$e_{B^{\prime}}:=1_{X}\otimes\frac{1_{G}+\epsilon^{\prime}_{P}(-1)^{p}g}{2}+1_{Y}\otimes\frac{1_{G}+\epsilon^{\prime}_{Y}(-1)^{q-1}g}{2}$.
A direct computation yields

$\mu^{2}(e_{B},\mu^{2}(s^{\|c\|}\otimes c\otimes
1_{G},e_{B^{\prime}}))=(-1)s^{\|c\|}\otimes\left(c+\epsilon_{P}\epsilon^{\prime}_{Q}(-1)^{p+q-1}gc\right)\otimes\frac{1_{G}+\epsilon^{\prime}_{P}(-1)^{p}g}{4}.$

Therefore using the relation in $H^{1}$ and the fact that $p+|c|=q$ we obtain
a non zero morphism if and only if $\epsilon_{Q}=-\epsilon^{\prime}_{Q}.$

∎

######  Remark 4.18.

In the case where the surface is a cylinder with an involution as in the proof
of Proposition 4.8, then the category
$H^{0}(\operatorname{Tw}\mathcal{A}*G)_{+}$ is equivalent to the derived
category of the path algebra of type $\widetilde{\mathbf{D}_{n}}$ (cf Section
5). One can check that the two objects $B^{\epsilon_{P},\epsilon_{Q}}$ and
$B^{-\epsilon_{P},-\epsilon_{Q}}$ corresponds to objects which are on the
mouth of tubes of rank $2$ in the Auslander-Reiten quiver (compare also with
[AP21, Subsection 6.1]). They are indeed bricks, but there is a non trivial
extension between them.

####  4.5.3. Morphisms between a double tagged object and a tagged object

######  Proposition 4.19.

Let $(\gamma,\mathfrak{f})$ and $\mathbb{S}$ as in Figure 12. In the category
$H^{0}(\operatorname{Tw}\mathcal{A}_{\mathbb{S}}*G)_{+}$ we have the following
equalities

$\dim_{k}\operatorname{Hom}(B^{\epsilon_{P},\epsilon_{Q}}_{\mathbb{S}},X^{\pm}[\ell])=\left\\{\begin{array}[]{ll}1&\textrm{if
}\pm\epsilon_{P}=(-1)^{p}\textrm{ and }\ell=p\\\
0&\textrm{else.}\end{array}\right.$

$\dim_{k}\operatorname{Hom}(X^{\pm},B^{\epsilon_{P},\epsilon_{Q}}_{\mathbb{S}}[\ell])=\left\\{\begin{array}[]{ll}1&\textrm{if
}\pm\epsilon_{P}=(-1)^{1-p}\textrm{ and }\ell=1-p\\\
0&\textrm{else.}\end{array}\right.$

$\dim_{k}\operatorname{Hom}(B^{\epsilon_{P},\epsilon_{Q}}_{\mathbb{S}},Y^{\pm}[\ell])=\left\\{\begin{array}[]{ll}1&\textrm{if
}\pm\epsilon_{Q}=(-1)^{q}\textrm{ and }\ell=q\\\
0&\textrm{else.}\end{array}\right.$

$\dim_{k}\operatorname{Hom}(Y^{\pm},B^{\epsilon_{P},\epsilon_{Q}}_{\mathbb{S}}[\ell])=\left\\{\begin{array}[]{ll}1&\textrm{if
}\pm\epsilon_{Q}=(-1)^{1-q}\textrm{ and }\ell=1-q\\\
0&\textrm{else.}\end{array}\right.$

Hence we have the following morphisms linking $X^{\pm}$ and
$B^{\epsilon_{P},\epsilon_{Q}}_{\mathbb{S}}$

$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces
B^{+_{P},\pm_{Q}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p\textrm{
even }}$$\scriptstyle{{\rm
deg}p}$$\textstyle{X^{+}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\rm
deg}(1-p)}$$\textstyle{X^{-}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\rm
deg}(1-p)}$$\textstyle{B^{-_{P},\pm_{Q}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\rm
deg}p}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces
B^{+_{P},\pm_{Q}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p\textrm{
odd }}$$\scriptstyle{{\rm
deg}p}$$\textstyle{X^{-}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\rm
deg}(1-p)}$$\textstyle{X^{+}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\rm
deg}(1-p)}$$\textstyle{B^{-_{P},\pm_{Q}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\rm
deg}p}$

###### Proof.

We have $H^{\ell}(\operatorname{Hom}_{\operatorname{Tw}\mathcal{A}}(B,X))=0$
for $\ell\neq p$ and
$H^{\ell}(\operatorname{Hom}_{\operatorname{Tw}\mathcal{A}}(B,X)={\rm
vect}(s^{-p}\otimes 1_{X})$. Therefore it is enough to compute
$\mu^{2}(e_{X}^{\pm},\mu^{2}(s^{-p}\otimes 1_{X}\otimes 1_{G},e_{B}))$ with
$e_{B}$ as above. A direct computation yields

$\mu^{2}(e_{X}^{\pm},\mu^{2}(s^{-p}\otimes 1_{X}\otimes
1_{G},e_{B}))=(-1)^{p}[1\pm\epsilon_{P}(-1)^{p}]s^{-p}\otimes
1_{X}\otimes\frac{1_{G}+(-1)^{p}\epsilon_{P}g}{2},$

which is not zero if and only if $\pm\epsilon_{P}=(-1)^{p}$.

Therefore, there is a morphism from $B^{\epsilon_{P},\epsilon_{Q}}$ to $X^{+}$
if and only if $\epsilon_{P}=(-1)^{p}$ and this morphism has degree $p$.

In the category $\operatorname{Tw}\mathcal{A}$ there are morphisms in degree
$-p$ and $1-p$ as follows

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
23.86598pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{X[0]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.57222pt\hbox{$\scriptstyle{s^{p}\otimes 1_{X}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 0.0pt\raise-29.5pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
37.50192pt\raise-12.60771pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.85973pt\hbox{$\scriptstyle{s^{q-1}\otimes c}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
67.72621pt\raise-29.35394pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
22.24872pt\raise-27.78116pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.47084pt\hbox{$\scriptstyle{s^{q-1}\otimes gc}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
65.89879pt\raise-33.25241pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern
36.8125pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 82.40446pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern-11.43922pt\raise-40.0pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{X[p\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
17.66711pt\raise-47.44722pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.30278pt\hbox{$\scriptstyle{(-1)^{\|c\|}\epsilon_{P}\epsilon_{Q}c+gc}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 66.8125pt\raise-40.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
36.8125pt\raise-40.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
66.8125pt\raise-40.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{Y[q-1]}$}}}}}}}\ignorespaces}}}}\ignorespaces.$

We compute

$\mu^{1}(s^{p}\otimes
1_{X})=(-1)^{\|c\|p}((-1)^{\|c\|}\epsilon_{P}\epsilon_{Q}s^{q-1}\otimes
c+s^{q-1}\otimes gc).$

Hence we obtain

$H^{1-p}(\operatorname{Hom}_{\operatorname{Tw}\mathcal{A}}(X,B))={\rm
vect}(s^{q-1}\otimes c,s^{q-1}\otimes
gc)/\langle(-1)^{\|c\|}\epsilon_{P}\epsilon_{Q}s^{q-1}\otimes c+s^{q-1}\otimes
gc\rangle.$

It is then enough to compute $\mu^{2}(e_{B},\mu^{2}(s^{q-1}\otimes c\otimes
1_{G},e_{X}^{\pm})).$ A direct computation gives

$\mu^{2}(e_{B},\mu^{2}(s^{q-1}\otimes c\otimes
1_{G},e_{X}^{\pm}))=(-1)^{p-1}s^{q-1}\otimes(c\pm(-1)^{q-1}\epsilon_{Q}gc)\otimes\frac{1\pm
g}{2}.$

Hence using the relation in
$H^{1-p}(\operatorname{Hom}_{\operatorname{Tw}\mathcal{A}}(X,B))$, one obtain
that $\mu^{2}(e_{B},\mu^{2}(s^{q-1}\otimes c\otimes 1_{G},e_{X}^{\pm}))$ is
non zero in cohomology if and only if $(-1)^{1-p}=\pm\epsilon_{P}.$

In the same way we have

$H^{1-q}(\operatorname{Hom}_{\operatorname{Tw}\mathcal{A}}(Y,B))={\rm
vect}(s^{q-1}\otimes 1_{Y}),$

and

$\mu^{2}(e_{B},\mu^{2}(s^{q-1}\otimes 1_{Y}\otimes
1_{G},e_{Y}^{\pm}))=(-1)^{q-1}(1\pm\epsilon_{Q}(-1)^{q-1})s^{q-1}\otimes
1_{Y}\otimes\frac{1_{G}\pm g}{4},$

which is not zero if and only if $\pm\epsilon_{Q}=(-1)^{1-q}$.

