If $n=-\infty$ then $f=0$ and
$\int_{\mathrm{sv}}\pi_{\partial_{z}}f\mathrm{d}z=0$. Likewise, we have
$G=\int_{\mathrm{sv}}\partial_{\overline{z}}f\mathrm{d}z=0$ and
$\int_{\mathrm{sv}}\pi_{\partial_{\overline{z}}}G\,\mathrm{d}{\overline{z}}=0$.

Now, assume that $n\geq 0$. By linearity and Corollary 60 we may assume that

(69) $f(z)=\frac{h(z)}{z-\beta({\overline{z}})},\quad
h\in\mathcal{G}_{\mathbb{F}}^{\mathbb{C}}.$

In particular (see Definition 23), $h$ is a ${\mathbb{C}}$-linear combination
of iterated integrals

$L_{v}({\overline{z}})L_{w({\overline{z}})}(z)=I(0,a_{1}\ldots
a_{\ell},{\overline{z}})I(0,\beta_{1}({\overline{z}})\ldots\beta_{m}({\overline{z}}),z),$

where $\ell+m\leq n$, $a_{i}\in{\mathbb{F}}$, $i=1,\ldots,\ell$,
$\beta_{k}({\overline{z}})\in\mathrm{FLT}\,_{\mathbb{F}}({\overline{z}})$,
$k=1,\ldots,m$. By I8 in Section 2 we get

$\displaystyle\partial_{\overline{z}}\frac{L_{v}({\overline{z}})L_{w({\overline{z}})}(z)}{z-\beta({\overline{z}})}$
$\displaystyle=$
$\displaystyle[\partial_{\overline{z}}\partial_{z}\log(z-\beta({\overline{z}}))]L_{v}({\overline{z}})L_{w({\overline{z}})}(z)$
$\displaystyle+\,[\partial_{z}\log(z-\beta({\overline{z}}))]\Bigg{[}\frac{L_{v^{\ell}}({\overline{z}})L_{w({\overline{z}})}(z)}{{\overline{z}}-a_{\ell}}+L_{v}({\overline{z}})\sum_{k=1}^{m}\Big{(}\partial_{\overline{z}}\log\frac{\beta_{k+1}-\beta_{k}}{\beta_{k}-\beta_{k-1}}\Big{)}L_{w({\overline{z}})^{k}}(z)\Bigg{]},$

where $\beta_{0}=0$ and $\beta_{m+1}=z$. The superscripts in $v^{\ell}$ and
$w({\overline{z}})^{k}$ refer to letters which have to be removed from the
words $v$ and $w({\overline{z}})$, respectively. For the empty word $e$ we
define $L_{e^{i}}=0$. Using Lemma 15 (2) we write the term
$[\partial_{z}\log(z-\beta({\overline{z}}))]L_{v}({\overline{z}})[\partial_{\overline{z}}\log(z-\beta_{m}({\overline{z}}))]L_{w({\overline{z}})^{m}}(z)$
in the above sum as

$\Bigg{[}[\partial_{\overline{z}}\log(z-\beta({\overline{z}}))][\partial_{z}\log(z-\beta_{m}({\overline{z}}))]+[\partial_{\overline{z}}\log(\beta({\overline{z}})-\beta_{m}({\overline{z}}))]\Big{[}\partial_{z}\log\frac{z-\beta({\overline{z}})}{z-\beta_{m}({\overline{z}})}\Big{]}\Bigg{]}L_{v}({\overline{z}})L_{w({\overline{z}})^{m}}(z).$

We get

(70)
$\partial_{\overline{z}}\frac{L_{v}({\overline{z}})L_{w({\overline{z}})}(z)}{z-\beta({\overline{z}})}=\partial_{z}[(\partial_{\overline{z}}\log(z-\beta({\overline{z}})))L_{v}({\overline{z}})L_{w({\overline{z}})}(z)]+g_{v,w({\overline{z}})}(z)$

with

(71) $\displaystyle g_{v,w({\overline{z}})}(z)$ $\displaystyle=$
$\displaystyle\sum_{k=1}^{m}\left(\partial_{\overline{z}}\log\left.\frac{\beta_{k+1}({\overline{z}})-\beta_{k}({\overline{z}})}{\beta_{k}({\overline{z}})-\beta_{k-1}({\overline{z}})}\right|_{\beta_{m+1}=\beta}\right)\frac{L_{v}({\overline{z}})L_{w({\overline{z}})^{k}}(z)}{z-\beta({\overline{z}})}+[\partial_{\overline{z}}\log({\overline{z}}-a_{\ell})]\frac{L_{v^{\ell}}({\overline{z}})I_{w({\overline{z}})}(z)}{z-\beta({\overline{z}})}$
$\displaystyle\quad-\,[\partial_{\overline{z}}\log(\beta({\overline{z}})-\beta_{m}({\overline{z}}))]\frac{L_{v}({\overline{z}})L_{w({\overline{z}})^{m}}(z)}{z-\beta_{m}({\overline{z}})}.$

We return to the function $f$ by summing over the words $v$ and
$w({\overline{z}})$ with their coefficients in ${\mathbb{C}}$. Equation (70)
becomes

(72)
$\partial_{\overline{z}}f(z)=\partial_{z}[(\partial_{\overline{z}}\log(z-\beta({\overline{z}})))h(z)]+g(z)=\partial_{z}[-f(z)\partial_{\overline{z}}\beta({\overline{z}})]+g(z),$

where $g$ is given by the sum over the $g_{v,w({\overline{z}})}$. Because
$\partial_{z}f,\partial_{\overline{z}}f\in\mathcal{S}\mathcal{V}_{\mathbb{C}}$
we get $g\in\mathcal{G}_{\mathbb{F}}$. By (71) we get
$g\in\mathcal{O}_{\mathbb{F}}({\overline{z}})\partial_{z}\mathcal{G}\mathcal{H}^{\mathbb{C}}_{\mathbb{F}}$.
Hence
$g\in\mathcal{O}_{\Sigma}^{\emptyset}(z,{\overline{z}})\partial_{z}\mathcal{G}\mathcal{H}^{\mathbb{C}}_{\Sigma}$
for some
$\Sigma({\overline{z}})\subset\mathrm{FLT}\,_{\mathbb{F}}({\overline{z}})$. We
use Proposition 58 and obtain that in the partial fraction basis of Remark 22

$g(z,{\overline{z}})=\sum_{a\in{\mathbb{F}}}\frac{g_{a}(z,{\overline{z}})}{{\overline{z}}-{\overline{a}}},\quad
g_{a}\in\mathcal{S}\mathcal{V}_{\mathbb{C}},$

where we used Lemma 15 (1) to restrict the (finite) sum to values in
${\mathbb{F}}$. From (71) we get
$g_{a}\in\partial_{z}\mathcal{G}\mathcal{H}_{\mathbb{F}}^{\mathbb{C}}$, hence
$g_{a}\in\partial_{z}\mathcal{G}_{\mathbb{F}}^{\mathbb{C}}$.

The weight of the $g_{a}$ is less than $n$. We use induction to obtain
primitives $G_{a}\in\mathcal{G}_{\mathbb{F}}^{\mathbb{C}}$ for the $g_{a}$.
Altogether we get

$\partial_{\overline{z}}f(z)=\partial_{z}G(z)\quad\text{with}\quad
G(z)=[\partial_{\overline{z}}\log(z-\beta({\overline{z}}))]h(z)+\sum_{a\in{\mathbb{F}}}\frac{G_{a}(z)}{{\overline{z}}-{\overline{a}}}.$

The first term in $G$ is in $\mathcal{S}\mathcal{V}_{\mathbb{C}}$, see (72).
By Lemma 14 it is in
$\partial_{\overline{z}}\mathcal{G}\mathcal{H}_{\mathbb{F}}^{\mathbb{C}}$ and
hence it is in $\partial_{\overline{z}}\mathcal{G}_{\mathbb{F}}^{\mathbb{C}}$.
The second term is in
$\partial_{\overline{z}}\mathcal{G}_{\mathbb{F}}^{\mathbb{C}}$ because
$G_{a}\in\mathcal{G}_{\mathbb{F}}^{\mathbb{C}}$. Therefore
$\int_{\mathrm{sv}}\partial_{\overline{z}}f\mathrm{d}z=G$.

Now, we construct a single-valued primitive of $\pi_{\partial_{z}}f$. Consider
$z$ and ${\overline{z}}$ as independent variables and determine a primitive
$F_{1}$ of $\pi_{\partial_{z}}f$ in
$\mathcal{G}\mathcal{H}_{\mathbb{F}}^{\mathbb{C}}$ by multivalued integration
with respect to $z$ using (11). With
$G_{1}=\partial_{\overline{z}}F_{1}\in\partial_{\overline{z}}\mathcal{G}\mathcal{H}_{\mathbb{F}}^{\mathbb{C}}$
we have

$\partial_{z}G_{1}(z,{\overline{z}})=\partial_{z}\partial_{\overline{z}}F_{1}(z,{\overline{z}})=\partial_{\overline{z}}\pi_{\partial_{z}}f(z,{\overline{z}})=\partial_{\overline{z}}f(z,{\overline{z}})=\partial_{z}G(z,{\overline{z}})=\partial_{z}\pi_{\partial_{\overline{z}}}G(z,{\overline{z}}).$

By Theorem 24 (2) we get

$G_{1}(z,{\overline{z}})=\pi_{\partial_{\overline{z}}}G(z,{\overline{z}})+G_{2}({\overline{z}}),\quad
G_{2}({\overline{z}})\in\partial_{\overline{z}}\mathcal{G}\mathcal{H}_{\mathbb{F}}^{\mathbb{C}}\cap\mathcal{H}\mathcal{L}_{\mathbb{F}}({\overline{z}}).$

Multivalued integration with respect to ${\overline{z}}$ provides an anti-
primitive
$F_{2}({\overline{z}})\in\mathcal{H}\mathcal{L}_{\mathbb{F}}^{\mathbb{C}}({\overline{z}})$
of $G_{2}({\overline{z}})$. We define

$F(z,{\overline{z}})=F_{1}(z,{\overline{z}})-F_{2}({\overline{z}})\in\mathcal{G}\mathcal{H}_{\mathbb{F}}^{\mathbb{C}}$

and get

$\partial_{z}F(z,{\overline{z}})=\pi_{\partial_{z}}f(z,{\overline{z}})\quad\text{and}\quad\partial_{\overline{z}}F(z,{\overline{z}})=G_{1}(z,{\overline{z}})-G_{2}({\overline{z}})=\pi_{\partial_{\overline{z}}}G(z,{\overline{z}})=\pi_{\partial_{\overline{z}}}\int_{\mathrm{sv}}\partial_{\overline{z}}f(z,{\overline{z}})\,\mathrm{d}z.$

We have to show that $F$ is single valued. Fix a point $a$ in
${\mathbb{C}}\cup\\{\infty\\}$. The function
$\partial_{z}F=\pi_{\partial_{z}}f\in\mathcal{S}\mathcal{V}_{\mathbb{C}}$ has
a single-valued log-Laurent expansion (59), (60) at $a$. By explicit
integration $\partial_{z}F$ has a primitive $\tilde{F}$ with a single-valued
log-Laurent expansion at $a$. The function
$F\in\mathcal{G}\mathcal{H}_{\mathbb{F}}^{\mathbb{C}}$ has a (possibly
multivalued) log-Laurent expansion at $(z,{\overline{z}})=(a,{\overline{a}})$
which has to coincide with $\tilde{F}$ up to an anti-analytic function.
Because $\partial_{\overline{z}}(F-\tilde{F})$ is single-valued at $a$ we get
for $a\in{\mathbb{C}}$,

$\partial_{\overline{z}}[F(z)-\tilde{F}(z)]=\sum_{{\overline{m}}=M_{a}}^{\infty}c_{\overline{m}}\,({\overline{z}}-{\overline{a}})^{\overline{m}},$

with an analogous formula for $a=\infty$. By explicit integration we get

(73)
$F(z)=\tilde{F}(z)+\sum_{{\overline{m}}\neq-1}\frac{c_{\overline{m}}}{{\overline{m}}+1}({\overline{z}}-{\overline{a}})^{{\overline{m}}+1}+c_{-1}\log({\overline{z}}-{\overline{a}})+d$

for some constant $d\in{\mathbb{C}}$. Because $\partial_{\overline{z}}F$ has
no anti-residues we get

$\overline{\mathrm{res}\,}_{a}\partial_{\overline{z}}\tilde{F}=-c_{-1}.$

By (62) this implies $\mathrm{res}\,_{a}\partial_{z}\tilde{F}=-c_{-1}$. If we
apply $\mathrm{res}\,_{a}\partial_{z}$ to both sides of (73) and use that
$\partial_{z}F$ is residue-free we obtain $0=-c_{-1}$. From (73) and the
single-valuedness of $\tilde{F}$ we get that $F$ is single-valued at $a$.

