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RIKEN-TH-181\\
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    \begin{Large}
       \begin{center}
         { \bf Triality in $SU(2)$ Seiberg-Witten theory \\
and \\
Gauss hypergeometric function
}
       \end{center}
        \end{Large}

\begin{center}

Ta-Sheng Tai\footnote[1]{
        e-mail address: tasheng@riken.jp
    }

\end{center}
\renewcommand{\thefootnote}{\arabic{footnote}}
\begin{small}
\begin{center}
{\it Theoretical Physics Laboratory, RIKEN,
                    Wako, Saitama 351-0198, JAPAN}
\end{center}
 \end{small}
\begin{abstract}
\noindent
{\normalsize
Through AGT conjecture,
we show how triality observed in $\N=2$ $SU(2)$ $N_f=4$ QCD can be interpreted geometrically as
the interplay among six of Kummer's twenty-four
solutions belonging to one fixed Riemann
scheme in the context of
hypergeometric differential equations.
We also stress that our
presentation is different from the usual crossing
symmetry of Liouville
conformal blocks, which is described by the
connection coefficient in the case
of hypergeometric functions. Besides, upon
solving hypergeometric differential
equations at the zeroth order
by means of the WKB method, a curve
(thrice-punctured Riemann sphere) emerges. The
permutation between these six
Kummer's solutions then boils down to the outer automorphism
of the associated curve.
}
\end{abstract}
\setstretch{1.1}

\section{Introduction}
Adding
$N_f$ massless fundamental hypermultiplets (flavors) to
pure $\N=2$ $SU(2)$ Yang-Mills theory results in
$SO(2N_f)$
flavor symmetry which gets
enhanced to $Spin(2N_f)$ at the quantum level. This is
due to the fact that
monopoles in the low-energy
Coulomb phase transform as spinors of
$Spin(2N_f)$ once
$2N_f$ fermionic zero-modes
(collective coordinate) on them are quantized.
In particular, when $2N_c=N_f=4$ where
both vanishing one-loop $\beta$-function
and exact scale invariance follow one sees that
the outer automorphism group
${\mathbf S}_3$ of $Spin(8)$, a double-cover of $SO(8)$,
gets realized as a kind of $S$-duality, i.e. $triality$.
Namely, its action in
Coulomb phase permutes three fundamental BPS objects
corresponding to three eight-dimensional
irreducible representations of $SO(8)$,
i.e. $(\mathbf{v,s,c})\equiv
{\text {(electron, monopole, dyon)}}$.
Our aim in this short letter is to clarify its geometric
origin in terms of Gauss hypergeometric functions.

In \cite{Seiberg:1994aj}
Seiberg and Witten wrote down explicitly
the $SU(2)$ $N_f=4$ Seiberg-Witten (SW) curve $\Sigma$
parameterized by
four bare flavor masses and
$\tau_0 \equiv\dfrac{\theta}{\pi} +
\dfrac{8\pi i}{g^2}$ (marginal bare gauge coupling)
\footnote{ In the presence of
fundamental flavors, $\tau_0$'s normalization
deviates from
the pure $\N=2$ one by a factor 2.}.
They discovered the following exotic transformation rule
attributable to the property of triality:
\begin{eqnarray}
S: ~\tau_0 \to -\frac{1}{\tau_0}, ~~~~~~
\begin{cases}
m_1 \to \frac{1}{2}(m_1 + m_2 + m_3 - m_4)\\
m_2 \to \frac{1}{2}(m_1 + m_2 - m_3 + m_4)\\
m_3 \to \frac{1}{2}(m_1 - m_2 + m_3 - m_4)\\
m_4 \to \frac{1}{2}(-m_1 + m_2 + m_3 - m_4)
\label{ma1}
\end{cases}
\end{eqnarray}
\begin{eqnarray}
T: ~\tau_0 \to {\tau_0 +1}, ~~~~~~
\begin{cases}
m_1 \to m_1\\
m_2 \to m_2\\
m_3 \to m_3\\
m_4 \to -m_4
\label{ma}
\end{cases}
\end{eqnarray}
under which $\Sigma$ is kept invariant.
This is because
combinations of $S$ and $T$ together
generate $SL(2,{\mathbb Z})/\Gamma(2)=\{ I, ~S, ~T,
~ST, ~TS, ~STS \}$ which is identical to
the outer automorphism group ${\mathbf S}_3$. Notice that
these rules arising from observation still lack
rigorous derivation. It is then seen that
full $SL(2,{\mathbb Z})$ invariance w.r.t. $\tau_0$
shrinks to a smaller $\Gamma(2)$ one by including flavor
mass deformations unless
\eqref{ma1} and
\eqref{ma} are taken into account.
Explaining triality in a more
geometric way has been attempted ever since
$\N=2$ $SU(2)$ low-energy Coulomb phase
dynamics got rephrased in, say,
Vafa's F-theory setup
\cite{Vafa:1996xn, Sen:1996vd} or Gaiotto's picture.
Either one seems promising
because totally geometric reformulations of a class of $\N=2$
theories were provided there.

The latter due to Gaiotto's seminal paper \cite{Gaiotto:2009we}
interpreted nicely $S$-duality group of a large family of
$\N=2$ superconformal quiver gauge theories
as physically equivalent ways of deforming certain genus-$g$
$n$-punctured Riemann surface $C_{g,n}$
\footnote{Though at first sight $C_{g,n}$ seems an
ultra-violet object, a $r$-sheeted cover of
it turns out to be the infra-red Seiberg-Witten curve
of $A_{r\text{-}1}$-type $SU(r)$ SCFTs. }. Also, $C_{g,n}$'s
moduli (Teichm{\"u}ller) space is accordingly identified with
the space of a set of ultra-violet coupling constants, say,
$q_{UV }=e^{\pi i\tau_{UV}}$.
Based on his idea,
the previous ${\mathbf S}_3$ has to be
thought of as
${\mathbf S}_4/({\mathbb Z}_2
\times {\mathbb Z}_2)$
(${\mathbb Z}_2
\times {\mathbb Z}_2$: Klein four-group or Vierergruppe)
which manages to permute marked punctures on
$C_{0,4}$ responsible for the $2N_c=N_f=4$ case.
More precisely, from two $SO(4)$ ($SO(4)\times SO(4)\sim SU(2)_a \times SU(2)_b \times
SU(2)_c \times SU(2)_d$) of $SO(8)$ one can
decompose $\mathbf{8}$ as
\begin{eqnarray*}
\mathbf{8}\sim (\mathbf{2}_a\otimes \mathbf{2}_b)
\oplus (\mathbf{2}_c\otimes \mathbf{2}_d);
\end{eqnarray*}
therefore the action of
triality exchanging three $\mathbf{8}$'s
results in permuting punctures
labeled by $SU(2)_\xi$
$(\xi=a,b,c,d)$ respectively.
As pointed out later by Alday, Gaiotto
and Tachikawa \cite{Alday:2009aq}
\footnote{See also
\cite{Alday:2010vg}-\cite{Nanopoulos:2009xe} for recent developments along AGT conjecture.}, further associating
every puncture with a mass parameter
$\mu$ subject to
\begin{eqnarray}
&&m_1=\mu_a +\mu_d - \frac{Q}{2},~~~~~~~
m_2=-\mu_a+\mu_d +\frac{Q}{2},\nonumber\\
&&m_3=\mu_c +\mu_b -\frac{Q}{2},
~~~~~~~m_4=-\mu_c+\mu_b +\frac{Q}{2},
\label{mu0}
\end{eqnarray}
one easily agrees
that \eqref{ma1} and \eqref{ma} can be completely accounted for
by permutations of $\mu$'s with $Q=0$. Henceforth,
\eqref{mu0} nowadays referred to as
``AGT dictionary" opens up
a new perspective for understanding triality geometrically
\footnote{See \cite{Eguchi} for another geometric interpretation of triality resulting
from E-string formalism.}.
In fact, by introducing F-theoretically a vev
of an $SO(8)$ adjoint scalar field $\Phi$ living on D7-branes, say,
\begin{eqnarray}
\langle \Phi \rangle=\left(
\begin{array}{cccc}
i\sigma_2 m_1&&&  \\
&i\sigma_2 m_2&& \\
&&i\sigma_2 m_3 & \\
&&&i\sigma_2 m_4 \\
\end{array}
\right), ~~~~~~~~~~
\sigma_2=\left(
\begin{array}{cc}
0&-i  \\
i&0\\
\end{array}
\right) ,
\end{eqnarray}
$\mu$'s (when $Q=0$) just stand for its diagonal
Cartan elements
w.r.t. $SO(4)\times SO(4)$ decomposition.
In addition, $\mu$'s get related to
momenta of 2D Liouville primary fields
$V_\mu=e^{2\mu \varphi}$ ($\varphi$: Liouville field)
with a conformal dimension
$\Delta(\mu)=\mu(Q-\mu)$. A zero background charge $Q=b+ \dfrac{1}{b}=\dfrac{\epsilon_1 + \epsilon_2}{\sqrt{\epsilon_1 \epsilon_2}}=0$ required here
corresponds to a 4D physical field theory limit  $\epsilon_1=-\epsilon_2$
\footnote{$\epsilon_{1,2}$ are related to the size
of a unit rectangle in Young tableaux appearing in Nekrasov's partition functions.}.

While these arguments do render a satisfactory
explanation of the geometric origin of triality, we instead
would like to explore another possibility using
Gauss hypergeometric functions.
