# Geometric Bordisms of the Accola-Maclachlan, Kulkarni and Wiman Type II
Surfaces

L. Ferrari
Institute of Mathematics
Polish Academy of Sciences
Sniadeckich 8
00-656 Warsaw, Poland<EMAIL_ADDRESS>
###### Abstract.

In this paper, we prove that the Accola-Maclachlan surface of genus $g$ bounds
geometrically an orientable compact hyperbolic $3$-manifold for every genus.
For infinitely many genera, these surfaces are also explicit examples of non-
arithmetic surfaces that embed geodesically into a closed non-arithmetic
manifold. We also provide explicit geodesic embeddings to the Wiman type II
and Kulkarni surfaces of every genus, and prove that these surfaces bound
geometrically a compact, orientable manifold for $g\equiv 1\,(\text{mod}\,2)$
or $g\equiv 3\,(\text{mod}\,8)$, respectively.

L.F. was partially supported by the Polish Center of Science Grant OPUS
2019/35/B/ST1/01120.

##  1\. Introduction

Bordism properties of closed manifolds have been a classical and important
topic in topology. Whilst classical results concern topological or smooth
boundaries, in [14] the question of bounding geometrically was introduced:
namely whether a connected closed orientable hyperbolic $n$–manifold $M$ could
arise as the totally geodesic boundary of a compact hyperbolic
$(n+1)$–manifold $W$. This problem is particularly complicated in dimension
$2$, given that there are uncountably many non-isometric surfaces for every
given genus. It follows from the Mostow’s rigidity theorem, moreover, that the
subset of surfaces of a given genus $g$ that do bound is countable. However,
so far only a few explicit examples of surfaces that do bound geometrically
have been constructed.

In [9] another problem was considered: whether a given connected closed
orientable hyperbolic $n$–manifold $M$ could embed geodesically, that is arise
as an embedded totally geodesic codimension $1$ submanifold of a hyperbolic
$(n+1)$–manifold $W$. The authors, in fact, do show that every arithmetic,
compact, hyperbolic manifold of even dimension embeds geodesically; and while
these two problems are connected (see Section 2), the methods in [9] do not
apply to non-arithmetic hyperbolic manifolds or provides explicit examples of
the embeddings themselves.111In [10], these methods were also extended to some
non-arithmetic manifolds, namely those that can be obtained by gluing
arithmetic pieces.

In this paper, we show an explicit construction of geometric bounding for a
famous family of Riemann surfaces, the Accola-Maclachlan surfaces $X_{g}$ of
genus $g\geq 2$. They are the surfaces described by the algebraic curve
equation

(1) $y^{2}=x^{2g+2}-1.$

These surfaces were shown independently by Accola [1] and Maclachlan [15] to
have full conformal automorphism group

(2) $\mathrm{Aut}(X_{g})=AM_{g}=\langle a,b\mid
a^{2g+2}=b^{4}=(ab)^{2}=[a,b^{2}]=1\rangle\cong(\mathbb{Z}_{2g+2}\times\mathbb{Z}_{2})\rtimes\mathbb{Z}_{2},$

a group of order $8g+8$. Later, Kulkarni [6] also showed that, for
$g\not\equiv 3(\text{mod}\,4)$, this is the only surface with a full
automorphism group of such order, except for a finite number of genera;222From
[7, Proposition 4.2] and [6, Sections 4,5], one can calculate that the
exceptional genera are precisely
$g=2,3,4,5,9,11,13,14,17,27,29,39,65,89,125,189$. but the only such group $G$
with this order and action signature $(0;2,4,2g+2)$, meaning that the quotient
$S_{g}/_{G}$ is an orbifold of such signature.

