mathfrak{g}$ and the matrix $D$ is defined as follows,

$\displaystyle D_{ij}=2{\delta_{ij}\over(\alpha_{i},\alpha_{i})}.$ (45)

The above index will play an important role only in the section 4, and readers
can see table 4 for the explicit values of $\Omega D$. Lastly, the vector
$\mathfrak{n}$ is a usual string charge vector except for $(C_{r})_{\pi}$ case
where the string charge at the last node is halved.

###  2.3 Geometric engineering

In this section, we want to understand the physics from the point of view of
geometry. To start with, the 6d $(2,0)$ SCFTs arise from F-theory on a
elliptic-fibered CY threefold, where the base is the ALE space, i.e, the
resolution of $\mathbb{C}^{2}/\Gamma$ with $\Gamma$ a finite subgroup of
$SU(2)$. Since the base itself is a CY twofold, the resulting 6d theory has
$(2,0)$ supersymmetry.222However, in order to turn on the M-string mass, one
needs to modify this picture, rendering the fibration non-trivial. The
resolution of the singularity in the base gives rise to a class of compact
divisors $S_{i}$ and a class of compact curves $\Sigma_{j}$, where the
negative of the intersection matrix

$\Omega_{ij}=-(S_{i}\cdot\Sigma_{j}),$ (46)

is exactly the Cartan matrix of $ADE$ type.

In the geometric picture, BPS strings arise naturally from D3-branes wrapping
on two-cycles in the base, which also inherits the self-duality condition of
the D3-branes in ten dimensions. The partition function under the 6d
$\Omega$-background localizes to the BPS strings wrapped on the $T^{2}$ in the
spacetime, giving us an expansion in terms of elliptic genera (37). Through
the F-theory/M-theory duality, the elliptic genus captures the essential part
of refined topological string partition function, i.e., $Z_{\text{GV}}$ in
(48), for the elliptically-fibered CY threefold.333For general
$\mathcal{N}=(1,0)$ SCFTs they are the same only up to overall factors
Gu:2017ccq ; DelZotto:2017mee .

This brings us another crucial ingredient of the story, i.e., topological
strings. Given a CY threefold $X$, if one denotes the Kähler parameter of two
cycles in $H_{2}(X,\mathbb{Z})$ as $\mathbf{t}$ with its exponential
$\mathbf{Q}=\exp(-\mathbf{t})$, we can associate the free energy as a formal
power series in $g_{s}$,

$\mathcal{F}_{\text{top}}(g_{s})=\sum\limits_{g\geq
0}\,F_{g}\,g_{s}^{2g-2}\,,$ (47)

where the genus $g$ free energy $\mathcal{F}_{g}$ can be expanded in terms of
$\mathbf{Q}$ with coefficients the celebrated Gromov-Witten invariants.
Furthermore, through lifting type IIA string theory to M-theory
Gopakumar:1998ii ; Gopakumar:1998jq ; Iqbal:2007ii ; Aganagic:2011mi , for $X$
non-compact, we are able to rewrite and refine topological string free energy
in the following way,

$\displaystyle\mathcal{F}_{\text{ref}}$ $\displaystyle=F_{\text{poly}}+\log
Z_{\text{GV}}(\epsilon_{1},\epsilon_{2},\mathbf{t})$ (48)
$\displaystyle=F_{\text{poly}}+\sum\limits_{2j_{\pm}\in\mathbb{N}}\sum\limits_{d\geq
1}\sum\limits_{\alpha\in
H_{2}(X,\mathbb{Z})}N^{\alpha}_{j_{-}j_{+}}{(-1)^{2(j_{-}+j_{+})}\,\chi_{j_{-}}(u^{d})\chi_{j_{+}}(v^{d})\over
v^{d}+v^{-d}-u^{d}-u^{-d}}\frac{\mathbf{Q}^{d\alpha}}{d}\,,$

where $\epsilon_{1}$ and $\epsilon_{2}$ are two equivariant parameters related
to the Cartan subalgebra of $SO(4)$, originated from instanton counting under
the $\Omega$-background in five dimensional gauge theory with $\mathcal{N}=1$
supersymmetry Nekrasov:2002qd ; Nekrasov:2003rj . $F_{\text{poly}}$ takes the
structure,

$F_{\text{poly}}=\frac{a_{ijk}t_{i}t_{j}t_{k}}{6\epsilon_{1}\epsilon_{2}}+b_{i}^{(0,1)}t_{i}+\frac{(\epsilon_{1}+\epsilon_{2})^{2}}{\epsilon_{1}\epsilon_{2}}b_{i}^{(1,0)}t_{i}\,.$
(49)

Moreover, for the second part in (48), the $\chi_{j}$ is the character of an
irreducible $SU(2)$ highest-weight representation with spin $j$,

$\chi_{j}(x)=x^{-2j}+x^{-2j+2}+\cdots+x^{2j}\,,$ (50)

and
$v=\exp(-\frac{\epsilon_{1}+\epsilon_{2}}{2}),u=\exp(-\frac{\epsilon_{1}-\epsilon_{2}}{2})$.
Last but not least, $N^{\alpha}_{j_{-}j_{+}}$ is the number of BPS states in
5d with spin $j_{-}$ and $j_{+}$, arising from M2-branes wrapped on curves in
$X$. They are integers known as the refined Gopakumar-Vafa (GV) or BPS
invariants.

For the type of CY threefold considered in this paper, we can divide the
Kähler parameters $\mathbf{t}$ into three different categories:

$\mathbf{t}=(-2\pi i\tau;\mathbf{v}=\\{v_{1},v_{2},\cdots v_{r}\\};m)\,,$ (51)

where they correspond to the volume of the elliptic fiber (instanton counting
parameter in 5d), the compact curves in the base (Coulomb parameters
associated to the simple roots of the gauge groups in 5d) and the M-string
mass (mass of the adjoint matter in 5d) respectively. This also provides a
unifying view on two types of expansion of
$Z(\tau,\epsilon_{1,2},m,\mathbf{v})$ in (34): If one sums up $\tau$ and $m$
while expands in terms of $\mathbf{v}$, this gives (37) which is an elliptic
genus expansion and naturally fits the 6d perspective; If one sums up
$\mathbf{v}$ and $m$ while expands in terms of $\tau$, this gives instead (35)
which is an instanton expansion and arises from the 5d perspective. This is
plausible since they are simply the UV/IR description of the same theory, and
are supposed to share the same protected quantities.

To put it in another way, the upshot of the above discussion is that the
elliptic genera or the instanton partition function enjoys a more constraining
expansion (48), henceforth referred as the BPS or GV expansion.

Finally, we sketch a heuristic picture to understand the twisted
compactification. We also consider the two compactification chains in figure
6, and let us first concentrate on the left one. 6d $\mathcal{N}=(2,0)$ SCFTs
compactified on $S^{1}$ is the same as M-theory compactification on the same
Calabi-Yau due to F-theory/M-theory duality. Furthermore, When compactifying
on a twisting circle, M-theory descends to type IIA theory, but the geometry
gets modified according to the action $\sigma$ depicted in section 2.1. The
first step is to identify the curve $\Sigma_{i}$ and $\Sigma_{\sigma(i)}$ in
the same orbit. Denote $\alpha,\beta$ the orbit of the action $\sigma$, we
select one node $i$ in the orbit $\alpha$ and the intersection matrix
naturally becomes Bhardwaj:2019fzv

$\Omega_{\alpha\beta}=-\sum_{j\in\beta}(S_{i}\cdot(\Sigma_{j}))\,.$ (52)

Remember that the elliptically fibered CY threefolds engineering 5d or 6d
theories of $ADE$ type have only one section. When we identify the curves, the
section over them corresponds to different points in the fiber in general,
hence the one section should be merged into an $N$-section in the terminology
of Cota:2019cjx . As a result, the threefold is transformed to a genus one
fibered CY manifold. This procedure is also explained, for example in
Bhardwaj:2019fzv .

Except for $A_{2l}$, the reduced intersection matrix becomes the Cartan matrix
of the invariant subalgebra of the twisting. Interestingly, this gives a
different result for $A_{2l}$ group, where the negative of the intersection
matrix becomes,444Notice that the last diagonal element is not two, so it
cannot possibly be identified as a Cartan matrix. In the terminology of
Bhardwaj:2019fzv , there is a self-edge attached to the last node in the
graph.

$\Omega=\left(\begin{array}[]{cccccc}2&-1&0&0&0&0\\\ -1&2&-1&0&0&0\\\
0&\ddots&\ddots&\ddots&0&0\\\ 0&0&-1&2&-1&0\\\ 0&0&0&-1&2&-1\\\
0&0&0&0&-1&1\\\ \end{array}\right).$ (53)

Remember that this does not give the 5d theory after twisted compactification.
Instead, as discussed in the section 2.1, they are S-dual to the real 5d
theory with adjoint matters, which lives on the right compactification chain.
Recall that S-duality includes exchanging the W-bosons and magnetic monopoles.
Therefore, in the geometric language, it is equivalent to exchanging the role
of divisors and curves, since D2-branes wrapping on compact curves give
W-bosons while D4-branes on compact divisors give magnetic monopoles. With
this in mind, after taking the S-dual the new intersection matrix becomes its
transpose, giving the same matrix $\Omega$ in (44). Here we are embedding the
intersection matrix into a larger intersection matrix of a putative genus one
fibered CY threefold, such that the column corresponds to rational curves
while the row represents toric divisors. For all cases except $(C_{r})_{\pi}$,
this is nothing but a rewriting of the procedure in the section 2.1.
Furthermore, for the $(C_{r})_{\pi}$ theory it gives the precise matrix
$\Omega$ that is identified quite indirectly in the sections 2.1 and 2.2.

##  3 Blowup equations

The twisted circle compactification of a 6d $(2,0)$ SCFT gives rise to a 5d
$\mathcal{N}=2$ theory, as explained in detail in 2.1. In section 3.1, we
first review the general setups for the $5d$ blowup equations. In section 3.2
we write down the blowup equations for 5d $\mathcal{N}=2$ theories of all
$ABCDEFG$ types, and discuss how to solve them in section 3.3. Then in section
3, we re-write them in the elliptic version, which also helps to fix/verify
the index of elliptic genera for twisted theories.

###  3.1 Review of the blowup equations

The idea of blowup equation for instanton counting is to consider the
equivariant localization on $\hat{\mathbb{C}}^{2}$, which is constructed from
$\mathbb{C}^{2}$ by blowing up the origin to create a compact 2-cycle
$\mathbb{P}^{1}$. One can regard the geometry as a total space of the
tautological line bundle $\mathcal{O}(-1)$ over the $\mathbb{P}^{1}$, which
can be parametrized by the homogeneous coordinates $(z_{0},z_{1},z_{2})$. We
are interested in the $U(1)^{2}$ equivariant partition function, with the
equivariant action on the homogeneous coordinates as

$(z_{0},z_{1},z_{2})\mapsto(z_{0},e^{\epsilon_{1}}z_{1},e^{\epsilon_{2}}z_{2}).$
(54)

In the blowup geometry, the instantons are located at two fix points, the
north and south poles of $\mathbb{P}^{1}$, whose coordinates are
$(z_{0},z_{1},z_{2})=(0,1,0)$ and $(0,0,1)$. Around these fix points, they can
be locally described by the $\mathbb{C}^{2}$ coordinates
$(z_{0}z_{1},z_{2}/z_{1})$ and $(z_{0}z_{2},z_{1}/z_{2})$ respectively,
leading to the equivalent action

$\begin{split}(z_{0}z_{1},z_{2}/z_{1})&\mapsto(e^{\epsilon_{1}}z_{0}z_{1},e^{\epsilon_{2}-\epsilon_{1}}z_{2}/z_{1}),\,\,\,\,\,\,\quad\text{near
the north pole},\\\
(z_{0}z_{2},z_{1}/z_{2})&\mapsto(e^{\epsilon_{2}}z_{0}z_{2},e^{\epsilon_{1}-\epsilon_{2}}z_{1}/z_{2}),\,\,\,\,\,\,\quad\text{near
the south pole}.\end{split}$ (55)

The full partition function $\hat{\mathcal{Z}}_{\hat{\mathbb{C}}^{2}}$ on
$\hat{\mathbb{C}}^{2}$ can be written as the product of the partition function
$\hat{\mathcal{Z}}^{\text{N}/\text{S}}$ around these two fix points,

$\hat{\mathcal{Z}}_{\hat{\mathbb{C}}^{2}}(\vec{v},\epsilon_{1},\epsilon_{2},q,\vec{m})=\sum_{\vec{n}\in
Q^{\vee}}(-1)^{|\vec{n}|}\hat{\mathcal{Z}}^{\text{N}}(\vec{n})\hat{\mathcal{Z}}^{\text{S}}(\vec{n}),$
(56)

by summing up all the fluxes $\vec{n}$ on the two-cycle $\mathbb{P}^{1}$. Here
$Q^{\vee}$ is the co-root lattice, $\vec{v}$ are the Coulomb parameters,
$\vec{m}$ are masses of matters and $q$ is the instanton counting parameter.
Around the North/South poles, they can be locally treated as $\mathbb{C}^{2}$.
In terms of $\mathbb{C}^{2}$ partition function
$\hat{\mathcal{Z}}_{{\mathbb{C}}^{2}}$, we have

$\begin{split}\hat{\mathcal{Z}}^{\text{N}}(\vec{n})&=\hat{\mathcal{Z}}_{\mathbb{C}^{2}}\left(\vec{v}+\vec{n}\epsilon_{1}+\vec{\lambda}_{G}\epsilon_{1},\epsilon_{1},\epsilon_{2}-\epsilon_{1},qe^{r_{b}\epsilon_{1}},\vec{m}+\vec{\lambda}_{F}\epsilon_{1}\right),\\\
\hat{\mathcal{Z}}^{\text{S}}(\vec{n})&=\hat{\mathcal{Z}}_{\mathbb{C}^{2}}\left(\vec{v}+\vec{n}\epsilon_{2}+\vec{\lambda}_{G}\epsilon_{2},\epsilon_{2}-\epsilon_{1},\epsilon_{2},qe^{r_{b}\epsilon_{2}},\vec{m}+\vec{\lambda}_{F}\epsilon_{2}\right).\\\
\end{split}$ (57)

Note that the partition function should be invariant under Weyl transformation
of its Lie algebra $\mathfrak{g}$, and $\vec{n}+\vec{\lambda}_{G}$ is actually
the co-weight lattice $P^{\vee}$. The number of nonequivalent choices
$\vec{\lambda}_{G}$ depends on the number of elements in the quotient
$|P^{\vee}/Q^{\vee}|=\det\Omega$, where $\Omega$ is the Cartan matrix of
$\mathfrak{g}$. For special cases, like $(C_{r})_{\pi}$, $\Omega$ should be
interpreted as the intersection matrix of compact divisors and curves in the
CY geometry.