We also have

$H^{q}(\operatorname{Hom}_{\operatorname{Tw}\mathcal{A}}(B,Y))={\rm
vect}(s^{-p}\otimes c,s^{-p}\otimes
gc)/\langle\epsilon_{P}\epsilon_{Q}(-1)^{\|c\|}c+gc\rangle,$

and

$\mu^{2}(e_{Y}^{\pm},\mu^{2}(s^{-p}\otimes c\otimes
1_{G},e_{B}))=(-1)^{q}s^{-p}\otimes(c\pm(-1)^{p}\epsilon_{P}gc)\otimes\frac{1_{G}+\epsilon_{P}(-1)^{p}g}{4}.$

So this does not vanish in cohomology if and only if
$\pm\epsilon_{Q}=(-1)^{q}.$

∎

######  Proposition 4.20.

Let $(\gamma,\mathfrak{f})$ be a graded simple curve such that
$g\gamma=\gamma^{-1}$, and let $\mathbb{S}$ and $\mathbb{S}^{\prime}$ be two
full system of graded arcs as in Figure 12. Then we have isomorphisms in
$H^{0}(\operatorname{Tw}\mathcal{A}_{\mathbb{S}^{\prime}}*G)_{+}$

$\Psi_{\mathbb{S}\to\mathbb{S}^{\prime}}(B_{\mathbb{S}}^{\epsilon_{P},\epsilon_{Q}})\simeq
B_{\mathbb{S}^{\prime}}^{\epsilon_{P},\epsilon_{Q}}.$

###### Proof.

We have

$B^{\epsilon_{P},\epsilon_{Q}}_{\mathbb{S}}\oplus
B^{-\epsilon_{P},-\epsilon_{Q}}_{\mathbb{S}}=B_{(\gamma,\mathfrak{f},\epsilon_{P}\epsilon_{Q},\mathbb{S})},\textrm{
and }B^{\epsilon_{P},\epsilon_{Q}}_{\mathbb{S}^{\prime}}\oplus
B^{-\epsilon_{P},-\epsilon_{Q}}_{\mathbb{S}^{\prime}}=B_{(\gamma,\mathfrak{f},\epsilon_{P}\epsilon_{Q},\mathbb{S}^{\prime})}.$

Moreover by Theorem 3.15 we know that there is an isomorphism in
$H^{0}(\operatorname{Tw}\mathcal{A}_{\mathbb{S}^{\prime}})$ between
$\Psi_{\mathbb{S}\to\mathbb{S}^{\prime}}(B_{(\gamma,\mathfrak{f},-\epsilon_{P}\epsilon_{Q},\mathbb{S})})$
and $B_{(\gamma,\mathfrak{f},-\epsilon_{P}\epsilon_{Q},\mathbb{S}^{\prime})}$.
Therefore we have either

$\Psi_{\mathbb{S}\to\mathbb{S}^{\prime}}(B_{\mathbb{S}}^{\epsilon_{P},\epsilon_{Q}})\simeq
B_{\mathbb{S}^{\prime}}^{\epsilon_{P},\epsilon_{Q}}\textrm{ or
}\Psi_{\mathbb{S}\to\mathbb{S}^{\prime}}(B_{\mathbb{S}}^{\epsilon_{P},\epsilon_{Q}})\simeq
B_{\mathbb{S}^{\prime}}^{-\epsilon_{P},-\epsilon_{Q}}.$

The arcs $X$ and $X^{\prime}$ intersects the curve $\gamma$ in $P$. Assume
that they intersect as follows :

$\times$$P$$\gamma$$X$$X^{\prime}$$p^{\prime}$$p$ Figure 14.

Denote by $p$ the index from $(\gamma,\mathfrak{f})$ to $X$, and by
$p^{\prime}$ the index from $(\gamma,\mathfrak{f})$ to $X^{\prime}$. Then
there is non zero morphisms in $\mathcal{F}(\mathcal{S})$

$\operatorname{Hom}_{\mathcal{F}}(B,X^{\prime}[p^{\prime}])\textrm{ and
}\operatorname{Hom}_{\mathcal{F}}(X^{\prime}[p^{\prime}],X[p])$

whose composition gives rise to a non zero morphism in
$\operatorname{Hom}_{\mathcal{F}}(B,X[p]).$ Note that here, because of Theorem
3.15, we can avoid subscript notation with respect to the system $\mathbb{S}$
or $\mathbb{S}^{\prime}$.

Denote by $\epsilon^{\prime}_{P}$ and $\epsilon^{\prime}_{Q}$ the signs such
that
$\Psi_{\mathbb{S}\to\mathbb{S}^{\prime}}(B^{+,\epsilon_{Q}}_{\mathbb{S}})\simeq
B^{\epsilon^{\prime}_{P},\epsilon^{\prime}_{Q}}_{\mathbb{S}^{\prime}}.$ Then
we have $\epsilon_{Q}=\epsilon^{\prime}_{P}\epsilon^{\prime}_{Q}$.

We first assume that $p$ is even to avoid heavy sign notation. Now by
Proposition 4.19 there is a non zero morphism in
$H^{0}(\operatorname{Tw}\mathcal{A}_{\mathbb{S}}*G)_{+}$ from
$B_{\mathbb{S}}^{+,\epsilon_{Q}}$ to $X^{+}_{\mathbb{S}}[p]$. Therefore there
is a non zero morphism from
$\Psi_{\mathbb{S}\to\mathbb{S}^{\prime}}(B_{\mathbb{S}}^{+,\epsilon_{Q}})$ to
$\Psi_{\mathbb{S}\to\mathbb{S}^{\prime}}(X_{\mathbb{S}}^{+}[p])$ in the
category $H^{0}(\operatorname{Tw}\mathcal{A}_{\mathbb{S}^{\prime}}*G)_{+}$. By
Proposition 4.5 we have
$\Psi_{\mathbb{S}\to\mathbb{S}^{\prime}}(X^{+}_{\mathbb{S}})=X^{+}_{\mathbb{S}^{\prime}}$,
so there is a non zero morphism from
$\Psi_{\mathbb{S}\to\mathbb{S}^{\prime}}(B_{\mathbb{S}}^{+,\epsilon_{Q}})\simeq
B^{\epsilon^{\prime}_{P},\epsilon^{\prime}_{Q}}_{\mathbb{S}^{\prime}}$ to
$X^{+}_{\mathbb{S}^{\prime}}[p]$. This morphism factors through
$X^{\prime}[p^{\prime}]=X^{\prime+}[p^{\prime}]\oplus
X^{\prime-}[p^{\prime}]$. But by Proposition 4.6 we have

$\operatorname{Hom}(X^{\prime\epsilon}[p^{\prime}],X^{+}[p])\neq 0$

if and only if $\epsilon=(-1)^{p-p^{\prime}}$. And by Proposition 4.19 we have

$\operatorname{Hom}(B^{\epsilon_{P},\epsilon_{Q}}_{\mathbb{S}^{\prime}},X^{\epsilon}[p^{\prime}])\neq
0$

if and only if $\epsilon=\epsilon_{P}(-1)^{p^{\prime}}.$ Therefore we have

$\operatorname{Hom}(B_{\mathbb{S}^{\prime}}^{\epsilon_{P},\epsilon_{Q}},X^{+}[p])\neq
0$

if and only if $\epsilon_{P}(-1)^{p^{\prime}}=(-1)^{p-p^{\prime}}.$ Since $p$
is even this is equivalent to $\epsilon_{P}=1$, hence we have

$\Psi_{\mathbb{S}\to\mathbb{S}^{\prime}}(B^{+,\epsilon_{Q}}_{\mathbb{S}})\simeq
B^{+,\epsilon^{\prime}_{Q}}_{\mathbb{S}^{\prime}}.$

Since $\epsilon^{\prime}_{Q}\epsilon^{\prime}_{P}=\epsilon_{Q}$ we conclude.
The case where $p$ is odd is completely similar. The case where the curves
$X$, $X^{\prime}$ and $B$ intersect in the order $(B,X,X^{\prime})$ is also
similar.