For $g\equiv 3(\text{mod}\,4)$ Kulkarni also showed there’s a second surface,
now called Kulkarni surface $Y_{g}$ of genus $g$, with a conformal
automorphism group $K_{g}$ of order $8g+8$, with group presentation

(3) $\mathrm{Aut}(Y_{g})=K_{g}=\langle a,b\mid
a^{2g+2}=b^{4}=(ab)^{2}=1,b^{2}ab^{2}=a^{g+2}\rangle.$

He also showed that $AM_{g}$ and $K_{g}$ are always distinct and the only two
such conformal automorphism groups with order $8g+8$ for any genus excluding
the exceptional genera, but still the only such groups with signature
$(0;2,4,2g+2)$ in any genus $g\equiv 3(\text{mod}\,4)$.

From the classification of extendability of Fuchsian signatures by Singerman
[18, Theorems 1, 2], it follows that these two group are also maximal for any
$g\neq 3$; while in $g=3$ it follows from [3, Case T11] that in $AM_{3}$ is
also maximal while $K_{3}$ is extendable to a group of order $96$, which is
unique among the non-hyperelliptic genus $3$ curves [12]. Since the Kulkarni
surfaces are non-hyperelliptic [20, Equation 2], we have that the Accola-
Maclachlan and the Kulkarni surfaces are entirely determined their conformal
automorphism groups containing $AM_{g}$ or $K_{g}$, and by these subgroups
acting with signature $(0;2,4,2g+2)$.

Using this fact, we will prove the following for the Accola-Maclachlan
surfaces:

######  Theorem A.

The Accola-Maclachlan surface $X_{g}$ of genus $g$ bounds geometrically a
compact, orientable, hyperbolic $3$-manifold $M_{g}$ for every $g\geq 2$.

We note that, since the surfaces $X_{g}$ are commensurable with the triangle
groups $(2,4,2g+2)$, they are arithmetic only for $g=2,3,4,5,8$ (with classes
II, III, VI, V, XII, respectively, by Takeuchi’s classification [19]).
Moreover, from the proof of Corollary 4.3, the manifolds $M_{g}$ are
commensurable with $R(2g+2)$, the right-angled Löbell polytope with two
$(2g+2)$-gons. More precisely, $M_{g}$ are obtained by the face-pairing of $8$
copies of $R(2g+2)$, and thus have volume333The actual volume can be
calculated numerically for every $g$ by [21, Theorem 3.19]. asymptotically
equivalent to $20v_{3}(g+1)$ as $g\rightarrow\infty$ [21, Corollary 3.18],
where $v_{3}\sim 1.0149$ is the volume of the ideal regular hyperbolic
tetrahedron. Also, by [21, Lemma 3.8], the reflection groups
$\Gamma\big{(}R(2g+2)\big{)}$ are only arithmetic in the cases $g=2,3$.444[9]
also provides lattice embeddings in this case. We then also have the
following:

######  Theorem B.

For $g>8$, the surfaces $X_{g}$ are explicit examples of non-arithmetic
surfaces that embed geodesically into compact non-arithmetic orientable
hyperbolic $3$-manifolds and bound geometrically compact orientable hyperbolic
$3$-manifolds.

We will also prove the following for the Kulkarni surfaces:

######  Theorem C.

The Kulkarni surface $Y_{g}$ of genus $g$ embeds geodesically for every
$g\equiv 3(\text{mod}\,4)$, and bounds geometrically an orientable manifold
for every odd $g\equiv 3(\text{mod}\,8)$.

We note that, from the proof of Corollary 6.3, the geodesic embedding and the
geometric bounding are both into compact, orientable manifolds obtained from
the face-pairings of $8$ copies of the Löbell polytopes $R(2g+2)$, from which
the same comments on volume and arithmeticity follow as above.

We will finally show an explicit geodesic embedding, and geometric bounding
for odd genus, for another famous family of surfaces, the Wiman type II
surfaces $Z_{g}$ of genus $g\geq 2$. These surfaces were described by Wiman
[23] by the algebraic curve equation

(4) $y^{2}=x^{2g+1}-x,$

and they are entirely classified as the unique surface up to isometry
admitting a conformal automorphism of order $4g$ [16, Theorem 2] for any
$g\neq 3$. In the case $g=3$, there is a second surface with an automorphism
of order $4g=12$, but the Wiman type II is the only surface with a cyclic
conformal automorphism subgroup of order $4g$ acting with signature
$(0;2,4g,4g)$ [16, p. 439].