Another remark here is that the partition function
$\hat{\mathcal{Z}}_{{\mathbb{C}}^{2}}$ is not the usual partition function
$\mathcal{Z}_{{\mathbb{C}}^{2}}$ used in gauge theories. They are the same
only up to a shift. As reviewed in section 2.3, a five dimensional gauge
theory can be engineered from the M-theory compactification on a CY manifold,
where the total partition function $\mathcal{Z}=\exp\mathcal{F}$ can be
written as the product of the GV or BPS expansion $Z_{\text{GV}}$ (48) plus
singular terms $e^{F_{\text{poly}}}$ (49) at large volume limit. Then we
define $\hat{\mathcal{Z}}=e^{F_{\text{poly}}}\hat{Z}_{\text{GV}}$, with555Note
that $r_{b}$ could be a half integer, so that $(-1)^{2r_{b}}$ is not
necessarily $1$.

$\begin{split}\hat{Z}_{\text{GV}}(\vec{v},\epsilon_{1},\epsilon_{2},q,\vec{m})&={Z}_{\text{GV}}(\vec{v},\epsilon_{1}+2\pi
i,\epsilon_{2},q,\vec{m})\\\ &={Z}_{\text{GV}}(\vec{v}+2\pi
i\vec{\lambda}_{G},\epsilon_{1},\epsilon_{2},q(-1)^{2r_{b}},\vec{m}+2\pi
i\vec{\lambda}_{F}),\end{split}$ (58)

where the second line comes from the checker-board pattern for the spin
$(j_{L},j_{R})$ BPS invariants at degree $(d_{G},d_{F},d_{b})$ in the BPS
expansion,666This is also known as the B-field condition in Hatsuda:2013oxa ;
Wang:2015wdy .

$(-1)^{2\vec{\lambda}_{G}\cdot d_{G}+2\vec{\lambda}_{F}\cdot d_{F}+2r_{b}\cdot
d_{b}}=(-1)^{2j_{L}+2j_{R}+1}.$ (59)

As a remark, the checkerboard pattern means that for a given curve class
$\\{d_{G},d_{F},d_{b}\\}$, if one enumerates all non-zero BPS invariants with
two axes $2j_{+}$ and $2j_{-}$, any two occupied blocks are either
disconnected or connected through a diagonal. In other words, the combination
$\left(2j_{-}+2j_{+}\right)$ is always even or odd. Clearly this is necessary
for the the equation (59) to hold. This pattern was first noticed in
Choi:2012jz  and holds true for all non-compact CY threefolds that the authors
know of.

Next, in order to get the blowup equation for ordinary gauge theory partition
function, one has to shift back the extra fluxes. This will give rise to an
extra phase $(-1)^{|\phi|}$, coming from rearranging the singular part
$e^{F_{\text{poly}}}$ in the blowup equation. The explicit expression is a bit
complicated and for 6d SCFTs it can be found in Gu:2020fem . Here we do not
need the explicit form so we omit its expression. Now the blowup equation
takes the form,

${\mathcal{Z}}_{\hat{\mathbb{C}}^{2}}(\vec{v},\epsilon_{1},\epsilon_{2},q,\vec{m})=\sum_{\vec{n}\in
Q^{\vee}}(-1)^{|\vec{n}|+|\phi|}{\mathcal{Z}}^{\text{N}}(\vec{n}){\mathcal{Z}}^{\text{S}}(\vec{n})\,.$
(60)

If one consider the limit where the size of blow-up $\mathbb{P}^{1}$ is small,
the resulting geometry is nothing but the original $\mathbb{C}^{2}$, so one
concludes immediately that
${{\mathcal{Z}}}_{\hat{\mathbb{C}}^{2}}\sim{\mathcal{Z}}_{{\mathbb{C}}^{2}}$.
In most cases, the two partition functions are exactly identical. However, in
certain special cases, there could be an extra factor $\Lambda$ coming from
the perturbative part. It was further explained in Huang:2017mis  that
$\Lambda$ does not depend on Coulomb VEVs $\vec{v}$, so we have

${\mathcal{Z}}_{\hat{\mathbb{C}}^{2}}(\vec{v},\epsilon_{1},\epsilon_{2},q,\vec{m})=\Lambda(\epsilon_{1},\epsilon_{2},q,\vec{m}){{\mathcal{Z}}_{\mathbb{C}}^{2}}(\vec{v},\epsilon_{1},\epsilon_{2},q,\vec{m}).$
(61)

If such an equation holds, we could define

$\Lambda(\epsilon_{1},\epsilon_{2},q,\vec{m})=\lim_{Q_{i}\rightarrow
0}\frac{{{\mathcal{Z}}}_{\hat{\mathbb{C}}^{2}}(\vec{v},\epsilon_{1},\epsilon_{2},q,\vec{m})}{{{\mathcal{Z}}}_{{\mathbb{C}}^{2}}(\vec{v},\epsilon_{1},\epsilon_{2},q,\vec{m})},$
(62)

where $Q_{i}$ is the exponential of the Coulomb VEVs $Q_{i}=e^{-v_{i}}$. For a
5d/6d SCFT that can be engineered from geometry, the instanton partition
function admits a BPS expansion, counting the BPS states of M2-brane wrapping
curves inside the CY such that the degree of $Q_{i}$’s is always non-negative.
This indicates that most of the instanton contributions in (62) vanish,
leaving only the neutral part
$Z^{\text{inst}}_{\text{neutral}}=\left.Z^{\text{inst}}\right|_{Q_{i}\rightarrow
0}$, times the contribution from $e^{F_{\text{poly}}}$.

It seems that one can use arbitrary $\vec{\lambda}_{G/F},r_{b}$ in (57) to
define an equation. Nevertheless, notice that the perturbative part is
divergent in this limit, so one has to choose $\vec{\lambda}_{G/F},r_{b}$
properly to cancel these divergences. The cancellation condition gives a
strong constraint on $\vec{\lambda}_{G/F},r_{b}$, which turns out to give all
the possible choices. Among them, we choose those that satisfy the
aforementioned condition (59) for all BPS states.

###  3.2 Blowup equations for $G+1{\text{\bf{Adj}}}$

Now let us focus on 5d theories with an adjoint matter $G+1{\text{\bf{Adj}}}$.
Recall that the adjoint representation is the same as the root system of its
Lie algebra $\mathfrak{g}$. We denote $m$ the mass of the adjoint matter,
$v_{\alpha}$ the Coulomb VEVs $\alpha\cdot\vec{v}$, $R^{+}=\\{(v_{\alpha},\pm
m)|\alpha\in\Delta^{+}\\}$ the positive weights where $\Delta^{+}$ is the set
of positive roots. Following the gauge theory language, we separate the
partition function into the classical, 1-loop and instanton part. We modify
slightly the notations in Gu:2020fem  to adapt to our present convention,

$Z^{\text{class}}=\exp\left[-\frac{2\pi
i\tau}{2\epsilon_{1}\epsilon_{2}}\sum_{ij}v_{i}v_{j}h_{ij}\right],$ (63)
$\begin{split}Z^{\text{1-loop}}=&\exp\left[\frac{1}{2}m^{2}\sum_{{{\alpha}}\in\Delta^{+}}v_{\alpha}+\frac{(\epsilon_{1}+\epsilon_{2})^{2}}{8\epsilon_{1}\epsilon_{2}}\sum_{{{\alpha}}\in\Delta^{+}}v_{\alpha}\right]\\\
&\times\mathrm{PE}\left(\frac{\sum_{\alpha\in\Delta^{+}}\left(Q_{\alpha}(Q_{m}+Q_{m}^{-1})-\left(v+v^{-1}\right)Q_{\alpha}\right)}{\left(q_{1}^{1/2}-q_{1}^{1/2}\right)\left(q_{2}^{1/2}-q_{2}^{1/2}\right)}\right),\end{split}$
(64)
$Z^{\text{inst}}=\text{PE}{\Bigg{[}\sum_{j_{-},j_{+}=0}^{\infty}\sum_{\beta}(-1)^{2(j_{-}+j_{+})}{N^{\beta}_{j_{-},j_{+}}}f_{(j_{-},j_{+})}(q_{1},q_{2})Q_{\beta}\Bigg{]}}.$
(65)

where PE stands for the plethystic exponential and $h_{ij}$ is the Killing
form of $\mathfrak{g}$,

$f_{(j_{-},j_{+})}(q_{1},q_{2})=\frac{\chi_{j_{+}}(v)\chi_{j_{-}}(u)}{(q_{1}^{1/2}-q_{1}^{-1/2})(q_{2}^{1/2}-q_{2}^{-1/2})}\,.$
(66)

One can easily recognize that $Z^{\text{inst}}$ is nothing but the
$Z_{\text{GV}}$ defined in (48).

As discussed in the last section, the number of choices of $\vec{\lambda}_{G}$
is determined by the determinant of the Cartan matrix or the negative of the
intersection matrix. Therefore we can immediately get the number of flux
$\vec{\lambda}_{G}$, summarized in table 3.

$G$ | $A_{r}$ | $B_{r}$ | $(C_{r})_{0}$ | $(C_{r})_{\pi}$ | $D_{r}$ | $E_{6}$ | $E_{7}$ | $E_{8}$ | $F_{4}$ | $G_{2}$
---|---|---|---|---|---|---|---|---|---|---
#$\vec{\lambda}_{G}$ | $r+1$ | 2 | 2 | 1 | 4 | 3 | 2 | 1 | 1 | 1
Table 3: The number of possible fluxes $\vec{\lambda}_{G}$ for simple Lie
algebras.

Among all the cases except for $(C_{r})_{\pi}$, there is a common one:
$\vec{\lambda}_{G}=0,\lambda_{F}=\pm\frac{1}{2},r_{b}=0$. For simplicity, we
only consider $\lambda_{F}=-\frac{1}{2}$. Together with the neutral part in
(2.2), a short computation shows that the expression for $\Lambda$ is

$\Lambda(m,\epsilon_{1},\epsilon_{2},\tau)=\mathfrak{f}(\tau)\sum_{{\vec{n}\in\mathbb{Z}^{g}}}{e}^{i\pi\tau{h_{ij}}n_{i}n_{j}+(m-\epsilon_{+})|\vec{n}|},$
(67)

where ${h_{ij}}=(\Omega D^{-1})_{ij}$ and $D$ is the matrix
$D_{ij}=2\frac{\delta_{i,j}}{\langle\alpha_{i},\alpha_{i}\rangle}$. One can
easily verify from the expression that $\phi=2\pi i\lambda_{F}|\vec{n}|$
cancels with the phase $(-1)^{|\vec{n}|}$. The prefactor $\mathfrak{f}(\tau)$
comes from the neutral part of instanton partition function described in
section 2.2, for classical Lie groups,

$\mathfrak{f}(\tau)=\begin{cases}\prod_{k=1}^{\infty}(1-q^{k})^{-r},&\text{for
}A_{r},D_{r}\ \text{and}\ E_{r},\\\
\prod_{k=1}^{\infty}(1-q^{2k})^{-r+1}(1-q^{k})^{-1},&\text{for
}(C_{r})_{0},\\\ \prod_{k=1}^{\infty}(1-q^{2k})^{-r},&\text{for
}(C_{r})_{\pi},\\\
\prod_{k=1}^{\infty}(1-q^{k})^{-1}(1-q^{3k})^{-1},&\text{for }G_{2},\\\
\prod_{k=1}^{\infty}(1-q^{k})^{-2}(1-q^{2k})^{-2},&\text{for
}F_{4},\end{cases}$ (68)

Note that $\mathfrak{f}(\tau)$ is not important in our current cases, so one
can set it to be one if one does not care about the neutral part.