∎

####  4.5.4. Morphisms between double tagged objects

######  Proposition 4.21.

Let $\gamma_{1}$ and $\gamma_{2}$ be two graded simple loops on $\mathcal{S}$
with $[g\gamma_{i}]=[\gamma^{-1}_{i}]$ and intersecting in an orbifold point
$P$. Denote by $B_{1}^{\epsilon_{P},\epsilon_{Q}}$ and
$B_{2}^{\epsilon^{\prime}_{P},\epsilon^{\prime}_{R}}$ the corresponding double
tagged objects in $\mathcal{F}^{G}_{+}$ and by $p$ the index between
$\gamma_{1}$ and $\gamma_{2}$ at $P$. Then we have

$\dim_{k}\operatorname{Hom}(B_{1}^{\epsilon_{P},\epsilon_{Q}},B_{2}^{\epsilon^{\prime}_{P},\epsilon^{\prime}_{R}}[\ell])=\left\\{\begin{array}[]{ll}1&\textrm{if
}\epsilon_{P}\epsilon^{\prime}_{P}=(-1)^{p}\textrm{ and }\ell=p\\\
0&\textrm{else.}\end{array}\right.$

$\dim_{k}\operatorname{Hom}(B_{2}^{\epsilon^{\prime}_{P},\epsilon^{\prime}_{R}},B_{1}^{\epsilon_{P},\epsilon_{Q}}[\ell])=\left\\{\begin{array}[]{ll}1&\textrm{if
}\epsilon^{\prime}_{P}\epsilon_{P}=(-1)^{1-p}\textrm{ and }\ell=1-p\\\
0&\textrm{else.}\end{array}\right.$

###### Proof.

The proof is similar to the previous one. We denote by $X$ a $G$-invariant arc
passing through the point $P$ as in the following picture and such that the
index from $\gamma_{1}$ to $X$ is $0$, and we work in a full $G$-invariant
system of graded arcs containing $X$.

$\times$$P$$\gamma_{1}$$\gamma_{2}$$X$$0$$p$ Figure 15.

Assume that $p$ is even. We prove that there is no morphism from
$B_{1}^{+_{P},+_{Q}}$ to $B_{2}^{-_{P},-_{R}}$ and that the space of morphism
from $B_{1}^{+_{P},+_{Q}}$ to $B_{2}^{+_{P},+_{R}}$ is one dimensional.

We know that the intersection induces a degree $p$ morphism from $(B_{1})_{1}$
to $(B_{2})_{1}$ in $H^{0}(\operatorname{Tw}\mathcal{A})$ factorising through
$X$. Hence the space of morphism from $B_{1}^{+_{P},+_{Q}}\oplus
B_{1}^{-_{P},-_{Q}}$ to $B_{2}^{+_{P},+_{R}}\oplus B_{2}^{-_{P},-_{R}}$ in
$H^{0}(\operatorname{Tw}\mathcal{A}*G)_{+}$ is $2$-dimensional and of degree
$p$, and these morphisms factor through $X^{+}\oplus X^{-}$. The space of
morphism from $B_{1}^{+_{P},+_{Q}}$ to $B_{2}^{+_{P},+_{R}}\oplus
B_{2}^{-_{P},-_{R}}$ is then one dimensional and factors through $X^{+}\oplus
X^{-}$. By proposition 4.19, there is no morphism from $B_{1}^{+_{P},+_{Q}}$
to $X^{-}$, and no morphism from $X^{+}$ to $B_{2}^{-_{P},-_{Q}}$, hence there
is no morphism from $B_{1}^{+_{P},+_{Q}}$ to $B_{2}^{-_{P},-_{R}}$. Thus the
space of morphism from $B_{1}^{+_{P},+_{Q}}$ to $B_{2}^{+_{Q},+_{R}}$ is one
dimensional. The other cases can be treated in a similar fashion by
considering the morphisms from $(B_{1})_{1}$ and $(B_{1})_{-1}$ to
$(B_{2})_{1}$ and $(B_{2})_{-1}$ in $H^{0}(\operatorname{Tw}\mathcal{A})$.

∎

##  5\. Tilting objects and skew-gentle algebras

In this section, $(\mathcal{S},\mathbb{M},\eta,g)$ is a $G$-graded marked
surface as in Definition 4.1. Then, we can naturally consider the orbifold
surface $\mathcal{S}/G$, it is an orbifold surface with orbifold points
correpponding to the points fixed by the automorphism $g$. The aim is to
describe some tilting objects in the category
$H^{0}\operatorname{Tw}\big{(}(\mathcal{A}*G)_{+}\big{)}$ in terms of
collections of (tagged) arcs on the orbifold surface $\mathcal{S}/G$.

For the graded surface $\mathcal{S}$, we denote by $\mathbf{g}$ its genus, by
$\mathbf{b}$ its number of boundary components and by $\mathbf{m}$ the number
of boundary segments. For the orbifold surface $\mathcal{S}/G$ we denote by
$\bar{\mathbf{g}}$ its genus, by $\bar{\mathbf{b}}$ its number of boundary
components and by $\bar{\mathbf{m}}=\frac{\mathbf{m}}{2}$ its number of marked
segments, and by $\mathbf{x}$ the number of orbifold points, that is the
number of fixed points of $\mathcal{S}$ by the action of $g$.

###  5.1. Formal generators and skew-gentle algebras

######  Definition 5.1.

A _$G$ -graded dissection_ $\mathbb{D}$ on $(\mathcal{S},\mathbb{M},\eta,g)$
is a system of graded arcs globally invariant by $g$ on $\mathcal{S}$ cutting
out the surface $\mathcal{S}$ into discs with exactly one non marked boundary
component in its boundary, or annuli with one non marked boundary component.

A $G$-graded dissection always exists and contains exactly
$\mathbf{m}+\mathbf{b}+2\mathbf{g}-2$ arcs by Proposition 1.12 in [APS23].

Our use of $G$-graded dissections will be giving us formal generators in the
category of twisted complexes over the skew-group $A_{\infty}$-algebra using
the following general facts.

######  Definition 5.2.

Let $\mathcal{A}$ and $\mathcal{B}$ be $A_{\infty}$-categories with a strict
action of a finite group $G$. A strictly unital strict $A_{\infty}$-functor
$F:\mathcal{A}\to\mathcal{B}$ is  _$G$ -equivariant_ if $g\cdot FX=F(g\cdot
X)$ for all objects $X$ of $\mathcal{A}$ and $g\cdot Fa=F(g\cdot a)$ for all
morphisms $a$ of $\mathcal{A}$.

######  Proposition 5.3.

Let $F:\mathcal{A}\to\mathcal{B}$ be a $G$-equivariant strictly unital strict
$A_{\infty}$-functor.

  1. (1)

There is an induced strictly unital strict $A_{\infty}$-functor
$F*G:\mathcal{A}*G\to\mathcal{B}*G$ defined by $F*G(\\{X\\})=\\{FX\\}$ and
$F*G(a\otimes g)=Fa\otimes g$.

  2. (2)

It $F$ is a quasi-equivalence, then so is $F*G$.

######  Corollary 5.4.

Let $\mathcal{A}$ be an $A_{\infty}$-category with a strict action of a finite
group $G$, and let $T$ be a $G$-invariant formal generator in
$\operatorname{Tw}\mathcal{A}$. Assume further that there exists a
$G$-equivariant quasi-equivalence
$H^{*}\operatorname{End}(T)\to\operatorname{End}(T)$. Then $\\{T\\}$ is a
formal generator in $\big{(}(\operatorname{Tw}\mathcal{A})*G\big{)}_{+}$. In
particular, the triangulated categories
$H^{0}\big{(}(\operatorname{Tw}\mathcal{A})*G\big{)}_{+}$ and
$\operatorname{per}(H^{*}\operatorname{End}(T))$ are equivalent.

###### Proof.

Apply Proposition 5.3 to the quasi-equivalence
$H^{*}\operatorname{End}(T)\to\operatorname{End}(T)$, noting that, if
considered as $A_{\infty}$-categories with one object, we have that
$\operatorname{End}(\\{T\\})$ and $\operatorname{End}(T)*G$ are the same. ∎

Let $\mathbb{S}$ be a full $G$-invariant arc system on
$(\mathcal{S},\mathbb{M},\eta,g)$, and denote by
$\mathcal{A}:=\mathcal{A}_{\mathbb{S}}$ the corresponding
$A_{\infty}$-category. Let $\mathbb{D}$ be a $G$-graded dissection of
$(\mathcal{S},\mathbb{M},\eta,g)$. Then by [HKK17, Section 3.4], the object
corresponding object $\mathbb{D}$ is a formal generator of the category
$\operatorname{Tw}\mathcal{A}$. This implies that there is an equivalence of
triangulated categories

$H^{0}(\operatorname{Tw}\mathcal{A})\simeq\operatorname{per}(A_{DG}),$

where
$A=\bigoplus_{\ell\in\mathbb{Z}}\operatorname{Hom}_{H^{0}\operatorname{Tw}\mathcal{A}}(\mathbb{D},\mathbb{D}[\ell])$
is the graded endomorphism algebra of the object $\mathbb{D}$ seen a
differential graded algebra with zero differential. Since $\mathbb{D}$ is
$G$-invariant, the algebra $A$ naturally inherits an action of the group $G$.
Then we have the following.