What makes this accessible is again
due to AGT conjecture
which proposes an equivalence between a
2D Liouville
conformal block ${\cal B}$ defined on $C_{g,n}$
and the instanton part of Nekrasov's
partition function $Z_{\text {inst}} [C_{g,n}]$ of
a 4D $\N=2$ $A_1$-type SCFT. Under special
circumstances, the four-point spherical
${\cal B}[C_{0,4}]$
satisfies a hypergeometric differential equation (HDE).
Therefore, based on the equality $Z_{\text {inst}} [C_{0,4}]
={\cal B}[C_{0,4}]$ with $q_{UV}$ regarded as the
cross-ration of four punctures, one can interpret
\eqref{ma1} and \eqref{ma} as
interchanging solutions of a HDE
fixed by some Riemann scheme because
$Z_{\text {inst}}({\bf a}, \vec{m}, q_{UV}(\tau_0),
\epsilon_{1},\epsilon_{2})$ (${\bf a}$:
Coulomb phase parameter) itself
contains the exact solution (SW curve) to
infra-red dynamics.
To conclude, we find
that triality generates six out of
Kummer's twenty-four solutions.
By grouping them properly into three pairs, each pair
just spans the basis of solutions belonging to
respectively $(0,1,\infty)$
known as regular singularities of a second-order HDE.

This letter is organized as
follows.
In section 2, we review necessary aspects
about hypergeometric functions.
Especially, the elliptic $lambda$ function relating
$\tau_{UV}$ to $\tau_0$
will play
a quite profound role in latter
discussions. In section 3, we show how triality can be interpreted as
the interplay among six of Kummer's twenty-four solutions
via AGT conjecture.
A summary is given in section 4.

\section{Preliminaries}

\subsection{Hypergeometric function }
Let us first recall some main features of Gauss hypergeometric functions and
their relation to the modular curve
$X_2={\mathbb H}/\Gamma(2)$ being isomorphic to
${\mathbb C} \backslash \{ 0,1\}$. See for example \cite{MY} for
details. Here, $X_N$ is in general a
noncompact Riemann surface whilst ${\mathbb H}$
stands for the upper half-plane. $\Gamma(N)$ denotes the
level $N$ principal congruence subgroup of $SL(2,\mathbb{Z})$:
\begin{eqnarray*}
\Gamma(N) \ni \left(
\begin{array}{cc}
 a& b  \\
 c & d
\end{array}
\right)\equiv
\left(
\begin{array}{cc}
 1& 0  \\
 0 & 1
\end{array}
\right) ~ ~ {\text {mod}}~ ~ N, ~ ~~ ~~ ~ad-bc=1.
\label{}
\end{eqnarray*}
Getting familiar with these stuffs serves as our cornerstone of clarifying
the role of triality observed
in the $2N_c=N_f=4$ SW theory
by means of Gauss hypergeometric functions $y(z)$, solutions of
a second-order linear ODE:
\begin{eqnarray}
z(1-z)y(z)'' + \big( c-(a+b+1)z\big) y(z)' -aby(z)=0, ~ ~~ ~~
z\in \mathbb{C}.
\label{gh}
\end{eqnarray}
Meanwhile, there are three $regular$ singularities
$(0,1,\infty)$ near each of which two linearly
independent solutions to \eqref{gh} exist
\footnote{At points other than $(0,1,\infty)$,
\eqref{gh} can be simplified to $y(z)''=0$ by changes of variables.}. That is, at $z=0$
\begin{eqnarray*}
\begin{cases}
y_{01}= {}_2F_1 (a,b,c;z), \\
y_{02}= z^{1-c}{}_2F_1 (a-c+1, b-c+1,2-c;z);
\end{cases}
\end{eqnarray*}
at $z=1$
\begin{eqnarray*}
\begin{cases}
y_{11}= {}_2F_1 (a,b,a+b-c+1;1-z) , \\
y_{12}= (1-z)^{c-a-b}{}_2F_1 (c-a , c-b, c-a-b+1; 1-z);
\end{cases}
\end{eqnarray*}
at $z=\infty$
\begin{eqnarray*}
\begin{cases}
y_{\infty 1}= (-z)^{-a} {}_2F_1 (a,a-c+1,a-b+1;z^{-1}), \\
y_{\infty 2}= (-z)^{-b} {}_2F_1 (b, b-c+1, b-a+1;z^{-1}).
\end{cases}
\end{eqnarray*}
According to the local exponent of $z$ around each singularity,
three pairs of solutions listed above can be summarized by
Table \ref{t1} (Riemann scheme) in the context of Fuchsian linear differential equations.
Also, due to Fuchs relation summing up all
entries inside the last two rows of
Table \ref{t1} gives zero.
\begin{table}[h]
\caption{Riemann scheme}
  \label{t1}
  \begin{center}
    \begin{tabular}{|c|c|c|} \hline
$z=0$& $z=1$& $z=\infty$ \\ \hline
0&0&$a$\\ \hline
$1-c$& $c-a-b$ & $b$ \\ \hline
    \end{tabular}
  \end{center}
\end{table}
As a matter of fact, each pair of solutions
can be transformed to one another
through suitable two by two matrices (connection coefficients); for instance,
\begin{eqnarray}
\label{cc}
&&(y_{01}, y_{02})= (y_{11}, y_{12})P_{01}, \nonumber \\
&&P_{01} =\left(
\begin{array}{cc}
 \dfrac{\Gamma(c)\Gamma(c-a-b)}{\Gamma(c-a)\Gamma(c-b)}&
 \dfrac{\Gamma(2-c)\Gamma(c-a-b)}{\Gamma(1-a)\Gamma(1-b)}  \\
 \dfrac{\Gamma(c)\Gamma(a+b-c)}{\Gamma(a)\Gamma(b)}
 &\dfrac{\Gamma(2-c)\Gamma(a+b-c)}{\Gamma(a-c+1)\Gamma(b-c+1)}
\end{array}
\right).
\end{eqnarray}
Furthermore,
$y(z)$ suffers monodromies like
\begin{eqnarray*}
&&(y_{\ell 1}, y_{\ell 2})\to (y_{\ell 1}, y_{\ell 2})M ,
~~~~~~M \in \pi_1 ({\mathbb C} \backslash \{ 0,1\})
\end{eqnarray*}
when winding around each singularity.
That there exists a group homomorphism between
the fundamental group
$\pi_1 ({\mathbb C} \backslash \{ 0,1\})$ and
$GL(2, {\mathbb{C}})$ leads to
$M(\gamma, z_0)\in GL(2, {\mathbb{C}})$ (modulo conjugation)
for each path $\gamma$ given a reference point $z_0$.
Due to $\gamma_1 \cdot \gamma_2=\gamma_\infty$ one is able to
establish $M(\gamma_1, z_0) \cdot M(\gamma_2, z_0)=
M(\gamma_\infty, z_0)$
where
$\gamma_1$ $(\gamma_2)$ is designated to
surround the singularity $z=0$ ($z=1$)
counterclockwise.

\subsection{ Schwarz map}

Next, we proceed to consider
the ratio
${\cal D}_\ell=\dfrac{y_{\ell 1}}{y_{\ell 2}}$
defining the famous triangle Schwarz map, a special
case of conformal Schwarz-Christoffel maps which
bring the upper half-plane ${\mathbb H}$ to certain
$n$-vertex polygon. In fact, the setup under consideration can be cast into
the so-called uniformization problem for the simplest case--
a three-punctured sphere ${\mathbb C} \backslash \{ 0,1\}$.
One can arrange \eqref{gh} into a $Q$-form
\begin{eqnarray*}
\partial_z^2 y + \dfrac{1}{2}\{ \rho ,z \}y=0, ~~~~~~
\{ \rho ,z \}:~ \text {Schwarzian derivative of}~ \rho, ~~~~~~\rho=\dfrac{y_\varrho}{y_\varsigma}.
\end{eqnarray*}
There, the multi-valued $\rho$ (ratio of two independent solutions ) induces
a map ${\mathbb C} \backslash \{ 0,1\} \to$ (unit disc)
/${ \cal G}$ with branching points $(0,1,\infty)$.
${\cal G} \subset SU(1,1)$
denotes the monodromy group of
$\rho$ as will soon be seen.

While one takes $\rho={\cal D}_{\ell}$, it naively
maps ${\mathbb H}$
to a triangle on a Riemann sphere ${\mathbb P}^1$ bounded by circular arcs. Connection coefficients $P$'s can thus be thought of as applying
M{\"o}bius transformations
(automorphism group of ${\mathbb P}^1$) to the triangle.
Meanwhile, $(0,1,\infty)$ on ${\mathbb H}$
are brought to three vertices whose angles are
$\pi \nu_\ell $ respectively:
\begin{eqnarray}
\label{nu}
\nu_0= 1-c=\dfrac{1}{p}, ~~~~~~
\nu_{1}= c-a-b=\dfrac{1}{q}, ~~~~~~
\nu_{\infty}= b-a=\dfrac{1}{r} .
\end{eqnarray}
$(p,q,r)$ are natural numbers greater than one.
The relation between \eqref{nu} and Table \ref{t1} can be made clear if one looks into the local behavior of ${\cal D}_\ell$
near each responsible singularity:
\begin{eqnarray*}
\label{}
{\cal D}_0 \simeq z^{\nu_0} \big( 1+{\cal O}(z)\big), ~~~~
{\cal D}_1 \simeq (1-z)^{\nu_1} \big( 1+{\cal O}(1-z)\big), ~~~~
{\cal D}_{\infty} \simeq z^{-\nu_{\infty}}
\big( 1+{\cal O}(z^{-1})\big) .