Finally, for the $(C_{r})_{\pi}$ theory, the negative of the intersection
matrix is not the Cartan matrix of $C_{r}$ and $\vec{\lambda}_{G}=0$ breaks
the flux quantization condition (59). It turns out that there is only one
blowup equation in this case. The correct choice is to simply replace the co-
root lattice in (67) with another co-weight lattice by the shift,

$n_{i}\rightarrow n_{i}+\frac{i}{2},\quad i=1,\cdots,r,$ (69)

which gives the blowup equation for both $(C_{r})_{0,\pi}$. As a summary, the
blowup equation for 5d $G+1{\text{\bf{Adj}}}$ theory is

$\begin{split}\Lambda&(m,\epsilon_{1},\epsilon_{2},\tau)Z^{\mathrm{inst}}\left(\epsilon_{1},\epsilon_{2},v_{i},\tau,m\right)=\sum_{\vec{n}\in\mathbb{Z}^{g}}{e}^{i\pi\tau{h_{ij}}n_{i}n_{j}+(m-\epsilon_{+})|\vec{n}|}\mathcal{A}^{0}(\epsilon_{1},\epsilon_{2},v_{i},m,n_{i})\\\
&\times
Z^{\mathrm{inst}}\left(\epsilon_{1}-\epsilon_{2},\epsilon_{2},v_{i}+\Omega_{ij}n^{j}\epsilon_{2},\tau,m-\frac{\epsilon_{2}}{2}\right)Z^{\mathrm{inst}}\left(\epsilon_{1},\epsilon_{2}-\epsilon_{1},v_{i}+\Omega_{ij}n^{j}\epsilon_{1},\tau,m-\frac{\epsilon_{1}}{2}\right).\\\
\end{split}$ (70)

For a given root $\alpha$, define
$Q_{\alpha}=e^{-v_{\alpha}}=\prod_{i}Q_{i}^{\alpha_{i}}$ and
$R_{\alpha}=-\alpha_{i}\Omega_{ij}n_{j}$,

$\mathcal{A}^{0}=\begin{cases}{\displaystyle\prod_{\alpha\in\Delta^{+}}\prod_{\displaystyle\begin{subarray}{c}m,n\geq
0\\\
m+n\leq|R_{\alpha}|-1\end{subarray}}}\frac{\left(1-Q_{\alpha}Q_{m}^{-1}q_{1}^{m+1/2}q_{2}^{n+1/2}\right)}{\left(1-Q_{\alpha}q_{1}^{m}q_{2}^{n}\right)}{\displaystyle\prod_{\displaystyle\begin{subarray}{c}m,n\geq
0\\\
m+n\leq|R_{\alpha}|-2\end{subarray}}}\frac{\left(1-Q_{\alpha}Q_{m}q_{1}^{m+1/2}q_{2}^{n+1/2}\right)}{\left(1-Q_{\alpha}q_{1}^{m+1}q_{2}^{n+1}\right)},&R_{\alpha}\geq
0\\\
{\displaystyle\prod_{\alpha\in\Delta^{+}}\prod_{\displaystyle\begin{subarray}{c}m,n\geq
0\\\
m+n\leq|R_{\alpha}|-1\end{subarray}}}\frac{\left(1-Q_{\alpha}Q_{m}q_{1}^{-m-1/2}q_{2}^{-n-1/2}\right)}{\left(1-Q_{\alpha}q_{1}^{-m}q_{2}^{-n}\right)}{\displaystyle\prod_{\displaystyle\begin{subarray}{c}m,n\geq
0\\\
m+n\leq|R_{\alpha}|-2\end{subarray}}}\frac{\left(1-Q_{\alpha}Q_{m}^{-1}q_{1}^{-m-1/2}q_{2}^{-n-1/2}\right)}{\left(1-Q_{\alpha}q_{1}^{-m-1}q_{2}^{-n-1}\right)},&R_{\alpha}<0.\end{cases}$
(71)

For the $(C_{r})_{\pi}$ theory, we should make the replacement as in (69). We
verify (70) for $A_{1}+1\text{\bf{Adj}}$ up to 4-instantons with instanton
partition function from ADHM description, and for $(C_{r})_{0,\pi}$, $r\leq 4$
at the one-instanton level.

###  3.3 Solving blowup equations

Here we give a description of our recipe to solve the blowup equation.

#### Solving BPS invariants

To solve the BPS invariants, one can expand the blowup equation with instanton
counting parameter $q$ and the Coulomb parameter $\mathbf{v}$. With the ansatz
on the BPS expansion with unknown BPS invariants, we can bootstrap all the BPS
invariants, except for the spin $(0,0)$ and $(0,1/2)$ invariants. See
Huang:2017mis ; Gu:2019pqj  for the proof and Kim:2020hhh  for a recent
discussion on this approach.

Following the definition in (66), define

$Bl_{(j_{-},j_{+},R)}(q_{1},q_{2})=f_{(j_{-},j_{+})}(q_{1},q_{2}/q_{1})q_{1}^{R}+f_{(j_{-},j_{+})}(q_{1}/q_{2},q_{2})q_{2}^{R}-f_{(j_{-},j_{+})}(q_{1},q_{2}),$
(72)

which is a Laurent polynomial of $q_{1},q_{2}$. It is proved in Huang:2017mis
; Gu:2019pqj  that for a fixed spin $R$, $Bl_{(j_{-},j_{+},R)}$ are mutually
independent for all possible $(j_{-},j_{+})$, except for $Bl_{(0,0,\pm
1/2)}(q_{1},q_{2})=Bl_{(0,1/2,0)}(q_{1},q_{2})=0$. Then finding the BPS
invariants becomes a problem of decomposing a Laurent polynomial into the
basis $Bl_{(j_{-},j_{+},R)}(q_{1},q_{2})$. In our case, we set
$\vec{\lambda}_{G}=0,\vec{\lambda}_{F}=-\frac{1}{2},r_{b}=0$, so that the only
unfixed BPS invariants are $N_{(0,0)}^{d_{m}=\pm 1}$ and
$N_{(0,1/2)}^{d_{m}=0}$. In principle, one can go to higher degrees to fix
these invariants, but here we use another condition to fix them more
conveniently.

It is known that if we set the mass of the adjoint matter to $\epsilon_{-}$,
the partition function is trivially one. In this limit,

$f_{(0,0)}Q_{m}^{\pm 1}\rightarrow\frac{u^{\pm
1}}{{(q_{1}^{1/2}-q_{1}^{-1/2})(q_{2}^{1/2}-q_{2}^{-1/2})}},\
f_{(0,1/2)}Q_{m}^{0}\rightarrow\frac{v+v^{-1}}{{(q_{1}^{1/2}-q_{1}^{-1/2})(q_{2}^{1/2}-q_{2}^{-1/2})}}\,,$
(73)

which are linearly independent. This means that we can use the triviality
condition to fix the remaining BPS invariants. In practice, we extract the BPS
invariants up to 10-instantons, at lower degree expansion of the Coulomb
parameters, which are compatible with the results using the modular bootstrap
in section 4. Part of the BPS invariants we solved can be found in appendix B.

#### Solving instanton partition functions

As reviewed in section 2.2, from the 5d point of view, the partition function
can be expanded in terms of instanton sectors. To begin with, we expand the
blowup equation at the $k$-instanton order and the resulting equation is

$\begin{split}&\sum_{\begin{subarray}{c}\mathbf{n}\in\mathbb{Z}^{g}\\\
k_{1}+k_{2}+\frac{1}{2}{h_{ij}}n_{i}n_{j}=k\end{subarray}}\left(\frac{Q_{m}}{\sqrt{q_{1}q_{2}}}\right)^{|\vec{n}|}\mathcal{A}^{0}(\epsilon_{1},\epsilon_{2},v_{i},m,n_{i})\\\
&\times
Z_{k_{1}}^{\mathrm{inst}}\left(\epsilon_{1}-\epsilon_{2},\epsilon_{2},v_{i}+\Omega_{ij}n_{j}\epsilon_{2},m-\frac{\epsilon_{2}}{2}\right)Z_{k_{2}}^{\mathrm{inst}}\left(\epsilon_{1},\epsilon_{2}-\epsilon_{1},v_{i}+\Omega_{ij}n_{j}\epsilon_{1},m-\frac{\epsilon_{1}}{2}\right)\\\
=&\sum_{\begin{subarray}{c}\mathbf{n}\in\mathbb{Z}^{g}\\\
l+k^{\prime}+\frac{1}{2}\Omega_{ij}n_{i}n_{j}=k\end{subarray}}\left(\frac{Q_{m}}{\sqrt{q_{1}q_{2}}}\right)^{|\vec{n}|}\mathfrak{f}_{l}Z_{k^{\prime}}^{\mathrm{inst}}\left(\epsilon_{1},\epsilon_{2},v_{i},m\right)\,.\end{split}$
(74)

Here $\mathfrak{f}_{l}$ is the $l$-th Fourier coefficient of
$\mathfrak{f}(\tau)$, which can be defined from the coefficients of the $-r$
power of Dedekind eta function $p_{-r}(l)$

$\eta(\tau)^{-r}:=q^{-\frac{r}{24}}\prod_{n=1}^{\infty}(1-q^{n})^{-r}=q^{-\frac{r}{24}}\sum_{l=0}^{\infty}p_{-r}(l)q^{l}\,.$
(75)

When $r>0$, $p_{-r}(l)$ are always positive integers for $l\geq 0$. For the
first few terms, we have

$p_{-r}(0)=1,\,p_{-r}(1)=r,\,p_{-r}(2)=\frac{1}{2}r(r+3),\,p_{-r}(3)=\frac{1}{6}r(r^{2}+9r+8),\cdots.$
(76)

(74) can be alternatively written as

$I_{k}=Z_{k}^{\mathrm{inst}}\left(\epsilon_{1},\epsilon_{2},v_{i},m\right)-Z_{k}^{\mathrm{inst}}\left(\epsilon_{1}-\epsilon_{2},\epsilon_{2},v_{i},m-\frac{\epsilon_{2}}{2}\right)-Z_{k}^{\mathrm{inst}}\left(\epsilon_{1},\epsilon_{2}-\epsilon_{1},v_{i},m-\frac{\epsilon_{1}}{2}\right),$
(77)

where $I_{k}$ is the contribution from lower instanton numbers

$\begin{split}I_{k}=&-\sum_{\begin{subarray}{c}l+k^{\prime}+\frac{1}{2}{h_{ij}}n_{i}n_{j}=k\\\
\mathbf{n}^{\prime}\in\mathbb{Z}^{g},k^{\prime}<k\end{subarray}}\left(\frac{Q_{m}}{\sqrt{q_{1}q_{2}}}\right)^{|\vec{n}|}p_{-r}(l)Z_{k^{\prime}}^{\mathrm{inst}}\left(\epsilon_{1},\epsilon_{2},v_{i},m\right)\\\
&+\sum_{\begin{subarray}{c}k_{1}+k_{2}+\frac{1}{2}{h_{ij}}n_{i}n_{j}=k\\\
\mathbf{n}\in\mathbb{Z}^{g},k_{1},k_{2}<k\end{subarray}}\left(\frac{Q_{m}}{\sqrt{q_{1}q_{2}}}\right)^{|\vec{n}|}\mathcal{A}^{0}(\epsilon_{1},\epsilon_{2},v_{i},m,n_{i})\\\
&\times
Z_{k_{1}}^{\mathrm{inst}}\left(\epsilon_{1}-\epsilon_{2},\epsilon_{2},v_{i}+\Omega_{ij}n_{j}\epsilon_{2},m-\frac{\epsilon_{2}}{2}\right)Z_{k_{2}}^{\mathrm{inst}}\left(\epsilon_{1},\epsilon_{2}-\epsilon_{1},v_{i}+\Omega_{ij}n_{j}\epsilon_{1},m-\frac{\epsilon_{1}}{2}\right).\\\
\end{split}$ (78)

In order to solve the equation, we can first make the ansatz

$Z_{k}^{\text{inst}}=\frac{\mathcal{N}_{k}(Q_{i},Q_{m},q_{1},q_{2})}{\mathcal{D}_{k}(Q_{i},q_{1},q_{2})}.$
(79)

where $\mathcal{N}_{k}(Q_{i},Q_{m},q_{1},q_{2})$ is a Laurent polynomial of
$Q_{i},Q_{m},q_{1},q_{2}$ with unknown coefficients, and
$\mathcal{D}_{k}(Q_{i},q_{1},q_{2})$ can be determined from the pole structure
of $I_{k}$. Then substituting the ansatz (79) into the $k$-instanton blowup
equation (77), we can solve the coefficients in the
$\mathcal{N}_{k}(Q_{i},Q_{m},q_{1},q_{2})$ with extra information. At one-
instanton level, the problem becomes much simpler and easier to solve.