Using this characterization, we will prove the following:

######  Theorem D.

The Wiman type II surface $Z_{g}$ of genus $g$ embeds geodesically for every
$g\geq 2$, and bounds geometrically an orientable manifold $N_{g}$ for every
odd $g\geq 2$.

We also note that, since the surfaces $Z_{g}$ are commensurable with the
triangle groups $(2,4,4g)$, they are arithmetic only for $g=2,3$ (with classes
III, V, respectively [19]). Moreover, from the proof of Corollary 5.3, the
manifolds $N_{g}$ are commensurable with $R(4g)$, obtained by the face-pairing
of $4$ copies of the polyhedron, and have volume asymptotically equivalent to
$20v_{3}g$. In addition, the reflection groups $\Gamma\big{(}R(4g)\big{)}$ are
only arithmetic in the case $g=2$.

The constructions in Theorems 4.2, 5.1, 6.1 will also give the following
characterization of the above surfaces:

######  Theorem E.

The Accola-Maclachlan, Wiman type II and Kulkarni surfaces of genus $g$ can be
obtained by the side-pairing of $4$ right-angled $(2g+2)$-gons, $2$ right-
angled $4g$-gons and $4$ right-angled $(2g+2)$-gons, respectively. Labelling
the edges of the polygons in clockwise order with numbers $1,2,...$ and the
copies of polygons by vectors in $\mathbb{Z}_{2}^{2}$ ($\mathbb{Z}_{2}$,
$\mathbb{Z}_{2}^{2}$), we have the following description of the orientation-
preserving edge pairings:

  1. AM

$(i,v)$ is paired with $(i,v+e_{1})$ if $i$ is odd; and with $(i,v+e_{2})$ if
$i$ is even.

  2. W2

$(i,0)$ is paired with $(i,1)$ if $i$ is odd; and with $(2g+i,1)$ if $i$ is
even.

  3. K

$(i,v)$ is paired with $(i,v+e_{1})$ if $i\equiv 1(\,\text{mod}\,4)$; with
$(i,v+e_{2})$ if $i\equiv 2(\,\text{mod}\,4)$; with $(g+1+i,v+e_{1})$ if
$i\equiv 3(\,\text{mod}\,4)$; and with $(g+1+i,v+e_{2})$ if $i\equiv
0(\,\text{mod}\,4)$,

where the edge labels $N=\\{1,2,...\\}$ are seen in
$\mathbb{Z}\,(\text{mod}\,|N|)$.

We note that this characterization is not unique. [22, Theorem 6.3], for
instance, characterizes these families and others by face-pairings of a single
suitable regular polygon with suitable inner angles; and the Wiman type II
curve in genus $g=2$, also known as the Bolza curve, has an additional
characterization as side-pairings of $6$ regular octagons with inner angles
$\frac{2\pi}{3}$. The above characterization, however, has the advantage of
using only right-angled polygons.

We’d like to stress, however, that this paper does not answer the question
posed by Zimmermann [24] and [9] regarding the geometric bounding of the Klein
quartic, the surface which realizes the Hurwitz bound
$|\mathrm{Aut}(S_{g})|\leq 84(g-1)$ in the lowest possible genus $g=3$. In
fact, having algebraic curve equation $y^{3}+yx^{3}+x=0$ [13], the Klein
Quartic does not belong to the above families. It also does not admit fixed-
point free, orientation-reversing involutions [17].

###  1.1. Structure of the paper

Section 2 defines geodesic embeddings and how to promote them to geometric
boundings. Section 3 recalls some useful basic results from the theory of
colourings of right-angled polytopes. Section 4 describes the explicit
construction of geometric boundings for the Accola-Maclachlan surfaces $X_{g}$
of every genus. Section 5 described the construction for the Wiman surfaces
$Z_{g}$. Finally, Section 6 describes the construction for the the Kulkarni
surfaces $Y_{g}$. The sections were organized by the complexity of the
constructions and each uses results from the previous ones.