Because ${\mathbb{C}}\cup\\{\infty\\}$ is compact $F$ can only have a finite
number of singularities, hence $F\in\mathcal{S}\mathcal{V}_{\mathbb{C}}$. Then
($F(0)$ is the regularized limit of $F(z)$ at $z=0$)

$\int_{\mathrm{sv}}\pi_{\partial_{z}}f(z)\,\mathrm{d}z=\int_{\mathrm{sv}}\pi_{\partial_{\overline{z}}}G(z)\,\mathrm{d}{\overline{z}}=F(z)-F(0)\in\mathcal{G}_{\mathbb{F}}^{\mathbb{C}}$

is the (unique) single-valued primitive of $\pi_{\partial_{z}}f$ and anti-
primitive of $\pi_{\partial_{\overline{z}}}G$. ∎

Note that the proof of the existence of single-valued primitives is fully
constructive, see Section 7. By symmetry under complex conjugation (Theorem 24
(4)) we also get single-valued anti-primitives for functions in
$\partial_{\overline{z}}\mathcal{G}_{\mathbb{F}}^{\mathbb{C}}$. In general,
the set of letters $\Sigma({\overline{z}})$ is not stable under taking single-
valued (anti-)primitives. It may be augmented by constants related to
factorizing differences of FLTs in Lemma 15 (1).

######  Corollary 64.

Every $f\in\partial_{z}\mathcal{G}_{\mathbb{F}}^{\mathbb{C}}$
($\overline{f}\in\partial_{\overline{z}}\mathcal{G}_{\mathbb{F}}^{\mathbb{C}}$)
has a single-valued (anti-)primitive $\int_{\mathrm{sv}}f\mathrm{d}z$
($\int_{\mathrm{sv}}\overline{f}\mathrm{d}{\overline{z}}$). In particular,

(74)
$\partial_{z}\mathcal{G}_{\mathbb{F}}^{\mathbb{C}}=\partial_{z}(\mathcal{G}_{\mathbb{F}}^{\mathbb{C}}),\quad\partial_{\overline{z}}\mathcal{G}_{\mathbb{F}}^{\mathbb{C}}=\partial_{\overline{z}}(\mathcal{G}_{\mathbb{F}}^{\mathbb{C}}),\quad\partial_{\overline{z}}\partial_{z}\mathcal{G}_{\mathbb{F}}^{\mathbb{C}}=\partial_{\overline{z}}\partial_{z}(\mathcal{G}_{\mathbb{F}}^{\mathbb{C}})=\partial_{z}(\partial_{\overline{z}}\mathcal{G}_{\mathbb{F}}^{\mathbb{C}}).$

###### Proof.

Let $f\in\partial_{z}\mathcal{G}_{\Sigma}^{\mathbb{C}}$ for some finite
$\Sigma({\overline{z}})\subset\mathrm{FLT}\,_{\mathbb{F}}({\overline{z}})$. We
have $f\in\mathcal{S}\mathcal{V}_{\Sigma_{0}}$ for some finite set
$\Sigma_{0}\subset{\mathbb{C}}$. Define

$F(z)=\int_{\mathrm{sv}}\pi_{\partial_{z}}f(z)\,\mathrm{d}z+\sum_{a\in\Sigma_{0}}(\mathrm{res}\,_{a}f)[L_{a}(z)+L_{\overline{a}}({\overline{z}})].$

The first term exists by Theorem 63. We get
$F\in\mathcal{G}_{\mathbb{F}}^{\mathbb{C}}$, $\partial_{z}F=f$ by (41), and
$F(0)=0$. Hence $F=\int_{\mathrm{sv}}f\mathrm{d}z$. Equations (74) follow. ∎

###  6.3. Examples

######  Example 65.

For $\Sigma\subset{\mathbb{F}}$ we inductively define single-valued
hyperlogarithms $\mathcal{L}_{w}(z)$ for $w\in\Sigma^{\ast}$ by (see
Definition 75 for an extension to GSVHs)

(75)
$\mathcal{L}_{e}(z)=1,\quad\mathcal{L}_{wa}(z)=\int_{\mathrm{sv}}\frac{\mathcal{L}_{w}(z)}{z-a}\,\mathrm{d}z,\quad
w\in\Sigma^{\ast},\;a\in\Sigma.$

For every word $w\in\Sigma^{\ast}$ we get that
$\mathcal{L}_{w}\in\mathcal{G}_{\Sigma}^{\mathbb{C}}$ is a sum of terms
$L_{\overline{u}}({\overline{z}})L_{v}(z)$ with $u,v\in\Sigma^{\ast}$ (see [4,
5] and Example 26).

As in the commutative hexagon in Figure 2 the subtraction of residues vital.

######  Example 66.

Let $f(z)=z^{-1}\in\partial_{z}\mathcal{G}_{\\{0\\}}^{\mathbb{C}}$ (see
Example 10). Then
$\int_{\mathrm{sv}}f(z)\,\mathrm{d}z=\mathcal{L}_{0}(z)=\log(z{\overline{z}})$
whereas $\partial_{\overline{z}}f=0$. The hexagon commutes because
$\pi_{\partial_{z}}f=0$.

######  Example 67.

Let
$f(z)=\mathcal{L}_{a}(z)/(z-b)\in\partial_{z}\mathcal{G}_{\\{a,b\\}}^{\mathbb{C}}$
for $a,b\in{\mathbb{F}}$, $a\neq b$. Then
$\pi_{\partial_{z}}f(z)=[\mathcal{L}_{a}(z)-\mathcal{L}_{a}(b)]/(z-b)=[L_{a}(z)+L_{\overline{a}}({\overline{z}})-\mathcal{L}_{a}(b)]/(z-b)$
and

$\int_{\mathrm{sv}}\pi_{\partial_{z}}f(z)\mathrm{d}z=\mathcal{L}_{ab}(z)-\mathcal{L}_{a}(b)\mathcal{L}_{b}(z)\in\mathcal{G}_{\\{a,b\\}}^{\mathbb{C}}.$

We use the commutative hexagon in Figure 1 to obtain the representation (49)
for $\mathcal{L}_{ab}(z)$. Multivalued integration of $\pi_{\partial_{z}}f(z)$
gives

(76)
$L_{ab}(z)+L_{\overline{a}}({\overline{z}})L_{b}(z)-\mathcal{L}_{a}(b)L_{b}(z)+g({\overline{z}})$

for some anti-analytic function $g({\overline{z}})$.

On the other hand we have
$\partial_{\overline{z}}f(z)=1/[({\overline{z}}-{\overline{a}})(z-b)]$ and
$\int_{\mathrm{sv}}\partial_{\overline{z}}f(z)\,\mathrm{d}z=\mathcal{L}_{b}(z)/({\overline{z}}-{\overline{a}})$.
Subtraction of the anti-residue gives
$(\mathcal{L}_{b}(z)-\mathcal{L}_{b}(a))/({\overline{z}}-{\overline{a}})$.
Multivalued integration with respect to ${\overline{z}}$ leads to

(77)
$L_{\overline{a}}({\overline{z}})L_{b}(z)+L_{\overline{b}{\overline{a}}}({\overline{z}})-\mathcal{L}_{b}(a)L_{{\overline{a}}}({\overline{z}})+h(z)$

for some analytic function $h(z)$. The term
$L_{\overline{a}}({\overline{z}})L_{b}(z)$ has both, a $z$\- and a
${\overline{z}}$-dependence and is hence reproduced in (76). This is a general
feature of the approach: the calculation of these terms in either of the
integrations is redundant. From the anti-analytic terms in the above equation
we read off $g({\overline{z}})$ while the analytic terms in (76) provide
$h(z)$,

$g({\overline{z}})=L_{\overline{b}{\overline{a}}}({\overline{z}})-\mathcal{L}_{b}(a)L_{{\overline{a}}}({\overline{z}}),\qquad
h(z)=L_{ab}(z)-\mathcal{L}_{a}(b)L_{b}(z).$

The hexagon commutes. We get

$\mathcal{L}_{ab}(z)=L_{ab}(z)+L_{\overline{a}}({\overline{z}})L_{b}(z)+L_{\overline{b}{\overline{a}}}({\overline{z}})+\mathcal{L}_{a}(b)L_{b}({\overline{z}})-\mathcal{L}_{b}(a)L_{{\overline{a}}}({\overline{z}})$

with $\mathcal{L}_{ab}(0)=0$ (as required). We see that the projections
$\pi_{\partial_{z}}$, $\pi_{\partial_{\overline{z}}}$ are vital for the
hexagon to commute. Note that for the calculation of $\mathcal{L}_{ab}$ it
suffices to consider $z$ and ${\overline{z}}$ as independent variables and
extract $g({\overline{z}})$ from the limit $z=0$ in (77), see (7).

The above examples are single-valued hyperlogarithms [4, 5]. In the following
examples we construct genuine GSVHs of weights 1, 2, and 3.

######  Example 68.

Consider
$f(z)=(z+1/{\overline{z}})^{-1}\in\partial_{z}\mathcal{G}_{\\{-{\overline{z}}^{-1}\\}}^{\mathbb{C}}$,
see Example 39. Because $z{\overline{z}}\geq 0$ the function $f$ has no
singularities in ${\mathbb{C}}$. So, $\pi_{\partial_{z}}f=f$ and

$\int_{\mathrm{sv}}f(z)\,\mathrm{d}z=\log(z+1/{\overline{z}})+g({\overline{z}}).$

Differentiation of $f$ with respect to ${\overline{z}}$ yields
$(z{\overline{z}}+1)^{-2}$ which integrates (with
$\int_{\mathrm{sv}}\mathrm{d}z$) to
$({\overline{z}}+1/z)^{-1}\in\partial_{\overline{z}}\mathcal{G}_{\mathbb{F}}^{\mathbb{C}}$.
The result has no anti-residue, so that

$\int_{\mathrm{sv}}f(z)\,\mathrm{d}z=\log({\overline{z}}+1/z)+h(z).$

Comparison of both equations leads to

$\int_{\mathrm{sv}}\frac{1}{z{\overline{z}}+1}\,\mathrm{d}z=\frac{\log(z{\overline{z}}+1)}{{\overline{z}}}.$

######  Example 69.

Consider
$f(z)=\log(z{\overline{z}})/(z-1/{\overline{z}})\in\partial_{z}\mathcal{G}_{\\{0,{\overline{z}}^{-1}\\}}^{\mathbb{C}}$,
see Example 48, Section 8.1, and Example 4.11 in [18]. The singularity at
$z=1/{\overline{z}}$ is lifted by the numerator. We have
$\pi_{\partial_{z}}f=f$ and with
$\log(z{\overline{z}})=\mathcal{L}_{0}(z)=L_{0}(z)+L_{0}({\overline{z}})$ we
obtain

$\int_{\mathrm{sv}}f(z)\,\mathrm{d}z=L_{0,{\overline{z}}^{-1}}(z)+L_{0}({\overline{z}})L_{{\overline{z}}^{-1}}(z)+g({\overline{z}}).$

Differentiation of $f$ with respect to ${\overline{z}}$ yields
$-\log(z{\overline{z}})/(z{\overline{z}}-1)^{2}+1/(z{\overline{z}}-1)$. The
sum is in $\mathcal{G}_{\mathbb{F}}$ although the individual terms are not. We
use integration by parts for the first term to obtain

(78)
$\int_{\mathrm{sv}}\left(-\frac{\log(z{\overline{z}})}{(z{\overline{z}}-1)^{2}}+\frac{1}{z{\overline{z}}-1}\right)\mathrm{d}z=\frac{\log(z{\overline{z}})}{{\overline{z}}-1/z}\in\partial_{\overline{z}}\mathcal{G}_{\mathbb{F}}^{\mathbb{C}}.$

The result has no anti-residues and we get

(79)
$\int_{\mathrm{sv}}f(z)\,\mathrm{d}z=L_{0,z^{-1}}({\overline{z}})+L_{0}(z)L_{z^{-1}}({\overline{z}})+h(z).$

The result is not in the $z,{\overline{z}}$ basis of (49). With the role of
$z$ and ${\overline{z}}$ interchanged, we seem to need a transformation into
the $z,{\overline{z}}$ basis before we are able to extract $g({\overline{z}})$
and $h(z)$. More efficiently, we use the redundancy of the procedure, set
$z=0$ in (78) (for general ${\overline{z}}$), and determine
$g({\overline{z}})$ by multivalued integration with respect to
${\overline{z}}$, see (7). Because $\lim_{z\to
0}\log(z{\overline{z}})/({\overline{z}}-1/z)=0$ for all ${\overline{z}}$, we
get $g({\overline{z}})=0$,

$\int_{\mathrm{sv}}\frac{\log(z{\overline{z}})}{z{\overline{z}}-1}\,\mathrm{d}z=\frac{L_{0,{\overline{z}}^{-1}}(z)+L_{0}({\overline{z}})L_{{\overline{z}}^{-1}}(z)}{{\overline{z}}}.$

######  Example 70.

To show that we can also use the commutative hexagon to integrate with
$\int_{\mathrm{sv}}\mathrm{d}{\overline{z}}$ in the $z,{\overline{z}}$ basis,
we now consider the complex conjugate of the previous example. We take
$f(z)=\log(z{\overline{z}})/({\overline{z}}-1/z)\in\partial_{\overline{z}}\mathcal{G}_{\mathbb{F}}^{\mathbb{C}}$.
The complex conjugate commutative hexagon gives the complex conjugates of the
previous equations. We use the complex conjugate of (79) to obtain

$\int_{\mathrm{sv}}f(z)\,\mathrm{d}{\overline{z}}=L_{0,{\overline{z}}^{-1}}(z)+L_{0}({\overline{z}})L_{{\overline{z}}^{-1}}(z)+h({\overline{z}}).$