When $k=1$, (77) becomes

$\begin{split}&-\mathfrak{f}_{1}+\sum_{\begin{subarray}{c}\mathbf{n}\in\mathbb{Z}^{g}\\\
\frac{1}{2}{h_{ij}}n_{i}n_{j}=1\end{subarray}}\left(\frac{Q_{m}}{\sqrt{q_{1}q_{2}}}\right)^{|\vec{n}|}\left(\mathcal{A}^{0}(\epsilon_{1},\epsilon_{2},v_{i},m,n_{i})-1\right)\\\
=&Z_{1}^{\mathrm{inst}}\left(\epsilon_{1},\epsilon_{2},v_{i},m\right)-Z_{1}^{\mathrm{inst}}\left(\epsilon_{1}-\epsilon_{2},\epsilon_{2},v_{i},m-\frac{\epsilon_{2}}{2}\right)-Z_{1}^{\mathrm{inst}}\left(\epsilon_{1},\epsilon_{2}-\epsilon_{1},v_{i},m-\frac{\epsilon_{1}}{2}\right)\,,\end{split}$
(80)

where $\mathfrak{f}_{1}$ coincides with the number of long simple roots
according to (68). Here the sum in the first line can be understood as a sum
over length-2 (co-)root. In our convention they correspond precisely to long
roots, hence there are only poles at the long roots in the first line of (80).
This analysis is enough to conjecture the denominator of
$Z_{1}^{\mathrm{inst}}\left(\epsilon_{1},\epsilon_{2},v_{i},m\right)$ for a
simple Lie algebra $\mathfrak{g}$ as

$\mathcal{D}_{1}(Q_{i},q_{1},q_{2})=(1-q_{1})(1-q_{2})\prod_{\alpha\in\Delta_{l}}(1-Q_{\alpha}q_{1}q_{2}),$
(81)

where $\Delta_{l}$ is the set of long roots. Based on the zeros and properties
of one-instanton partition function, one can further make an ansatz for the
numerator

$\mathcal{N}_{1}(Q_{i},Q_{m},q_{1},q_{2})=(\sqrt{q_{2}}-\sqrt{q_{1}}Q_{m})(\sqrt{q_{2}}-\sqrt{q_{1}}Q_{m}^{-1})\sum_{n,i,k,c_{i}}a_{n,i,k,c_{i}}(Q_{m}^{n}+Q_{m}^{-n})Q_{i}^{c_{i}}(q_{1}q_{2})^{\frac{k}{2}}.$
(82)

Define
$Z_{1}^{\text{inst}}=\frac{\left(\sqrt{q_{1}}Q_{m}-\sqrt{q_{2}}\right)\left(\sqrt{q_{1}}-\sqrt{q_{2}}Q_{m}\right)}{\left(1-q_{1}\right)\left(1-q_{2}\right)Q_{m}}\tilde{Z}_{1}$
and $v=(q_{1}q_{2})^{-\frac{1}{2}}$, we solve $\tilde{Z}_{1}$ for gauge groups
$A_{1},A_{2}$ and $A_{3}$ model with gauge fugacities turning off by setting
$Q_{i}=1$,

$\displaystyle\tilde{Z}_{1,A_{1}}$ $\displaystyle=$
$\displaystyle\frac{v^{2}}{\left(1-v^{2}\right)^{2}}$
$\displaystyle\left(-4\chi_{(0)}+\chi_{(1)}(v+v^{-1})\right),$
$\displaystyle\tilde{Z}_{1,A_{2}}$ $\displaystyle=$
$\displaystyle\frac{v^{4}}{\left(1-v^{2}\right)^{4}}$
$\displaystyle\left(\chi_{(0)}(-v^{2}+32-v^{-2})+\chi_{(1)}(v^{3}-13v-13v^{-1}+v^{-3})\right.$
$\displaystyle\left.+\chi_{(2)}(v^{2}+4+v^{-2})\right),$
$\displaystyle\tilde{Z}_{1,A_{3}}$ $\displaystyle=$
$\displaystyle\frac{v^{6}}{\left(1-v^{2}\right)^{6}}$
$\displaystyle\left(\chi_{(0)}(-v^{4}-14v^{2}-250-14v^{-2}-v^{-4})\right.$
$\displaystyle\left.+\chi_{(1)}(v^{5}-7v^{3}+146v+146v^{-1}-7v^{-3}+v^{-5})\right.$
$\displaystyle\left.+\chi_{(2)}(v^{4}-22v^{2}-78-22v^{-2}+v^{-4})+\chi_{(3)}(v^{3}+9v+9v+v^{-3})\right),$

and for $(C_{1})_{\pi}$,

$\displaystyle\begin{aligned}
\tilde{Z}_{1,(C_{1})_{\pi}}&=\frac{v^{2}}{\left(1-v^{2}\right)^{2}}\Big{(}2\chi_{(0)}({v}+v^{-1})-2\chi_{(1)}\Big{)}.\end{aligned}$

For all cases, they have the structure

$\tilde{Z}_{1,G}=\frac{v^{2h_{G}^{\vee}-2}}{(1-v^{2})^{2h_{G}^{\vee}-2}}P(v,\chi_{(n)}),\,$
(83)

with $P(v,\chi_{(n)})$ always a Laurent polynomial in $v$, which is symmetric
under $v\leftrightarrow v^{-1}$. Here $\chi_{(n)}:=\chi_{(n)}(m)$ is the
character of the spin $\frac{n}{2}$ representation of the $SU(2)$ flavor
group. Furthermore, the structure (83) still holds for other gauge groups, and
in particular for $ABCD$ type, we verified that the results agree with those
computed from ADHM descriptions. For theories without an ADHM description, we
list $\tilde{Z}_{1,G}$ in appendix C.

Notice that for most cases, we turn off the gauge fugacities $Q_{i}=1$, but
our method works also with all gauge fugacities turned on. For $G_{2}$, we
first compute the exact result and then expand it in terms of $v$,

$\begin{split}\tilde{Z}_{1,G_{2}}=&\chi_{(3)}^{SU(2)}\left(\sum_{n=0}^{\infty}\chi_{(0,n)}v^{3+2n}\right)-\chi_{(2)}^{SU(2)}\left(\sum_{n=0}^{\infty}\left(\chi_{(0,n)}+\chi_{(0,n+1)}+\chi_{(3,n)}v^{2}\right)v^{4+2n})\right)\\\
&+\chi_{(1)}^{SU(2)}\left(-\chi_{(1,0)}v^{3}+\sum_{n=0}^{\infty}\left(\chi_{(0,n+1)}+\chi_{(3,n)}(1+v^{2})+\chi_{(4,n)}v^{2}\right)v^{5+2n}\right)\\\
&+\chi_{(0)}^{SU(2)}\left(\chi_{(1,0)}(v^{2}+v^{4})-\sum_{n=0}^{\infty}\left(\chi_{(4,n)}(1+v^{2})+\chi_{(3,n)}\right)v^{6+2n}\right),\\\
\end{split}$ (84)

where $\chi_{(n_{1},n_{2})}$ is the character of $G_{2}$ with highest weight
$(n_{1},n_{2})$. As a remark, all the constant coefficients are $\pm 1$ in the
expansion, which agrees with the conjecture in Gu:2020fem . Similarly, for
$F_{4}$ we have

$\begin{split}\tilde{Z}_{1,F_{4}}=&-\chi_{(8)}^{SU(2)}\left(\sum_{n=0}^{\infty}\chi_{(n,0,0,0)}v^{8+2n}\right)\\\
&+\chi_{(7)}^{SU(2)}\left(\sum_{n=0}^{\infty}\left(\chi_{(n,1,0,0)}v^{2}+\chi_{(n+1,0,0,0)}+\chi_{(n,0,0,0)}\right)v^{9+2n}\right)\\\
&+\cdots.\\\ \end{split}$ (85)

For $E$ type gauge groups, the computation becomes too lengthy to reproduce
here.

### Elliptic blowup equations

In our 5d KK theory description, there is no cubic terms for the Coulomb
branch parameter $a_{i}$ in the prepotential. According to a theorem in
MR1228584 ; MR1314743 , geometrically, they are described by the genus one
fibered CY three-folds on twisted ALE singularities. This already indicates
that this is actually a 6d theory and the Coulomb branch becomes the tensor
branch in 6d. One can then re-write the blowup equation to its elliptic
version

$\begin{split}&\sum_{\mathfrak{n}^{\prime}+\mathfrak{n}{{}^{\prime\prime}}=\mathfrak{n}}\sum_{\vec{n}\in\mathbb{Z}^{r}}{e}^{i\pi\tau{h_{ij}}n_{i}n_{j}+(m-\epsilon_{+})|\vec{n}|+\mathfrak{n}_{i}^{\prime}\Omega_{ij}n_{j}\epsilon_{1}+\mathfrak{n}_{i}{{}^{\prime\prime}}\Omega_{ij}n_{j}\epsilon_{2}}\\\
&\times\mathbb{E}_{\mathfrak{n}^{\prime}}\left(\epsilon_{1}-\epsilon_{2},\epsilon_{2},m-\frac{\epsilon_{2}}{2}\right)\mathbb{E}_{\mathfrak{n}{{}^{\prime\prime}}}\left(\epsilon_{1},\epsilon_{2}-\epsilon_{1},m-\frac{\epsilon_{1}}{2}\right)\\\
=&\sum_{{\vec{n}\in\mathbb{Z}^{r}}}{e}^{i\pi\tau{h_{ij}}n_{i}n_{j}+(m-\epsilon_{+})|\vec{n}|}\mathbb{E}_{\mathfrak{n}}\left(\epsilon_{1},\epsilon_{2},m\right).\end{split}$
(86)

By comparing with the elliptic blowup equation in Gu:2019pqj , one observes
that for general simple groups the formula is the same if we do the
replacement $\Omega_{ij}\rightarrow{h_{ij}}=(\Omega D^{-1})_{ij},\
k\rightarrow k\cdot D$. We conclude that the index
$\mathfrak{i}_{\mathfrak{n}}$ for $Z_{\mathfrak{n}}$ is

$\mathfrak{i}_{\mathfrak{n}}={\epsilon_{1}\epsilon_{2}\over
2}(\mathfrak{n}\cdot D)(\Omega D^{-1})(\mathfrak{n}\cdot
D)^{T}+({m^{2}-\epsilon_{+}^{2}})\sum_{a=1}^{r}D_{aa}\mathfrak{n}_{a},$ (87)

agree with the results in section 2.2. For $(C_{r})_{\pi}$ theory, the blowup
equation is the same as the second blowup equation of $(C_{r})_{0}$, so they
should have the same index. The only difference between them is that there are
half-integer string charges, such that we need to normalize
$\mathfrak{n}\rightarrow\frac{\mathfrak{n}}{2}$ in the definition. Again, we
recover the index for $(C_{r})_{\pi}$ theories as described in section 2.2.

If we focus on a single node in the base, i.e., $\mathfrak{n}_{I}>0\
\text{and}\ \mathfrak{n}_{J}=0,J\neq I$, (86) becomes

$\begin{split}&\sum_{\mathfrak{n}_{I}^{\prime}+\mathfrak{n}_{I}^{{}^{\prime\prime}}=\mathfrak{n}}\sum_{{n}\in\mathbb{Z}}{e}^{i\pi
D_{II}\tau
n^{2}+(m-\epsilon_{+})n-2\mathfrak{n}_{I}^{\prime}n\epsilon_{1}-2\mathfrak{n}_{I}^{{}^{\prime\prime}}n\epsilon_{2}}\\\
&\times\mathbb{E}_{\mathfrak{n}^{\prime}_{I}}\left(\epsilon_{1}-\epsilon_{2},\epsilon_{2},m-\frac{\epsilon_{2}}{2},\tau\right)\mathbb{E}_{\mathfrak{n}_{I}^{{}^{\prime\prime}}}\left(\epsilon_{1},\epsilon_{2}-\epsilon_{1},m-\frac{\epsilon_{1}}{2},\tau\right)\\\
=&\sum_{{\vec{n}\in\mathbb{Z}^{r}}}{e}^{i\pi D_{II}\tau
n^{2}+(m-\epsilon_{+})n}\mathbb{E}_{\mathfrak{n}_{I}}\left(\epsilon_{1},\epsilon_{2},m,\tau\right).\end{split}$
(88)

One can observe that equation (88) is the same as $A_{1}$ M-string blowup
equation with the replacement $\tau\rightarrow D_{II}\tau$, such that the
solution of (88) is

$\mathbb{E}_{\mathfrak{n}_{I}}\left(\epsilon_{1},\epsilon_{2},m,\tau\right)=\mathbb{E}^{A_{1}}_{\mathfrak{n}_{I}}\left(\epsilon_{1},\epsilon_{2},m,D_{II}\tau\right)$
(89)

##  4 Modular bootstrap

In this section, we will talk about the modular bootstrap approach towards the
twisted elliptic genera. In section 4.1, we give a quick overview of the
modular group of twisted elliptic genus. In section 4.2, we discuss in detail
how to bootstrap all non-simply-laced cases except $(C_{r})_{\pi}$. Section
4.3 is devoted to the exceptional case $(C_{r})_{\pi}$.