Acknowledgements: The author is grateful to Alan Reid for the insightful talks
and comments on the paper; to Benson Farb for the suggestions and references
that allowed this work; to Tadeusz Januszkiewicz for comments and suggestions
on an earlier version of the paper; and to Leone Slavich and Stefano Riolo for
corrections and fruitful discussions.

##  2\. Geodesic embeddings and geometric boundaries

Throughout this article, we will consider only manifolds that are closed,
orientable, hyperbolic and complete, unless explicitly stated.

We say that a manifold $N^{n}$ embeds geodesically if there is an embedding
$i:N\to M^{n+1}$ such that $i(N)$ is totally geodesic in $M$, that is, if the
geodesics in $(M,g)$ are still geodesic in $(i(N),g)$.

We say that a manifold $N^{n}$ bounds geometrically if there is a compact
manifold $M^{n+1}$ with totally geodesic boundary such that $\partial M$ is
isometric to $N$.

Clearly, bounding geometrically is equivalent to embedding geodesically as a
separating totally geodesic submanifold: from the embedding you can cut along
to produce two compact components with a connected totally geodesic boundary;
and from the geometric bounding you can double the manifold along the boundary
to produce a geodesic embedding.

A non-separating embedding, however, is not enough to produce a geometric
boundary: if $N$ embeds geodesically in $M$ in a non-separating way, then by
cutting $M$ along $N$ we obtain a manifold $M^{\prime}$ with boundary
components $N^{+}\sqcup N^{-}$, where each component is given opposite
orientation.

In some cases, nevertheless, it is still possible to promote a geodesic
embedding to a geometric bounding. We recall the following [5, Lemma 3.1]:

######  Lemma 2.1.

Let $N$ be an orientable hyperbolic $n$-manifold that has a fixed point free
involution $\varphi\in\mathrm{Isom}(N)$. If $N$ embeds geodesically then it
also bounds geometrically.

###### Proof.

Let $N$ embed into an orientable manifold $M$ as a totally geodesic
submanifold of codimension $1$. Denote by $M^{\prime}$ the manifold obtained
by cutting $M$ along $N$ and taking a connected component. Then either
$\partial M^{\prime}=N$ and we are done, or $\partial M^{\prime}=N\sqcup N$,
and we can quotient out one copy of $M$ in $\partial N$ by self-identifying it
via $\varphi$. Given that $\varphi$ is a fixed point free involution, the
resulting metric space $M^{\prime}_{\varphi}$ will be a hyperbolic manifold
with a single boundary component isometric to $N$. Moreover,
$M^{\prime}_{\varphi}$ is orientable or not depending on whether $\varphi$ is
orientation-reversing or not. $\sqcup\\!\\!\\!\\!\sqcap$

In this article, we will also need the following useful fact concerning the
immersion of isometry groups of quotient manifolds, specially in the case that
the action group $H$ is generated by the fixed-point free involution:

######  Lemma 2.2.

Let $M$ be a hyperbolic manifold and $H<\mathrm{Isom}(M)$ such that $H$ acts
freely. Then $M/_{H}$ is also a hyperbolic manifold and

$N_{\mathrm{Isom}(M)}(H)/_{H}<\mathrm{Isom}\big{(}M/_{H}\big{)}.$

If $M$ is orientable, the statement is also true when replacing
$\mathrm{Isom}$ with $\mathrm{Isom}^{+}$, and in this case it also follows
that $M/_{H}$ is orientable.

###### Proof.

Since $M$ is hyperbolic, $\mathrm{Isom}(M)$ is finite, from which it follows
that $M\mapsto M/_{H}$ is a regular cover for any $H<\mathrm{Isom}(M)$ acting
freely.