\end{eqnarray*}
Of course, extending ${\cal D}_\ell({\mathbb H})$ to
${\cal D}_\ell({\mathbb C})$
is totally possible and one encounters
\begin{eqnarray}
\label{mono}
{\cal D}_{\ell} \to \dfrac{a {\cal D}_{\ell} + b}{c
{\cal D}_{\ell} + d},
~~~~~~~~~~~\left(
\begin{array}{cc}
 a& b  \\
 c & d
\end{array}
\right)\in GL(2,\mathbb{C})
\end{eqnarray}
just because of monodromies when winding around each responsible singularity. The resulting image
${\cal D}_{\ell} ({\mathbb C})$ becomes two sets of triangles
\footnote{For instance, one can paint each triangle certain
color according to which one of two half-planes they come from.} which
tile the entire ${\mathbb P}^1$ if \eqref{nu} is assumed.  Certainly, after an automorphism (pattern-preserving) group $\Gamma(p,q,r)$ is
divided, one is able to claim that
\begin{eqnarray}
{\cal D}_\ell: ~{\mathbb C} \backslash \{ 0,1\} \to  \big( {\mathbb P}^1-{
{\text {triangle~vertices}}} \big)/\Gamma(p,q,r).
\label{MO}
\end{eqnarray}

As will be justified below,
in view of \eqref{mono} we cannot help but
regard ${\cal D}_\ell$ as the complex moduli of some
elliptic curve $E_{{\cal D}_\ell}$ with
${\cal D}_\ell\in {\mathbb H}/g$ (modular curve).
This way of thinking is also inspired by
the definition of ${\cal D}_\ell$ being a ratio of two
hypergeometric functions both of which solve
Fuchsian equations and are
identified with period integrals over
an algebraic curve. That generators of $g$ must be those of
$\pi_1 ( {\mathbb C} \backslash \{ 0,1\},z_0 )$ w.r.t.
${\cal D}_\ell$ confirms that there exists
a group homomorphism between
$\pi_1 ( {\mathbb C}\backslash \{ 0,1\},z_0 )$,
$\Gamma(p,q,r)$ and $g$. Indeed, we will
find that the above picture is realized
when $(p,q,r)=(\infty, \infty, \infty)$
and $g=\Gamma(2)$.

\subsection{Elliptic curve and $\lambda$-function}
Let us proceed to clarify the appearance of an elliptic curve $E_{\cal D}$ mentioned above.
An integral representation of
${}_2F_1 (a,b,c;z)$
for $z\ne (0, 1, \infty)$ is given by
\footnote{Usual normalizations
like Beta factors are omitted. Note also that
with
$(z_1, z_2, z_3,z_4)=(0,1,z,\infty)$ a simplification occurs, i.e.
\begin{eqnarray*}
\label{}
\int_{\gamma}  \prod^4_{i=1} (u-z_i)^{-\mu_i} du~~~\to ~~~
  \int_{\gamma}  \prod^3_{i=1} (u-z_i)^{-\mu_i} du
\end{eqnarray*}
due to the term involving $z_4=\infty$ dropped. }
\begin{eqnarray}
\label{ig}
\int_{\gamma} u ^{-\mu_0} (u-1)^{-\mu_1} (u-z)^{-\mu_z} du
= \int_{\gamma} \eta(z), ~~~~~~~~\mu_0+ \mu_1+ \mu_z+ \mu_{\infty}=2
\end{eqnarray}
where all $\mu$'s are simple linear combinations of
$(a,b,c)$ and assumed to be rational.
$\eta(z) \equiv \dfrac{du}{x}$ is defined w.r.t. an
algebraic curve
\begin{eqnarray}
\label{ac}
X: ~x^\kappa = u^{\kappa\mu_0} (u-1)^{\kappa\mu_1} (u-z)^{\kappa\mu_z}
\end{eqnarray}
with
$\kappa$ being the least common denominator of $\mu$'s.
$\gamma$ known as Pochhammer's contour now becomes
some homology cycle of $X$, i.e.
$\gamma\in H_1 (X, {\mathbb Z})$. Inequivalent $\gamma$'s will
lead to independent hypergeometric functions.
$X$ turns out to be an elliptic curve
of the standard Legendre form:
$x^2=4u(u-1)(u-z)$
when $\kappa=2$ and $\mu_0= \mu_1= \mu_z= \mu_{\infty}=\frac{1}{2}$. This soon implies
$(a,b,c)=(\frac{1}{2},\frac{1}{2},1)$
\footnote{Equivalently,
$\nu_0= \nu_1= \nu_{\infty}=0$ or
$p=\lambda=r=\infty$.} from the parameterization:
\begin{eqnarray*}
\begin{cases}
\label{}
\mu_0=\frac{1}{2} (1-\nu_0+\nu_1 -\nu_{\infty}), ~~~~~~~~
\mu_{1}=\frac{1}{2} (1+\nu_0-\nu_1 -\nu_{\infty}),\\
\mu_{z}=\frac{1}{2} (1-\nu_0-\nu_1 +\nu_{\infty}), ~~~~~~~~
\mu_{\infty}=\frac{1}{2} (1+\nu_0+\nu_1 +\nu_{\infty}). \\
\end{cases}
\end{eqnarray*}
Now, consider the ratio
\begin{eqnarray}
\label{ddd}
\tilde{\cal D}(z)=\dfrac{{}_2F_1 (\frac{1}{2},\frac{1}{2},1;z)}{{}_2F_1 (\frac{1}{2},\frac{1}{2},1;1-z)}
=\dfrac{K(\sqrt{z})}{K'(\sqrt{z})} , ~~~~~~~
K'(\sqrt{z})=K(\sqrt{1-z})
\end{eqnarray}
where $K$ denotes the complete elliptic integral
of the first kind. Conventionally, $\tilde{\cal D}(z)/2$
is called the $aspect$ ratio of a rectangle
yielded by performing
a Schwarz-Christoffel map
over ${\mathbb H}$.
Through defining
\begin{eqnarray*}
\label{}
\tau\equiv i\tilde{\cal D}(z)
=\dfrac{\int_{\gamma_1} \eta(z)}{\int_{\gamma_2} \eta(z)},
~~~~~~\gamma \in H_1 \big(X(\mu_i), {\mathbb Z}\big),
~~~~~~\forall\mu_i={1}/{2},
\end{eqnarray*}
the famous isomorphism:
\begin{eqnarray}
\label{iso}
 i\tilde{\cal D}: ~\mathbb{C} \backslash \{0,1\}\to  {\mathbb H}/\Gamma(2)
\end{eqnarray}
is induced. In particular,
the appearance of $\Gamma(2)$ is due to the choice of
$(a,b,c)$ as explained around \eqref{mmm}.
${\tau}$ becomes exactly
the complex moduli of a torus
${\mathbb C}/\Lambda_{\tau}$ ($\Lambda_{\tau} \equiv
{\mathbb Z}{\tau}+ {\mathbb Z}$) which is
isomorphic to $X$ in \eqref{ac} with
$\forall\mu_i={1}/{2}$.
In addition, the inverse of
$\tilde{\cal D}(z)$ is known as
the elliptic $lambda$ function:
\begin{eqnarray}
\label{la}
\lambda\equiv z=\frac{\theta_2^4 (q)}{\theta_3^4 (q)}
\end{eqnarray}
where $\theta_i(q)$'s are theta constants whilst
$q=e^{i\pi \tau}=e^{-\pi \tilde{\cal D}}$ is called the $nome$.
From now on, we will not especially
distinguish between
$\lambda$ and $z$ which eventually represent
the cross-ratio of four points on ${\mathbb P}^1$
\footnote{The cross-ratio of four points on
${\mathbb P}^1$ is given by \begin{eqnarray}
\lambda=(x_2, x_1; x_3,x_4)=\frac{(x_2-x_3)(x_1-x_4)}{(x_1-x_3)(x_2-x_4)}.\nonumber
\end{eqnarray} }.
By definition $\lambda$ should be invariant under $\Gamma(2)$
or, equivalently, subject to
\begin{eqnarray}
\lambda(\tau+2)=\lambda(\tau)
,~~~~~~~~~~~\lambda\Big( \frac{\tau}{1-2\tau}
\Big)=\lambda(\tau).\nonumber
\end{eqnarray}

Furthermore, since \eqref{iso} is a bijective map
what is inferred is that
six distinct $\lambda$'s
define the same elliptic curve
because of same
Klein's absolute $j$-invariants they will provide. Namely,
a six-to-one relation does follow owing to
\begin{eqnarray}
j=\frac{4}{27}\frac{\big(1-\lambda+\lambda^2 \big)^3}{\lambda^2
\big(1-\lambda \big)^2}=\frac{g_2^3}{g_2^3 - 27g_3^2}
\label{jj}
\end{eqnarray}
where $g_2$ and $g_3$ are
modular invariants of an
elliptic curve expressed in Weierstrass form:
\begin{eqnarray*}
y^2=4x^3 -g_2 x -g_3
\end{eqnarray*}
whose three distinct roots are $(e_1,e_2,e_3):=\Big(
\wp(\frac{1}{2}), \wp(\frac{\tau}{2}),\wp(\frac{\tau}{2}+\frac{1}{2})
\Big)$. Notice that $\wp$ is Weierstrass's doubly-periodic
function. Consequently,
\begin{eqnarray*}
\lambda=\frac{\wp(\frac{\tau}{2}+\frac{1}{2})-\wp(\frac{1}{2})}{
\wp(\frac{\tau}{2})-\wp(\frac{\tau}{2}+\frac{1}{2})}.