The right hand side is in the desired $z,{\overline{z}}$ basis and we
determine $h({\overline{z}})$ by taking the limit $z=0$ in $f(z)$ before the
multivalued integration with respect to ${\overline{z}}$. Because
$f(0,{\overline{z}})=0$ we obtain

$\int_{\mathrm{sv}}\frac{\log(z{\overline{z}})}{z{\overline{z}}-1}\,\mathrm{d}{\overline{z}}=\frac{L_{0,{\overline{z}}^{-1}}(z)+L_{0}({\overline{z}})L_{{\overline{z}}^{-1}}(z)}{z}.$

######  Example 71.

Consider
$[\mathcal{L}_{01}(z)-\mathcal{L}_{10}(z)]/(z-{\overline{z}})=4\mathrm{i}D(z)/(z-{\overline{z}})$,
see Example 51 and Section 8.3. The single-valued primitive of
$D(z)/(z-{\overline{z}})\in\partial_{z}\mathcal{G}_{\\{0,1,{\overline{z}}\\}}^{\mathbb{C}}$
was first constructed (using the Galois coaction) in [11]. The construction
with the commutative hexagon is analogous to the previous examples. We obtain

$\displaystyle\int_{\mathrm{sv}}\frac{\mathcal{L}_{01}(z)-\mathcal{L}_{10}(z)}{z-{\overline{z}}}\,\mathrm{d}z$
$\displaystyle=$ $\displaystyle
L_{01{\overline{z}}}(z)-L_{10{\overline{z}}}(z)+L_{0}({\overline{z}})L_{1{\overline{z}}}(z)-L_{1}({\overline{z}})L_{0{\overline{z}}}(z)$
$\displaystyle\quad+\,L_{10}({\overline{z}})L_{\overline{z}}(z)-L_{01}({\overline{z}})L_{\overline{z}}(z)+L_{100}({\overline{z}})-L_{101}({\overline{z}})\in\mathcal{G}_{\\{0,1,{\overline{z}}\\}}^{\mathbb{C}}.$

This is the first (by weight) GSVH in the alphabet $0,1,{\overline{z}}$ which
is not a single-valued multiple polylogarithm. The alphabet
$0,1,{\overline{z}}$ is of particular importance in pQFT because it is
ubiquitous in dimensionally regularized amplitudes [18, 22].

###  6.4. The structure theorem for GSVHs

We use the partial fraction basis in Remark 22 to extend single-valued
integration to $\mathcal{G}_{\mathbb{F}}$.

######  Definition 72.

Let $f=(z-\beta({\overline{z}}))^{m}h(z)\in\mathcal{G}_{\mathbb{F}}$,
$\beta\in\mathrm{FLT}\,_{\mathbb{F}}$,
$h\in\mathcal{G}_{\mathbb{F}}^{\mathbb{C}}$. If $m=-1$ then
$f\in\partial_{z}\mathcal{G}_{\mathbb{F}}^{\mathbb{C}}$ and
$\int_{\mathrm{sv}}f\mathrm{d}z$ is (uniquely) defined in Definition 62.
Otherwise we use integration by parts to inductively define

(80)
$\int_{\mathrm{sv}}(z-\beta({\overline{z}}))^{m}h(z)\,\mathrm{d}z=\frac{(z-\beta({\overline{z}}))^{m+1}}{m+1}h(z)-\frac{1}{m+1}\int_{\mathrm{sv}}(z-\beta({\overline{z}}))^{m+1}\partial_{z}h(z)\,\mathrm{d}z.$

We extend the definition to $\mathcal{G}_{\mathbb{F}}$ by linearity in
$\mathcal{O}_{\mathbb{F}}({\overline{z}})$ using the partial fraction basis in
Remark 22. The complex conjugate integral
$\int_{\mathrm{sv}}\mathrm{d}{\overline{z}}$ is defined such that complex
conjugation commutes with integration.

######  Proposition 73.

The single-valued integrals $\int_{\mathrm{sv}}\mathrm{d}z$ and
$\int_{\mathrm{sv}}\mathrm{d}{\overline{z}}$ in Definition 72 are well-defined
endomorphisms of $\mathcal{G}_{\mathbb{F}}$ which are consistent with the
single-valued integrals on
$\partial_{\overline{z}}\partial_{z}\mathcal{G}_{\mathbb{F}}^{\mathbb{C}}$ in
Definition 62.

For every $f\in\mathcal{G}_{\mathbb{F}}$ we get
$\partial_{z}\int_{\mathrm{sv}}f\mathrm{d}z=f$ and
$\partial_{\overline{z}}\int_{\mathrm{sv}}f\mathrm{d}{\overline{z}}=f$.

###### Proof.

We have $\int_{\mathrm{sv}}0\,\mathrm{d}z=0$. Using linearity in the partial
fraction basis of Remark 22 for $0\neq f\in\mathcal{G}_{\mathbb{F}}$ we may
assume that $f(z)=(z-\beta({\overline{z}}))^{m}h(z)$, with
$\beta\in\mathrm{FLT}\,_{\mathbb{F}}$, $m\in{\mathbb{Z}}$,
$h\in\mathcal{G}_{\mathbb{F}}^{\mathbb{C}}$. We can iterate (80) because
$(z-\beta({\overline{z}}))^{m+1}\partial_{z}h(z)=((z-\beta({\overline{z}}))\partial_{z}-m)f(z)\in\mathcal{G}_{\mathbb{F}}$.
With every iteration the weight of $h$ is reduced until $h=0$. This ensures
that the integration algorithm terminates after a finite number of steps.

We show that $\int_{\mathrm{sv}}\mathrm{d}z$ maps into
$\mathcal{G}_{\mathbb{F}}$. For $m=-1$ this is proved in Theorem 63. For
$m\neq-1$ it suffices to show that both terms on the right hand side of (80)
are single-valued. The first term equals
$f(z)(z-\beta({\overline{z}}))/(m+1)\in\mathcal{G}_{\mathbb{F}}$ while the
second term is single-valued by induction over the weight of $h$. So, for
every $f\in\mathcal{G}_{\mathbb{F}}$ we get a well-defined function
$F=\int_{\mathrm{sv}}f\mathrm{d}z\in\mathcal{G}_{\mathbb{F}}$.

By induction over the weight of $f$ we get $\partial_{z}F=f$ because (80)
becomes an identity upon differentiation with respect to $z$.

Let
$f\in\partial_{\overline{z}}\partial_{z}\mathcal{G}_{\mathbb{F}}^{\mathbb{C}}$
and $F=\int_{\mathrm{sv}}f\mathrm{d}z$ according to Definition 72. We need to
show that $F\in\partial_{\overline{z}}\mathcal{G}_{\mathbb{F}}^{\mathbb{C}}$.
By Corollary 64 we have
$f\in\partial_{z}(\partial_{\overline{z}}\mathcal{G}_{\mathbb{F}}^{\mathbb{C}})$,

$f(z)=\partial_{z}\sum_{\beta\in\mathrm{FLT}\,_{\mathbb{F}}}\frac{h_{\beta}(z)}{{\overline{z}}-\beta(z)}=\sum_{\beta\in\mathrm{FLT}\,_{\mathbb{F}}}\Big{[}[\partial_{\overline{z}}\partial_{z}\log({\overline{z}}-\beta(z))]h_{\beta}(z)+\frac{\partial_{z}h_{\beta}(z)}{{\overline{z}}-\beta(z)}\Big{]},$

where the sum is finite and
$h_{\beta}\in\mathcal{G}_{\mathbb{F}}^{\mathbb{C}}$. From (49) we get
$\partial_{z}h_{\beta}(z)=g_{\beta}(z)/(z-\gamma({\overline{z}}))$ with
$g_{\beta}\in\mathcal{G}_{\mathbb{F}}^{\mathbb{C}}$ and
$\gamma\in\mathrm{FLT}\,_{\mathbb{F}}$. The terms with $\beta\in{\mathbb{F}}$
integrate to
$F_{0}(z)=\sum_{\beta\in{\mathbb{F}}}h_{\beta}(z)/({\overline{z}}-\beta)$. In
the other terms $\beta(z)=(az+b)/(cz+d)$ is invertible and we get from Lemma
14,

$\partial_{z}\frac{h_{\beta}(z)}{{\overline{z}}-\beta(z)}=\frac{[\partial_{\overline{z}}\beta^{-1}({\overline{z}})]h_{\beta}(z)}{(z-\beta^{-1}({\overline{z}}))^{2}}+\frac{g_{\beta}(z)}{({\overline{z}}-\beta(z))(z-\gamma({\overline{z}}))}.$

Using Lemma 14 with $(z,\beta)\mapsto({\overline{z}},\beta^{-1})$ we get

$-\frac{\partial_{\overline{z}}\beta^{-1}({\overline{z}})}{z-\beta^{-1}({\overline{z}})}=\frac{1}{{\overline{z}}-\beta(z)}-\frac{1}{{\overline{z}}-a/c},$

where the second term is absent if $c=0$. With this equation, integration by
parts in (80) gives for the integral over the terms with
$\beta\notin{\mathbb{F}}$,

$F_{1}(z)=\sum_{\beta\in\mathrm{FLT}\,_{\mathbb{F}}\backslash{\mathbb{F}}}\Big{[}\frac{h_{\beta}(z)}{{\overline{z}}-\beta(z)}-\frac{h_{\beta}(z)}{{\overline{z}}-a/c}+\int_{\mathrm{sv}}\frac{g_{\beta}(z)}{({\overline{z}}-a/c)(z-\gamma({\overline{z}}))}\,\mathrm{d}z\Big{]}=\sum_{\beta\in\mathrm{FLT}\,_{\mathbb{F}}\backslash{\mathbb{F}}}\frac{h_{\beta}(z)}{{\overline{z}}-\beta(z)},$

where we used
$\int_{\mathrm{sv}}\frac{g_{\beta}(z)}{z-\gamma({\overline{z}})}\,\mathrm{d}z=h_{\beta}(z)$
according to the case $m=-1$ in Definition 72. Altogether,
$F(z)=F_{0}(z)+F_{1}(z)=\sum_{\beta\in\mathrm{FLT}\,_{\mathbb{F}}}h_{\beta}/({\overline{z}}-\beta(z))\in\partial_{\overline{z}}\mathcal{G}_{\mathbb{F}}^{\mathbb{C}}$.

By complex conjugation Proposition 73 also holds for integration with respect
to ${\overline{z}}$. ∎

From Theorem 63 we obtain the following structure theorem for GSVHs.

######  Theorem 74.

Let ${\mathbb{F}}=\overline{{\mathbb{F}}}$ be a quadratically closed field,
see (45). Let
$\Sigma({\overline{z}})\subset\mathrm{FLT}\,_{\mathbb{F}}({\overline{z}})$ and
$\Sigma_{0}\subset{\mathbb{F}}$ be finite sets of letters and singular points,
respectively.

  1. (1)

Let $\mathcal{O}_{\Sigma}^{\emptyset}(z,{\overline{z}})$ be the ring of
regular functions, see Definition 55. Then $\mathcal{G}_{\Sigma}$ is a free
$\mathcal{O}_{\Sigma}^{\emptyset}(z,{\overline{z}})$-module which is closed
under multiplication. In particular, $\mathcal{G}_{\Sigma}^{\mathbb{C}}$ is a
${\mathbb{C}}$-algebra.

  2. (2)

The sequence

$0\longrightarrow\mathcal{O}_{\Sigma_{0}}({\overline{z}})\longrightarrow\mathcal{G}_{\mathbb{F}}\cap\mathcal{S}\mathcal{V}_{\Sigma_{0}}\stackrel{{\scriptstyle\partial_{z}}}{{\longrightarrow}}\mathcal{G}_{\mathbb{F}}\cap\mathcal{S}\mathcal{V}_{\Sigma_{0}}\longrightarrow
0$

is exact. I.e. the kernel of $\partial_{z}$ in
$\mathcal{G}_{\mathbb{F}}\cap\mathcal{S}\mathcal{V}_{\Sigma_{0}}$ is
$\mathcal{O}_{\Sigma_{0}}({\overline{z}})$ and every
$f\in\mathcal{G}_{\mathbb{F}}\cap\mathcal{S}\mathcal{V}_{\Sigma_{0}}$ has a
primitive
$F\in\mathcal{G}_{\mathbb{F}}\cap\mathcal{S}\mathcal{V}_{\Sigma_{0}}$ with
$\partial_{z}F=f$.

  3. (3)

$\mathcal{G}_{\Sigma}$ is differentially simple. I.e. for every $0\neq
f\in\mathcal{G}_{\Sigma}$ there exists a differential operator $D$ such that
$Df=1$.

  4. (4)

$\mathcal{G}_{\mathbb{F}}$ and $\mathcal{G}_{\mathbb{F}}^{\mathbb{C}}$ are
stable under complex conjugation,

(81)
$\overline{\mathcal{G}_{\mathbb{F}}}=\mathcal{G}_{\mathbb{F}},\quad\overline{\mathcal{G}_{\mathbb{F}}^{\mathbb{C}}}=\mathcal{G}_{\mathbb{F}}^{\mathbb{C}}.$

In particular, the sequence

$0\longrightarrow\mathcal{O}_{\Sigma_{0}}(z)\longrightarrow\mathcal{G}_{\mathbb{F}}\cap\mathcal{S}\mathcal{V}_{\Sigma_{0}}\stackrel{{\scriptstyle\partial_{\overline{z}}}}{{\longrightarrow}}\mathcal{G}_{\mathbb{F}}\cap\mathcal{S}\mathcal{V}_{\Sigma_{0}}\longrightarrow
0$

is exact.