Take one such $H$ and any $\varphi\in\mathrm{Isom}(M)$. If $\varphi
H=H\varphi$, we can naturally define a map $\tilde{\varphi}:M/_{H}\to M/_{H}$
by putting $\tilde{\varphi}([x])=[\varphi(x)]$, and this is an isometry of
$M/_{H}$. We thus have a map
$\Psi:N_{\mathrm{Isom}(M)}(H)\to\mathrm{Isom}\big{(}M/_{H}\big{)}$ which maps
$\varphi$ to $\tilde{\varphi}$, and whose kernel is precisely $H$. It follows
that there is an injection $\Psi/_{\text{ker}\,\Psi}$ from
$N_{\mathrm{Isom}(M)}(H)/_{H}$ into $\mathrm{Isom}\big{(}M/_{H}\big{)}$.
$\sqcup\\!\\!\\!\\!\sqcap$

##  3\. Colourings

We now introduce a very useful technique for the construction of explicit
examples.

###  3.1. Definitions

A compact hyperbolic polytope $\mathcal{P}\subset\mathbb{H}^{n}$ is called
right–angled if any two codimension $1$ faces (or facets, for short) are
either intersecting at a right angle or disjoint.

Let $\mathcal{P}\subset\mathbb{X}^{n}$ be a compact, right-angled polytope
with the set of facets $\mathcal{F}$. A colouring of $\mathcal{P}$ is a map
$\lambda:\mathcal{F}\rightarrow V$, where $V$ an
$\mathbb{Z}/2\mathbb{Z}$–vector space. The map $\lambda$ is called proper if,
for every vertex $v=F_{1}\cap\ldots\cap F_{n}$, the vectors
$\lambda(F_{1}),\ldots,\lambda(F_{n})$ are linearly independent.

The rank of $\lambda$ is the $\mathbb{Z}/2\mathbb{Z}$–dimension of
$\mathrm{im}\,\lambda$. We will always assume in this article that colourings
are surjective, in the sense that the image of the map $\lambda$ is a
generating set of vectors for $V$.

A colouring of a right-angled $n$–polytope $\mathcal{P}$ naturally defines a
homomorphism, which we still denote by $\lambda$ without much ambiguity, from
the associated right–angled reflection group $\Gamma(\mathcal{P})$ into $V$
with its natural group structure. Being a Coxeter polytope, $\mathcal{P}$ has
a natural orbifold structure as the quotient
$\mathbb{H}^{n}/_{\Gamma(\mathcal{P)}}$.

######  Proposition 3.1 ([4], Proposition 1.7).

If the colouring $\lambda$ is proper, then $\ker\lambda<\Gamma(\mathcal{P})$
is torsion-free, and $\mathcal{M}_{\lambda}=\mathbb{H}^{n}/_{\ker\lambda}$ is
a closed hyperbolic manifold. Moreover, $\mathcal{M}_{\lambda}\to\mathcal{P}$
is an orbifold covering of degree $|V|=2^{\text{rk}\,\lambda}$.

######  Remark 3.2.

$\mathcal{M}_{\lambda}$ is homeomorphic to the topological manifold
$(\mathcal{P}\times\mathbb{Z}_{2}^{m})/_{\sim}$, where

(5) $(p,g)\sim(q,h)\Leftrightarrow p=q\text{ and }g-h\in G_{f}$

$f=F_{i_{1}}\cap...\cap F_{i_{l}}$ being the unique face that contains $p$ as
an interior point and $G_{f}$ is the subgroup of $\mathbb{Z}_{2}^{k}$
generated by $\lambda(F_{i_{1}})$,…, $\lambda(F_{i_{l}})$ (if
$p\in\mathrm{int}(\mathcal{P})$, we put $G_{p}=\\{0\\})$.

More practically, this means that
$(\mathcal{P}\times\mathbb{Z}_{2}^{k})/_{\sim}$ can be obtained from
$\mathcal{P}\times\mathbb{Z}_{2}^{k}$ by identifying distinct copies
$F\times\\{g\\}$ and $F\times\\{h\\}$ through the identity on
$F\in\mathcal{F}$ whenever $g-h=\lambda(F)$. It is clear then that
$\mathcal{M}_{\lambda}$ can be tessellated by $2^{k}$ copies of $\mathcal{P}$.