######  Corollary 5.5.

In the setup above, the object $\\{\mathbb{D}\\}$ in
$H^{0}(\operatorname{Tw}\mathcal{A}*G)_{+}$ is a formal generator and we have
an equivalence of triangulated categories

$H^{0}(\operatorname{Tw}\mathcal{A}*G)_{+}\simeq\operatorname{per}((A*G)_{DG}).$

###### Proof.

Using the work of [HKK17], up to Morita-equivalence, we can assume that
$\mathbb{D}$ is part of the $G$-invariant full arc system $\mathbb{S}$. Then
by definition, $\operatorname{End}{\mathbb{D}}$ has vanishing $\mu^{1}$ and
$\mu^{\geq 3}$. Thus we have a canonical quasi-equivalence
$H^{*}\operatorname{End}{\mathbb{D}}\to\operatorname{End}{\mathbb{D}}$ which
is simply the identity map. Since it is $G$-equivariant, we can apply
Corollary 5.4. ∎

Note that the endormophism algebra of the object $\\{\mathbb{D}\\}$ in the
category $H^{0}(\operatorname{Tw}\mathcal{A}*G)_{+}$ is not basic, indeed each
arc of $\mathbb{D}$ which is not $G$-invariant give rise to two indecomposable
summand in $\\{\mathbb{D}\\}$ which are isomorphic. There is no canonical
choice of a basic representant of the object $\\{\mathbb{D}\\}$.

Since the algebra $A$ is graded gentle in the sense of [AS87], we immediately
obtain the following.

######  Corollary 5.6.

The category $H^{0}(\operatorname{Tw}\mathcal{A}*G)_{+}$ is triangle
equivalent to the perfect derived category of a graded skew-gentle algebra.

###  5.2. Indecomposable objects and tagged curves on orbifold surfaces

Let $(\mathcal{S},\mathbb{M},\eta,g)$ be a $G$-graded marked surface, and
denote by $\pi:\mathcal{S}\to\mathcal{S}/G$ the natural projection.

######  Definition 5.7.

A  _tagged arc_ on $\mathcal{S}/G$ is an arc $\gamma$ whose endpoints are
either in $\mathbb{M}$ or in the set of orbifold points, and which does not
cut out a disc containing nothing but an orbifold point. The  _tagging_ is a
choice of a sign $\pm$ at each orbifold endpoint of $\gamma$. A  _graded
tagged arc_ is a tagged arc $\gamma$ with a choice of grading on the preimage
of $\gamma$ on $\mathcal{S}$ via $\pi$.

######  Remark 5.8.

If two graded tagged arcs $\gamma$ and $\delta$ intersect, we can define the
index of the intersection as the index of the lift of this intersection point
on $\mathcal{S}$. Note that this applies also in the case where the
intersection point is an orbifold point: in that case, the index in one
direction is some integer $p$, while the index in the other is $1-p$.

$p$$1-p$$p$$1-p$$\mathcal{S}$$\times$$p$$1-p$$\times$$\mathcal{S}/G$ Figure
16. Index between two graded arcs intersecting at an orbifold point on the
orbifold surface $\mathcal{S}/G$

We describe here the indecomposable objects discussed in Section 4 in terms of
graded tagged arcs on the orbifold $\mathcal{S}/G$.

######  Definition 5.9.

Let $\gamma$ be a graded simple tagged arc $\gamma$ on $\mathcal{S}/G$. We
exclude the case where both endpoints of $\gamma$ are the same orbifold point.
We define the object $\mathbb{X}(\gamma)$ of
$H^{0}\operatorname{Tw}(\mathcal{A}*G)_{+}$ as follows:

  * •

if $\gamma$ is not connected to an orbifold point, then its preimage on
$\mathcal{S}$ is a union of two (graded) arcs in the same $G$-orbit. These are
associated to objects $X$ and $gX$ in $\operatorname{Tw}\mathcal{A}$. We
define $\mathbb{X}(\gamma):=\\{X\\}\oplus\\{gX\\}$.

  * •

if $\gamma$ has one endpoint on the boundary and another on an orbifold point
with tag $\pm$, then its lift to $\mathcal{S}$ is a $G$-invariant curve
$\bar{\gamma}$ with $g\tilde{\gamma}=\tilde{\gamma}^{-1}$. The graded arc
$\gamma$ is associated to a $G$-invariant object $X$. Using the notation of
Section 4, we define $\mathbb{X}(\gamma):=X^{\pm}$.

  * •

if $\gamma$ has both endpoints on orbifold points $P$ and $Q$ with tags
$\pm_{P}$ and $\pm_{Q}$, then its lift is of the form $\alpha.g(\alpha^{-1})$
where $\alpha$ is a simple curve linking two fixed points $P$ and $Q$. Then we
define $\mathbb{X}:=B^{\pm_{P},\pm_{Q}}$, with the notations of Section 4.

######  Theorem 5.10.

Let $C:=(C_{1},\ldots,C_{n})$ be a collection of pairwise distinct and non
intersecting (except in the orbifold points) tagged simple graded arcs on
$\mathcal{S}/G$. We assume that none of the $C_{i}$ starts and ends at the
same orbifold point. The object $\bigoplus_{i=1}^{n}\mathbb{X}(C_{i})$ of
$H^{0}(\operatorname{Tw}\mathcal{A}*G)_{+}$ is rigid if and only if the
following three conditions are satisfied for the graded tagged arcs
$C_{i}=(\gamma_{i},\mathfrak{f}_{i})$ and
$C_{j}=(\gamma_{j},\mathfrak{f}_{j})$

  * •

if for $i\neq j$, $\gamma_{i}$ is isotopic to $\gamma_{j}$, (then there exists
$p\in\mathbb{Z}$ with $\mathfrak{f}_{i}=\mathfrak{f}_{j}[p]$), then

    * –

either $\gamma_{i}$ and $\gamma_{j}$ are arcs linking the boundary to an
orbifold point, and the sign attached to $C_{i}$ is $(-1)^{p-1}$ times the
sign attached to $C_{j}$ ;

    * –

or they are double tagged arcs linking orbifold points, and on one orbifold
point, the sign attached to $C_{i}$ is $(-1)^{p}$ times the sign attached to
$C_{j}$, and on the other orbifold point, the sign attached to $C_{i}$ is
$(-1)^{p-1}$ times the sign attached to $C_{j}$;

  * •

if $\gamma_{i}$ and $\gamma_{j}$ are not isotopic and have an endpoint on the
same boundary segment, then the corresponding $\mathbb{M}$-segment has degree
$0$ (here $i$ and $j$ may coincide);

  * •

if $C_{i}$ and $C_{j}$ are not isotopic and intersect at an orbifold point,
then the signs at this point are the same, and the index at this orbifold
point is $0$ and $1$ (cf Figure 16);

If moreover we have
$n=\bar{\mathbf{m}}+\bar{\mathbf{b}}+2\bar{\mathbf{g}}-2+2\mathbf{x}$, then
$C$ correspond to a tilting object in the triangulated category
$H^{0}(\operatorname{Tw}\mathcal{A}*G)_{+}$.

###### Proof.

By Propositions 4.3, 4.17, each of these objects corresponds to a rigid
object. The graded morphisms between $C_{i}$ and $C_{j}$ are computed in
Propositions 4.6, 4.19 and 4.21.

Moreover, if $C:=(\gamma,\mathfrak{f})$ is a $G$-invariant graded arc on
$\mathcal{S}$ corresponding to the objects $C^{+}$ and $C^{-}$, then
$C^{+}\oplus C^{-}[p]$ is rigid for any $p\in\mathbb{Z}$. The first subitem
comes from the fact that $C^{-}[p]=(\gamma,\mathfrak{f}[p])^{(-1)^{p-1}}$ (cf
Remark 4.7). The second subitem is analogous since $B^{+_{P},+_{Q}}\oplus
B^{-_{P},+_{Q}}[p]$ is rigid for any $p\in\mathbb{Z}$.

For the second statement, we can combine the fact that the cardinality of a
$G$-invariant dissection is $\mathbf{m}+\mathbf{b}+2\mathbf{g}-2$, and the
fact that if $A$ is an algebra with a $G$-action then the rank of the algebra
$A*G$ is $\frac{{\rm rk}A+3\mathbf{x}}{2}$ where $\mathbf{x}$ is the number of
$G$-invariant projective modules. ∎

######  Remark 5.11.

If $X=(\gamma,\mathfrak{f})$ is a $G$-invariant arc on $\mathcal{S}$, then for
$p\neq 0$, the object $X^{+}\oplus X^{-}[p]$ is rigid, but cannot be completed
into a collection of
$n=\bar{\mathbf{m}}+\bar{\mathbf{b}}+2\bar{\mathbf{g}}-2+2\mathbf{x}$ arcs as
in Theorem 5.10, and hence into a tilting object.