\end{eqnarray*}

We are led to the following conclusion.
Upon defining Hom$\big( \pi_1(
\mathbb{C} \backslash \{0,1\}, z_0), SL(2, \mathbb{Z}) \big)$, because generators of
the monodromy group w.r.t.
${}_2 F_1 (a,b,c;z)$ are determined
by $(a,b,c)$ as
\begin{eqnarray*}
&&M_1=\left(
\begin{array}{cc}
 1& 0  \\
 -1+e^{-2\pi i b} & e^{-2\pi ic}
\end{array}
\right) ~~~{\text {and}}~~~\nonumber\\
&&M_2=\left(
\begin{array}{cc}
1& 1-e^{-2\pi ia}  \\
 0 & e^{-2\pi i(a+b-c)}
\end{array}
\right)~~~{\text {with}}~~~
(a, b, c-a ,c-b) \notin {\mathbb Z},
\end{eqnarray*}
when $(a,b,c)=(\frac{1}{2},\frac{1}{2},1)$ they
are just those of $\Gamma(2)$, i.e.
\begin{eqnarray}
\label{mmm}
M_1 =\left(
\begin{array}{cc}
 1& 0  \\
 -2& 1
\end{array}
\right) ~~~{\text {and}}~~~
M_2=
\left(
\begin{array}{cc}
 1& 2  \\
 0&1
\end{array}
\right)
\end{eqnarray}
as used in \eqref{iso}.
As a remark, the relation
\eqref{ddd} is completely not new
since it has long been known as the
infra-red gauge coupling
$\tau_{IR}$ in the pure $SU(2)$
SW theory if one equates
${(2-z)}/{z}$ with its Coulomb phase parameter there.

All in all, we have just wandered quite a lot from the conventional
interpretation of
\eqref{iso}, i.e. one can
always express an elliptic curve
in terms of a two-sheeted cover of
a sphere with branching points $(0,1,\lambda,\infty)$
such that the equivalence between their moduli spaces
naturally introduces the underlying isomorphism
\eqref{iso} or its inverse--$\lambda$-function \eqref{la}.

\section{Triality in $SU(2)$ $N_f=4$ Seiberg-Witten theory}
A standard $2N_c=N_f=4$ SW curve
is of a rather
sophisticated form
parameterized by
four bare flavor masses and $\tau_0$ \cite{Seiberg:1994aj}:
\begin{eqnarray}
y^2=4\Big[W_1W_2W_3+A\big(W_1T_1(e_2-e_3)+W_2T_2(e_3-e_1) +W_3T_3(e_1-e_2)\big)-A^2 N \Big]
\label{swc}
\end{eqnarray}
where ($u$: Coulomb phase parameter)
\begin{eqnarray*}
{W_i}=x-{e_i}u-{{e_i}}^{2}R,~~~~~
A= \left( {e_1}-{e_2} \right)  \left( {e_2}-{e_3} \right)
 \left( {e_3}-{e_1} \right)
\end{eqnarray*}
and
\begin{eqnarray*}
&&R=\dfrac{1}{2}\sum_{i=1}^4m_i^2, ~~~~~
N=\dfrac{3}{16}\sum_{i>j>k}m_i^2m_j^2m_k^2-\dfrac{1}{96}
\sum_{i\neq j}m_i^2m_j^4+\dfrac{1}{96}\sum_{i=1}^4m_i^6,
\nonumber\\
&&T_1=\dfrac{1}{12}\sum_{i>j}m_i^2m_j^2-\dfrac{1}{24}\sum_{i=1}^4m_i^4,  ~~~~~T_2=-\dfrac{1}{2}\prod_{i=1}^4 m_i-\dfrac{1}{24}\sum_{i>j}m_i^2m_j^2+\dfrac{1}{48}\sum_{i=1}^4m_i^4,\nonumber\\
&&T_3=\dfrac{1}{2}\prod_{i=1}^4m_i-\dfrac{1}{24}\sum_{i>j}
m_i^2m_j^2+\dfrac{1}{48}\sum_{i=1}^4m_i^4.
\nonumber
\end{eqnarray*}
$e_i$'s are functions of $\tau_0$:
\begin{eqnarray*}
e_1=\dfrac{1}{12}({\theta}_3^4+\theta_4^4),~~~~~ e_2=\dfrac{1}{12}(\theta_2^4-\theta_4^4), ~~~~~
 e_3=\dfrac{1}{12}(-\theta_2^4-\theta_3^4).\nonumber
\end{eqnarray*}
Here,
$\tau_0$ must be regarded as the asymptotic value of
$\tau_{IR}= \tau_0 +\dfrac{1}{2\pi i}\Big(
\sum_{i=1}^4 \log(u-m_i^2)
-4\log u \Big)+\cdots$ expanded at large
$u$. Its reduction to asymptotically-free
counterparts ($N_f\le 3$)
is easily achieved via tuning $\tau_0$ and $m_i$
in order to yield a suitable dynamical scale $\Lambda_{N_f}$.

Seiberg and Witten found that \eqref{swc}
is invariant under elements of
\begin{eqnarray}
SL(2,\mathbb{Z})/\Gamma(2)=\mathbf{S}_3=\{ I, ~S, ~T,
~ST, ~TS, ~STS \}
\label{TT}
\end{eqnarray}
if and only if \eqref{ma1} and \eqref{ma}
are taken into account simultaneously. This phenomenon
is often referred to as triality whose origin may
be owing to
the outer automorphism group ${\mathbf S}_3$ of
$Spin(8)$, the quantum
flavor symmetry in the superconformal case ($m_i=0$).
Because we want to interpret triality
as interchanging Kummer's solutions,
our strategy is to think of
$\Gamma(1)/\Gamma(2)=\mathbf{S}_3$ here as
$\mathbf{S}_4/({\mathbb Z}_2 \times {\mathbb Z}_2 )$ on
a four-punctured ${\mathbb P}^1$ via $\lambda$-function introduced in \eqref{la}
\footnote{In AGT's Appendix (B.29)
$\lambda$-function has already shown up. }.

In other words, under $\mathbf{S}_3$ $\tau_0$ enlarges
its ``fundamental" domain
\footnote{As \eqref{swc} reduces to merely an usual
Weierstrass elliptic curve characterized by $\tau_0$
when all $m_i=0$, so basically $\tau_0 \in {\mathbb H}/SL(2,{\mathbb Z})$.} instead to
${\mathbb H}/\Gamma(2)$
and by \eqref{la} we will now map it
bijectively
to $\lambda$-space defined on
$C_{0,4}$ where
six distinct cross-ratios are
caused by applying $\mathbf{S}_4/({\mathbb Z}_2 \times {\mathbb Z}_2 )$. That is, the action of ${\bf S}_3$
is translated to interchanging four marked punctures.
The next step is to know how
\eqref{ma1} and \eqref{ma} can be
incorporated
into the four-point spherical conformal block
${\cal B}[C_{0,4}]$
with
\begin{eqnarray}
{\cal B}
(\alpha, \vec{\mu}, Q|\lambda)=
Z^{2N_c =N_f =4}_{inst} ({\bf a}, \vec{m},  \epsilon_{1,2}|
q_{UV}),
~~~~~~
\alpha: ~\text {internal ~momentum}
\label{ttt}
\end{eqnarray}
where in addition to $\lambda\equiv q_{UV}(\tau_0)$ the
dictionary between parameters on two sides
has been spelt out by AGT.
All in all, making use of properties of
Gauss hypergeometric
functions we will arrive at a new unifying
understanding of this mysterious
part of $S$-duality--triality for $2N_c=N_f=4$.

\subsection{Gaiotto's picture and AGT conjecture}
Gaiotto's idea arises from rearranging old SW curves and
leads to another way of engineering a huge class of $\N=2$ $SU(r)$ SCFTs by wrapping $r$ M5-branes on $C_{g,n}$, i.e. compactifying
6D $A_{r\text{-}1}$-type $(2,0)$ theories
on $C_{g,n}$ accompanied by a partial twisting.
There are various ways of decomposing
$C_{g,n}$ into trinions and tubes so
$S$-duality (mapping class) group gets identified with such
physically equivalent surgeries. In addition,
weak-coupling limits are attained intuitively
by elongating extremely tubes joining two
punctures. This kind of
$degenerate$ limits correspond to $cusps$
in the moduli space of $C_{g,n}$.
Total $3g$-3+$n$ tubes contained in
$C_{g,n}$ correspond to
the number of gauge groups of a weakly-coupled
quiver SCFT equipped with a Lagrangian description.