We say that two colourings $\lambda_{1},\lambda_{2}:\mathcal{F}\to V$ are
equivalent if there is a symmetry $g\in\,\mathrm{Sym}(\mathcal{P})$, which
acts on $\mathcal{F}$ as a permutation; and an isomorphism $\varphi\in GL(V)$
such that $\lambda_{2}=\varphi\circ\lambda_{1}\circ g$. Clearly, equivalent
colourings produce isometric manifolds.

[8, Lemma 2.4] also gives us the following criterion for orientability:

######  Proposition 3.3.

$\mathcal{M}_{\lambda}$ is orientable if and only if $\lambda$ is equivalent
to a colouring that assigns to each facet a colour with an odd number of
entries 1.

######  Example 3.4 ($2n$-gons).

The lowest-rank colouring $\lambda$ on a right-angled $2n$-gon $P$ is given by
associating alternately the canonical vectors $e_{1}$, $e_{2}$ to its edges.
Since the polygon has Euler characteristic $\chi(P)=1-\frac{n}{2}$, we have
from Proposition 3.1 that $\chi(\mathcal{M}_{\lambda})=4-2n$, that is,
$\mathcal{M}_{\lambda}$ is an orientable surface of genus $g=n-1$. It is not
hard to see that this is the only orientable rank $2$ colouring for $P$ up to
equivalence.

###  3.2. Induced Colourings and Geodesic Embeddings

Now, a facet $F_{0}\in\mathcal{F}$ of a right-angled, compact polytope
$\mathcal{P}\subset\mathbb{H}^{n}$ is also a right-angled, compact, convex
polytope in the hypersurface $\mathcal{H}\cong\mathbb{H}^{n-1}$ such that
$\mathcal{H}\cap\mathcal{P}=F_{0}$. A $V$-colouring $\lambda$ of $\mathcal{P}$
then naturally induces a $W$-colouring $\mu$ of $F_{0}$, with $W\subset V$
generated by $\\{\lambda(F)\mid F\,\text{adjacent to}\,F_{0}\\}$: it suffices
to define $\mu$ by $\mu(F_{0}\cap F)=\lambda(F)$, where $F$ runs over
$\mathcal{F}(F_{0})=\\{F\in\mathcal{F}\mid F\,\text{is adjacent
to}\,F_{0}\\}$.

We have the following:

######  Proposition 3.5 ([4],[11]).

The manifold $\mathcal{M}_{\mu}$ is contained (in a non-canonical way) in
$\mathcal{M}_{\lambda}$ as a totally geodesic hypersurface so that the cover
$\pi:\mathcal{M}_{\lambda}\to\mathcal{P}$ restricts to a cover
$\mathcal{M}_{\mu}\to F_{0}$. Moreover, the number of connected components of
$\pi^{-1}(F_{0})$ is equal to $2^{k-s-1}$, where $k$ and $s$ are the ranks of
$\lambda$ and $\mu$, respectively. Each connected component is a copy of
$\mathcal{M}_{\mu}$ geodesically embedded in $M_{\lambda}$.

######  Example 3.6 (Löbell polyhedra).

The Löbell polyhedron $R(2n)$, which admits a hyperbolic, right-angled
realization for $n\geq 3$, has a lowest-rank proper
$\mathbb{Z}_{2}^{3}$-colouring $\Lambda_{g}$ described in [21, Proposition
3.6], which assigns $e_{1}$ to one $P_{2n}$ face, $e_{2}$, $e_{3}$ alternately
to its adjacent pentagonal faces, $e_{3}$ to the other $P_{2n}$ face and
$e_{1}$, $(e_{1}+e_{2}+e_{3})$ alternately to its adjacent faces (see Figure
1). The colouring induced by $\Lambda$ in one of the $P_{2n}$ faces, then, is
equivalent to the colouring $\lambda$ of $P_{2n}$ described in Example 3.4. By
Proposition 3.5, it follows that $\mathcal{M}_{\lambda}$ embeds geodesically
in $\mathcal{M}_{\Lambda}$, which is a closed, hyperbolic $3$-manifold, and
also orientable by Proposition 3.3).