###  5.3. Tagged dissections on orbifold surfaces

Using our description of tilting objects from Section 5.2, we can use the
combinatorics of collections of tagged arcs on an orbifold surface to
construct algebras derived-equivalent to graded skew-gentle algebras.

######  Definition 5.12.

Let $\mathcal{S}$ be a surface with an action of $G$, and let
$\pi:\mathcal{S}\to\mathcal{S}/G$ be the projection. A _tagged dissection_ of
$\mathcal{S}/G$ is a collection $\Delta$ of tagged arcs on $\mathcal{S}/G$
satisfying the following conditions.

  * •

The arcs of $\Delta$ are simple and are pairwise non-intersecting in the
interior of $\mathcal{S}/G$.

  * •

If $C_{i},C_{j}\in\Delta$ are distinct but isotopic, then they are either arcs
linking the boundary to an orbifold point, and their signs at this point are
distinct or they are double tagged arcs with common signs at one orbifold
point and opposite signs at the other.

  * •

If $C_{i},C_{j}\in\Delta$ are not isotopic and meet at an orbifold point, then
their signs at this point are equal.

  * •

The arcs of $\Delta$ cut $\mathcal{S}/G$ into pieces of three kinds:

    1. (1)

discs whose sides are all arcs in $\Delta$ except for one, which is a boundary
segment with exactly one unmarked segment;

    2. (2)

discs containing exactly one unmarked boundary component and whose sides are
all in $\Delta$;

    3. (3)

discs with three sides which are all arcs in $\Delta$, at least one of which
is connected to an orbifold point for which all the tagged arcs incident to it
have the same sign.

  * •

each orbifold point is the endpoint of at least two tagged arcs.

  * •

there is a bijection between the set of orbifold points for which all the
tagged arcs incident to it have the same sign and the set of pieces of type
(3) sending an orbifold point to a piece of which it is part of the boundary.
This bijection is part of the data of a tagged dissection.

..$-$$+$$+$$\pm$$\pm$$\times$$\times$$\times$$\times$$\times$ Figure 17.
Example of a dissection. The circle arcs around orbifold points illustrate the
bijection with pieces of type (3).

######  Proposition 5.13.

The number of tagged arcs in a tagged dissection is
$n=\bar{\mathbf{m}}+\bar{\mathbf{b}}+2\bar{\mathbf{g}}-2+2\mathbf{x}$.

###### Proof.

Let $\Delta$ be a tagged dissection. If it contains isotopic arcs, we draw
them parallel to each other so that they form a digon. We note that if two
isotopic arcs join two orbifold points, then one of these orbifold points has
unequal signs, while the other has equal signs; if both had unequal signs,
then no arcs could be added to either orbifold points, and the surface cannot
be cut into discs.

We turn $\mathcal{S}/G$ into a compact surface with marked points by turning
all orbifold points into marked points, shrinking each marked segment on the
boundary into a marked point, and filling each boundary component with a disc,
keeping the boundary edges between marked points (unmarked boundary components
thus disappear). If $\chi$ is the Euler characteristic of the surface, then
$\chi=2-2\bar{\mathbf{g}}$. Moreover, if the surface is cut into discs, then
$\chi$ is the number of faces, minus the number of edges, plus the number of
vertices. Now, $\Delta$ provides a cutting of the surface into discs, with
$\bar{\mathbf{m}}+\mathbf{x}$ vertices, $n+\bar{\mathbf{m}}$ edges (one per
arc in $\Delta$ and one per boundary edge, which are in bijection with marked
points on the boundary) and $\bar{\mathbf{b}}+\bar{\mathbf{m}}+\mathbf{x}$
faces (one face per “filled” marked boundary component, one face containing
each of the “filled” unmarked boundary components (pieces of type (2)), one
face per unmarked boundary segment (pieces of type (1), one triangle per
orbifold point with equal signs (pieces of type(3), and one digon per orbifold
point with unequal signs). This proves the formula. ∎

######  Proposition 5.14.

Let $\Delta$ be a tagged dissection of $\mathcal{S}/G$. There exists a line
field $\eta$ on $\mathcal{S}$ and a grading on each of the $C_{i}$ such that
the resulting graded tagged arcs $C_{1},\ldots,C_{n}$ of $\Delta$ satisfy the
conditions of Theorem 5.10.

###### Proof.

We construct $\eta$ as follows. First, let $\overline{\Delta}$ be the
collection of all lifts of arcs in $\Delta$ via
$\pi:\mathcal{S}\to\mathcal{S}/G$ (forgetting the taggings). If two arcs in
$\Delta$ are isotopic, we only keep the lifts of one of them in
$\overline{\Delta}$. Then $\Delta$ is a collection of arcs and closed curves
on $S$ which may intersect at $G$-fixed points. It cuts $S$ into polygons
whose vertices are either on the boundary or are $G$-fixed points, and whose
arcs are portions of the curves in $\overline{\Delta}$.

We now build $\eta$ by defining a line field on each of the polygons, in such
a way that the foliation of the line field is transverse to the edges of the
polygon. Thus $\eta$ is obtained by gluing these foliations together.

There are two types of polygons.

_Case 1: polygons containing exaclty one unmarked boundary segment._. This
corresponds to pieces of types (1) and (2) in Definition 5.12. In this case,
the foliation is as follows:

Figure 18. Foliations on polygons of types (1) and (2)

We see that in each case, the $\mathbb{M}$-path between two adjacent arcs is
of degree 0.

_Case 2: triangles with at least one orbifold point in their boundary._ Those
triangles are lifts of pieces of type (3) in Definition 5.12. Such a triangle
is associated to one of the orbifold point in its boundary by the bijection in
the definition of a dissection. If we denote by $\times$ this orbifold point,
then the foliation in this triangle is given as follows:

$\times$ Figure 19. Foliation on polygon of type (3)

All these foliations glue together and thus define a line field $\eta$ on
$\mathcal{S}$. Moreover, by contruction and up to homotopy, we can assume that
$\eta$ is $G$-invariant. Then the map $\mathbb{X}$ of Definition 5.9 sends
$\Delta$ to an object which can be seen to be tilting using Theorem 5.10.

∎

We now define an algebra from a tagged dissection.

######  Definition 5.15.

Let $\mathcal{S}/G$ be an orbifold surface with a tagged dissection $\Delta$.
We define an algebra $A(\Delta)$ is several steps:

  1. (1)

Let $\eta$ be the line field of Proposition 5.14.

  2. (2)

Choose a full $G$-invariant arc system $\mathcal{A}$ on $\mathcal{S}$.

  3. (3)

By Proposition 5.14 and Theorem 5.10, the arcs of $\Delta$ correspond to the
indecomposable summands of a tilting object $T$ in
$H^{0}\operatorname{Tw}\big{(}(\mathcal{A}*G)_{+}\big{)}$.

  4. (4)

Let $A(\Delta)$ be the endomorphism algebra of $T$ in
$H^{0}\operatorname{Tw}\big{(}(\mathcal{A}*G)_{+}\big{)}$.

######  Remark 5.16.

The algebra $A(\Delta)$ is generally not basic: indeed, any arc joining two
boundary segments will correspond to two isomorphic objects $\\{X\\}$ and
$\\{gX\\}$.

######  Corollary 5.17.

Let $\mathcal{S}/G$ be an orbifold surface with a tagged dissection $\Delta$.
Then $A(\Delta)$ is derived equivalent to a graded skew-gentle algebra.

###### Proof.

By construction, $A(\Delta)$ is the endomorphism algebra of a tilting object
in $H^{0}\operatorname{Tw}\big{(}(\mathcal{A}*G)_{+}\big{)}$, which is the
perfect derived category of a skew-gentle algebra. ∎

######  Remark 5.18.

Corollay 5.17 implies that the algebras $A(\Delta)$ are in the derived-
equivalence class of the class of graded skew-gentle algebra.

######  Corollary 5.19.

Let $\Delta$ and $\Delta^{\prime}$ be tagged dissections on $\mathcal{S}/G$.
If the corresponding line fields $\eta$ and $\eta^{\prime}$ (see Proposition
5.14) are homotopic, then $A(\Delta)$ and $A(\Delta^{\prime})$ are derived-
equivalent.

###### Proof.

If $\mathcal{A}$ is a full $G$-invariant arc system on $\mathcal{S}$, then
$A(\Delta)$ and $A(\Delta^{\prime})$ are the endormorphism algebras of tilging
objects in $H^{0}\operatorname{Tw}\big{(}(\mathcal{A}*G)_{+}\big{)}$.
Therefore they are derived-equivalent. ∎

##  6\. The $\mathbf{D}_{n}$-case: a complete classification

In this section, we apply the results of Section 5 to the case where the
$G$-graded marked surface $(\mathcal{S},\mathbb{M},\eta,g)$ is a disc with
$2(n-1)$ marked segment, and the automorphism $g$ is the central symmetry.
Denote by $\mathbb{S}$ a $G$-invariant full graded system on
$(\mathcal{S},\mathbb{M},\eta,g)$, and by $\mathcal{A}_{\mathbb{S}}$ the
corresponding Fukaya category. Then we have the following.