  5. (5)

$\mathcal{G}_{\mathbb{F}}$ and $\mathcal{G}_{\mathbb{F}}^{\mathbb{C}}$ are
stable under transformations in $\mathrm{FLT}\,_{\mathbb{F}}$. I.e. for any
$f\in\mathcal{G}_{\mathbb{F}}$, $g\in\mathcal{G}_{\mathbb{F}}^{\mathbb{C}}$,
and $\beta\in\mathrm{FLT}\,_{\mathbb{F}}$,

(82) $f(\beta(z))\in\mathcal{G}_{\mathbb{F}},\quad\text{and}\quad
g(\beta(z))\in\mathcal{G}_{\mathbb{F}}^{\mathbb{C}}.$

  6. (6)

Every $f\in\mathcal{G}_{\Sigma}$ is in
$\mathcal{S}\mathcal{V}_{\Sigma\cap{\mathbb{F}}}$. If
$f\in\mathcal{G}_{\Sigma}^{\mathbb{C}}$, the single-valued log-Laurent
expansions (59), (60) at
$a\in(\Sigma({\overline{z}})\cap{\mathbb{F}})\cup\\{\infty\\}$ have $M_{a}=0$.

###### Proof.

It is clear from Theorem 24 (1) and Remark 36 (2) that
$\mathcal{G}_{\Sigma}^{\mathbb{C}}$ is a ${\mathbb{C}}$-algebra. Lemma 56
states that the weight zero part of $\mathcal{G}_{\Sigma}$ is
$\mathcal{O}_{\Sigma}^{\emptyset}(z,{\overline{z}})$. Hence
$\mathcal{G}_{\Sigma}$ is a
$\mathcal{O}_{\Sigma}^{\emptyset}(z,{\overline{z}})$-module. It is free by
Theorem 24 (1).

By Theorem 24 (2) and Proposition 8 the kernel of $\partial_{z}$ in
$\mathcal{G}_{\mathbb{F}}\cap\mathcal{S}\mathcal{V}_{\Sigma_{0}}$ is
$\mathcal{O}_{\mathbb{F}}({\overline{z}})\cap\mathcal{S}\mathcal{V}_{\Sigma_{0}}=\mathcal{O}_{\Sigma_{0}}({\overline{z}})$.
In Proposition 73 it was proved that every
$f\in\mathcal{G}_{\mathbb{F}}\cap\mathcal{S}\mathcal{V}_{\Sigma_{0}}$ has a
primitive $\tilde{F}\in\mathcal{G}_{\tilde{\Sigma}}$ for some finite set
$\tilde{\Sigma}({\overline{z}})\subset\mathrm{FLT}\,_{\mathbb{F}}({\overline{z}})$.
By Theorem 54 the function $\tilde{F}$ is ${\mathbb{C}}$-analytic outside
$\tilde{\Sigma}_{0}:=\tilde{\Sigma}({\overline{z}})\cap{\mathbb{F}}$. If
$\tilde{\Sigma}_{0}\subset\Sigma_{0}$ then
$F:=\tilde{F}\in\mathcal{S}\mathcal{V}_{\Sigma_{0}}$. Otherwise let
$a\in\tilde{\Sigma}_{0}\backslash\Sigma_{0}$. Because
$f\in\mathcal{G}_{\Sigma_{0}}$, the single-valued log-Laurent expansion of
$\tilde{F}$ at $a$ differentiates (with respect to $z$) to a
${\mathbb{C}}$-analytic function at $a$. By explicit differentiation of its
expansion we see that $\tilde{F}$ is the sum of a ${\mathbb{C}}$-analytic
function at $a$ and a pole part
$G_{a}({\overline{z}})\in{\mathbb{C}}[({\overline{z}}-{\overline{a}})^{-1}]$.
We consider all points in $\tilde{\Sigma}_{0}\backslash\Sigma_{0}$ and set
$F:=\tilde{F}-\sum_{a\in\tilde{\Sigma}_{0}\backslash\Sigma_{0}}G_{a}$. We get
$\partial_{z}F=\partial_{z}\tilde{F}=f$ and
$F\in\mathcal{S}\mathcal{V}_{\Sigma_{0}}$.

To prove (3) we use the differential operator $D$ that was constructed in the
proof of Theorem 24 (3).

Statement (4) follows from Theorem 24 (4) and Remark 36 (4).

Statement (5) follows from Theorem 24 (5) and Remark 36 (5).

Statement (6) is Theorem 54. ∎

Theorem 74 (5) and (6) suggest that GSVHs should be considered as objects on
the Riemann sphere ${\mathbb{C}}\cup\\{\infty\\}$.

##  7\. Single-valued integration

The commutative hexagon in Figure 1 establishes a bootstrap algorithm for the
construction of GSVHs. The crucial step is the single-valued integration in
$\partial_{z}\partial_{\overline{z}}\mathcal{G}_{\mathbb{F}}^{\mathbb{C}}$ on
the bottom right sector. After using integration by parts there is always a
weight-drop in this integration. The subtraction of residues with
$\pi_{\partial_{z}}$ needs to be compensated by adding single-valued
logarithms. With the strategy of Examples 69 and 70 we get the following
inductive formulae for the single-valued integral of a function
$f\in\partial_{z}\mathcal{G}_{\Sigma}^{\mathbb{C}}$,
$\Sigma({\overline{z}})\subset\mathrm{FLT}\,({\overline{z}})$, in the
$z,{\overline{z}}$ basis of (49) (in terms of the multivalued integrals
$\int_{0}\mathrm{d}z$ and $\int_{0}\mathrm{d}{\overline{z}}$, see (3.5)),

$\displaystyle\int_{\mathrm{sv}}f(z)\,\mathrm{d}z$ $\displaystyle=$
$\displaystyle\int_{0}\pi_{\partial_{z}}f(z)\,\mathrm{d}z+\int_{0}\Big{(}\pi_{\partial_{\overline{z}}}\int_{\mathrm{sv}}\partial_{\overline{z}}f(z)\,\mathrm{d}z\Big{)}\Big{|}_{z=0}\mathrm{d}{\overline{z}}+\sum_{a\in\Sigma({\overline{z}})\cap{\mathbb{F}}}(\mathrm{res}\,_{a}f)\mathcal{L}_{a}(z),$
(83) $\displaystyle\int_{\mathrm{sv}}f(z)\,\mathrm{d}{\overline{z}}$
$\displaystyle=$
$\displaystyle\int_{0}\Big{(}\pi_{\partial_{\overline{z}}}f(z)\Big{)}\Big{|}_{z=0}\mathrm{d}{\overline{z}}+\int_{0}\pi_{\partial_{z}}\int_{\mathrm{sv}}\partial_{z}f(z)\,\mathrm{d}{\overline{z}}\,\mathrm{d}z+\sum_{a\in\Sigma({\overline{z}})\cap{\mathbb{F}}}(\overline{\mathrm{res}\,}_{a}f)\mathcal{L}_{a}(z),$

where we consider $z$ and ${\overline{z}}$ as independent variables. Both
integrations in (7) preserve the $z,{\overline{z}}$ basis.

###  7.1. Extension of GSVHs

Equation (7) was used for first calculations with GSVHs in a predecessor of
[19]. Because the algorithm was programmed in Maple which is not ideal for
handling large expressions it turned out that the following single-valued
representation is more efficient.

We use (7) to extend single-valued hyperlogarithms $\mathcal{L}_{w}$ in
Example 65 to words
$w({\overline{z}})\in\mathrm{FLT}\,_{\mathbb{F}}({\overline{z}})^{\ast}$. By
linearity, GSVHs can be expressed as sums of such extended single-valued
hyperlogarithms (ESVHs).

######  Definition 75.

Let $\beta({\overline{z}})\in\mathrm{FLT}\,_{\mathbb{F}}({\overline{z}})$ and
$w({\overline{z}})\in\mathrm{FLT}\,_{\mathbb{F}}({\overline{z}})^{\ast}$. We
inductively define $\mathcal{L}_{w({\overline{z}})}(z)$ by
$\mathcal{L}_{e}(z)=1$ for the empty word $e$ and (see (75))

(84)
$\mathcal{L}_{w({\overline{z}})\beta({\overline{z}})}(z)=\int_{\mathrm{sv}}\frac{\mathcal{L}_{w({\overline{z}})}(z)}{z-\beta({\overline{z}})}\,\mathrm{d}z,$

where $\int_{\mathrm{sv}}\mathrm{d}z$ is defined by the first identity in (7)
with zero (anti-)residues at non-constant points
($\mathrm{res}\,_{\beta}=\overline{\mathrm{res}\,}_{\beta}=0$ if
$\beta({\overline{z}})\notin{\mathbb{F}}$). Integration by parts in the bottom
right sector of the commutative hexagon is defined via (80). The space of
ESVHs is the free $\mathcal{O}_{\Sigma}(z,{\overline{z}})$ module

$\mathcal{E}_{\Sigma}=\langle\mathcal{L}_{w({\overline{z}})}(z),\,w({\overline{z}})\in\mathrm{FLT}\,_{\mathbb{F}}^{\ast}({\overline{z}})\rangle_{\mathcal{O}_{\Sigma}(z,{\overline{z}})}.$

The ${\mathbb{C}}$-algebra of ESVHs with constant coefficients is
$\mathcal{E}_{\Sigma}^{\mathbb{C}}$. Moreover,

$\mathcal{E}_{\mathbb{F}}=\bigcup_{\Sigma({\overline{z}})\subset\mathrm{FLT}\,_{\mathbb{F}}({\overline{z}})}\mathcal{E}_{\Sigma}\quad\text{and}\quad\mathcal{E}_{\mathbb{F}}^{\mathbb{C}}=\bigcup_{\Sigma({\overline{z}})\subset\mathrm{FLT}\,_{\mathbb{F}}({\overline{z}})}\mathcal{E}_{\Sigma}^{\mathbb{C}}.$

Note that ESVHs are not always single-valued in the sense of Definition 35.
They may fail to have single-valued log-Laurent expansions and they may have
non-trivial monodromies.

######  Example 76.

Consider the function $f(z)=1/(z-2{\overline{z}})$ which is
${\mathbb{C}}$-analytic in ${\mathbb{C}}\backslash\\{0\\}$ but fails to have a
single-valued log-Laurent expansion at $0$, see Example 43. Using Definition
75 we obtain

$\mathcal{L}_{2{\overline{z}}}(z)=L_{2{\overline{z}}}(z)+L_{0}({\overline{z}})\in\mathcal{E}_{\\{2{\overline{z}}\\}}^{\mathbb{C}}.$

Because $|z/(2{\overline{z}})|=1/2<1$ we find that
$L_{2{\overline{z}}}(z)=\log(1-z/(2{\overline{z}}))$ has trivial monodromy at
0. Hence
$\mathcal{M}_{0}\mathcal{L}_{2{\overline{z}}}(z)=\mathcal{L}_{2{\overline{z}}}(z)-2\pi\mathrm{i}$.
We have
$\mathcal{L}_{2{\overline{z}}}(z)\notin\mathcal{G}_{\\{2{\overline{z}}\\}}^{\mathbb{C}}$.

######  Example 77.

For the weight three GSVH in the letters 0, 1, ${\overline{z}}$ in Example 71
we have

(85)
$\int_{\mathrm{sv}}\frac{\mathcal{L}_{01}(z)-\mathcal{L}_{10}(z)}{z-{\overline{z}}}\,\mathrm{d}z=\mathcal{L}_{01{\overline{z}}}(z)-\mathcal{L}_{10{\overline{z}}}(z)\in\mathcal{G}_{0,1,{\overline{z}}}^{\mathbb{C}}$

with

$\displaystyle\mathcal{L}_{01{\overline{z}}}(z)$ $\displaystyle=$
$\displaystyle
L_{01{\overline{z}}}(z)+L_{0}({\overline{z}})L_{1{\overline{z}}}(z)+L_{10}({\overline{z}})L_{\overline{z}}(z)$
$\displaystyle\quad+\,2L_{100}({\overline{z}})+2L_{010}({\overline{z}})+2L_{001}({\overline{z}})-L_{101}({\overline{z}})-L_{110}({\overline{z}})-\zeta(2)L_{1}({\overline{z}}),$
$\displaystyle\mathcal{L}_{10{\overline{z}}}(z)$ $\displaystyle=$
$\displaystyle
L_{10{\overline{z}}}(z)+L_{1}({\overline{z}})L_{0{\overline{z}}}(z)+L_{01}({\overline{z}})L_{\overline{z}}(z)$
$\displaystyle\quad+\,L_{100}({\overline{z}})+2L_{010}({\overline{z}})+2L_{001}({\overline{z}})-L_{110}({\overline{z}})-\zeta(2)L_{1}({\overline{z}}),$

where $\zeta(2)=\pi^{2}/6$ is the Riemann zeta function at 2. Neither
$\mathcal{L}_{01{\overline{z}}}(z)$ nor $\mathcal{L}_{10{\overline{z}}}(z)$
are GSVHs because each one is singular on the real axis. Note that many terms
cancel in
$\mathcal{L}_{01{\overline{z}}}(z)-\mathcal{L}_{10{\overline{z}}}(z)$.