Let us elaborate arguments about aforementioned
$\lambda$-space on $C_{0,4}$. Upon viewing $\lambda$ as the coordinate on ${\mathbb P}^1 \backslash
(0,1,\infty)$ (up to a M{\"o}bius transformation),
six distinct values generated from it by
$\Gamma(1)/\Gamma(2)= \mathbf{S}_3$ are
referred to as six different cross-ratios:
\begin{eqnarray}
\begin{array}{cccccccc}
\mathrm{element} ~\mathrm{in}~ \mathbf{S}_3  &&I &T &S &ST &TS &STS   \\
\mathrm{cross}\text{-}\mathrm{ratio}&&\lambda&\dfrac{\lambda}{\lambda-1}&1-\lambda&\dfrac{1}{1-\lambda}&\dfrac{\lambda-1}{\lambda}&\dfrac{1}{\lambda}\\
\end{array}
\label{6q}
\end{eqnarray}
\eqref{6q} can be
derived directly based on
either footnote 10 or \eqref{la} with modular properties
of theta constants listed below:
\begin{eqnarray*}
&&\theta_{2}(q)\equiv \vartheta_{10} (0,\tau) = \frac{1}{\sqrt{-i\tau}}
\vartheta_{01} (0,-\frac{1}{\tau}), ~~~~~~
\theta_{3}(q) \equiv\vartheta_{00} (0,\tau) = \frac{1}{\sqrt{-i\tau}}
 \vartheta_{00} (0,-\frac{1}{\tau}),\nonumber\\
&&\theta_4 (q)\equiv \vartheta_{01} (0,\tau) =
\vartheta_{00} (0,{\tau+1}), ~~~~~~~~~~~~
\vartheta_{10} (0,\tau) = e^{- \frac{i\pi}{4}}
 \vartheta_{10} (0,{\tau +1}).
\end{eqnarray*}
Replacing
${\mathbb H}/\Gamma(2)$ by
$\Big( {\mathbb H}/ SL(2, \mathbb{Z})\Big) \times {\mathbf S}_3$ in
\eqref{iso} and recalling
that another famous
isomorphism
\begin{eqnarray*}
\label{}
j: ~{\mathbb H}/ SL({2, \mathbb{Z}})
 \to {\mathbb C} \backslash \{0,1\}
\end{eqnarray*}
is induced by Klein's $j$-invariant, one agrees
that
the identity \eqref{jj} between $j$ and $\lambda$ describes a six-to-one relation.

Next, we focus on ${\cal B}[C_{0,4}]$ in \eqref{ttt}.
In general, because it satisfies
Zamolodchikov's recursion relation \cite{Zamolodchikov:1995aa}
an expansion over $\lambda$
to any desired order is possible. However,
a closed-form expression of it is still missing.
Nevertheless, if
one of four inserted primary fields
becomes degenerate, say, $V_{\mu_3}=\Phi_{2,1}$ it is well-known
\cite{BPZ, Fateev:2009me} that
by means of the null-state condition:
\begin{eqnarray*}
\Big( L_{-2} -\frac{3}{2\big(
2\Delta(h_{2,1}) +1 \big)}L^2_{-1} \Big)
\Phi_{2,1}=0 ,
~~~~~~  h_{r,s}=\frac{1-r}{2}b +
\frac{1-s}{2b},~~~~~~ \mu_3\equiv h_{2,1}=-\dfrac{b}{2},
\end{eqnarray*}
one is led to
\begin{eqnarray}
\label{cbhy}
{\cal B}[C_{0,4}]\equiv \langle V_{\mu_1}(0) V_{\mu_2}
(1) V_{\mu_3}(\lambda) V_{\mu_4}(\infty)\rangle=
\lambda^{b\mu_1} (1-\lambda)^{b\mu_2} {}_2F_1(a,b,c;\lambda)
\end{eqnarray}
where
\footnote{We adhere to conventions used in
\cite{Schiappa:2009cc}. Also, we wish
``$b$" used
in both ${}_2F_1(a,b,c; z)$ and Liouville theory side
will cause no confusion.}
\begin{eqnarray}
\label{abcn}
\begin{cases}
a=-N,\\
b=\dfrac{1}{\beta}(-\dfrac{2\mu_1}{\epsilon_1}
-\dfrac{2\mu_2}{\epsilon_1}+2) +N-1,\\
c=\dfrac{1}{\beta}(-\dfrac{2\mu_1}{\epsilon_1}+1),\\
N=-\epsilon_1(\mu_1+\mu_2+\mu_3-\mu_4),~~~~~
\beta=-\dfrac{\epsilon_2}{\epsilon_1}, ~~~~~\epsilon_1=b, ~~~~~
\epsilon_2=\dfrac{1}{b}.
\end{cases}
\end{eqnarray}
The internal momentum $\alpha$ is set
to be
\begin{eqnarray*}
\alpha=\frac{Q}{2}+{{\bf a}}=\mu_{4}+\frac{b}{2}, ~~~~~~~~~~
Q=b + \frac{1}{b}.
\end{eqnarray*}
Adopting a 4D physical field theory limit
$\epsilon_{1}+\epsilon_2=0$ ($\beta=1$) may give rise to
a further simplification
\footnote{$\epsilon_{1}=-\epsilon_2$ serves as the
$genus$-expansion parameter inside
$Z_{{\text{inst}}}=\exp({{\cal F}})$ since
${\cal F}=-\frac{1}{\epsilon_{1}\epsilon_{2}}{\cal F}_0+\cdots$ is referred to as
the A-model topological string free energy
w.r.t. a responsible
Calabi-Yau three-fold in Type IIA theory.}.
Note that $N$ is designated to characterize
the size of a $hermitian$ matrix appearing in
the recent Dijkgraaf-Vafa proposal \cite{Dijkgraaf:2009pc}.
There, an ($A_1$-type) $n$-point spherical
${\cal B}$ was rewritten in terms of a Penner-type
matrix integral
(or Selberg-Kaneko integral \cite{ka}).
As a remark,
from \eqref{abcn} ${}_2F_1(a,b,c;\lambda)$ also
stands for
a Jacobi polynomial defined by
\begin{eqnarray*}
&&G_N(\xi, \zeta;\lambda)={}_2F_1(-N,\xi+N, \zeta;\lambda)
=1+\sum_{r=1}^N (-)^r {}_N {\text C}_r
\dfrac{\Gamma(\xi+N+r)\Gamma(\zeta)}{\Gamma(\xi+N)\Gamma(\zeta+r)}\lambda^r, \\
&&\text{for} ~
\zeta\ne 0, -1, -2, \cdots,-N+1.
\end{eqnarray*}
This sounds quite consistent with the fact pointed out by
Schiappa and Wyllard \cite{Schiappa:2009cc}
that a three-point DV matrix model $Z^{DV}_{3pt}$
for ${\cal T}_{0,3}(A_1)$ is
exactly solved by its orthogonal polynomial--Jacobi
polynomial; namely,
$\langle \det(\lambda-M)\rangle_{Z^{DV}_{3pt}}$ is equal to
\eqref{cbhy} without the factor
$\lambda^{b\mu_1} (1-\lambda)^{b\mu_2}$ as shown in \cite{Schiappa:2009cc}.

\subsubsection{Relation to ${\cal N}=2^\ast$ $A_1$ system}
Inspired by the appearance of a Jacobi
polynomial said above, we strongly
expect that its reduction to $A_1$-Jack
(or Gegenbauer) polynomials by further
constraining three $\mu$'s can be given
a physical interpretation
\footnote{I thank Hirotaka Irie, Yutaka Matsuo and
Akitsugu Miwa with whom I have communicated about these stuffs.}.
Namely, having in mind that an $A_1$-Jack polynomial
gets closely related to a specialized hypergeometric function
\begin{eqnarray}
{}_2F_1(-A,A+2B,B+\frac{1}{2};x)
\label{jjjj}
\end{eqnarray}
and is the eigenstate
of $A_1$-type Calogero-Sutherland model,
a limiting case of
$A_1$-type Calogero-Moser model as
$p=\exp(2\pi i \tau)$ of
Weierstrass's $\wp$-function
goes to zero (or $\tau \to i\infty$), we cannot help
suspecting that the constraint imposed on
three $\mu$'s leading to \eqref{jjjj} should result from
a one-punctured $pinched$ torus. In other words, one may think of \eqref{jjjj} as a two-point conformal block ${\cal B}[C_{1,2}]$ with
one insertion being $\Phi_{1,2}$ defined on a pinched torus.  Notice that redefining
$x\sim \exp(i\lambda)\in{\mathbb C}^\ast$
makes the periodicity $\lambda\sim \lambda+2\pi$ explicit.

To carry out the check, one also needs to know the degenerating process:
${\cal T}_{1,1}\to {\cal T}_{0,3}$.
Given the fact that
in the physical ${\cal T}_{0,3}$ theory four free hypermultiplets
have their masses
$\mu_1 \pm \mu_2 \pm \mu_4$ yielded
from assigned momenta of
three inserted Liouville primary fields \cite{Alday:2009aq},
plausibly $\mu$'s will now not be independent
because in the former there are
only two independent variables
$({\mathfrak a}, {\mathfrak m})$,
i.e. ${\cal N}=2^\ast$ $SU(2)$ Coulomb branch
parameter and adjoint hypermultiplet mass.
Further, from a toric diagram associated with
a Calabi-Yau three-fold
engineering $\N=2^{\ast}$ $SU(2)$ theory,
one is able to read off masses of four free
hypermultiplets in terms of
$({\mathfrak a}, {\mathfrak m})$. Then, the constraint
for $\mu$'s gained from comparing
\eqref{cbhy} with \eqref{jjjj} can be directly contrasted
with what is derived above via an $\N=2^\ast$ toric diagram.

An even interesting
direction is to consider connections between various
orthogonal polynomials by means of AGT picture.
Serving as eigenstates of
distinct Schr{\"o}dinger equations
(or two-body integrable systems),
they are nonetheless transformed to one another
by performing some limit which may acquire
suitable geometric meaning in terms of
Riemann surfaces.
We wish to report these topics in an upcoming paper.