###  4.1 Modular group of twisted elliptic genus

A 6d SCFT can be described by F-theory compactification on an elliptically
fibered Calabi-Yau threefold. After twisted circle compactification, since the
$\mathbb{Z}_{N}$ automorphism should act on the fiber as well, the geometry
becomes a geometry with $N$-sections. The geometry is expected to be a genus
one fibered Calabi-Yau threefold Bhardwaj:2019fzv . The genus one fibration is
a fibration with $N$-sections, which is defined by $N$ points that can be
identified with each other and are transformed to each other by monodromies in
the base. As proved in Schimannek:2019ijf ; Cota:2019cjx , the monodromy is an
element in $\Gamma_{1}(N)$,

$\displaystyle\Gamma_{1}(N)=\Bigr{\\{}\left(\begin{array}[]{cc}a&b\\\
c&d\end{array}\right)\in\mathrm{SL}(2,\mathbb{Z}):a,d\equiv 1(\text{mod}\
N),\quad c\equiv 0\ (\text{mod}\ N)\Bigr{\\}}\,.$ (92)

which is a subgroup of $\Gamma_{0}(N)$. Moreover, it is known that

$\Gamma_{0}(2)=\Gamma_{1}(2),\quad\Gamma_{0}(N)=\langle\
\Gamma_{1}(N),\begin{pmatrix}-1&0\\\ 0&-1\\\ \end{pmatrix}\ \rangle,\
\text{for}\ 3\leq N\leq 4\,.$ (93)

Although the minus of identity matrix acts trivially on $\tau$, its presence
eliminates non-trivial modular forms of odd weights. In other words, if one
considers modular forms invariant under $\Gamma_{1}(N)$ for $3\leq N\leq 4$,
in additional to those invariant under $\Gamma_{0}(N)$ one in principle also
has to include odd-weight modular forms. But since those are absent in the
elliptic genera, we think it better to choose the modular group to be
$\Gamma_{0}(N)$ rather than $\Gamma_{1}(N)$. This should be understood as
applying only to particular CY threefolds considered in this paper.777We thank
Thorsten Schimannek for answering a relevant question concerning this point.

In short, we expect the elliptic genus to be a Jacobi form in $\Gamma_{0}(N)$,
where $N=2$ for $B_{r}$, $(C_{r})_{0}$, $F_{4}$ algebras, $N=3$ for $G_{2}$
algebra and $N=4$ for $(C_{r})_{\pi}$.

In order to bootstrap the elliptic genera, one should use the ring of
$\Gamma_{0}(N)$, which is reviewed in appendix A. However, the full ring of
$\Gamma_{0}(N)$ modular forms contains object $E_{2}^{(N)}$ with smaller
weight, which makes the number of generators growing too fast as we increase
the string numbers. In order to simplify our computation, we treat the
geometry after twisted circle compactification as a gluing of single nodes in
the base, such that along each node the modular group is either
$\Gamma_{0}(N)$ or $\text{SL}(2,\mathbb{Z})$. Another key observation from the
end of last section is that, restricting to a single base node, the elliptic
genera always resemble the elliptic genera of $A_{1}$ M-strings. If the node
is a long simple root of the Lie algebra, then they are the same, while for a
short simple root we simply make the replacement $\tau\rightarrow N\tau$. Such
an observation makes us conclude that along a single short node, the
$E_{2}^{(N)}$ actually disappear and we could just use the modular forms
arising from $\text{SL}(2,\mathbb{Z})$ Jacobi forms instead. For general
string numbers not necessarily restricted to a single node, except for
$(C_{r})_{\pi}$, we further conjecture that the numerator of the elliptic
genera can still be written in terms of the following set of elliptic modular
forms and Jacobi forms,

$\displaystyle\Bigr{\\{}E_{4}(\tau),\ E_{6}(\tau),\ A(\tau|z),\ B(\tau|z),\
E_{4}(N\tau),\ E_{6}(N\tau),\ A(N\tau|z),\ B(N\tau|z)\Bigr{\\}},$
$\displaystyle z\in\\{m,\epsilon_{+},\epsilon_{-}\\}.$ (94)

Recall that $E_{4}(\tau)$ and $E_{6}(\tau)$ are Eisenstein series with weight
4 and 6 under $\Gamma_{0}(1)=\text{SL}(2,\mathbb{Z})$. Second,
$A(\tau|z)=\varphi_{0,1}(\tau,z)$ and $B(\tau|z)=\varphi_{-2,1}(\tau,z)$ are
weak Jacobi forms with weight 0, -2 and index 1. Those functions generate the
ring of elliptic modular forms and weak Jacobi forms for $\Gamma_{0}(1)$
respectively. For more details, one can refer appendix A. In order to keep the
expression simple, we will use the following notation

$\displaystyle\widehat{E}_{4}\equiv E_{4}(N\tau),$
$\displaystyle\widehat{E}_{6}\equiv E_{6}(N\tau),$
$\displaystyle\widehat{A}(z)\equiv A(N\tau|z),$
$\displaystyle\widehat{B}(z)\equiv B(N\tau|z)$ (95)

from now on. Nevertheless, for the $(C_{r})_{\pi}$ case, we propose a
different modular ansatz, which will be discussed in section 4.3.

As a remark, notice that the generators in (94) are not independent. For all
base degrees computed in this paper, we find that the unknown part of the
elliptic genera indeed lies in the subring (94), and is completely fixed after
imposing all the identities among them. For definiteness, we list those
identities explicitly in appendix A for $N=2$ and $3$ which are used in this
paper. We also indicate how to prove them for the case $N=2$. Again the $N=4$
case is very different and will be discussed separately in section 4.3.

Finally, we explain how the elliptic genus generated from (94) transforms
under $\Gamma_{0}(N)$. In general, one can express the elliptic genus in the
following form,

$\displaystyle\mathbb{E}_{\mathfrak{n}}(\tau,z)=\sum_{k}f^{(k)}(\tau,z)\cdot\widehat{f}^{(k)}(N\tau,z).$
(96)

Here, we consider the monomial expansion of the elliptic genus such that the
$k$’th monomial is given by $f^{(k)}(\tau,z)\cdot\widehat{f}^{(k)}(N\tau,z)$.
The functions $f^{(k)}(\tau,z)$ and $\widehat{f}^{(k)}(\tau,z)$ are modular
functions with weight $w$, $\hat{w}$ and index $\mathfrak{i}$,
$\hat{\mathfrak{i}}$ respectively. They transforms as follows under
$\Bigr{(}\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\Bigr{)}\in\text{SL}(2,\mathbb{Z})$,

$\displaystyle f^{(k)}({a\tau+b\over c\tau+d})$
$\displaystyle=(c\tau+d)^{w}\cdot\exp\Bigr{[}-{\pi ic\over
c\tau+d}\mathfrak{i}(z)\Bigr{]}\cdot f^{(k)}(\tau,z),$
$\displaystyle\hat{f}^{(k)}({a\tau+b\over c\tau+d})$
$\displaystyle=(c\tau+d)^{\hat{w}}\cdot\exp\Bigr{[}-{\pi ic\over
c\tau+d}\hat{\mathfrak{i}}(z)\Bigr{]}\cdot\widehat{f}^{(k)}(\tau,z).$ (97)

Then, under $\Bigr{(}\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\Bigr{)}\in\Gamma_{0}(N)$, the elliptic genus (96)
transforms as follows,

$\displaystyle\mathbb{E}_{\mathfrak{n}}({a\tau+b\over c\tau+d},{z\over
c\tau+d})$ $\displaystyle=\sum_{k}f^{(k)}({a\tau+b\over c\tau+d},{z\over
c\tau+d})\cdot\widehat{f}^{(k)}({a(N\tau)+Nb\over{c\over
N}(N\tau)+d},{z\over{c\over N}(N\tau)+d})$
$\displaystyle=(c\tau+d)^{w+\hat{w}}\cdot\exp\Bigr{[}{-\pi ic\over
c\tau+d}\Bigr{(}\mathfrak{i}(z)+{\hat{\mathfrak{i}}(z)\over
N}\Bigr{)}\Bigr{]}\cdot\mathbb{E}_{\mathfrak{n}}(\tau,z).$ (98)

Note that $c\equiv 0$ (mod $N$) plays a critical role in the transformation
above. As a result, the full elliptic genus has a weight $w+\hat{w}$ and index
$\mathfrak{i}+{\hat{\mathfrak{i}}\over N}$ under $\Gamma_{0}(N)$. This also
shows that the index of the hatted Jacobi forms should be divided by $N$.

###  4.2 Bootstrap from the vanishing bound

Recall that from the end of section 2.2, the weight and index of the elliptic
genus are

$\displaystyle w_{\mathfrak{n}}$
$\displaystyle=0,\quad\mathfrak{i}_{\mathfrak{n}}={\epsilon_{1}\epsilon_{2}\over
2}\mathfrak{n}^{T}(\Omega
D)\mathfrak{n}+(m^{2}-\epsilon_{+}^{2})\sum_{a=1}^{r}D_{aa}\mathfrak{n}_{a}\,,$
(99)

where $\Omega$ is the Cartan matrix for the Lie algebra $\mathfrak{g}$ and $D$
is the matrix
$D_{ij}=2\frac{\delta_{i,j}}{\langle\alpha_{i},\alpha_{i}\rangle}$. For
$\mathfrak{g}$ simply-laced, $D$ is the identity matrix and we are back to the
case considered in Duan:2020cta , while for $\mathfrak{g}$ non-simply-laced
which is the focus of this paper, it is non-trivial such that the combined
matrix $\Omega^{\text{s}}=\Omega D$ becomes

$\Omega^{\text{s}}_{ij}=4\frac{\langle\alpha_{i},\alpha_{j}\rangle}{\langle\alpha_{i},\alpha_{i}\rangle\langle\alpha_{j},\alpha_{j}\rangle}\,,$
(100)

which is manifestly symmetric. For reader’s convenience, we give the explicit
form of $\Omega^{\text{s}}$ for $B_{n}$, $(C_{r})_{0}$, $F_{4}$ and $G_{2}$ in
table 4.

$B_{n}$ | $(C_{r})_{0}$ | $F_{4}$ | $G_{2}$
---|---|---|---
$\scriptstyle\begin{pmatrix}2&-1&0&\cdots&0&0\\\ -1&2&-1&\cdots&0&0\\\ 0&-1&2&\cdots&0&0\\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\\ 0&\cdots&\cdots&0&2&-1\\\ 0&\cdots&\cdots&0&-1&1\\\ \end{pmatrix}$ | $\scriptstyle\begin{pmatrix}1&-\frac{1}{2}&0&\cdots&0&0\\\ -\frac{1}{2}&1&-\frac{1}{2}&\cdots&0&0\\\ 0&-\frac{1}{2}&1&\cdots&0&0\\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\\ 0&\cdots&\cdots&0&1&-1\\\ 0&\cdots&\cdots&0&-1&2\\\ \end{pmatrix}$ | $\scriptstyle\begin{pmatrix}2&-1&0&0\\\ -1&2&-1&0\\\ 0&-1&1&-\frac{1}{2}\\\ 0&0&-\frac{1}{2}&1\\\ \end{pmatrix}$ | $\scriptstyle\begin{pmatrix}\frac{2}{3}&-1\\\ -1&2\\\ \end{pmatrix}$
Table 4: $\Omega^{\text{s}}$ for non-simply-laced $\mathfrak{g}$.

Before doing actual computation, we emphasize that our modular ansatz is
motivated by the picture of twisting at the level of the elliptically fibered
CY threefold, briefly explained in section 2.3. Note that the number $N$
should coincide with the order of twisting mentioned in section 2.1.

Although we do not have a detailed understanding of how the geometry
transforms, and for the cases we computed we actually do not need the full
ring of $\Gamma_{0}(N)$ modular forms, we believe that this geometric picture
is supposed to support our modular ansatz for the twisted elliptic genera.

Without further ado, we claim that the modular ansatz takes the following
form,

$\displaystyle\mathbb{E}_{\mathfrak{n}}$
$\displaystyle={\mathcal{N}_{\mathfrak{n}}\over\mathcal{D}_{\mathfrak{n}}},\quad\mathcal{D}_{\mathfrak{n}}=\prod_{a=1}^{r}\prod_{k=1}^{n_{a}}{\theta_{1}(2\tau/\Omega^{\text{s}}_{aa},k\epsilon_{1})\over\eta^{3}(2\tau/\Omega^{\text{s}}_{aa})}\cdot{\theta_{1}(2\tau/\Omega^{\text{s}}_{aa},k\epsilon_{2})\over\eta^{3}(2\tau/\Omega^{\text{s}}_{aa})}\,,$
(101)

and the numerator $\mathcal{N}_{n}$ is an element in the ring of
$\Gamma_{0}(N)$ Jacobi modular forms,

$\mathcal{N}_{\mathfrak{n}}=\sum_{\mathfrak{p},\widehat{\mathfrak{p}}}C_{\mathfrak{p},\widehat{\mathfrak{p}}}\left(E_{4}^{p_{4}}E_{6}^{p_{6}}\prod_{i=1}^{3}A(z_{i})^{p_{-2,i}}B(z_{i})^{p_{0,i}}\right)\times\left(\widehat{E}_{4}^{\widehat{p}_{4}}\widehat{E}_{6}^{\widehat{p}_{6}}\prod_{i=1}^{3}\widehat{A}(z_{i})^{\widehat{p}_{-2,i}}\widehat{B}(z_{i})^{\widehat{p}_{0,i}}\right)\,,$
(102)

where as before the overall hat means $\widehat{f}(\tau,...)=f(N\tau,...)$ and
$z_{i}$ represent the chemical potentials as
$(z_{1},z_{2},z_{3})=(\epsilon_{+},\epsilon_{-},m)$.