######  Example 78.

For
$\Sigma({\overline{z}})\subset\mathrm{FLT}\,_{\mathbb{F}}^{\emptyset}({\overline{z}})$,
see (61), we have $\mathcal{E}_{\Sigma}=\mathcal{G}_{\Sigma}$. In particular,
the dimension $d_{n}$ of GSVHs in $\Sigma({\overline{z}})$ with weight $n$ is
$|\Sigma({\overline{z}})|^{n}$. This case also contains the class of single-
valued hyperlogarithms defined in [4, 5] where $\Sigma\subset{\mathbb{F}}$.

Note that the weight zero piece of $\mathcal{E}_{\Sigma}$ is
$\mathcal{O}_{\Sigma}(z,{\overline{z}})$, see (47), in contrast to the weight
zero piece $\mathcal{O}_{\Sigma}^{\emptyset}(z,{\overline{z}})$ of
$\mathcal{G}_{\Sigma}$ (see Lemma 56). In analogy to Theorem 74 (2) we get the
exact sequence

$0\longrightarrow\mathcal{O}_{\mathbb{F}}({\overline{z}})\longrightarrow\mathcal{E}_{\mathbb{F}}\stackrel{{\scriptstyle\partial_{z}}}{{\longrightarrow}}\mathcal{E}_{\mathbb{F}}\longrightarrow
0,$

where the existence of primitives uses integration by parts as in Definition
72. The kernel of $\partial_{z}$ in $\mathcal{E}_{\mathbb{F}}$ is given by
anti-analytic functions in the weight zero piece
$\mathcal{O}_{\mathbb{F}}(z,{\overline{z}})$. This is
$\mathcal{O}_{\mathbb{F}}({\overline{z}})$.

In $\mathcal{E}_{\mathbb{F}}$ single-valued integration with respect to $z$ is
trivial. We need the commutative hexagon in Figure 1 to find anti-derivatives
$\partial_{\overline{z}}f$ and anti-primitives
$\int_{\mathrm{sv}}f\mathrm{d}{\overline{z}}$ of functions
$f\in\mathcal{E}_{\mathbb{F}}$. The bottleneck of most calculations in
$\mathcal{E}_{\mathbb{F}}$ is the evaluation of ESVHs at certain points. This
evaluation, in general, demands the conversion into the $z,{\overline{z}}$
basis (49). Only for constant letters there exists a significant shortcut: The
evaluation $\mathcal{L}_{w}(a)$, $a\in{\mathbb{C}}$, can be obtained from the
multivalued evaluation $L_{w}(a)$ by the single-valued map which has a simple
formula in the $f$-alphabet [8].

Expressing GSVHs in terms of ESVHs as in (85) makes expressions much more
compact. In the Maple implementation [19] this advantage of using ESVHs
clearly outweights the drawbacks.

###  7.2. The $f$-alphabet

A third option to present GSVHs (in addition to the $z,{\overline{z}}$ basis
and ESVHs) is the conversion into the $f$-alphabet. One can use F. Brown’s
decomposition algorithm (for motivic numbers (periods) defined in [7]) to
convert GSVHs into the $f$-alphabet [9]. This representation has the simplest
structure at the expense of more lengthy expressions. It has not yet been used
and all results are experimental [23]. However, in future implementations of
GSVHs the $f$-alphabet should be considered an option.

##  8\. Construction of some GSVHs

By Corollary 60, the space $\mathcal{G}_{\mathbb{F}}^{\mathbb{C}}$ is
(inductively) spanned by GSVHs

(86)
$f(z)=\int_{\mathrm{sv}}\frac{h(z)}{z-\beta({\overline{z}})}\,\mathrm{d}z,$

with $\beta({\overline{z}})\in\mathrm{FLT}\,_{\mathbb{F}}({\overline{z}})$ and
$h(z)/(z-\beta({\overline{z}}))\in\partial_{z}\mathcal{G}_{\mathbb{F}}^{\mathbb{C}}$.
This leads to the following problem.

######  Problem 79.

Fix $\beta\in\mathrm{FLT}\,_{\mathbb{F}}$. Construct functions
$h\in\mathcal{G}_{\mathbb{F}}^{\mathbb{C}}$ such that
$h(z)/(z-\beta({\overline{z}}))\in\partial_{z}\mathcal{G}_{\mathbb{F}}^{\mathbb{C}}$.

######  Remark 80.

  1. (1)

If $\beta\in\mathrm{FLT}\,_{\mathbb{F}}^{\emptyset}$ then every
$h\in\mathcal{G}_{\mathbb{F}}^{\mathbb{C}}$ gives
$h(z)/(z-\beta({\overline{z}}))\in\partial_{z}\mathcal{G}_{\mathbb{F}}^{\mathbb{C}}$
(Example 78). So, we may restrict ourselves to the case
$\beta\notin\mathrm{FLT}\,_{\mathbb{F}}^{\emptyset}$.

  2. (2)

Graphical functions live in Euclidean space whereas QFT is built on Minkowski
metric. For the transition from Euclidean to Minkowski metric one needs to
analytically continue ${\overline{z}}$ away from the complex conjugate of $z$
such that $z$ and ${\overline{z}}$ become independent real variables. In this
situation, letters in $\mathrm{FLT}\,_{\mathbb{F}}^{\emptyset}$ quickly
develop singularities (see, e.g., Example 68). The situation is more stable
for $\beta\notin\mathrm{FLT}\,_{\mathbb{F}}^{\emptyset}$ where singularities
are only attained on higher monodromy sheets. Accordingly we observe in
graphical functions letters which are not in
$\mathrm{FLT}\,_{\mathbb{F}}^{\emptyset}$.

###  8.1. Involutions

A special case arises when $z\mapsto\beta({\overline{z}})$ is an involution.

######  Definition 81.

Let $\overline{\beta}$ be $\beta$ with complex conjugated coefficients and

$\mathrm{FLT}\,_{\mathbb{F}}^{2}=\\{\beta\in\mathrm{FLT}\,_{\mathbb{F}},\beta\circ\overline{\beta}=\mathrm{id}\\}.$

Equivalently, $\mathrm{FLT}\,_{\mathbb{F}}^{2}$ is the subset of
$\mathrm{FLT}\,_{\mathbb{F}}$ for which the map
$\tau:z\mapsto\beta({\overline{z}})$ is an involution, $\tau^{2}=\mathrm{id}$.

######  Example 82.

The identity $\mathrm{id}\in\mathrm{FLT}\,_{\mathbb{F}}$ gives
$\tau:z\mapsto{\overline{z}}$, see Example 71 and Section 8.3.

In Examples 69 and 70 we have $z\mapsto
1/z\in\mathrm{FLT}\,_{\mathbb{F}}^{2}$.

######  Lemma 83.

Every $\beta\in\mathrm{FLT}\,_{\mathbb{F}}^{2}$ has a representation

$\beta(z)=\frac{az+b}{cz+{\overline{a}}},\quad\text{with}\quad\overline{b}=-b,\,\overline{c}=-c.$

Every such $\beta$ is uniquely represented by the projective point
$(\mathop{\mathrm{missing}}{Re}a:\mathop{\mathrm{missing}}{Im}a:\mathop{\mathrm{missing}}{Im}b:\mathop{\mathrm{missing}}{Im}c)\in{\mathbb{P}}{\mathbb{R}}^{3}$.

###### Proof.

For any $\beta(z)=(az+b)/(cz+d)\in\mathrm{FLT}\,_{\mathbb{F}}^{2}(z)$ an
inverse is given by $\beta^{-1}(z)=(dz-b)/(-cz+a)$ which has to equal
$\overline{\beta}$ up to a non-zero common factor $\alpha$ in the
coefficients. We get

${\overline{a}}=d\alpha,\overline{b}=-b\alpha,\overline{c}=-c\alpha,\overline{d}=a\alpha.$

By complex conjugation we get $\alpha\overline{\alpha}=1$. If we multiply the
coordinates with $\alpha^{-1/2}$ we obtain the desired representation of
$\beta$. The representation is unique up to a common factor $\eta$ with
$\eta=\overline{\eta}\in{\mathbb{R}}^{\times}$. ∎

######  Proposition 84.

Let $h\in\mathcal{G}_{\mathbb{F}}^{\mathbb{C}}$ and
$\beta\in\mathrm{FLT}\,_{\mathbb{F}}^{2}$. Then

(87) $f(z)=\frac{h(z)-h(\beta({\overline{z}}))}{z-\beta({\overline{z}})}$

is a symmetric function (under $\tau:z\mapsto\beta({\overline{z}})$) in
$\partial_{z}\mathcal{G}_{\mathbb{F}}^{\mathbb{C}}$.

###### Proof.

It is clear by Theorem 24 (4), (5) that
$f\in\mathcal{G}\mathcal{H}_{\mathbb{F}}$. The symmetry of $f$ is evident from
(87). We need to show that $f\in\mathcal{S}\mathcal{V}_{\mathbb{C}}$.

Fix $a\in{\mathbb{C}}\cup\\{\infty\\}$. We have
$h(z)-h(\beta({\overline{z}}))\in\mathcal{S}\mathcal{V}_{\Sigma}$ for some
finite set $\Sigma$, see Remark 36 (4), (5). Because
$\beta\in\mathrm{FLT}\,_{\mathbb{F}}^{2}$ the function
$h(z)-h(\beta({\overline{z}}))\in\mathcal{G}_{\mathbb{F}}$ vanishes at
$z=\beta({\overline{z}})$ for any (independent) values of ${\overline{z}}$.

If $a\neq\beta({\overline{a}})$ then $1/(z-\beta({\overline{z}}))$ is
${\mathbb{C}}$-analytic at $a$ and $f$ has a single-valued log-Laurent
expansion at $a$. By Lemmas 49 and 50 (note that $\beta\notin{\mathbb{F}}$)
this statement extends to $a=\beta({\overline{a}})$. Because
${\mathbb{C}}\cup\\{\infty\\}$ is compact, the function $f$ has a finite
number of singular points. Hence $f\in\mathcal{S}\mathcal{V}_{\mathbb{C}}$. ∎

######  Example 85.

For $h(z)=\mathcal{L}_{0}(z)/2$, $\beta(z)=z^{-1}$ we obtain the function
$\log(z{\overline{z}})/(z-1/{\overline{z}})$ in Example 69.

The complex conjugate of $h(z)=\mathcal{L}_{01}(z)$ is $\mathcal{L}_{10}(z)$.
With $\beta=\mathrm{id}$ we get the integrand
$(\mathcal{L}_{01}(z)-\mathcal{L}_{10}(z))/(z-{\overline{z}})$ in Example 71.

###  8.2. Exceptional GSVHs

In the notion of Section 7.1, the function $f(z)$ in (86) is a linear
combination of ESVHs which end in the letter $\beta({\overline{z}})$.

If the zero locus of the denominator in the integrand of (86) is empty, $h$
can be any GSVH in $\mathcal{G}_{\mathbb{F}}^{\mathbb{C}}$. If
$\beta\in\mathrm{FLT}\,_{\mathbb{F}}^{2}$ and
$h(z)=[h_{1}(z)-h_{1}(\beta({\overline{z}}))]h_{2}(z)$ with
$h_{1},h_{2}\in\mathcal{G}_{\mathbb{F}}^{\mathbb{C}}$, the construction of $f$
is a trivial extension of Proposition 84. If none of these cases is true then
$f$ is an exceptional GSVH.

######  Definition 86.

Assume $f\in\mathcal{G}_{\mathbb{F}}^{\mathbb{C}}$ as is (86). If
$\beta\notin\mathrm{FLT}\,_{\mathbb{F}}^{\emptyset}$ and either
$\beta\notin\mathrm{FLT}\,_{\mathbb{F}}^{2}$ or there exist no
$h_{1},h_{2}\in\mathcal{G}_{\mathbb{F}}^{\mathbb{C}}$ such that
$h(z)=[h_{1}(z)-h_{1}(\beta({\overline{z}}))]h_{2}(z)$ then $f$ is an
exceptional GSVH.

Note that factorization is readily checked using an algebra basis of ESVHs in
Lyndon words.

0.5 Black (176,89)(80,41) (80,41)3 (112,41)3 (144,41)3 (176,41)3 (80,89)3
(80,-7)3 (176,89)3 (176,-7)3 (80,89)(80,41) (80,89)(112,41) (80,89)(176,41)
(176,89)(176,41) (112,41)(80,-7) (144,41)(80,-7) (80,41)(176,-7)
(144,41)(176,-7) (176,41)(176,-7) (80,41)(176,41)
(66,87)[lb]$0$(185,87)[lb]$1$(66,-9)[lb]$z$(185,-9)[lb]$\infty$(123,-9)[lb]$\overline{G}$

Figure 3. A graphical function which contains an exceptional GSVH.