\subsection{Triality and Kummer's 24 solutions}
Mathematically speaking, multiplying prefactors like $\lambda^{A}(1-\lambda)^{B}$ just brings ${}_2F_1(a,b,c;\lambda)$ to
another Riemann scheme containing solutions like
${}_2F_1(a',b',c';\lambda)$ and so on.
To say which scheme is more preferable seems
not so essential.
We decide to exclude
these prefactors below also
because in \cite{Fujita:2009gf} this choice of
${\cal B}$
did reproduce the gravitationally-corrected
asymptotically-free SW prepotential
${\cal F}_0$.

By applying AGT dictionary \eqref{mu0} together with
\eqref{ma1}, \eqref{ma} and
\eqref{abcn} specialized at $\beta=1$, it is seen that
other five of Kummer's twenty-four solutions can be
generated from
${}_2F_1(a,b,c;\lambda)$ by
elements $\{ S,T,TS,STS,ST \}$:
\\
\\
$\underline{\mathbf{(1)~ S:} ~~
\mathbf{\mu_1 \leftrightarrow \mu_2}}$
\begin{eqnarray*}
\begin{cases}
a\to a\\
b\to b\\
c\to a+b-c+1\\
\lambda \to 1-\lambda
\end{cases}
\end{eqnarray*}
$\underline{\mathbf{(2)~ T:} ~~
\mathbf{ \mu_4 \leftrightarrow \mu_2}}$
\begin{eqnarray*}
\begin{cases}
a\to -a+c\\
b\to b\\
c\to c\\
\lambda\to \dfrac{\lambda}{\lambda-1}
\end{cases}
\end{eqnarray*}
$\underline{\mathbf{(3)~ TS:} ~~
\mathbf{(\mu_1, \mu_2, \mu_4) \rightarrow (\mu_2, \mu_4, \mu_1)}}$
\begin{eqnarray*}
\begin{cases}
a\to b-c+1\\
b\to b\\
c\to a+b-c+1\\
\lambda\to \dfrac{\lambda-1}{\lambda}
\end{cases}
\end{eqnarray*}
$\underline{\mathbf{(4)~ STS:} ~~
\mathbf{ \mu_4 \leftrightarrow \mu_1}}$
\begin{eqnarray*}
\begin{cases}
a\to b+c-1\\
b\to b\\
c\to -a+b+1\\
\lambda\to \dfrac{1}{\lambda}
\end{cases}
\end{eqnarray*}
$\underline{\mathbf{(5)~ ST:} ~~
\mathbf{
 (\mu_1, \mu_2, \mu_4) \rightarrow (\mu_4, \mu_1, \mu_2)}}$
\begin{eqnarray*}
\begin{cases}
a\to -a+c\\
b\to b\\
c\to -a+b+1\\
 \lambda\to \dfrac{1}{1-\lambda}
\end{cases}
\end{eqnarray*}
Finally, all of them are collected below:
\begin{eqnarray*}
\begin{cases}
\text{(I)}~{}_2F_1(a, b,c; \lambda)\\
\text{(II)}~{}_2F_1(a, b,a+b-c+1; 1-\lambda)\\
\text{(III)}~(1-\lambda)^{-b} {}_2F_1(c-a, b,c; \dfrac{\lambda}{\lambda-1})\\
 \text{(IV)}~\lambda^{-b} {}_2F_1(b-c+1, b,a+b-c+1; \dfrac{\lambda-1}{\lambda})\\
 \text{(V)}~ \lambda^{-b} {}_2F_1(b-c+1, b,b-a+1; \dfrac{1}{\lambda})\\
 \text{(VI)}~ (1-\lambda)^{-b} {}_2F_1(-a+c, b,b-a+1; \dfrac{1}{1-\lambda})
\end{cases}
\end{eqnarray*}
Although strictly speaking (III)--(V) have been
dressed by either $\lambda^{b}$ or $(1-\lambda)^{b}$,
it can merely be
added by hand in order to preserve the given
Riemann scheme. As a matter of fact,
according to \cite{iwa} we see that
\begin{eqnarray*}
\begin{cases}
\text{(I)~(III) }
\end{cases}
\lambda=0 ~{\text{basis,}}~~~~~~
\begin{cases}
\text{(II)~(IV) }
\end{cases}
\lambda=1 ~{\text{basis,}}~~~~~~
\begin{cases}
\text{(V)~(VI) }
\end{cases}
\lambda=\infty ~{\text{basis,}}~~
\label{51}
\end{eqnarray*}
where by ``basis" we mean spanning a basis of
solutions around there.
Therefore, triality plays the role of
interchanging solutions around three regular singularities $(0,1,\infty)$. This
manipulation thus manifests
how triality is just understood in terms of
another mathematical object--hypergeometric function.

\section{Discussion}

\subsection*{(I) Crossing symmetry and triality}

Correlators $\Omega$ of
some rational CFT including Liouville field theory (LFT)
defined on a four-punctured Riemann sphere
are endowed with the $crossing$ $symmetry$.
Usually, this fact is expressed as follows:
\begin{eqnarray}
\Omega^{(4)}
(
\begin{array}{cc}
\mu_2 & \mu_3   \\
\mu_4 & \mu_1
\end{array}
\Big|\lambda)
=\Omega^{(4)}
(
\begin{array}{cc}
\mu_1 & \mu_3   \\
\mu_4 & \mu_2
\end{array}
\Big|1-\lambda)
\end{eqnarray}
where $\Omega^{(4)}$ is analytic over
${\mathbb P}^1_{\ast}: {\mathbb P}^1 \backslash \{ 0,1,\infty\}$.
When it comes to conformal blocks ${\cal B}$
(main component constituting the above $\Omega^{(4)}$) though,
the crossing symmetry is respected to a $covariant$ extent:
\begin{eqnarray}
{\cal B}_{\mu_{31}}^{s}
(
\begin{array}{cc}
\mu_2 & \mu_3   \\
\mu_4 & \mu_1
\end{array}
\Big|\lambda)
=\int_{\frac{Q}{2}+i{\mathbb R}^+} d\mu_{23}~
F_{\mu_{31}\mu_{23}}^{L}
\left(
\begin{array}{cc}
\mu_2 & \mu_3   \\
\mu_4 & \mu_1
\end{array}
\right)
{\cal B}_{\mu_{23}}^{t}
(
\begin{array}{cc}
\mu_1 & \mu_3   \\
\mu_4 & \mu_2
\end{array}
\Big|1-\lambda).
\label{fu}
\end{eqnarray}
Note that $F_{\mu_{31}\mu_{23}}^{L}$ specifies the invertible
fusion matrix in between $s$- and $t$-channel LFT
conformal blocks.
When one of four primary fields gets degenerate then
${\cal B}$ reduces to a hypergeometric function such that
\eqref{fu} reads
\begin{eqnarray*}
 &&{}_2F_1 (a,b,c;\lambda)=\dfrac{\Gamma(c)\Gamma(c-a-b)}{\Gamma(c-a)\Gamma(c-b)}{}_2F_1 (a,b,a+b-c+1;1-\lambda) \\
&&+
\dfrac{\Gamma(c)\Gamma(a+b-c)}{\Gamma(a)\Gamma(b)}(1-z)^{c-a-b}{}_2F_1 (c-a , c-b, c-a-b+1; 1-\lambda)
\end{eqnarray*}
as having been indicated by
connection coefficients listed in (2.2).

Notice that (I)-(VI) collected in Sec. 3.2 is somewhat unexpected and independent of (2.2).
One can just hardly regard
(2.2) as any evidence showing triality (or $S$-duality)
as AGT conjecture relates Liouville conformal blocks to
Nekrasov's instanton partition functions. This is because the latter never splits
into two compatible (and linearly independent) components.
On the other hand, given certain Riemann scheme dictated by
four flavor masses, the permutation between
six Kummer's solutions upon imposing triality ultimately amounts to another geometric
realization of ${\mathbf S}_3$ outer automorphism of
${\mathbb P}^1_{\ast}$. This picture will never
be obtained by merely applying the crossing symmetry though.

\subsection*{(II) Geometric realization
of triality}

In order to make clearer the above statement, let us first
write down the
following expansion:
\begin{eqnarray*}
\Psi(\lambda)={}_2F_1 \big(
\frac{a}{\hbar},\frac{b}{\hbar},\frac{c}{\hbar};\lambda \big)=
\exp -\Big(\hbar^{-1} W_0 + W_1 + \hbar W_2 +\cdots \Big).
\end{eqnarray*}
The zeroth-order term $W_0$ is obtained by means of the WKB
approximation
\begin{eqnarray*}
\exp(-\hbar^{-1} W_0)= \exp\big(i\hbar^{-1}\int^{\lambda} dz \sqrt{T(z)}\big)
\end{eqnarray*}
w.r.t. the $Q$-form
$\Big( \hbar^2\partial^2_{\lambda} + T({\lambda})\Big)\Psi({\lambda})=0$
of an usual hypergeometric equation
\footnote{$\hbar$ is restored momentarily for computational
convenience during applying the WKB method.}. It has been
found in [10] that
\begin{eqnarray*}
&&W_0= \int^{\lambda} dz \sqrt{\frac{-a_1^2 z(1-z) +
a_2^2(1-z) +
a_3^2 z}{z^2(1-z)^2}},\\
&& a= -a_1+a_2-a_3, ~~~~~b=a_1+a_2+a_3,~~~~~
c=2a_2.