As promised above, further constraints can be imposed to reduce the number of
terms in (102). The basic idea behind is that since we have roots of different
length, the BPS string wrapping the corresponding curves should to some extent
see the difference. For extreme cases, we observe experimentally that the
$\mathcal{N}_{n}$ with non-zero $n$ only for roots dual to unfolded roots
(corresponding to those with $\Omega^{\text{s}}_{aa}\neq 2$) only depends on
$\text{SL}(2,\mathbb{Z})$ forms, while with only non-vanishing wrapping
degrees for folded roots it only depends on hatted forms.

Although in general cases both forms are needed, we can still try to separate
their contributions. Let us look again at (101). Note that the numerator
$\mathcal{N}_{\mathfrak{n}}$ has weight and index

$\displaystyle w_{\mathcal{N}_{\mathfrak{n}}}$
$\displaystyle=-2\sum_{a=1}^{r}\sum_{k=1}^{n_{a}}k\,,$ (103) $\displaystyle
i_{\mathcal{N}_{\mathfrak{n}}}$
$\displaystyle=\frac{\epsilon_{+}^{2}-\epsilon_{-}^{2}}{2}\mathfrak{n}^{T}\Omega^{\text{s}}\mathfrak{n}+(m^{2}-\epsilon_{+}^{2})D\cdot\mathfrak{n}+(\sum_{a=1}^{r}D_{aa}\sum_{k=1}^{n_{a}}k^{2})(\epsilon_{+}^{2}+\epsilon_{-}^{2})\,.$

The weight $w_{\mathcal{N}_{\mathfrak{n}}}$ is naturally factorized, hence we
can demand that the total weight for two types of roots should match the
weight of the two products in (102) separately. The index
$i_{\mathcal{N}_{\mathfrak{n}}}$ is a bit more complicated. Since $D$ is a
diagonal matrix, the index for $m$ is factorized, and we can ask for
separation as above. However, because of the non-diagonal form of
$\Omega^{\text{s}}$, the index for $\epsilon_{\pm}$ mixes $n_{a}$ for both
types. The best we can demand is to separate the sum of indices
$i_{\epsilon_{+}}+i_{\epsilon_{-}}$ into roots different types. For all the
cases computed in appendix B, this set of constraints is satisfied.

To give a concrete example, let us consider the case $\mathfrak{g}=F_{4}$
($N=2$). From table 4 we learn that nodes 1 and 2 are long roots dual to the
folded roots of $E_{6}$, while nodes 3 and 4 correspond to short roots dual to
the unfolded roots of $E_{6}$. A potentially confused reader can look up the
table 2 for clarification. We rewrite the weight and index polynomial in the
following way,

$\displaystyle w_{\mathcal{N}_{\mathfrak{n}}}$
$\displaystyle=-2\left(\frac{n_{1}(n_{1}+1)}{2}+\frac{n_{2}(n_{2}+1)}{2}\right)-2\left(\frac{n_{3}(n_{3}+1)}{2}+\frac{n_{4}(n_{4}+1)}{2}\right)\,,$
(104) $\displaystyle i_{\mathcal{N}_{\mathfrak{n}}}$
$\displaystyle=(m^{2}-\epsilon_{+}^{2})(n_{1}+n_{2})+\frac{1}{2}(m^{2}-\epsilon_{+}^{2})(n_{3}+n_{4})$
$\displaystyle+\frac{\epsilon_{+}^{2}-\epsilon_{-}^{2}}{2}\mathfrak{n}^{T}\Omega^{\text{s}}\mathfrak{n}+(\sum_{k=1}^{n_{1}}k^{2}+\sum_{k=1}^{n_{2}}k^{2})(\epsilon_{+}^{2}+\epsilon_{-}^{2})+\frac{1}{2}(\sum_{k=1}^{n_{3}}k^{2}+\sum_{k=1}^{n_{4}}k^{2})(\epsilon_{+}^{2}+\epsilon_{-}^{2})\,.$

Then we require the weight and index $i_{m}$ involving $n_{1}$ and $n_{2}$ to
match the first parenthesis in (102), and those of $n_{3}$ and $n_{4}$ to
match the second. Moreover, the sum of indices
$i_{\epsilon_{+}}+i_{\epsilon_{-}}$ containing $n_{1}$ and $n_{2}$ should
match the sum in the first parenthesis in (102) while those of $n_{3}$ and
$n_{4}$ should match the second one.

Given the modified ansatz (102) and the weight and index polynomial (103), one
can impose constraints to determine the coefficients which are finite. In
earlier works Gu:2017ccq ; DelZotto:2017mee ; Duan:2020cta , the authors
successfully employed the vanishing conditions to fix them in many 6d SCFTs.
To be more precise, the elliptic genus of BPS strings in 6d
$\mathcal{N}=(2,0)$ SCFTs can be identified with the topological string
partition function on the elliptically fibered CY manifold, thanks to the
M-theory/F-theory duality. The latter is known to have a GV expansion,
encapsulating integral enumerative invariants known as the GV invariants.
Those invariants turn out to satisfy vanishing conditions, which immediately
give us infinitely many constraints on the coefficients. After twisted circle
compactification, we expect the elliptically-fibered CY manifold to change to
an $N$-section geometry, and the twisted elliptic genus should still be
identified with the topological string partition function. Therefore, it’s
still legitimate to apply the vanishing condition to determine the elliptic
genera. As a final remark, we stress that from the point of view of
topological strings, the existence of the structure of Jacobi forms (101) is
non-trivial and signals special $Sp(n,\mathbb{Z})$ monodromies in the extended
Kähler moduli space. The latter is demonstrated beautifully in Cota:2019cjx
based on homological mirror symmetry.

Here we present a self-contained summary of the vanishing bound for GV
invariants. For more details, the reader can refer Gu:2017ccq ;
DelZotto:2017mee ; Duan:2020cta . First, recall that the refined free energy
enjoys the GV expansion (48),

$\displaystyle\mathcal{F}_{\text{GV}}$ $\displaystyle=\log
Z_{\text{GV}}(\epsilon_{1},\epsilon_{2},\mathbf{t})$ (105)
$\displaystyle=\sum\limits_{2j_{\pm}\in\mathbb{N}}\sum\limits_{d\geq
1}\sum\limits_{\alpha\in
H_{2}(X,\mathbb{Z})}N^{\alpha}_{j_{-}j_{+}}{(-1)^{2(j_{-}+j_{+})}\,\chi_{j_{-}}(u^{d})\chi_{j_{+}}(v^{d})\over
v^{d}+v^{-d}-u^{d}-u^{-d}}\frac{\mathbf{Q}^{d\alpha}}{d}\,,$

with $N^{\alpha}_{j_{-}j_{+}}$ the refined GV invariants. They count the
number of BPS states with spin $j_{\pm}$ in five dimensions, so they are
always non-negative integers. Mathematically they can be defined in terms of
the motivic enumerative invariants MR2927365 ; Choi:2012jz , but they are
quite complicated to understand. To argue for the vanishing property, we adopt
the route taken in Duan:2020cta  which instead starts from the unrefined GV
invariants $I^{\alpha}_{g}$. The unrefined or self-dual limit sets $v=1$, and
we have

$\sum_{j_{\pm}\in\mathbb{N}/2}(-1)^{2j_{+}}(2j_{+}+1)\chi_{j_{-}}(u)\,N^{\alpha}_{j_{-}j_{+}}=\sum_{g=0}^{\infty}I^{\alpha}_{g}(u^{\frac{1}{2}}+u^{-\frac{1}{2}})^{2g}\,,$
(106)

where the symbol $\chi_{j}(x)$ denotes the character of an irreducible $SU(2)$
highest-weight representation with spin $j\in\mathbb{N}/2$ (50).

In mathematics, $I^{\alpha}_{g}$ is closely related to enumerative invariants
known as stable pair or Pandharipande-Thomas (PT) invariants $P_{n,\alpha}$
MR2545686 . To define the latter, we start from a stable pair consisting of
torsion sheaves $F$ having dimension one support and a non-trivial holomorphic
section $s$, which altogether captures the D6-D2-D0 bound states in physics.
The moduli space of all possible stable pairs with a fixed holomorphic Euler
characteristic $n$ and a fixed class of support $\alpha$ carries a perfection
obstruction theory MR2545686 , and the virtual fundamental cycle happens to be
zero dimensional for a CY threefold. Hence the PT invariants $P_{n,\alpha}$
just count the number of points inside the virtual fundamental cycle. For a
fixed $\alpha$, there exists an $n$ small enough such that $P_{m,\alpha}$
vanishes for any $m\leq n$. This is simply because a curve with given homology
class can not have arbitrarily small Euler characteristic, hence the moduli
space is empty.

On the other hand, $P_{n,\alpha}$ can be expressed in terms of
$I_{g}^{\alpha}$ MR2545686 . For simplicity, if we consider an irreducible
class $\beta\in H_{2}(X,\mathbb{Z})$, they are related in the following way,

$\sum_{n}P_{n,\beta}\,q^{n}=\sum_{g\geq 0}I_{g}^{\beta}q^{1-g}(q+1)^{2g-2}\,.$
(107)

In order to satisfy the vanishing property of $P_{n,\beta}$, $I_{g}^{\beta}$
must vanish when $g$ is sufficiently large. Inductively, we can extend the
vanishing property for $I_{g}^{\beta}$ to all curve classes in
$H_{2}(X,\mathbb{Z})$.

Furthermore, we assume that the refined GV invariants satisfy the so-called
checkerboard pattern Choi:2012jz : for a given curve class $\alpha$, the
combination $\left(2j_{-}+2j_{+}\right)$ is always even or odd. This pattern
also plays an important role in the blowup equations, as explained in the
section 3.1.

Let us look back at the equation (106). Decompose $\chi_{j_{-}}(u)$ into a
polynomial with variable $(u^{\frac{1}{2}}+u^{-\frac{1}{2}})^{2}$, based on
the above assumption it is not hard to derive a generic vanishing bound for
the refined GV invariants,

$N^{\alpha}_{j_{-}j_{+}}=0\quad\text{for}\ j_{-}\ \text{or}\ j_{+}\gg 0\,.$
(108)

However, the precise vanishing bound for $j_{\pm}$ depends on the detailed
knowledge of the geometry. At this stage, it is not clear to the authors how
to construct a useful presentation of the genus-one fibered CY manifolds
needed in this paper, let alone deriving the precise bound from the first
principle. Therefore, we adopt the same strategy as in Duan:2020cta . Namely,
we make some guess on the bound by looking at the BPS data available from
section 3. Moreover, since the CY threefolds consider here are obtained from
certain twist of the geometries that engineer 6d $\mathcal{N}=(2,0)$ SCFTs, we
expect the vanishing conditions to be mostly unchanged. In fact, we
essentially use the same ones found in Duan:2020cta , with only minor
simplifications.

To fix our notation, we specialize the general formula (105),

$\mathcal{F}_{\text{GV}}=\sum\limits_{\mathfrak{n},k,\mu}\sum\limits_{2j_{\pm}\in\mathbb{N}}\sum\limits_{d\geq
1}\frac{N_{j_{-}j_{+}}^{\,\mathfrak{n},k,\mu}}{k}{(-1)^{2(j_{-}+j_{+})}\,\chi_{j_{-}}(u^{d})\chi_{j_{+}}(v^{d})\over
v^{d}+v^{-d}-u^{d}-u^{-d}}Q_{m}^{\,d\mu}\,q^{dk}\,\mathbf{Q}^{d\mathfrak{n}}\,,$
(109)

with $Q_{m}=e^{-m}$, $q=e^{2\pi i\tau}$ and $\mathbf{Q}=e^{-\mathbf{v}}$
formal exponential of Kähler parameters associated to two-cycle of the
M-string mass, elliptic fiber and exceptional curves inside the base
respectively. We choose the parametrization such that $\mu$ can be negative
but with a $\mathbb{Z}_{2}$ symmetry
$N_{j_{-}j_{+}}^{\,\mathfrak{n},k,\mu}=N_{j_{-}j_{+}}^{\,\mathfrak{n},k,-\mu}$.
Our conjecture for the vanishing bound of the refined GV invariants goes as
follows:

######  Conjecture.

For genus one fibered CY threefolds that engineer 5d $\mathcal{N}=2$ gauge
theories arising from twisted circle compactification of 6d
$\mathcal{N}=(2,0)$ $ADE$ SCFTs, their refined GV invariants obey uniformly
the vanishing bound888This bound is slightly smaller than that used in
Duan:2020cta .,

$N_{j_{-}j_{+}}^{\,\mathfrak{n},k,\mu\geq 0}=0\ {\rm for}\
\left\\{\begin{array}[]{rl}2j_{-}&>k\cdot\max\\{n_{i}\\}-\frac{\mu-k-1}{2}\cdot
H(\mu-k-2),\\\ &{\rm\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ or}\\\
2j_{+}&>(k+1)\cdot\max\\{n_{i}\\}-(\mu-k)\cdot H(\mu-k-1),\\\ &{\rm\ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ or}\\\ n_{i}&=0\ \forall\,i\ \text{with}\
\Omega^{\text{s}}_{ii}=2\ \text{and}\ N\nmid k.\end{array}\right.$ (110)

The function $H(x)$ is the Heaviside step function,

$\displaystyle H(x)=\begin{cases}0,&\text{if}\ x<0\\\
1,&\text{otherwise}\,,\end{cases}$ (111)

and the last condition comes simply from the observation that the free energy
$\mathcal{F}_{\text{ref}}$ is always a power series in $q^{N}$ in that case.