######  Example 87.

Consider the ESVH of weight six that emerges from integrating the square of
the Bloch-Wigner dilogarithm (1) twice with respect to
$\int_{\mathrm{sv}}\mathrm{d}z/(z-{\overline{z}})$,

$\displaystyle f_{6}(z)=$
$\displaystyle\;3\mathcal{L}_{0011{\overline{z}}{\overline{z}}}-3\mathcal{L}_{0110{\overline{z}}{\overline{z}}}-3\mathcal{L}_{1001{\overline{z}}{\overline{z}}}+3\mathcal{L}_{1100{\overline{z}}{\overline{z}}}+\mathcal{L}_{01010{\overline{z}}}-\mathcal{L}_{01100{\overline{z}}}-3\mathcal{L}_{01110{\overline{z}}}$
$\displaystyle\quad+\,2\mathcal{L}_{10010{\overline{z}}}-2\mathcal{L}_{10100{\overline{z}}}-3\mathcal{L}_{11001{\overline{z}}}+6\mathcal{L}_{11100{\overline{z}}}+6\zeta(3)\mathcal{L}_{01{\overline{z}}}.$

Expansions at $0,1,\infty$ to high orders suggest that
$f_{6}\in\mathcal{G}_{\\{0,1,{\overline{z}}\\}}^{\mathbb{C}}$, see Section
8.3. In this case $f_{6}$ is exceptional.

######  Example 88.

Consider the four-dimensional graphical function $f_{G}(z)$ for the graph $G$
whose completion $\overline{G}$ is depicted in Figure 3 [1, 17]. An explicit
calculation with HyperlogProcedures [19] gives
$f_{G}(z)\in\mathcal{O}_{\Sigma}^{\emptyset}(z,{\overline{z}})\partial_{z}\mathcal{G}\mathcal{H}_{\Sigma}^{\mathbb{C}}$
for
$\Sigma({\overline{z}})=\\{0,1,{\overline{z}},-{\overline{z}},1-{\overline{z}},{\overline{z}}/({\overline{z}}-1)\,z-{\overline{z}},z+{\overline{z}}\\}$.
Every graphical function is single-valued [14]. The function $f_{G}(z)$ has
denominators $z-{\overline{z}}$ and $z+{\overline{z}}$. By Proposition 58,
single-valued integration of the terms with denominator $z+{\overline{z}}$
gives a GSVH of weight nine. It is exceptional. The numerator $h$ in (86) does
not factorize. Hence $h_{2}=1$ in Definition 86. An explicit calculation with
HyperlogProcedures shows that $h(z)\neq-h(-{\overline{z}})$.

######  Question 89.

  1. (1)

Does there exist a general construction method for (some) exceptional GSVHs?

  2. (2)

What is the smallest (by weight) exceptional GSVH?

  3. (3)

Do GSVHs with letters
$\notin\mathrm{FLT}\,_{\mathbb{F}}^{\emptyset}({\overline{z}})\cup\mathrm{FLT}\,_{\mathbb{F}}^{2}({\overline{z}})$
exist? If yes, what is the smallest (exceptional) example of such a GSVH?

###  8.3. The alphabet 0, 1, ${\overline{z}}$

Consider the alphabet $\Sigma({\overline{z}})=\\{0,1,{\overline{z}}\\}$. This
mathematically interesting case is important in dimensionally regularized pQFT
[11, 18, 22].

Because every GSVH in $\mathcal{G}_{\\{0,1,{\overline{z}}\\}}^{\mathbb{C}}$ is
a ${\mathbb{C}}$-linear combination of ESVHs in $\\{0,1,{\overline{z}}\\}$ we
obtain finite dimensional vector-spaces at each weight

$d_{n}=\dim\mathcal{G}_{\\{0,1,{\overline{z}}\\}}^{\mathbb{C}}(\text{weight}\,=n)\leq
3^{n}.$

The actual dimensions are much smaller than $3^{n}$. At weights one and two we
only have single-valued multiple polylogarithms whereas at weight three one
genuine GSVH exists, see Examples 71, 77.

weight | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9
---|---|---|---|---|---|---|---|---|---|---
dimension $d_{n}$ | 1 | 2 | 4 | 9 | 21 | 52 | 134? | 358? | 986? | 2781?
# generators | 0 | 2 | 1 | 3 | 6 | 16 | 38? | 105? | 284? | 805?
Table 1. (Conjectured) dimensions $d_{n}$ and numbers of generators in
$\mathcal{G}_{\\{0,1,{\overline{z}}\\}}^{\mathbb{C}}$ for weights $\leq 9$.

######  Theorem 90.

The dimensions and numbers of generators in
$\mathcal{G}_{\\{0,1,{\overline{z}}\\}}^{\mathbb{C}}$ for weights $\leq 5$ are
as given in Table 1.

###### Proof.

The numbers of generators follow from the dimensions $d_{n}$. We get a lower
bound for $d_{n}$ by explicit construction of GSVHs in
${\\{0,1,{\overline{z}}\\}}$ using Proposition 84. Up to weight $n\leq 3$ we
obtain $2^{n}$ single-valued multiple polylogarithms plus
$\mathcal{L}_{01{\overline{z}}}-\mathcal{L}_{10{\overline{z}}}$ in Example 77.
By integration with denominators $z$ and $z-1$ the nine GSVHs at weight three
give rise to 18 GSVHs of weight four. The three missing GSVHs of weight four
can be constructed with (87). The result is
$\mathcal{L}_{001{\overline{z}}}-\mathcal{L}_{100{\overline{z}}}$,
$\mathcal{L}_{011{\overline{z}}}-\mathcal{L}_{110{\overline{z}}}$, and
$2\mathcal{L}_{01{\overline{z}}{\overline{z}}}-2\mathcal{L}_{10{\overline{z}}{\overline{z}}}-\mathcal{L}_{010{\overline{z}}}+\mathcal{L}_{101{\overline{z}}}$
(calculations were done with HyperlogProcedures [19]).

At weight 5 we need $52-2\times 21=10$ new GSVHs. Nine of them are direct
applications of (87),

$\mathcal{L}_{0001{\overline{z}}}-\mathcal{L}_{1000{\overline{z}}}$, |  $\mathcal{L}_{0010{\overline{z}}}-\mathcal{L}_{0100{\overline{z}}}$,
---|---
$\mathcal{L}_{0011{\overline{z}}}-\mathcal{L}_{1100{\overline{z}}}+2\zeta(3)\mathcal{L}_{1{\overline{z}}}$, |  $\mathcal{L}_{0101{\overline{z}}}-\mathcal{L}_{1010{\overline{z}}}-4\zeta(3)\mathcal{L}_{1{\overline{z}}}$,
$\mathcal{L}_{0111{\overline{z}}}-\mathcal{L}_{1110{\overline{z}}}-2\zeta(3)\mathcal{L}_{1{\overline{z}}}$, |  $\mathcal{L}_{1011{\overline{z}}}-\mathcal{L}_{1101{\overline{z}}}+6\zeta(3)\mathcal{L}_{1{\overline{z}}}$,
$2\mathcal{L}_{001{\overline{z}}{\overline{z}}}-2\mathcal{L}_{100{\overline{z}}{\overline{z}}}+\mathcal{L}_{01{\overline{z}}0{\overline{z}}}-\mathcal{L}_{10{\overline{z}}0{\overline{z}}}-\mathcal{L}_{0010{\overline{z}}}-\mathcal{L}_{0100{\overline{z}}}+\mathcal{L}_{1001{\overline{z}}}+\mathcal{L}_{1010{\overline{z}}}+2\zeta(3)\mathcal{L}_{1{\overline{z}}}$,
$2\mathcal{L}_{011{\overline{z}}{\overline{z}}}-2\mathcal{L}_{110{\overline{z}}{\overline{z}}}+\mathcal{L}_{01{\overline{z}}1{\overline{z}}}-\mathcal{L}_{10{\overline{z}}1{\overline{z}}}-\mathcal{L}_{0101{\overline{z}}}-\mathcal{L}_{0110{\overline{z}}}+\mathcal{L}_{1011{\overline{z}}}+\mathcal{L}_{1101{\overline{z}}}+2\zeta(3)\mathcal{L}_{1{\overline{z}}}$,
$4\mathcal{L}_{01{\overline{z}}{\overline{z}}{\overline{z}}}-4\mathcal{L}_{10{\overline{z}}{\overline{z}}{\overline{z}}}-2\mathcal{L}_{010{\overline{z}}{\overline{z}}}+2\mathcal{L}_{101{\overline{z}}{\overline{z}}}-\mathcal{L}_{01{\overline{z}}0{\overline{z}}}-\mathcal{L}_{01{\overline{z}}1{\overline{z}}}+\mathcal{L}_{10{\overline{z}}0{\overline{z}}}+\mathcal{L}_{10{\overline{z}}1{\overline{z}}}+\mathcal{L}_{0101{\overline{z}}}-\mathcal{L}_{1010{\overline{z}}}$.

The tenth GSVH has the product
$(\mathcal{L}_{01{\overline{z}}}-\mathcal{L}_{10{\overline{z}}})^{2}/4$ in the
numerator (see Definition 86) yielding

$\mathcal{L}_{0011{\overline{z}}}-\mathcal{L}_{0110{\overline{z}}}-\mathcal{L}_{1001{\overline{z}}}+\mathcal{L}_{1100{\overline{z}}}.$

Because $\mathcal{E}_{\\{0,1,{\overline{z}}\\}}$ is a free
$\mathcal{O}_{\\{0,1,{\overline{z}}\\}}(z,{\overline{z}})$ module (see Section
7.1), linear independence is easy to check.

For an upper bound consider the generic linear combination of all
$(3^{n+1}-1)/2$ ESVHs in $\\{0,1,{\overline{z}}\\}$ up to weight $n$. We
expand each ESVH at $z=0$, $z=1$, and $z=\infty$ in $z$ and ${\overline{z}}$.
In these expansions we consider the ESVHs as generalized hyperlogarithms
(Section 4.3) and expand first in $z$ and then the coefficient in
${\overline{z}}$. For any word which has at least one letter ${\overline{z}}$
we obtain increasing pole orders in ${\overline{z}}$ (with increasing degree
in $z$). In linear combinations which are GSVHs all poles have to cancel
(Theorem 74 (6)). This condition gives increasing systems of linear equations
whose rank $R$ stabilizes at $(3^{n+1}-1)/2-\sum_{k=0}^{n}d_{k}$. We
(inductively) get $d_{n}\leq(3^{n+1}-1)/2-R-\sum_{k=0}^{n-1}d_{k}$. The
procedure zzdims in HyperlogProcedures [19] gives the desired result. ∎

In practice, the upper bound $(3^{n+1}-1)/2-R-\sum_{k=0}^{n-1}d_{k}$ for
$d_{n}$ drops rapidly with the size of the linear system (the orders in $z$
and ${\overline{z}}$) until it stabilizes. The conjectured dimensions in Table
1 are these stabilized upper bounds. At weight six we conjecturally encounter
three exceptional GSVHs (see Example 87).

For $n\geq 2$ the dimensions in Table 1 are approximated by the recursion

$d_{n}\approx 5d_{n-1}-6d_{n-2}$

with surplus elements in dimensions three and five and a missing element in
dimension nine. An ansatz for a generating function is

$\sum_{n=0}^{\infty}d_{n}x^{n}\stackrel{{\scriptstyle?}}{{=}}\,\frac{1}{1-2x}+\frac{x^{3}+x^{5}-x^{9}+\ldots}{(1-2x)(1-3x)},$

where the first term counts the number of single-valued multiple
polylogarithms in the letters 0 and 1.

The number contents of GSVHs in $\\{0,1,{\overline{z}}\\}$ is given by
multiple zeta values (MZVs).

######  Definition 91.

The ${\mathbb{Q}}$-algebra of MZVs is the ${\mathbb{Q}}$-span of MZV sums

(88) $\zeta(n_{1},n_{2},\ldots,n_{r})=\sum_{1\leq
k_{1}<k_{2}<\ldots<k_{r}}n_{1}^{-k_{1}}n_{2}^{-k_{2}}\cdots
n_{r}^{-k_{r}},\quad n_{i}=\\{1,2,3,\ldots\\},\,n_{r}\geq 2.$

The weight of the tuple $(n_{1},n_{2},\ldots,n_{r})$ is
$n_{1}+n_{2}+\ldots+n_{r}$. For any finite
$\Sigma({\overline{z}})\subset\mathrm{FLT}\,_{\mathbb{F}}({\overline{z}})$ we
define
$\mathcal{G}_{\Sigma}^{\mathrm{MZV}}\subset\mathcal{G}_{\Sigma}^{\mathbb{C}}$
as the ${\mathbb{Q}}$-algebra of GSVHs in $\Sigma({\overline{z}})$ with MZV
coefficients. Likewise, for any finite set $\Sigma_{0}\subset{\mathbb{C}}$,

$\displaystyle\mathcal{O}_{\Sigma}^{\mathrm{MZV}}(z,{\overline{z}})$
$\displaystyle=\mathrm{MZV}[z,((z-\beta({\overline{z}}))^{-1})_{\beta({\overline{z}})\in\Sigma({\overline{z}})},{\overline{z}},(({\overline{z}}-{\overline{b}})^{-1})_{b\in\Sigma({\overline{z}})\cap{\mathbb{F}}}],$
$\displaystyle\mathcal{O}_{\Sigma_{0}}^{\mathrm{MZV}}(z)$
$\displaystyle=\mathrm{MZV}[z,((z-a)^{-1})_{a\in\Sigma_{0}}]$

are the subrings of $\mathcal{O}_{\Sigma}(z,{\overline{z}})$ and
$\mathcal{O}_{\Sigma_{0}}(z)$ which are defined over MZVs (respectively).