\end{eqnarray*}
In addition,
the curve
\begin{eqnarray*}
{\cal P}:~ y^2=T(z)= \frac{-a_1^2 z(1-z) +
a_2^2(1-z) +
a_3^2 z}{z^2(1-z)^2}
\end{eqnarray*}
turns out to be dual to ${\mathbb P}^1_{\ast}$
[10, 28] since it stands for the real two-dimensional portion of
M5-branes occupying ${\mathbb R}^{1,3} \times
{\cal P}$ which arise from the uplift of
a type IIA brane configuration (hence $\epsilon_1 + \epsilon_2=0$) engineering
${\cal T}_{0,3}(A_1)$ theory of
four free hypermultiplets. Therefore, choosing one Riemann scheme means fixing all parameters of
${\cal P}$ (pants diagram), and interchanging
solutions to its corresponding $Q$-form is
equivalent to
manifesting the outer automorphism ${\mathbf S}_3$ of ${\cal P}$.

In summary, ${\mathbf S}_3$ originally stemming from
the outer automorphism of $spin(8)$
(as seen from its Dynkin diagram) at the low-energy regime has been
once interpreted as
${\mathbf S}_4/{\mathbb Z}_2 \times {\mathbb Z}_2$ built
on a four-punctured sphere via AGT dictionary (1.3).
Through studying (1.1) and (1.2) under the
hypergeometric nature of this sort of $SU(2)$
instanton partition functions, we see the revival of
${\mathbf S}_3$
in a fashion naturally encoded in ${\cal P}$
dual to ${\mathbb P}^1_{\ast}$.
This presentation dose provide a new geometric though simple viewpoint of (1.1) and (1.2) spelt out by Seiberg and
Witten.

\section{Summary}

Let us briefly summarize our
main results. The
geometric origin of triality stemming from
the outer automorphism group
${\mathbf S}_3$ of
the quantum flavor symmetry $Spin(8)$ has long
been pursuit. In Vafa's F-theory setup,
a $D_4$-type singularity on an elliptically
fibered K3 can be used to engineer
an $\N=2$ $A_1$-type $N_f=4$ SCFT
due to an arbitrary string coupling.
While Vafa's picture compactified down to IIB theory stresses
a geometric realization of $u$-plane parameterizing
Coulomb branch, triality, namely \eqref{ma1} and
\eqref{ma}, connecting physically
equivalent theories seems
not immediately visible. This is
because now one is
confined nearby a slightly deformed
$D_4$-type singularity whereas
in addition to bare mass parameters (positions of D7-branes
located on $u$-plane) triality involves further an asymptotic piece of information,
say, $\tau_0$ at $u\to \infty$.
This problem of $\tau_0$ can be once
remedied if one notices a
bijection between the
``fundamental" domain
${\mathbb H}/{\Gamma(2)}$ of $\tau_0$ and
moduli space of four
marked points on a Riemann sphere
by means of the celebrated $\lambda$-function \eqref{la}.
The latter object denoted as $C_{0,4}$ emerges in
Gaiotto's revolutionary description of an
$\N=2$ $SU(2)$ $N_f=4$ SCFT. Instead, how to encode
mass transformation rules into $C_{0,4}$
now turns out to be invisible.

What comes to one's rescue is AGT conjecture which
states precisely \eqref{ttt}.
Equipped with it, \eqref{ma1} and \eqref{ma}
performed onto
bare masses contained in
$Z_{\text{inst}}$ as well as $\tau_0$ are then
translated into interchanging six
hypergeometric functions belonging to three
regular singularities under
certain Riemann scheme, provided one primary insertion
of the four-point spherical conformal block gets degenerate.
These arguments do provide another insight
into capturing triality geometrically, e.g.
permutation around vertices of a Schwarz triangle.
Note that solutions in \eqref{51} are not equal to one another
echoes the fact that ${\cal B}[C_{0,4}]$ along is basically
not $S$-duality invariant or Nekrasov's partition function
on ${\mathbb R}^4$ transforms nontrivially
under $S$-duality as stressed in \cite{Alday:2009fs}.

\section*{Acknowledgements}
I thank two Japanese mathematicians
Masaaki Yoshida and Keiji Matsumoto for
their e-mail correspondence and providing me
with many valuable references. I am
grateful to
organizers of the workshop ``Recent Advances in Gauge Theories and CFTs" held at YITP Kyoto. I am also indebted to
Toru Eguchi, Kazuhiro Sakai and Yuji Tachikawa for
encouragement and helpful discussions. I am supported in part by the postdoctoral program at RIKEN.


TITLE: A well-defined binary operation on a class of functions (Eudoxus magnitudes) from $\mathbb N$ to $\mathbb N$?
QUESTION [0 upvotes]: Update: The answer shows that some tweaking is necessary to get this to work. The problem are those $f$ where there exist an $N$ such that $f(n)$ is always odd for $n \ge N$. But this can be remedied in a simple way -condition $\text{(2)}$ below is dropped and we just handle these 'overloaded' functions by equating them to 'there equal'.
If $f$ is overloaded there is a least $k$ (that might be $0$ with $f(0)$ also odd) so that $f(m)$ is odd for all $m \gt k$. We define a function $g$ that agrees with $f$ for $n \lt k$, has $g(k)=f(k)+1$ and for $m \ge k$, we recursively define $g(m + 1) = 2 * g(m)$. This function $g(x)$ is not overloaded.
Then, considering these two functions as equivalent the problem goes away.
So, we can define a binary operation that is obviously commutative. Interestingly, when adding numbers, if one is overloaded we can't use that representation in the definition of the binary operation - you must use that other representation in the two element block of the (really fine-grained) partition. So the necessity of condition $\text{(2)}$ below 'melts away;.
But showing that the operation is associative is not immediate. I think we can do this by using the fact that the binary operation is 'continuous'; c.f.
Prove Addition is Continuous (without epsilon-delta!)
When I work out the details I will post them in a another proof-checking question.
Incidentally, the purpose of this is to create a model of the positive real numbers under addition. Intuitively, you can think of the function $f$ as representing a canonical Cauchy sequences; for every $x$ there is one and only one $f$ with
$\quad x = \lim\limits_{n \to +∞} f(n) \, 2^{-n}$
But note that this is quite different - the real numbers with $+$ are being directly constructed ab initio; it is not necessary to define or use the rational numbers as an intermediate step. A computer running an artificial intelligence program might be able to 'understand' this construction easier than the usual methods.

Here $0 \in \mathbb N$.
Definition: A function $f: \mathbb N \to \mathbb N$ is called a Eudoxus magnitude if it satisfies the following:
$\tag 1 \forall n \; f(n+1) = 2f(n) \text{ or } f(n+1) = 2f(n) + 1$
$\tag 2 \forall\, m \; \exists \, n \ge m \; \text{ such that } f(n+1) = 2f(n)$
One interesting thing about a Eudoxus magnitude $f$ is that if we know the value of $f$ at $n$, then the value at all smaller numbers is determined:
$\tag 3 f(n-1) = \text{the quotient (Euclidean division) of dividend } f(n) \text{ by divisor } 2$.
Also,  given any number $k$, we can create a Eudoxus magnitude with $f(n) = k$ using $\text{(3)}$.
I've come up with a way of adding two Eudoxus magnitudes (not pointwise addition) $f$ and $g$ as follows.
For any integer $n \ge 0$ such that $f(n+1) = 2f(n)$ or $g(n+1) = 2g(n)$, set $k = f(n) + g(n)$ and create a (partial) Eudoxus magnitude $h$ with $h(n) = k$ and defined on $[0,n]$.
In words, we look for an even output of either function at $n + 1$, then working on the initial segment $[0,n]$ take point-wise addition on $n$ and then 'ripple back' with $\text{(3)}$.
Claim: The value of the partial functions obtained by using different integers $n$ where $f(n+1) = 2f(n)$ or $g(n+1) = 2g(n)$ always agree on their common initial integer intervals.
If this is true, we've defined a unique function $h: \mathbb N \to \mathbb N $ by just 'going out further towards' $+\infty$. A bonus is that we can then immediately assert that the 'Eudoxus sum' $\oplus$ of two Eudoxus magnitudes is a Eudoxus magnitude.
I would like to prove this using only elementary number theory and logic. But besides writing a Python program to check out the coherence of this endeavor, at this point I don't have any proof at all.

Question 1: Can the above claim be established using any mathematical
  theories?

REPLY [4 votes]: The claim$^1$ is false: the operation you've described doesn't always output a Eudoxus magnitude. In particular - and shifting from functions to sequences for ease of writing - take $$f=(1,2,5,10,21,...),\quad g=(1,3,6,13,26,...)$$ (the important feature being that each doubles infinitely often, but they never double at the same time). Then when we try to build their "Eudoxus sum" according to your rule, we have good news and bad news. On the plus side, the "initial segment sums" we get agree with each other; however, on the minus side we get the sequence $$(2,5,11,23,37,...)$$ which never doubles and hence fails to satisfy condition $(2)$.
Dropping condition $(2)$ from the definition doesn't help: the sequence $$(1,3,7,15,31,...)$$ would then be considered a Eudoxus magnitude, but there's no good way to define "Eudoxus sum" so as to not have its Eudoxus sum with itself violate $(1)$.
What you need is a way to ensure frequent simultaneous doubling. You can ensure this by strengthening condition $(2)$; e.g. you'll get better results if you demand that $f$ is a Eudoxus magnitude only if its set of doubling points has asymptotic density $1$ (since the intersection of two asymptotic-density-$1$ sets has asymptotic density $1$). But I suspect such a requirement will ruin things elsewhere, namely the original motivation for these.