Then we explain how to use (110) to determine the elliptic genus recursively.
This starts from the following obvious relation,

$\ln\mathcal{Z}_{\text{GV}}=\ln\left(1+\sum_{\mathfrak{n}\neq
0\,\in\mathbb{Z}_{\geq
0}^{r}}\mathbb{E}_{\mathfrak{n}}\,\mathbf{Q}^{\mathfrak{n}}\right)=\mathcal{F}_{\text{GV}}=\sum_{\mathfrak{n}\neq
0\,\in\mathbb{Z}_{\geq 0}^{r}}F_{\mathfrak{n}}\mathbf{Q}^{\mathfrak{n}}\,,$
(112)

with both sides regarded as power series in $\mathbf{Q}$. Expand the logarithm
in (112), we have

$F_{\mathfrak{n}_{\alpha}}=\mathbb{E}_{\mathfrak{n}_{\alpha}}+\sum_{j=2}^{\infty}\frac{1}{j!}\sum_{\mathfrak{n}_{1},\ldots,\mathfrak{n}_{j}>\boldsymbol{0}\atop\sum\mathfrak{n}_{i}=\mathfrak{n}_{\alpha}}\kappa_{(\mathfrak{n}_{\alpha})}^{\mathfrak{n}_{1},\ldots,\mathfrak{n}_{j}}\prod_{i=1}^{j}\mathbb{E}_{\mathfrak{n}_{i}}\,,$
(113)

where
$\kappa_{(\mathfrak{n}_{\alpha})}^{\mathfrak{n}_{1},\ldots,\mathfrak{n}_{j}}$
are some integers that can be computed order by order. Note that the second
term in the right-hand side only involves $Z_{\mathfrak{n}}$ with base degree
strictly smaller than $\mathfrak{n}_{\alpha}$.

It turns out that this equation is very constraining. We first supply a few
refined GV invariants at the lowest base degrees as the input data, which
e.g., can be obtained from the blowup equations in section 3. Then we impose
simultaneously the vanishing bound in the left-hand side and the modular
ansatz in the right-hand side. This already gives us enough constraints to
completely fix $F_{\mathfrak{n}_{\alpha}}$ and hence
$\mathbb{E}_{\mathfrak{n}_{\alpha}}$. Increasing the degree one step at each
time, we are able to bootstrap the elliptic genus for any value of
$\mathfrak{n}_{\alpha}$ in principle.

As always, we need to clarify one issue: is the solution to the above
bootstrap equation unique? Surely enough, the uniqueness of the solution is
equivalent to determining all the unknowns in the ansatz. For the case of 6d
$\mathcal{N}=(2,0)$ SCFTs, we have the following powerful criterion, which is
discussed in detail in Gu:2017ccq .

######  Criterion.

If either $i_{\epsilon_{+}}$ or $i_{\epsilon_{-}}$ is strictly smaller than
$-1$, then the solution to (113) must be unique. Otherwise, the ambiguous term
can be enumerated explicitly.

After looking at their proof carefully, it is not difficult to show that the
above criterion still holds for the modified ansatz and the difference just
lies in the set of ambiguous terms.

In our situation, the index of $\varepsilon_{-}$ can be written as

$i_{\epsilon_{-}}=-\frac{1}{2}\mathfrak{n}\cdot\Omega^{\text{s}}\cdot\mathfrak{n}^{\text{T}}\,,\quad\Omega^{\text{s}}_{ij}=\langle\alpha_{i}^{\vee},\alpha_{j}^{\vee}\rangle\,,$
(114)

where $\alpha^{\vee}_{i}$ correspond to the coroots of $\mathfrak{g}$. Then
the symmetric matrix $\Omega^{\text{s}}$ defines a positive-definite lattice
and hence the exceptional cases are always finite. For them, the number of
ambiguous terms is always very small after we further separate the
contribution for different length of roots, explained earlier in this
subsection. So we just need to supply a few BPS invariants if necessary.

Finally, it is worth emphasizing that finding the suitable ansatz to make the
most of the vanishing bound is nontrivial. For example, if one uses the
generators of the $\Gamma_{0}(N)$ modular forms mentioned in section 4.1, at
some low base degrees where we cannot invoke the above criterion, there appear
to be many coefficients not determined by the vanishing condition (110). On
the contrary, for cases listed in appendix B, we find that the ansatz used
here can fix all the unknowns after imposing identities among them, although
the number of terms in both ansatze are of roughly the same order.

In table 5, we list some useful information about the numerator of the
elliptic genera at low base degrees for gauge groups $G_{2}$, $B_{3}$,
$(C_{3})_{0}$ and $F_{4}$. The numerator is determined by the modular
bootstrap whose explicitly form is shown in appendix B. As an independent
check, the BPS invariants extracted from the elliptic genera is in perfect
agreement with those obtained from the blowup equations.

Gauge group | Base degree | Index | Weight | Unknowns
---|---|---|---|---
$G_{2}$ |  [reverse arrows, labels*=0,1]G2 | $\epsilon_{+}^{2}+m^{2}$ | -2 | 2
$G_{2}$ |  [reverse arrows, labels*=1,0]G2 | $\frac{1}{3}\epsilon_{+}^{2}+\frac{1}{3}m^{2}$ | -2 | 2
$G_{2}$ |  [reverse arrows, labels*=1,1]G2 | $\frac{1}{3}\epsilon_{+}^{2}+\epsilon_{-}^{2}+\frac{4}{3}m^{2}$ | -4 | 4
$G_{2}$ |  [reverse arrows, labels*=1,2]G2 | $\frac{16}{3}\epsilon_{+}^{2}+3\epsilon_{-}^{2}+\frac{7}{3}m^{2}$ | -6 | 132
$G_{2}$ |  [reverse arrows, labels*=2,1]G2 | $\frac{4}{3}\epsilon_{+}^{2}+\frac{7}{3}\epsilon_{-}^{2}+\frac{5}{3}m^{2}$ | -6 | 226
$B_{3}$ |  [labels*=1,1,1]B3 | $\frac{1}{2}\epsilon_{+}^{2}+2\epsilon_{-}^{2}+\frac{5}{2}m^{2}$ | -6 | 8
$B_{3}$ |  [labels*=2,1,1]B3 | $\frac{11}{2}\epsilon_{+}^{2}+4\epsilon_{-}^{2}+\frac{7}{2}m^{2}$ | -8 | 220
$B_{3}$ |  [labels*=1,2,1]B3 | $\frac{9}{2}\epsilon_{+}^{2}+5\epsilon_{-}^{2}+\frac{7}{2}m^{2}$ | -8 | 220
$(C_{3})_{0}$ |  [labels*=1,1,1]C3 | $\frac{9}{2}\epsilon_{+}^{2}+5\epsilon_{-}^{2}+\frac{7}{2}m^{2}$ | -6 | 10
$(C_{3})_{0}$ |  [labels*=1,1,2]C3 | $\frac{11}{2}\epsilon_{+}^{2}+\frac{7}{2}\epsilon_{-}^{2}+3m^{2}$ | -8 | 330
$F_{4}$ |  [labels*=1,1,1,1]F4 | $\frac{1}{2}\epsilon_{+}^{2}+\frac{5}{2}\epsilon_{-}^{2}+3m^{2}$ | -8 | 20
$F_{4}$ |  [labels*=2,1,1,1]F4 | $\frac{11}{2}\epsilon_{+}^{2}+\frac{9}{2}\epsilon_{-}^{2}+4m^{2}$ | -10 | 550
$F_{4}$ |  [labels*=1,2,1,1]F4 | $\frac{9}{2}\epsilon_{+}^{2}+\frac{11}{2}\epsilon_{-}^{2}+4m^{2}$ | -10 | 550
Table 5: The index polynomial, weight and unknown coefficients for the
numerator of the elliptic genera except $(C_{r})_{\pi}$ case.

###  4.3 $(C_{r})_{\pi}$ theories

For the last case $(C_{r})_{\pi}$, which is called non-geometric in
Bhardwaj:2019fzv , the situation becomes much subtler, which is the reason why
we treat it in a separate subsection.

From (44) at the end of section 2.2, we learn that its weight and index can be
written as

$\displaystyle w_{\mathfrak{n}}$
$\displaystyle=0,\quad\mathfrak{i}_{\mathfrak{n}}={\epsilon_{1}\epsilon_{2}\over
2}\mathfrak{n}^{T}(\Omega^{\text{s}})\mathfrak{n}+\frac{1}{2}(m^{2}-\epsilon_{+}^{2})\sum_{a=1}^{r}\mathfrak{n}_{a}\,,$
(115)

where the $\Omega^{\text{s}}$ takes the form,

$\Omega^{\text{s}}=\begin{pmatrix}1&-\frac{1}{2}&0&\cdots&0&0\\\
-\frac{1}{2}&1&-\frac{1}{2}&\cdots&0&0\\\ 0&-\frac{1}{2}&1&\cdots&0&0\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\\
0&\cdots&\cdots&0&1&-\frac{1}{2}\\\
0&\cdots&\cdots&0&-\frac{1}{2}&\frac{1}{2}\\\ \end{pmatrix}\,.$ (116)

In this case, all the nodes come from $\mathbb{Z}_{2}$ outer-automorphism of
$A_{2r}$, we would expect that for all possible base degrees the $\tau$
variable in the ansatz will be multiplied by two. As a result, we propose its
modular ansatz should be,

$\displaystyle\mathbb{E}_{\mathfrak{n}}$
$\displaystyle={\mathcal{N}_{\mathfrak{n}}\over\mathcal{D}_{\mathfrak{n}}},\quad\mathcal{D}_{\mathfrak{n}}=\prod_{a=1}^{r}\prod_{k=1}^{n_{a}}{\theta_{1}(2\tau,k\epsilon_{1})\over\eta^{3}(2\tau)}\cdot{\theta_{1}(2\tau,k\epsilon_{2})\over\eta^{3}(2\tau)}\,,$
(117)

It’s interesting to remark that the $\mathcal{D}_{\mathfrak{n}}$ does not
follow the general rule in (101) which works perfectly for other cases.

The structure of the numerator $\mathcal{N}_{\mathfrak{n}}$ is different,
which should enjoy the $\Gamma_{0}(4)$ modular group rather than the naive
$\Gamma_{0}(2)$ from the Dynkin diagram. The main reason is that, as explained
in section 2.1, the whole automorphism group is actually $\mathbb{Z}_{4}$,
leading to the Jacobi form of $\Gamma_{0}(4)$. Following the above logic, we
conjecture that the numerator $\mathcal{N}_{\mathfrak{n}}$ is an element

$\displaystyle\mathcal{N}_{\mathfrak{n}}\in$
$\displaystyle\mathbb{C}[E^{(2)}_{2}(\tau),E^{(4)}_{2}(\tau),E_{4}(\tau),E_{6}(\tau),A(4\tau|z_{i}),B(4\tau|z_{i})]\cdot\mathbb{C}[E_{4}(2\tau),E_{6}(2\tau),A(2\tau|z_{i}),B(2\tau|z_{i})],$

such that the total weight and index of $Z_{\mathfrak{n}}$ satisfy (115). Here
$z_{i}\in(\epsilon_{+},\epsilon_{-},m)\,.$

Unlike other cases, this ansatz contains too many unfixed terms at low base
degrees after imposing the previous vanishing conditions. Also, the number of
coefficients in the ansatz is growing so fast that it is already quite hard to
determine the case of base degree two. The main reason is due to the full
generators we are using for $\Gamma_{0}(4)$ elliptic modular forms. In
practice, we fix all of them consistently by further comparing them with the
result obtained from the blowup equations. Below, we will explain this in more
detail for the $(C_{1})_{\pi}$ theory. Still, it serves as strong evidence for
the existence of elliptic genera invariant under the $\Gamma_{0}(4)$ modular
group. In table 6, we list some useful information about the the
$(C_{r})_{\pi}$ elliptic genera at low base degrees, whose explicitly form is
shown in appendix B.

Base degree | Index | Weight | Unknowns
---|---|---|---
[labels*=1]A1 | $\frac{1}{4}\epsilon_{+}^{2}+\frac{1}{4}\epsilon_{-}^{2}+\frac{1}{2}m^{2}$ | -2 | 30
[labels*=2]A1 | $\frac{5}{2}\epsilon_{+}^{2}+\frac{3}{2}\epsilon_{-}^{2}+m^{2}$ | -4 | 21707
[labels*=1,0]C2 | $\frac{1}{2}\epsilon_{+}^{2}+\frac{1}{2}m^{2}$ | -2 | 2
[labels*=1,1]C2 | $\frac{1}{4}\epsilon_{+}^{2}+\frac{3}{4}\epsilon_{-}^{2}+m^{2}$ | -4 | 60
Table 6: The index polynomial, weight and unknown coefficients for the
numerator of the $(C_{r})_{\pi}$ elliptic genera.