######  Proposition 6.1.

Let $(\mathcal{S},\mathbb{M},\eta,g)$ and $\mathbb{S}$ as above. Then we have
a one to one correspondence between graded tagged dissections on
$\mathcal{S}/G$ satisfying conditions of Theorem 5.10 and basic tilting
objects of $H^{0}(\operatorname{Tw}\mathcal{A}*G)_{+}$.

###### Proof.

Any graded tagged dissection satisfying conditions of Theorem 5.10 gives rise
to a tilting object. Now let $T$ be a tilting object in
$H^{0}(\operatorname{Tw}\mathcal{A}*G)_{+}$. Since the surface $\mathcal{S}$
is a disc, then the indecomposable objects of
$H^{0}(\operatorname{Tw}\mathcal{A})$ are in bijection with the graded simple
arcs on $(\mathcal{S},\mathbb{M},\eta)$. Therefore the indecomposable objects
of $H^{0}(\operatorname{Tw}\mathcal{A})$ are in bijection with graded tagged
arcs on $\mathcal{S}/G$. Therefore the tilting object $T$ corresponds to a
collection $(C_{1},\ldots,C_{\ell})$ of pairwise non interesecting (except in
the orbifold point) simple graded tagged arcs. We conclude by Theorem 5.10.

∎

######  Corollary 6.2.

Let $(\mathcal{S},\mathbb{M},g)$ be a disc with $2(n-1)$ marked points and $g$
be the central symmetry. Then for any tagged dissection $\Delta$ on
$\mathcal{S}/G$ the algebra $A(\Delta)$ is an algebra derived equivalent to
the path algebra of a quiver $\bar{Q}$ of type $\mathbf{D}_{n}$. Moreover each
algebra derived equivalent to $k\bar{Q}$ is Morita equivalent to an algebra of
the form $A(\Delta)$.

###### Proof.

By Definition 5.11, the algebra $A(\Delta)$ is the endomorphism algebra of a
tilting object $T$ in $H^{0}(\mathcal{A}*G)_{+}$ where $\mathcal{A}$ is a
$G$-invariant full system of graded arc defined on the graded surface
$(\mathcal{S},\mathbb{M},\eta,g)$ where $\eta$ is a well-chosen line field on
$\mathcal{S}$.

Hence we have a triangle equivalence
$H^{0}(\operatorname{Tw}\mathcal{A}*G)_{+}\simeq\operatorname{per}A(\Delta)$.

Now let $\mathbf{D}$ be the following $G$-invariant dissection on
$\mathcal{S}$.

$X_{1}$$X_{2}$$X_{3}$$X_{4}$$gX_{2}$$gX_{3}$$gX_{4}$$\times$ Figure 20.

Since the surface is a disc, there is only one line field up to homotopy, so
it is possible to choose the grading on each arc in such a way that the graded
algebra corresponding to the dissection $\mathbb{D}$ is concentrated in degree
$0$. The endomorphism algebra of the object $\mathbb{D}$ in
$H^{0}\operatorname{Tw}\mathcal{A}$ is isomorphic to the path algebra of the
quiver $Q$ of type $\mathbb{A}_{2n+1}$ with symmetric orientation.

$X_{1}$$X_{2}$$X_{3}$$X_{4}$$gX_{2}$$gX_{3}$$gX_{4}$

The corresponding formal generator $\\{\mathbb{D}\\}$ has its endomorphism
algebra concentrated in degree $0$ and is Morita equivalent to the path
algebra of a quiver $\bar{Q}$ of type $\mathbb{D}_{n}$.

$X_{1}^{+}$$X_{1}^{-}$$\\{X_{2}\\}$$\\{X_{3}\\}$$\\{X_{4}\\}$

Hence we have a triangle equivalence
$H^{0}(\operatorname{Tw}\mathcal{A}*G)_{+}\simeq\operatorname{per}k\bar{Q}$.

Conversely, if $A$ is an algebra derived equivalent to $k\bar{Q}$, then by
[Ric89] it is the endomorphism algebra of a tilting object $T$ in
$H^{0}(\operatorname{Tw}\mathcal{A}*G)_{+}$. We conclude by Proposition 6.1.

∎

######  Example 6.3.

Here we give some examples of algebras which are derived equivalent to the
path algebra of a quiver of type $\mathbf{D}_{5}$. In this case, $\mathcal{S}$
is a disc with $8$ marked points, and $S/G$ is a disc with $4$ marked points
and one orbifold point.

Let $\Delta$ be a tagged dissection of $\mathcal{S}/G$. If $\Delta$ contains
two isotopic tagged arcs with opposite signs, then it does not contain any
other tagged arc with endpoint in the orbifold point. In that case, the
endomorphism algebra $A(\Delta)$ is skew-gentle.

We focus here on the case where $\Delta$ does not contain two isotopic arcs
with opposite signs, since the skew-gentle algebras have been already heavily
studied. In that case, since there is only one orbifold point, there is
exactly one piece of type (3) in the dissection, and no choice for the
bijection between pieces of type (3) and orbifold points.

For example, consider the following tagged dissection of $\mathcal{S}/G$.

$1$$2$$3$$4$$5$$\times$$+$$+$
Figure 21.

It corresponds to the following collection of graded arcs in the covering of
the orbifold (the indices between the objects $X_{1}$ and $X_{2}$ are
indicated in red):

$X_{1}$$X_{2}$$X_{3}$$X_{4}$$X_{5}$$gX_{3}$$gX_{4}$$gX_{5}$$\times$$0$$1$
Figure 22.

One can compute the endomorphism algebra of the object $X_{1}^{+}\oplus
X_{2}^{+}\oplus\\{X_{3}\\}\oplus\\{X_{4}\\}\oplus\\{X_{5}\\}$ in
$H^{0}(\operatorname{Tw}\mathcal{A}*G)_{+}$ and one obtains the following
quiver with relation (the dotted line indicates that the composition from
$X_{1}^{+}$ to $X_{4}$ vanishes).

$X_{1}^{+}$$X_{3}$$X_{2}^{+}$$X_{4}$$X_{5}$

Note that by Proposition 4.6, there is a non zero morphism from $X_{1}^{+}$ to
$X_{2}^{+}$. On can easily check that this morphism factors through the object
$\\{X_{3}\\}$.

In a similar way, the algebra $A(\Delta)$ corresponding to the following
tagged dissection

$1$$2$$3$$4$$5$$\times$$+$$+$$+$$+$ Figure 23.

gives rise to the following algebra (where the dotted line indicates a
commutativity relation).

$X_{1}^{+}$$X_{2}^{+}$$\\{X_{3}\\}$$X_{5}^{+}$$X_{4}^{+}$

This is (up to cyclic permutation, and change of tagging) the only tagged
dissection containig four tagged arcs. In a similar way we can list all the
algebras that are derived equivalent to the path algebra of a quiver of type
$\mathbf{D}_{5}$ and which are not skew-gentle. There are 25.

There are $6$ corresponding to a dissection containing exactly $3$ tagged
arcs.

$\times$$+$$+$$+$

$+$.$+$$+$.$+$.$+$$+$.$+$.$+$$+$.$+$.$+$$+$.$+$.$+$$+$.$+$.$+$$+$.

Figure 24.

There are exactly $4$ tagged dissections the following three arcs, and no
other tagged arc.

$\circ$$\times$$+$$+$ Figure 25.

The corresponding $4$ algebras are the following

.$+$$\circ$$+$..$+$$\circ$$+$.$+$$\circ$$+$..$+$$\circ$$+$..

There are $7$ dissections containing the following three arcs and no other
tagged arcs.

$\circ$$\times$ Figure 26.

The $7$ corresponding algebras are as follows :

$+$$\circ$..$+$$+$$\circ$..$+$$+$$\circ$..$+$$+$$\circ$..$+$$+$$\circ$..$+$$+$$\circ$..$+$$+$$\circ$..$+$

Finally there are 7 dissections containing the following three arcs and no
other tagged arcs.

$\circ$$\times$ Figure 27.

The corresponding algebras are as follows

$+$$\circ$$+$..$+$$\circ$$+$..$+$$\circ$$+$..$+$$\circ$$+$..$+$$\circ$$+$..$+$$\circ$$+$...$+$$\circ$$+$.

## Acknowledgements

Much of this work was completed while we were visiting the Centre for Advanced
Study in Oslo during the project _Representation Theory: Combinatorial Aspects
and Applications_. We are grateful to the organizers of the project, Aslak
Buan and Steffen Oppermann, for inviting us. We acknowledge the support of the
French ANR grant CHARMS (ANR-19-CE40-0017-02) and of the Institut
Universitaire de France (IUF). Some of the diagrams where produced using
q.uiver.app.