$^1$This is a bit unfair of me. Your actual claim as written is just that we never see disagreement between the "initial segment sums," and it's not hard to show that this is in fact true. But implicitly, you want the output to again be a Eudoxus magnitude ("A bonus is that we can then immediately assert that the [Eudoxus sum] of two Eudoxus magnitudes is a Eudoxus magnitude.") and this doesn't hold.


TITLE: Algebraic conjugates
QUESTION [4 upvotes]: Suppose $L/K$ is an algebraic field extension. Take $\alpha_1 \in L$. Then $\alpha_1$ has minimal polynomial $f(x)$ over $K$. Let $\alpha_2, ... \alpha_k$ be the other roots of $f$ in $L$. The $\alpha_i$ are known as algebraic conjugates.
Suppose I am given an $\alpha \in L$ and wish to find the minimal polynomial of $\alpha$ over $K$. It'd be helpful if I knew something about how the algebraic conjugates of $\alpha$ behaved.
Let's take a concrete example. Suppose $L = \mathbb Q(\sqrt{3})$ and $K = \mathbb Q$, and I want to know the minimal polynomial of $2 + \sqrt{3}$. I could work it out by writing $ x = 2 + \sqrt{3}$, squaring, rearranging and squaring again. But this would quickly become cumbersome if we chose a more complicated extension and complicated $\alpha$. I know its a good idea to look at the product $ \prod (x - \alpha_i) $, where the $\alpha_i$ are algebraic conjugates of $\alpha$. What are the algebraic conjugates of $ 2 + \sqrt{3} $? I 'know' they are $ \pm 2 \pm \sqrt{3}$, but why is this?
I suppose my question really comes down to:
i) Why are the conjugates of $2 + \sqrt{3}$ those mentioned above? Why do the symmetric sums and products involved in Vieta's formulas yield rationals in this case?
ii) Are there any other 'types' of $\alpha$ (and/or $L$) for which the form of its conjugates are known?
Thanks

REPLY [1 votes]: Why are the conjugates of $2+\sqrt{3}$ those mentioned above?

The elements of $\mathbb Q(\sqrt{3})$ are of the form $a + b\sqrt{3}$
where $a,b \in \mathbb Q$. Let $c + d\sqrt{3}$ be a conjugate (there
might be more than one) of
$a + b\sqrt{3}$, and consider the polynomial
$$\begin{align*}
(x-(a+b\sqrt{3})(x-(c+d\sqrt{3})) &= x^2 -x((a+c) + (b+d)\sqrt{3})
+ (a+b\sqrt{3})(c+d\sqrt{3})\\
&= x^2 -x((a+c) + (b+d)\sqrt{3}) + ((ac+3bd)+(ad+bc)\sqrt{3})
\end{align*}$$
This is a polynomial in $\mathbb Q[x]$ if $d = -b$ and $ad+bc = 0$,
that is, $c = a$.  In other words,
$$x^2 -2ax + (a^2-3b^2) = (x-(a+b\sqrt{3})(x-(a-b\sqrt{3}))$$
is a quadratic polynomial in $\mathbb Q[x]$ that has $a+b\sqrt{3}$
as a root, and so $x^2 -2ax + (a^2-3b^2)$ is the minimal polynomial
of $a+b\sqrt{3}$ over $\mathbb Q$. The other root of this polynomial
is the (unique) conjugate $a - b\sqrt{3}$ of $a+b\sqrt{3}$.
In particular, $2-\sqrt{3}$ is the conjugate of $2+\sqrt{3}$,
and $\pm 2 \pm \sqrt{3}$ are not conjugates as you claim.
The four elements constitute two sets of conjugate elements:
$\{2+\sqrt{3}, 2-\sqrt{3}\}$ and $\{-2+\sqrt{3}, -2-\sqrt{3}\}$


TITLE: Frobenius dihedral groups
QUESTION [2 upvotes]: How can we show by a direct group-theoretic proof that the dihedral group $D_{2n}$ is a Frobenius group iff $n$ is an odd number?

REPLY [1 votes]: Hint: My assertion is that $D_{2n}$ acts on the vertices of respective $n$-gon. Non-trivial rotations have no fixed points. Further: 1) If $n$ is odd then each reflection has precisely one fixed point. So, $D_{2n}$ is Frobenius. 2) If $n$ is even then every reflection has precisely two fixed points. So, $D_{2n}$ is not Frobenius.


TITLE: Show that the sequence $a_n = \frac{(n+1)^2 -n^2}{n}$ converges and give its limit.
QUESTION [1 upvotes]: can somebody tell me if I did this somewhat correctly? First I estimated the limit of the sequence by calculating the first few results of the sequence and it looks like it is converging towards $2$. Using the Cauchy criterion we then have: $$\forall \epsilon > 0 \exists N\in\mathbb{N}_0 \forall n \geq N: \left|\frac{(n+1)^2 -n^2}{n} - 2\right| < \epsilon$$
and after doing a bit of math we have $$\left|\frac{(n+1)^2 -n^2}{n} - 2\right| < \epsilon \Leftrightarrow \left|\frac{1}{n}+2-2\right| < \epsilon \Leftrightarrow \frac{1}{n} < \epsilon \Leftrightarrow \frac{1}{\epsilon} < n $$ which means that we need a $N\in\mathbb{N}$ with $N > \frac{1}{\epsilon}$ and we have $\forall n\geq N$:$$\left|\frac{(n+1)^2 -n^2}{n} - 2\right| = \dots=\frac{1}{n}\leq\frac{1}{N}<\epsilon\,.$$
And hence our sequence $a_n \longrightarrow a_\infty$ with $a_\infty = 2$$._{\,\,\square}$

Did I do this right?

REPLY [1 votes]: To do this more cleanly but also rigorously, note that given $\epsilon > 0$, it is possible to find $N$ such that $N > \dfrac{1}{\epsilon}$.  Then for all $n > N$, we have that $\dfrac{1}{n} < \epsilon$.  From here, we can achieve the desired result:
$$
\epsilon > |{\dfrac{1}{n}}|  = |\dfrac{2n + 1 - 2n}{n}| = |\dfrac{2n + 1}{n} - 2 | = |\dfrac{n^2 + 2n + 1 - n^2}{n} - 2 | = |\dfrac{(n + 1)^2 - n^2}{n} - 2|
$$
Thus, given any $\epsilon > 0$, if we choose any $n > N$, we have that
$$
|\dfrac{(n + 1)^2 - n^2}{n} - 2| < \epsilon
$$
Since such $N$ exists, this implies that the sequence $a_n$ converges to 2.


TITLE: Decomposing a sphere (or defomed sphere) into a vertex-transitive graph with fixed-length curved edges
QUESTION [1 upvotes]: Please see the original problem specification (which Joseph O'Rourke was responding to in his answer) below.
Motivation -
I'm interested in a particular case of the problem where one wants to pack equiradius spheres around a central sphere s.t. every sphere has the same kissing number (i.e. contacts with its nearest-neighbors) and its nearest-neighbors are arranged in a rotationally symmetric fashion.  In the problem specification (below), the centerpoint of each sphere represents a vertex, and edges represent kissing contacts between nearest-neighbors.
It troubles me that I have no intuitive understanding for why the set of solutions to this problem should be restricted to the platonic solids, and I'd really like a better appreciation for why this is so.  Intuitively it feels like there should be some set of solutions involving more vertices/spheres than the dodedecahedron or icosahedron allows.

I'd like to decompose a sphere (say, or radius $R_s$) into a fully connected graph where:
(1) - The degree of each vertex is fixed at $N$,
(2) - Each edge in the decomposition graph has a constant bending angle $\theta$ and fixed length $L$,
(3) - The $N$ edges around each vertex, $v_i$, are arranged in a rotationally symmetric pattern,
(4) - The vertices $v_i$ lie on the sphere one is attempting to approximate.
As a function of the bending angle/curvature of each edge $\theta$, the uniform length of each edge $L$, and the number of edges around each vertex $N$, when can I generate a fully-connected graph that approximates the closed surface of the sphere, and how many vertices will it contain?
Again, any and all feedback is greatly appreciated!
Note - This is similar to my earlier question (which disallowed curved edges): "Constructing a graph that approximates a sphere using rotationally symmetric building blocks with equal numbers of edges".  I now realize (thanks to André Henriques) that the requirement of having a vertex-transitive/isohedral graph, with fixed-length straight edges and rotational symmetry around vertices, should restrict one to the five platonic solids (though I'm not entirely sure if the rotational symmetry constraint matters in this case).
Update - I have now removed the constraint that the graph is vertex-transitive.

REPLY [3 votes]: I assume you are aware that the vertex-transitive graphs represent a relatively constrained list?
See this description.  I fear that adding your conditions (2,3,4) on top of
vertex-transitivity will
again confine you to the skeletons of the Platonic solids.
Are you familiar with Fuller's geodesic domes?
They fail to satisfy your criteria, but they get perhaps
as close as is feasible, and may serve, depending upon your application:

After seeing your addendum on motivation, perhaps you should investigate the
literature on packing disks on a sphere.
For example, you might explore
Neil Sloane's web pages on
"Nice arrangements of points on a sphere in various dimensions"
which address the problem of

placing $n$ points on a sphere in $d$ dimensions so as to maximize the minimal distance (or equivalently the minimal angle) between them.

For example, here is the best packing known for $n=50$, due to Székely (1974):

Image © Hugo Pfoertner, 2001