######  Lemma 92.

For any $f\in\mathcal{G}_{\\{0,1,{\overline{z}}\\}}^{\mathrm{MZV}}$ the
regularized limit $f(1)$ is an MZV.

###### Proof.

We use the representation of $f$ in the $z,{\overline{z}}$ basis (49),

(89)
$f(z)=\sum_{v,w({\overline{z}})}c_{v,w({\overline{z}})}L_{v}({\overline{z}})L_{w({\overline{z}})}(z),$

where $v\in\\{0,1\\}^{\ast}$,
$w({\overline{z}})\in\\{0,1,{\overline{z}}\\}^{\ast}$, and
$c_{v,w({\overline{z}})}$ is an MZV. We consider $z$ and ${\overline{z}}$ as
independent variables. Because $f$ is a GSVH with a single-valued log-Laurent
expansion at 1 we may determine $f(1)$ by first taking the limit
${\overline{z}}\to 1$ followed by the limit $z\to 1$. In
$L_{v}({\overline{z}})$ we obtain the regularized limit $L_{v}(1)$ which is an
MZV. In $L_{w({\overline{z}})}(z)$ the value ${\overline{z}}=1$ differs from
the lower bound $0$ and the upper bound $z$ of the iterated integral. The
limit ${\overline{z}}\to 1$ is regular yielding $L_{w(1)}(z)$. Because
$w(1)\in\\{0,1\\}^{\ast}$ the regularized limit $z\to 1$ gives the MZV
$L_{w(1)}(1)$. Therefore every summand in $f(1)$ is the product of three MZVs.
∎

######  Theorem 93.

Theorem 74 restricts to
$\mathcal{G}_{\\{0,1,{\overline{z}}\\}}(\mathrm{MZV})=\mathcal{O}_{\\{0,1,{\overline{z}}\\}}^{\mathrm{MZV}}(z,{\overline{z}})\mathcal{G}_{\\{0,1,{\overline{z}}\\}}^{\mathrm{MZV}}$,
where statement (5) restricts to the Möbius transformations
$z\to\\{z,1-z,1/z,1/(1-z),z/(z-1),(z-1)/z\\}$ that stabilize the singular
points $0,1,\infty$. In particular, the sequence

(90)
$0\longrightarrow\mathcal{O}_{\\{0,1\\}}^{\mathrm{MZV}}({\overline{z}})\longrightarrow\mathcal{G}_{\\{0,1,{\overline{z}}\\}}(\mathrm{MZV})\stackrel{{\scriptstyle\partial_{z}}}{{\longrightarrow}}\mathcal{G}_{\\{0,1,{\overline{z}}\\}}(\mathrm{MZV})\longrightarrow
0$

is exact. We have

(91)
$\mathcal{G}_{\\{0,1,{\overline{z}}\\}}^{\mathrm{MZV}}\otimes_{\mathbb{Q}}{\mathbb{C}}\cong\mathcal{G}_{\\{0,1,{\overline{z}}\\}}^{\mathbb{C}}.$

###### Proof.

For (90) it suffices to show that every
$f\in\mathcal{G}_{\\{0,1,{\overline{z}}\\}}(\mathrm{MZV})$ has a primitive.
The alphabet $0,1,{\overline{z}}$ is stable under partial fraction
decomposition because for $a=0,1$,

$\frac{1}{z-a}\frac{1}{z-{\overline{z}}}=\frac{1}{{\overline{z}}-a}\Big{(}\frac{1}{z-{\overline{z}}}-\frac{1}{z-a}\Big{)},$

and
$1/({\overline{z}}-a)\in\mathcal{O}_{0,1,{\overline{z}}}(z,{\overline{z}})$.
Moreover, the projections $\pi_{\partial_{z}}$ and
$\pi_{\partial_{\overline{z}}}$ filter (anti-)resides at 0 and 1. The
(anti-)residue at 0 trivially maps into MZVs. The (anti-)residue at 1 maps
into MZVs by Lemma 92.

The previous arguments are symmetric under complex conjugation so that we can
also use the commutative hexagon in Figure 1 to take anti-primitives in
$\mathcal{G}_{\\{0,1,{\overline{z}}\\}}(\mathrm{MZV})$, see (7).

We use induction over the weight of $f$ to show that
$\mathcal{G}_{\\{0,1,{\overline{z}}\\}}(\mathrm{MZV})$ is invariant under
complex conjugation. The weight zero piece
$\mathcal{O}^{\emptyset,\mathrm{MZV}}_{\\{0,1,{\overline{z}}\\}}(z,{\overline{z}})$
of $\mathcal{G}_{\\{0,1,{\overline{z}}\\}}(\mathrm{MZV})$ is
$\mathcal{O}_{\\{0,1\\}}^{\mathrm{MZV}}(z,{\overline{z}})$ which is invariant
under complex conjugation. Assume
$f\in\mathcal{G}_{\\{0,1,{\overline{z}}\\}}(\mathrm{MZV})$ has weight $n\geq
1$. Because
$\mathcal{O}_{\\{0,1,{\overline{z}}\\}}^{\mathrm{MZV}}(z,{\overline{z}})$ is
invariant under complex conjugation we may assume without restriction that
$f\in\mathcal{G}_{\\{0,1,{\overline{z}}\\}}^{\mathrm{MZV}}$ with $f(0)=0$.
Then $g=\partial_{z}f$ has weight $n-1$. By induction $g$ has a complex
conjugate $\overline{g}$. The anti-primitive
$\overline{f}=\int_{\mathrm{sv}}\overline{g}\,\mathrm{d}{\overline{z}}\in\mathcal{G}_{\\{0,1,{\overline{z}}\\}}(\mathrm{MZV})$
possibly differs from the complex conjugate of $f$ by a constant. Because
$f(0)=\overline{f}(0)=0$ the constant is zero.

For (5), consider the Möbius transformation $\phi(z)=z/(z-1)$. It is an
endomorphism of $\mathcal{O}_{\\{0,1\\}}^{\mathrm{MZV}}(z,{\overline{z}})$ and
because it maps $1/(z-{\overline{z}})$ to
$-(z-1)({\overline{z}}-1)/(z-{\overline{z}})$ it is also an endomorphism of
$\mathcal{O}_{\\{0,1,{\overline{z}}\\}}^{\mathrm{MZV}}(z,{\overline{z}})$. As
in the case of complex conjugation we show by induction over the weight that
$\mathcal{G}_{\\{0,1,{\overline{z}}\\}}(\mathrm{MZV})$ is invariant under
$\phi$. The integration constant is trivial because $\phi(0)=0$.

Consider the Möbius transformation $\tau(z)=1-z$ which is also an endomorphism
of $\mathcal{O}_{\\{0,1,{\overline{z}}\\}}^{\mathrm{MZV}}(z,{\overline{z}})$.
If we use induction over the weight of
$f\in\mathcal{G}_{\\{0,1,{\overline{z}}\\}}^{\mathrm{MZV}}$ with $f(0)=0$ we
run into the difficulty that $f(\tau(0))=f(1)$. By Lemma 92 the value $f(1)$
is an MZV and we obtain that
$f(\tau(z))\in\mathcal{G}_{\\{0,1,{\overline{z}}\\}}^{\mathrm{MZV}}$.

Statement (5) is true because the set of Möbius transformations that stabilize
$0,1,\infty$ is generated by $\phi$ and $\tau$.

To prove (91) we represent
$f\in\mathcal{G}_{\\{0,1,{\overline{z}}\\}}^{\mathbb{C}}$ as a
${\mathbb{C}}$-linear combination of ESVHs in $\\{0,1,{\overline{z}}\\}$. By
induction over the weight (using the commutative hexagon, see Section 7.1) we
see that ESVHs in $0,1,{\overline{z}}$ are defined over MZVs (as in (89)). The
space $\mathcal{G}_{\\{0,1,{\overline{z}}\\}}^{\mathbb{C}}$ is given by those
linear combinations of ESVHs which have single-valued log-Laurent expansions
at $0$, $1$, and $\infty$, see the proof of Theorem 90. These expansions
provide linear equations for the ESVHs in $\\{0,1,{\overline{z}}\\}$ of weight
$\leq n$. At 0 these equations have MZV-coefficients, see (13). By statement
(5) the expansions at 1 and at $\infty$ can be mapped to expansions at 0 by
Möbius transformations which stabilize
$\mathcal{G}_{\\{0,1,{\overline{z}}\\}}^{\mathrm{MZV}}$. So, these expansion
also give rise to linear equations with MZV-coefficients. For weight $\leq n$
the space $\mathcal{G}_{\\{0,1,{\overline{z}}\\}}^{\mathbb{C}}$ is obtained
from the $(3^{n+1}-1)/2$ ESVHs in $\\{0,1,{\overline{z}}\\}$ of weight $\leq
n$ by a finite number of these linear equations. Because the total weight (the
weight of the MZV plus the weight of the ESVH) is a filtration, the
coefficients of ESVHs with weight $n$ are rational (MZVs of weight 0). The
solution of the system provides
$\mathcal{G}_{\\{0,1,{\overline{z}}\\}}^{\mathbb{C}}$ at weight $n$ as MZV-
span of ESVHs. Therefore
$\mathcal{G}_{\\{0,1,{\overline{z}}\\}}^{\mathrm{MZV}}\otimes_{\mathbb{Q}}{\mathbb{C}}$
surjects onto $\mathcal{G}_{\\{0,1,{\overline{z}}\\}}^{\mathbb{C}}$. The
kernel is zero because the map is an embedding. ∎

A detailed understanding of GSVHs in $0,1,{\overline{z}}$ could lead to much
faster calculations in this alphabet. However, for most computations at modest
weights in pQFT the bottleneck is not handling these GSVHs.

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#  Vision-based Engagement Detection in Virtual Reality

###### Abstract

User engagement modeling for manipulating actions in vision-based interfaces
is one of the most important case studies of user mental state detection. In a
Virtual Reality environment that employs camera sensors to recognize human
activities, we have to know when user intends to perform an action and when
not. Without a proper algorithm for recognizing engagement status, any kind of
activities could be interpreted as manipulating actions, called ”Midas Touch”
problem. Baseline approach for solving this problem is activating gesture
recognition system using some focus gestures such as waiving or raising hand.
However, a desirable natural user interface should be able to understand
user’s mental status automatically.

In this paper, a novel multi-modal model for engagement detection, DAIA
111Disengagement, Attention, Intention, Action, is presented. using DAIA, the
spectrum of mental status for performing an action is quantized in a finite
number of engagement states. For this purpose, a Finite State Transducer (FST)
is designed. This engagement framework shows how to integrate multi-modal
information from user biometric data streams such as 2D and 3D imaging. FST is
employed to make the state transition smoothly using combination of several
boolean expressions. Our FST true detection rate is 92.3% in total for four
different states. Results also show FST can segment user hand gestures more
robustly.

Index Terms—  Gesture Recognition Systems, User Engagement Detection, Human
Activity Recognition, Vision-based Interface, Virtual Reality, Finite State
Machine

##  1 Introduction

User mental status detection has variety of applications in human activity
recognition systems. Without a proper algorithm for detecting human intention
to interact, the vision based system is always on, therefore any kind of
activity may interpreted as an interaction [1]. Group meetings which are
frequent business events is modeled as a case study. In this case study, among
all available data streams, a combination of tracking 3D skeleton data is
combined for user engagement detection in meetings. Multiple binary
classifiers are implemented to detect user intention for performing an action.
The output of these binary classifiers are used to create transition and guard
conditions in FST. Characteristics of engagement will be discussed and
biometric data which can be used for this purpose will be introduced. 3D
skeleton tracking will be introduced as one of the channels of biometric
information for engagement detection. Although we just use this only channel
of biometric data, experiment results show we still can predict engagement
with high accuracy.

In addition, DAIA, the FST of engagement detection helps system to flow among
states smoothly. FST is a predefined structure based on our knowledge of human
activities that helps system predict engagement state more accurately.
Furthermore, FST algorithm is computational efficient. This property of FST
allows achievable on-line and real-time performance.

###  1.1 Related Works

Engagement has been investigated in various fields such as education,
organizational behavior, work, or media. Engagement is defined as the value
that a participant in an interaction attributes to the goal of being together
with other participant(s) and continuing interaction [2, 3]. It is also
defined as the process by which two or more participants establish, maintain,
and end their perceived connection directly related to attention [3, 4] ).
“Effort without distress” [5], “A meaningful involvement”[4], Enabled through
vigor, dedication, and absorption[6]) are other definitions of engagement.

Body posture gives important information about engagement. Various approaches
have been investigated based on body language analysis to improve human
computer interaction. Intention to engage with an agent e.g. a robot [7, 8],
or interactive display [9], are some of these studies. Measuring the
engagement intent is used in service robots to identify relevant gestures from
irrelevant gestures which is known as Midas Touch Problem [7, 10]. Another
research interest is related to learner engagement with robotic companions or
interfaces[11].In addition, Intention to engage with a display for improving
user identification is addressed in Schwarz et al. [9].

The role of body pose and motion in user’s interest detection using body
tracking systems such as Kinect has been addressed in several research [12,
13, 14, 15].