As a non-trivial check of our ansatz, we fix the modular expression up to base
degree two for the $(C_{1})_{\pi}$ theory. In the rank one case, one would
expect that the numerator is generated by the following $\Gamma_{0}(4)$
modular and Jacobi forms

$\mathcal{N}_{\mathfrak{n}}\in\mathbb{C}[E^{(2)}_{2}(\tau),E^{(4)}_{2}(\tau),E_{4}(\tau),E_{6}(\tau),A(4\tau|z_{i}),B(4\tau|z_{i})]\,.$
(118)

Indeed, as shown in table 13 of appendix B, the one-string elliptic
genus999Here we use the notation
$\widehat{A}_{z_{i}}=\varphi_{-2,1}(4\tau,z_{i})$,
$\widehat{B}_{z_{i}}=\varphi_{0,1}(4\tau,z_{i})$

$\mathbb{E}_{1}(\tau,\epsilon_{1},\epsilon_{2},m)=\frac{(E_{2}^{(2)}-E_{2}^{(4)})(\widehat{A}_{m}\widehat{B}_{+}-\widehat{A}_{+}\widehat{B}_{m})(\widehat{A}_{m}\widehat{B}_{-}-\widehat{A}_{-}\widehat{B}_{m})}{2^{8}\,3^{2}\,\varphi_{-2,1}(2\tau,\epsilon_{1})\varphi_{-2,1}(2\tau,\epsilon_{2})}$
(119)

contains elements $E_{2}^{(2)}$ and $E_{2}^{(4)}$ of $\Gamma_{0}(4)$. We
verify that this expression coincides with the instanton partition function
from ADHM up to 5-instantons. And we verify further with the one-string
elliptic blowup equation

$\begin{split}\theta_{2}(&\tau,m-\epsilon_{+})\mathbb{E}_{1}(\tau,\epsilon_{1},\epsilon_{2},m)=\\\
&\theta_{2}(\tau,m+\epsilon_{-})\mathbb{E}_{1}(\tau,\epsilon_{1},\epsilon_{2}-\epsilon_{1},m-\epsilon_{1}/2)+\theta_{2}(\tau,m-\epsilon_{-})\mathbb{E}_{1}(\tau,\epsilon_{1}-\epsilon_{2},\epsilon_{2},m-\epsilon_{2}/2)\end{split}$

up to $q^{21}$ order.

For base degree two, from table 6 we have $21707$ possible generators, making
it hard to fix the coefficients. From the gauge theoretic point of view, at
the leading order the two strings contain the W-boson contribution, which is
the same as a single string for $(C_{1})_{0}$ theory. Further noticing that
the index of the elliptic genus at two-strings of $(C_{1})_{\pi}$ theory is
the same as that of the single $(C_{1})_{0}$ string, we may expect that the
two-string elliptic genus contains the following contribution101010It is not
difficult to show that this expression is invariant under the $\Gamma_{0}(4)$
modular group.

$\frac{1}{2}\left(\mathbb{E}_{1}^{(C_{1})_{0}}(\tau,\epsilon_{1},\epsilon_{2},m)+\mathbb{E}_{1}^{(C_{1})_{0}}(\tau+1/2,\epsilon_{1},\epsilon_{2},m))\right)\,.$
(120)

The statement is indeed true, and we observe that the remaining part can be
factorized as

$\begin{split}\mathbb{E}_{2}^{(C_{1})_{\pi}}(\tau,\epsilon_{1},\epsilon_{2},m)&=\frac{1}{2}\left(\mathbb{E}_{1}^{(C_{1})_{0}}(\tau,\epsilon_{1},\epsilon_{2},m)+\mathbb{E}_{1}^{(C_{1})_{0}}(\tau+1/2,\epsilon_{1},\epsilon_{2},m))\right)\\\
&+\frac{(\widehat{A}_{m}\widehat{B}_{+}-\widehat{A}_{+}\widehat{B}_{m})^{2}(\widehat{A}_{m}\widehat{B}_{-}-\widehat{A}_{-}\widehat{B}_{m})^{2}}{\prod_{k=1}^{2}\varphi_{-2,1}(2\tau,k\epsilon_{1})\varphi_{-2,1}(2\tau,k\epsilon_{2})}\mathcal{I}(\tau,\epsilon_{1},\epsilon_{2})\,.\end{split}$
(121)

We verify the structure of (121) by solving the two-string elliptic blowup
equation

$\begin{split}\theta_{2}(&\tau,m-\epsilon_{+})\mathbb{E}_{2}(\tau,\epsilon_{1},\epsilon_{2},m)=\\\
&\theta_{2}(\tau,m+\epsilon_{+})\mathbb{E}_{1}(\tau,\epsilon_{1},\epsilon_{2}-\epsilon_{1},m-\epsilon_{1}/2)\mathbb{E}_{1}(\tau,\epsilon_{1}-\epsilon_{2},\epsilon_{2},m-\epsilon_{2}/2)\\\
&+\theta_{2}(\tau,m-\epsilon_{+}+2\epsilon_{1})\mathbb{E}_{2}(\tau,\epsilon_{1},\epsilon_{2}-\epsilon_{1},m-\epsilon_{1}/2)+\theta_{2}(\tau,m-\epsilon_{+}+2\epsilon_{2})\mathbb{E}_{2}(\tau,\epsilon_{1}-\epsilon_{2},\epsilon_{2},m-\epsilon_{2}/2)\,,\\\
\end{split}$

and get the Fourier expansion of $\mathcal{I}(\tau,\epsilon_{1},\epsilon_{2})$
up to $q^{20}$ order. Now it is much easier to find the modular expression of
$\mathcal{I}(\tau,\epsilon_{1},\epsilon_{2})$, which is expected to be a
$\Gamma_{0}(4)$ Jacobi form with weight $0$ and index
$2\epsilon_{+}^{2}+\epsilon_{-}^{2}$, with only $2221$ generators. At the
$q^{14}$ order,111111The reason why we have to go to this order is that there
exist a combination of the generators which contains a factor
$(E_{2}^{(2)}-E_{2}^{(4)})^{13}\sim q^{13}$. we manage to fix all of the
unknowns and find

$\mathcal{I}(\tau,\epsilon_{1},\epsilon_{2})=(E_{2}^{(2)}-E_{2}^{(4)})^{2}\cdot\mathbb{C}\left[E_{2}^{(2)},E_{2}^{(4)},A(4\tau,\epsilon_{\pm}),B(4\tau,\epsilon_{\pm})\right]\,.$
(122)

Its expression is listed in appendix B. Interestingly, this does not depend on
$E_{4}(\tau)$ and $E_{6}(\tau)$. We further check it up to $q^{20}$ order with
the two-string elliptic blowup equation, and find a perfect agreement.

##  5 6d twisted theories in the Cardy limit

In this section, we explore the Cardy limit of 6d $(2,0)$ theories on
$\mathbb{R}^{4}\times T^{2}$ with an outer-automorphism twist. The Cardy limit
is defined as a limit where momenta are much larger than the central charge of
the CFT Cardy:1986ie ; Kim:2019yrz . The free energy in the Cardy limit is
usually called the Cardy formula, and it probes the high energy spectrum of
the CFT. Furthermore, Cardy formulas have been used to account the Bekenstein-
Hawking entropy of supersymmetric black holes in various dimensions
Strominger:1996sh ; Choi:2018hmj . In the case of 6d, the Cardy formulas of
$(2,0)$ $ADE$ theories on $\mathbb{R}^{4}\times T^{2}$ were studied in
Lee:2020rns ; Duan:2020cta . The authors obtain the Cardy free energy by
summing over the elliptic genera using continuum approximation and S-duality.
Here, we shall expand their approach to all 6d $(2,0)$ theories, including
twisted ones.

In order to explore the large momenta limit of the BPS partition function
(34), we need to set the momentum-conjugate chemical potentials to be small.
On $\mathbb{R}^{4}\times T^{2}$, this can be achieved by setting

$\displaystyle|\epsilon_{1,2}|\to 0,\quad|\tau|\to 0\,.$ (123)

First, $|\epsilon_{1,2}|\to 0$ is the well-known Seiberg-Witten prepotential
limit Seiberg:1994rs ; Nekrasov:2002qd . It is also called thermodynamic limit
where the volume of $\mathbb{R}^{4}$ becomes infinite. In this limit, the
leading free energy on $\mathbb{R}^{4}\times T^{2}$ is proportional to
$\text{vol}(\mathbb{R}^{4})\sim{1\over\epsilon_{1}\epsilon_{2}}$. Second,
$|\tau|\to 0$ is the limit where KK momentum becomes large. Recall that
$\tau=i{R_{\text{time}}\over R_{\text{KK}}}$ is inversely proportional to the
KK circle radius. Therefore, the KK circle is effectively decompactified, and
the full 6d physics is visible in $|\tau|\to 0$ limit. In this limit, the
leading free energy on $\mathbb{R}^{4}\times T^{2}$ is proportional to the
volume of the spatial circle in $T^{2}$, i.e.
$\text{vol}(R_{\text{KK}})\sim{1\over\tau}$ Cardy:1986ie . Therefore, the
leading free energy in the Cardy limit should be $\log
Z\sim\mathcal{O}({1\over\epsilon_{1}\epsilon_{2}\tau})$.

When taking the Cardy limit (123), we should decide the phase of the chemical
potentials which can generically have complex values. Here, we shall consider
the following regime of the chemical potentials,

$\displaystyle\epsilon_{1}>0,\quad\epsilon_{2}<0,\quad 2\pi i\tau<0.$ (124)

For simplicity, we set three fugacities $Q_{1}=e^{-\epsilon_{1}}$,
$Q_{2}=e^{-\epsilon_{2}}$, and $Q_{\tau}=e^{2\pi i\tau}$ to be purely real.
First, setting $2\pi i\tau<0$ corresponds to setting $Q_{\tau}<1$ which is
needed for the convergence of KK expansion. Second, setting different signs
for $\epsilon_{1,2}$ might seem peculiar, but it is nothing but a slight
deviation from the unrefined setting $\epsilon_{1}=-\epsilon_{2}$.

In order to investigate the Cardy limit, let us take $\tau\to 0$ limit first.
Recall that, according to our ansatz (102), the elliptic genus of twisted
M-string (except $(C_{r})_{\pi}$ theory) can be written as follows,

$\displaystyle\mathbb{E}_{\mathfrak{n}}(\tau,z)={1\over\mathcal{D}_{\mathfrak{n}}(\tau,z)\cdot\widehat{\mathcal{D}}_{\mathfrak{n}}(n_{G}\tau,z)}\times\sum_{a=1}^{|\mathcal{S}|}\mathcal{N}^{(a)}_{\mathfrak{n}}(\tau,z)\cdot\widehat{\mathcal{N}}^{(a)}_{\mathfrak{n}}(n_{G}\tau,z).$
(125)

Here, unhatted parts have modular parameter $\tau$, and hatted parts have
modular parameter $n\tau$, where $n$ is the order of the outer-automorphism.
Also, $a$ is a monomial index which enumerates the monomials in the numerator.
The asymptotic behavior of the elliptic genus at $\tau\to 0$ can be obtained
by exploiting its property under the S-transformation. Unlike to the simply-
laced case, we have to be careful about the S-transformation. The elliptic
genus $Z_{\mathfrak{n}}$ is a modular form under $\Gamma_{0}(n_{G})$ which is
a congruence subgroup of $\text{SL}(2,\mathbb{Z})$. If $n_{G}\neq 1$, the
S-transformation action $(\begin{smallmatrix}0&-1\\\ 1&0\end{smallmatrix})$ is
not an element of $\Gamma_{0}(n_{G})$. Therefore, the twisted elliptic genus
does not have the S-duality property.

However, one can still apply S-transformation to the elliptic genus and
observe how it behaves. The elliptic genus (125) has two parts that depend on
$\tau$ and $n_{G}\tau$ respectively. Instead of the conventional
S-transformation, we shall apply the following ‘separate S-transformation’ to
our elliptic genus of the form (125),

$\displaystyle\mathcal{D}(\tau,z),\ \mathcal{N}(\tau,z)\quad$
$\displaystyle\to\quad\mathcal{D}(-{1\over\tau},{z\over\tau}),\
\mathcal{N}(-{1\over\tau},{z\over\tau})\,,$
$\displaystyle\widehat{\mathcal{D}}(n_{G}\tau,z),\
\widehat{\mathcal{N}}(n_{G}\tau,z)\quad$
$\displaystyle\to\quad\widehat{\mathcal{D}}(-{1\over n_{G}\tau},{z\over
n_{G}\tau}),\ \widehat{\mathcal{N}}(-{1\over n_{G}\tau},{z\over n_{G}\tau}).$
(126)

Recall that $\tau$-dependent part and $n_{G}\tau$-dependent part both have
zero weights. Let us assume that $\tau$-part has index $\mathfrak{i}$, and
$n_{G}\tau$-part has index $\hat{\mathfrak{i}}$. By applying the separate
S-duality transformation above, the elliptic genus changes as follows,

$\displaystyle\mathbb{E}_{\mathfrak{n}}(\tau,z)$
$\displaystyle=\exp\Bigr{[}-{1\over 2\pi
i\tau}\Bigr{(}\mathfrak{i}+{\widehat{\mathfrak{i}}\over