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# The Case for RISP:
A Reduced Instruction Spiking Processor

James S. Plank, ChaoHui Zheng, Bryson Gullett, Nicholas Skuda,
Charles Rizzo, Catherine D. Schuman, Garrett S. Rose
EECS Department, University of Tennessee, Knoxville, TN, USA

###### Abstract

In this paper, we introduce RISP, a reduced instruction spiking processor.
While most spiking neuroprocessors are based on the brain, or notions from the
brain, we present the case for a spiking processor that simplifies rather than
complicates. As such, it features discrete integration cycles, configurable
leak, and little else. We present the computing model of RISP and highlight
the benefits of its simplicity. We demonstrate how it aids in developing hand
built neural networks for simple computational tasks, detail how it may be
employed to simplify neural networks built with more complicated machine
learning techniques, and demonstrate how it performs similarly to other
spiking neurprocessors.

##  I Introduction

Since the 1960s, researchers have turned to the brain for inspiration on how
to build novel computing devices. This has led to several waves of research on
spiking neuroprocessors [28]. In our previous research, we have explored all
facets of neuroprocessors, from fabrication and nanotechnology [2, 4], to
circuits and systems [11, 29] to applications and simulations [9, 15], to data
encoding and decoding [31], to machine learning [25, 30]. Informed especially
by our development of several neuroprocessors [8, 19, 20, 6], we have been
motivated to research what happens when we simplify, rather than complicate.
The result is RISP, a Reduced Instruction Spiking Processor.

RISP features integrate-and fire-neurons with discrete integration cycles.
Synaptic delays are also discrete, and synaptic weights and neuron thresholds
may be configured as either discrete or analog. Besides the aforementioned
parameters, the only other configurable feature of RISP is that neurons may
either retain their activation potentials indefinitely, or they may leak away
completely at each integration cycle. There are no other complicating features
of RISP that are in many other neuroprocessors, such as plasticity, refractory
periods, custom integration techniques, complicated leak models and learning
rules. In this paper we define RISP and detail some of its attractive
properties. Our goal is to demonstrate that a simple neuroprocessor like RISP
is viable alternative to more complicated neuroprocessors.

##  II Related Work

There are several neuroprocessor models, typically implemented in simulation,
that are commonly used in the literature to explore and evaluate spiking
neural networks and neuromorphic-style computation. These tend to be focused
on one particular domain at a time. For example, there are a set of simulators
that target computational neuroscience simulations, including NEST [12, 10],
NEURON [5], Brian [13], Brian2 [35], Nengo [3], GeNN [37] and Brian2GeNN [36].
Though these simulators tend to have significant flexibility in the types of
computation that can be achieved, with large variation in the types of neuron
and synapse models that can be implemented, that flexibility often comes at
the price of slow simulation times for even simple networks, making it
difficult to use those simulators to evaluate new types of algorithms [16].
There are also simulators that primarily target machine learning-style
computation with spiking neural networks, including BindsNET [14] and NengoDL
[27]. Because these simulation environments focus on the machine learning use
case of spiking neural networks, their functionality is often restricted to
accommodate that behavior. Finally, there are simulation frameworks that
primarily target particular neuromorphic hardware implementations, including
NEMO [22], which targets IBM’s TrueNorth [17], and NengoLoihi, which targets
Intel’s Loihi [7]. These simulators are focused on capturing the functionality
of the underlying hardware and can also restrict the types of evaluations that
can occur on those systems or perform prohibitively slowly [16].

##  III RISP Specification

In RISP, each neuron has two configurable parameters: a threshold, and whether
or not the neuron leaks. As with most spiking neuroprocessors, each neuron
stores an action potential, whose value may be increased or decreased by
spikes on incoming synapses. Processing works in discrete timesteps, where
incoming spikes are integrated over the timestep, and it is the neuron’s
action potential at the end of the timestep that determines whether or not the
neuron fires. When a neuron is configured with leak enabled, then if a neuron
does not fire at the end of the integration cycle, its action potential is
reset to zero. Without leak, neurons retain their action potentials across
time steps.

Because of the integration cycle of neurons, synapses only have unit delays.
Like neurons, synapses may be configured to either have discrete weights or
analog weights. There is no restriction on connectivity, or the number of
synapses that may come into or go out of a neuron.

Certain neurons may be designated as input neurons that may receive spikes
from the outside world. Similarly, neurons may be specified to be output
neurons, whose spikes may be monitored by the outside world.

To be precise, a RISP network is defined by 8 sets:

  1. 1.

$N$ Neurons: $V=\\{v_{0},v_{1},...v_{N-1}\\}$

  2. 2.

$N$ Thresholds: $T=\\{t_{0},t_{1},...t_{N-1}\\}|t_{i}\in\cal{R}\;{\rm
or}\;\cal{Z}$

  3. 3.

$N$ Leaks: $L=\\{l_{0},l_{1},...l_{N-1}\\}|t_{i}\in\\{T,F\\}$

  4. 4.

$M$ Synapses: $E=\\{e_{0},e_{1},...e_{M-1}\\}|e_{i}=(v_{j}\rightarrow v_{k})$

  5. 5.

$M$ Weights: $W=\\{w_{0},w_{1},...w_{M-1}\\}|w_{i}\in\cal{R}\;{\rm
or}\;\cal{Z}$

  6. 6.

$M$ Delays: $D=\\{d_{0},d_{1},...d_{M-1}\\}|d_{i}\in\cal{Z}$

  7. 7.

Input neurons: $I\subseteq V$.

  8. 8.

Ouptut neurons: $O\subseteq V$.

Figure 1 shows a very simple RISP network that computes the binary AND of two
inputs $A$ and $B$ [26]. When spikes are applied to both input neurons at
timestep 0, then the output neuron $X$’s action potential is increased to a
value of two at timestep 1. As such, it fires. If a spike is applied to one of
$A$ or $B$, but not to the other, then $X$’s action potential only reaches a
value of one, and therefore it does not fire. Moreover, since it is configured
with leak, its action potential is reset to zero, and the network is ready to
compute a new AND at the next timestep.

Figure 1:  An example RISP network that computes the binary AND of inputs $A$
and $B$. The output neuron must be configured with leak, so that when one
input spikes and the other doesn’t, the action potential is reset to zero for
the next problem.

##  IV Attractiveness of RISP

RISP’s simplicity is attractive in many pragmatic respects. We detail them in
the sections below.

###  IV-A Hand Constructing Networks

Because of its simplicity, it is a straightforward matter to construct RISP
networks by hand, rather than by machine learning or other optimization
techniques. This can be useful, either for solving simple problems, or for
composing larger neural networks.

Figure 2:  RISP network that computes which input, $I_{X}$ or $I_{Y}$, fires
more over an interval of $t$ timesteps. The synapses are colored to facilitate
reading the elements of the sets $E$, $W$ and $D$.

We present an example in Figure 2. This network calculates which input neuron,
$I_{X}$ or $I_{Y}$, spikes more over an interval of $t$ timesteps. If $I_{X}$
spikes more, then $O_{X}$ spikes exactly once, at timestep $t+1$. If $I_{Y}$
spikes more, then $O_{Y}$ spikes exactly once, at timestep $t+1$. If they
spike equally, then neither spikes, although it is a straightforward matter to
add an extra synapse to break ties in favor of $I_{X}$ or $I_{Y}$. There is a
bias neuron that is required to spike once at timestep 0. The network clears
itself, and may be reused for more inputs at timestep $t+1$.

This type of network is useful for composing spiking neural networks, perhaps
created by other methodologies (e.g., backpropagation [34, 33] or genetic
algorithms [30]). For example, some neural networks, like the one we show
later in section IV-B, require their outputs to be intepreted by voting among
two output neurons. The network in Figure 2 performs this interpretation,
which can in turn be used as inputs to other spiking neural networks, or even
as control signals for a physical device.

There have been other research projects that construct spiking neural networks
that may be implemented by RISP, either by targeting RISP explicitly, or by
targeting a neural network that conforms to the RISP specifications. For
example, in our previous work, we have constructed RISP networks that perform
binary operations on a variety of input/output encodings and decodings, and
RISP networks that convert between encodings/decodings [26]. The Whetstone
training methodology uses backpropagation and a “sharpening” activation
function to train deep spiking neural networks that may be implemented on RISP
[33], and to reduce the size of the resulting networks, we developed a
convolutional “kernel” on RISP [25]. Monaco and Vidiola’s integer
factorization sieve [21] may be implemented on RISP.