A variety of studies strives for a multi-modal approach using some features of
facial expression, body motion, voice, or seat pressure to elucidate on mental
states. Benkaouar et al. [16] is discussing gaze and upper body posture for
engagement detetion. Schwarz et. al [9] used combination of gaze, upper body
and arm position for the purpose of intention detction in engagement.
Vaufreydaz et. al [7] used gaze and proxemics and Salam et. al [3] employed
human state observation for engagement detection. Engell et al. used gaze and
facial expression[17] and Balaban et al. [18] employed weight, head, and upper
body motion; Scherer et al. [19] and Dael et. al [6] discuss voice, face,
posture. Using Finite State Machine (FSM) for multi-modal system modeling is
addressed in multiple research[20],[21],[22],[23].

Mota used both neural networks for posture detection and Hidden Markov Models
to detect engagement at an overall accuracy of 77% which needs an expensive
computation[24]. Michalowski et al. offered a spatial model combined with gaze
tracking to detect user engagement with a robot receptionist [25].

All of the studies mentioned in this section discuss a very small set of
potential classifiers. They also do not make full usage of the amount of
qualitative research on non-verbal body language and its indication of mental
states. In this research, we address these features for engagement modeling
and detection.

##  2 Engagement Modeling and Metrics

Frank et. al[26] have proposed a multi-modal engagement user state detection
process. In our intended scenario, multiple people are within the operating
range of the sensor, e.g. field of view of the camera. This module identifies
the state of the participants on the engagement scale as disengaged up to
involved action. Using the biometric information, the module identifies the
person who exhibits a specific combination of classifiers of all categories.
The classifiers, e.g. body posture components, are chosen based on research,
relating them to attentiveness and engagement. The analysis is occurring on a
frame-by-frame basis. Each frame is analyzed regarding all classifiers, e.g.
if the person is 1) raising a hand, 2) facing a display, 3) leaning forward,
4) leaning backward, 5) uttering a feedback, 6) slouching, or 7) changing
position in the last 60 frames. Classifiers are evaluated as being exhibited
or not exhibited in a specific frame as a binary value.

Among all available biometric information such as gaze, voice and gesture to
extend the proposed framework to detect engagement state of the user, we just
use 3D joint information provided by a depth camera and NiTE SDK by
Primesense. However, using more biometric channels of information will make
system more accurate, our experiment results showed even use this only channel
of information could result in a high performance user engagement detection
system.

Upper body joints play important role in engagement detection. Multiple
classifiers are designed to detect and classify upper body direction. In
addition, hand movements such as raising hand, pointing, swiping, pushing or
pulling are used to manipulate in vision-based interfaces. Therefore,
different classifiers are designed to detect various hand movements such as
raise hand above waist and also different levels of hand speed. These
classifiers help to detect user intention for performing an action.

An action video with $T$ frames and $N$ joints in each frame can be
represented as a set of 3D points sequence, written as
$p=\\{x^{t}_{n}\in\mathbb{R}^{3}|n=1,..,N,t=1,2,...,T\\}$. The 3D sensor
provides us fifteen joints, and $T$ varies for different sequences. However,
in our system $N=10$, because our classifiers only use ten upper body joints
from this set which are head, left and right shoulders, left and right elbows,
left and right hand, torso and left and right hips. For each point, 3
dimensional position ${X,Y,D}$ is obtained.

The first basic step of feature extraction is to compute basic feature for
each frame, which describes the pose information of every of these ten joints
in a single frame. The second step is calculating of left and right hand speed
information. This feature is obtained by calculating 3D distance that each
hand moves in two consecutive frames.

The binary values for the individual binary classifiers are weighted based on
the relative influence in the training data and summed up to . The engagement
score thereby assumes a value between 0 and 1.

Figure 1 shows how our system extract features from users and calculate
engagement levels of each users.

Fig. 1: System Approach

The engagement score is calculated using $W^{\prime}.G$ where $W$ is vector of
weights $[w_{1},w_{2},w_{3},...,w_{n}]$, and $G$ is the vector of binary
classifiers $[g_{1},g_{2},g_{3},...,g_{n}]$ such that
$g_{1},g_{2},g_{3},...,g_{n}$ are 0 or 1 based on the output of the binary
classifiers.

Table 1 gathers binary classifiers which are designed for this purpose. For
each posture status, multiple binary classifiers are designed. The 0 or 1
output of these 37 classifiers are used to make our feature vector for
intention to action classifier.

Posture Status | Binary Classifiers
---|---
Hand Horizontal | Right of Body, Close to Body, Left of Body
Hand Vertical | Below Hip, Below Torso, Below Shoulder, Below Head
Hand Depth | Back of Body, Close to Body, Front of Body
Hand Speed | Stopped, Slow, Fast, Too Fast
Body Direction | Facing Sensor
Leaning | Lean back, No Lean, Lean Forward
Special Postures | Hands folded, Hands on Head, Hands on Torso
Table 1: Binary classifiers for each posture status

This classifiers are mostly designed based on heuristic information extracted
from joints 3D location. For instance, body direction classifier is made using
the normal vector of the plane containing right and left shoulders, and torso
joint.

##  3 DAIA: FST for Engagement Detection

DAIA is a frame-based engagement detection system using an FST. State machines
are the description of a life cycle of a system. They can describe the
different states of the lifeline, the events which influence it, and what it
does when a particular event is detected at any states as the transition
condition for particular state change. They offer the complete specification
of the dynamic behavior of the system. Figure 2 shows the outline of this
framework.

Fig. 2: DAIA: FST for Engagement Detection

In order to increase efficiency and accuracy of our algorithm we implemented a
Finite State Transducer (FST). A state is a description of the mental state or
engagement of the user that is anticipated to change over time. A transition
is initialized by a change in condition that results in a change of state. For
example, when using a gesture recognition system to find out meaningful
gestures of users, swiping or pointing can happen in some states such as in
the Action state and similar gestures in the Attention state will be ignored
or interpreted differently. In this research, we model engagement states as a
finite state transducer with four different states: Disengagement: User is
disengaged from screen or the target person/object. Attention: User has
attention, but doesn’t have intention to do any actions. Intention: User has
intention to do some action, but still not doing it. Action: User is
performing an action.

A finite state transducer is a sextuple
$(\Sigma,\Gamma,S,s_{0},\delta,\omega)$, where: $\Sigma$ is the input alphabet
(a finite non-empty set of symbols). $\Gamma$ is the output alphabet (a
finite, non-empty set of symbols). $S$ is a finite, non-empty set of
states.$s_{0}$ is the initial state, an element of $S$. $\delta$ is the state-
transition function: $\delta:S\times\Sigma\rightarrow S$. $\omega$ is the
output function. If the output function is a function of a state and input
alphabet ($\omega:S\times\Sigma\rightarrow\Gamma$) that definition corresponds
to the Mealy model, and can be modeled as a Mealy machine.

Some hypotheses are considered in designing this FST: Engagement states change
in a specific order: This property describes the FST design. It starts with
disengagement (Initial State). All states can be transited to disengagement,
but there is always a chain of ordered state transition for other states of
engagement.

We may modify the detected state based on conditions of FST: States should
transit smoothly. FST smooths state transition that helps better gesture
segmentation. We don’t change state just based on Intention to act or
disengagement classifier. We know the human activities are continues, so,
using some protection which is called guard conditions we smoothly change
states. Furthermore, engagement state classifier is memoryless. it may report
wrong engagement state based on the current biometric properties of the
states. However, FST keeps record of engagement states and help relabeling the
frames more accurately.

Table 2 describes Transition Conditions, $C_{nm}$, for these state changes.
$C_{nm}$ is the combination of event triggering the transition, the target
state, guard and actions as follows:

State | $S_{1}$ | $S_{2}$ | $S_{3}$ | $S_{4}$
---|---|---|---|---
$S_{1}$ | $C_{11}$ | $C_{12}$ | - | -
$S_{2}$ | $C_{21}$ | $C_{22}$ | $C_{23}$ | $C_{24}$
$S_{3}$ | $C_{31}$ | $C_{32}$ | $C_{33}$ | $C_{34}$
$S_{4}$ | $C_{41}$ | $C_{42}$ | $C_{43}$ | $C_{44}$
Table 2: Transition table

$C_{11}$, $C_{21}$, $C_{31}$, $C_{41}$ : Body Direction is not facing sensor
or a Special Posture such as Hand Folded exists

$C_{22}$, $C_{33}$, $C_{44}$ : The output of binary classifiers and Intention
to Act  classifier doesn’t change.

$C_{12}$: Body Direction is facing sensor. $C_{23}$: Intention to Act
classifier is triggered and output is 1, but both Hand Speed classifiers are
stopped in at least one frame of each window of predefined number of
consecutive frames. This window frame is a guard to protect state from
transition because of small movements of hand which are not actions. $C_{32}$:
Intention to Act classifier converts from 1 to 0 for more than predefined
consecutive frames. This window of frames protects transition from action to
Intention for small position changes that make Intention to Act classifier
zero. $C_{34}$: Intention to Act classifier is triggered and output is 1, also
at least one of Hand Speed classifiers for detecting slow or fast movements is
1 for a predefined number of consecutive frames. $C_{43}$: Intention to Act
classifier is triggered and output is 1, but both of Hand Speed classifiers
for detecting stopped is 1 for a predefined number of consecutive frames.
After each transition, if FST recognizes the labels that is assigned to some
frames are wrong, it can change reconsider and modify them by relabling. In
addition, based on analysis of the speed signal of the hand, FST will relabel
the frames when user starts moving hand to raise his hand. That helps we have
correct segment of gesture for our gesture recognition system.

##  4 Experiment Results

DAIA framework was implemented in C++ on Windows workstation. We used ASUS
Xtion Pro to capture depth images and track skeletons using Primesense NiTE
SDK. The system ran at 30 frames per second. As it is mentioned in Table 1 we
created 37 binary classifiers. For each posture status, multiple binary
classifiers are designed. The 0 or 1 output of these 37 classifiers is used to
make our feature vector, $G$, for intention to action classifier. Furthermore,
we need to define $W$ or vector of weights to calculate engagement score and
afterwards we should define a threshold to classify the frame as intention to
act or disengagement similar to the procedure proposed in [9]. It needs
extensive research on different body postures to calculate these weights.
Furthermore, putting constant weight values for different classifiers may
result in wrong classification for complex body postures. Therefore, instead
of defining constant values for the weight vector, we used SVM[27] with linear
kernel for training our intention to act classification. We used $G$ as the
feature vector for training and testing our SVM.

To train the classifier, a simple ”Catch the Box!” game is designed. In this
game, we used hand tracking algorithm implemented in NiTE middle-ware by
Primesense to move the cursor on screen. A solid rectangle randomly appears on
the screen and user should move the cursor on rectangle area to receive
points. The game has 3 stages which are ”Getting Ready”, ”Play” and ”Stop”.
The binary classifier outputs of table 1 in ”Play” mode of the game are
combined as a series of 0 and 1 and used as ”Intention to Act” feature vectors
to train an binary SVM with linear kernel. The output of classifiers in
”Getting Ready” and ”Stop!” stages of game creates feature vectors of the SVM
when ”Intention to Act” is not present.

We captured and labeled 23,210 frames from 5 different subjects that played
the game separately. 5,000 frames are used for training and the remaining used
for testing the classifier. This classifier performance was 86.38%. The frame
is classified as Intention to Act or not. FST helps to relabel the frame based
on the current state properties and guards. In our experiment, we asked 30
different users to hear random order of commands from a list of actions such
as ”raising hand” or ”swiping right to left” and perform them in front of a
screen.

Afterwards, each recorded frame is labeled in four different engagement states
from Disengagement to Action by an expert and used as ground truth. Our FST
performance is calculated based on the number of correct states reported by
FST after relabeling the frames and also ground truth labeled manually. In
total, 165,422 frames are labeled to each engagement states. The results are
gathered in table 3. Figure 3 shows engagement state detection using FST and
combinations of boolean operations on raising and putting down right hand.

State | FST Performance
---|---
Disengagement | 97.3%
Attention | 87.2%
Intention | 90.8%
Action | 94.2%
Total | 92.3%
Table 3: Performance of FST Fig. 3: Engagement state detection using FST for
raising and putting down right hand in 600 frames: a) Facing classifier b)
Right hand speed value c) Intention to Act Classifier d) Engagement states for
raising and putting down right hand

##  5 Discussions and Conlutions

In this paper, a novel multi-modal model for engagement is introduced. Based
on this model, a combination of tracking 3D gesture data is employed for user
engagement detection. Therefore, the spectrum of mental status for performing
an action is quantized in a finite number of engagement states. Furthermore, a
finite state transducer (FST) with the following engagement states is
proposed: Disengagement, Attention, Intention, Action. Results show our
Intention to Act performance is 86.3%. In addition, FST relables some of those
labels based on the history of engagement states and guard conditions. The
performance of our FST for labeling the frames correctly is 92.3%.The
processing time for each frame is less than 10ms which indicates real-time
usability of our algorithm.

In future research, we expect using other channels of biometric information
such as voice and facial data such as gaze. We may reach even higher true
detection rates using extra channels of information. In addition, by using
multi-camera and calculating engagement state for each of the audience in a
meeting room we will be able to detect the main operator and give the control
of the vision based interface to that participant.

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