###  IV-B Network Simplification

RISP’s simplicity makes it straightforward to analyze network behavior to
simplify and improve the networks. As an example, consider the spiking neural
network in Figure 3. This network was trained by the genetic algorithm EONS
[30] to solve the cart-pole problem. In this problem, there is a cart enclosed
on a fixed-size road, and on the cart is a pole on a hinge that can fall left
or right. The goal of the problem is to push the cart left and right at fixed
power, and keep both the cart on the road and the pole balanced.

Figure 3:  RISP network to solve the cart-pole problem. There are four inputs,
$X$, $\theta$, $dX$, and $d\theta$, which report the cart’s horizontal
position, angle from vertical, horizontal speed and angle speed. Inputs are
split over two neurons, and spike one to four times with unit values. There
are two outputs, $O_{L}$ and $O_{R}$ for boosting the cart left and right, and
the decision is made by voting. This network was created by the genetic
algorithm, EONS.

To solve this with RISP and EONS, we selected an input encoding technique
where each input value corresponds to two input neurons, with negative values
spiked into one neuron (e.g. $I_{X-}$) and positive values spiked into another
(e.g. $I_{X+}$). The magnitide of the value is converted into a number of
spikes, from one to four, and the spikes all have unit weights.

There are two output neurons, $O_{L}$ and $O_{R}$, which correspond to pushing
the cart left and right respectively. The spiking neural network “runs” the
application by working in discrete simulated time intervals of 0.02 seconds.
At the beginning of each interval, the inputs are converted to spikes and
applied to the appropriate input neurons. The neural network then runs for 50
timesteps, and the number of spikes on $O_{L}$ and $O_{R}$ are compared. If
$O_{R}$ spikes more, then the cart is pushed right. Otherwise, the cart is
pushed left. (Interestingly, a network like the one in Figure 2 can be used to
turn the votes into a single spike for pushing the cart).

With EONS, we train the spiking neural network using ten random starting
positions for the cart and pole, and run the genetic algorithm until a network
has been trained to keep the cart on the road and the pole balanced for five
simulated minutes in each of the ten starting positions. In a testing phase,
the network in Figure 3 keeps the cart on the road and the pole balanced for
an average of 4 minutes and 42 seconds on 100 randomly chosen starting
positions. As shown in the figure, the network has four hidden neurons and 18
synapses. The neuron thresholds and synapse weights are floating point numbers
whose values are rounded on the picture.

Because of the simplicity of RISP and of the input encoding, we can simplify
this network with the goals being:

  * •

Reduce the number of neurons and synapses.

  * •

Set neuron thresholds and synapse weights to 1 or -1.

Both goals allow our networks to be implemented more simply, and to consume
less power while executing. An example of this simplification is to consider
the neurons $2$, $B$ and $8$ in Figure 3. Since the synapse $2\rightarrow B$
has a weight of 0.65 and delay of 2, and neuron $B$ has a threshold of -0.66,
we may deduce that whenever neuron 2 fires, neuron $B$ will fire one timestep
later. Further, because the synapse $B\rightarrow 8$ is the only post-synapse
to neuron $B$, and its delay is two, we may simply delete neuron $B$, and
replace its pre and post synapses with a synapse $2\rightarrow 8$ whose delay
is 3.

In a separate analysis, we observe that all four pre-synapses to neuron 8 have
weights that exceed the neuron’s threshold of -0.21. Therefore, we may set the
weights of all of these synapses equal to one, and set the threshold of neuron
8 to one as well.

By applying observations such as these, we may simplify the network in Figure
3 into the one in Figure 4. This network has only one hidden neuron, a
reduction of 75 percent, and 15 synapses, a reduction of 17 percent. Moreover,
seven of the neuron thresholds (64 percent) and nine of the synapse weights
(60 percent) have been set to one.

Figure 4:  RISP network that is equivalent to the network in Figure 3, but has
been simplified. It has three fewer neurons and three fewer synapses.
Moreover, seven of the neuron thresholds and nine of the synapse weights have
been set to one. Therefore, this network is simpler to implement and should
require less power consumption. Its performance on the application is the
same.

One of our current research projects is to implement these and similar
observations to simplify RISP networks. We are also attempting stochastic and
enumeration techniques to simplify the networks using empirical observations.
These techniques are enabled by the fact that RISP is simple, and does not
have features such as plasticity, complex learning rules and refractory
periods, that complicate the analysis of the networks.

###  IV-C Genetic Training and Simulation Speed

Over the past seven years, the TENNLab research group has performed many
explorations of real-time control applications with spiking neuroprocessors
[23, 1, 18, 31]. One of the easiest of these is the cart-pole application
mentioned in section IV-B. In this section, we perform a small experiment
using a genetic optimization of spiking neural networks on the cart-pole
application, to see whether RISP trains similarly to other spiking processors.
We use the same input enocoding as in section IV-B. We perform six sets of
optimizations using the EONS genetic algorithm and the TENNLab neuromorphic
computing framework [30, 24, 32]. These are listed in Table I.

Label | Processor | Values | Leak | Features
---|---|---|---|---
RISP-A | RISP | Analog | No | None
RISP-D | RISP | Discrete | No | None
RISP-A-L | RISP | Analog | Yes | None
RISP-D-L | RISP | Discrete | Yes | None
Caspian | Caspian | Discrete | Yes | None
DANNA2 | DANNA2 | Discrete | Yes | Refractory
TABLE I:  Neuroprocessors tested in a genetic training experiment on the cart-
pole problem.

The first four processors listed are the four variants of RISP. The fourth is
Caspian [20], developed at Oak Ridge National Laboratory and implemented on
FPGA’s, with a very fast simulator. The fifth is DANNA2, developed by TENNLab
and implemented on FPGA’s, with a VLSI workflow [19].

We performed a simple genetic optimization composed of populations of 500
networks optimizing over 100 epochs, with 10 random starting positions of the
cart and pole. For each neuroprocessor in Table I, we performed 100
independent optimizations.

In Figure 5, we show both training and testing results for the
neuroprocessors.

|
---|---
Figure 5:  Results of training and testing the cart-pole application using the
EONS genetic algorithm.

Because cart-pole is such a simple application, all of the neuroprocessors
train to a perfect fitness in nearly every run. The left graph in Figure 5
shows a tukey plot of the number of epochs until perfect training fitness is
achieved (keeping the pole balanced and the cart in position for 5 simulated
minutes for each of the 10 training episodes). RISP’s training success is
similar to the others, with RISP-D requiring more epochs than the others.

To test, we run the final trained network on 100 episodes of cart-pole with
different starting positions than the training runs, and record their average
values. A “perfect” training run lasts five simulated minutes. The values for
the 100 episodes are averaged, and the tukey plots for the 100 runs are shown
on the right side of Figure 5. RISP’s testing performance was superior to the
others, on average, but we are hesitant to draw conclusions without performing
a larger study.

The speed of simulation impacts the speed of training. Caspian, and the three
RISP processors with the exception of RISP-D took between 40 and 83 minutes to
complete all 100 of their training runs. RISP-D was slower, at nearly 4 hours,
most likely because of the increased number of training epochs. DANNA2’s
simulator is much slower, requiring 27 hours to complete. For reference, our
NengoLoihi runs took a day to complete an average of 5 epochs, while
dedicating 10 processors to each training run (the others used a single
processor).

We conclude from this study that RISP is viable alternative to the other
neuroprocessors for training via genetic algorithm. Its simple features do not
hinder the success of training, and its simulation speed is excellent.

##  V Conclusion

In this paper, we introduce RISP, a Reduced Instruction Spiking Processor,
whose goal is to explore the effectiveness of simplifying spiking
neuroprocessors. RISP features integrate-and-fire neurons, unit integration
cycles and therefore unit synaptic delays, and either analog or discrete
weights and thresholds. The only other feature is that neurons may leak their
action potentials at each cycle, or they may retain them.

In this paper, we demonstrate how RISP’s simplicity enables the hand-
construction of networks, and how it allows us to reason about pre-trained
networks to simplify them. Finally, we perform a very simple experiment with
genetic training on a control application to show that RISP trains on par with
other neuroprocessors that contain more features.

Our near-future plan with RISP is to implement a network compiler that renders
RISP networks on inexpensive microcontrollers like the RP2040 ($5) and FPGA’s
like the UPduino ($25). We anticipate that RISP’s simplicity should lead to
very clean implementations, featuring high density of neurons and synapses on
these inexpensive devices. The devices may then be used in physical
experiments such as the ones we performed with the robots NeoN and GRANT [18,
1].

The bottom line is that simplicity in a spiking neuroprocessor has multiple
advantages, and is worth exploring as an alternative to neuroprocessors with
more complicated, albeit potentially bio-realistic features. Our goal with
RISP is for it to be the “simple” option as a spiking neuroprocessor in
applications and scenarios that desire to explore spiking neuroprocessors as
their solutions.

##  VI Acknowledgments

This research was supported in part by an Air Force Research Laboraty
Information Directorate grant (FA8750-19-1-0025).

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