Figure 1. Colouring of the polyhedron $R(8)$ as described in Example 3.6. Here
the boundary of the polyhedron is seen in $\mathbb{R}^{2}\cup\\{\infty\\}$ and
the colours are represented in binary notation, associating the vector
$(a,b,c)$ to $a+2b+4c$.

###  3.3. Admissible symmetries and Coloured Isometries

Now, given a (not necessarily proper) colouring $\lambda$ of a right angled
polytope $\mathcal{P}$, there is a natural group of symmetries of the
associated orbifold $\mathcal{M}_{\lambda}$ called the coloured isometry group
of $\mathcal{M}_{\lambda}$. We briefly recall its definition as given in [11,
Section 2.1]:

######  Definition 3.7.

Let be a $V$–colouring of $\mathcal{P}$. A symmetry
$g\in\text{Sym}(\mathcal{P})$, which acts on $\mathcal{F}$ as a permutation,
is admissible with respect to $\lambda$ if:

  1. (1)

it induces a permutation of the colors assigned to the facets, by sending each
color $\lambda(F)$ to $\lambda\big{(}g(F)\big{)}$;

  2. (2)

such permutation is realised by an invertible linear automorphism $\phi\in
GL(V)$.

Equivalently, $g$ is admissible with respect to $\lambda$ if $\exists\,\phi\in
GL(V)$ such that $\phi^{-1}\circ\lambda\circ g=\lambda$.

Admissible symmetries, which we denote by
$\mathrm{Adm}_{\lambda}(\mathcal{P})$, are easily seen to form a subgroup of
the symmetry group $\text{Sym}(\mathcal{P})$, and there is a naturally defined
homomorphism $\Phi_{\lambda}:\mathrm{Adm}_{\lambda}(\mathcal{P})\to GL(V)$
such that $\Phi_{\lambda}(g)^{-1}\circ\lambda\circ g=\lambda$ for every
$g\in\text{Adm}_{\lambda}(\mathcal{P})$.

The coloured isometry group $\mathrm{Isom}_{c}(M_{\lambda})$ is then defined
as the group of symmetries of $\mathcal{M}_{\lambda}$ which are lifts of
admissible symmetries of $\mathcal{P}$ under the orbifold cover
$M_{\lambda}\to\mathcal{P}$.

######  Proposition 3.8 ([11], Section 2.1).

For any $V$-colouring $\lambda$ of $\mathcal{P}$, we have that
$\mathrm{Isom}_{c}(\mathcal{M}_{\lambda})\cong\mathrm{Adm}_{\lambda}(\mathcal{P})\ltimes
V$. Moreover, the action of $\mathrm{Adm}_{\lambda}(\mathcal{P})$ on $V$ is
precisely the one induced by the homomorphism $\Phi_{\lambda}$.

######  Corollary 3.9.

The orientation-preserving coloured isometry group
$\mathrm{Isom}_{c}^{+}(\mathcal{M}_{\lambda})$ is generated by the isometries
$(r,e_{i})$, where $r\in\mathrm{Adm}_{\lambda}(\mathcal{P})$ is orientation-
reversing; $(\mathrm{id},e_{i}+e_{j})$ with $i\neq j$; and $(s,0)$ for
$s\in\mathrm{Adm}_{\lambda}(\mathcal{P})$ orientation-preserving.

###### Proof.

Since $\mathrm{ker}\,\lambda\,\triangleleft\,\Gamma$, any reflection
$r_{F}\in\Gamma$ lifts to an isometry $\big{(}\mathrm{id},\lambda(F)\big{)}$
in $\mathcal{M}_{\lambda}\cong(\mathcal{P}\times V)/_{\sim}$, and it follows
that $(\mathrm{id},v)\in\mathrm{Isom}_{c}(\mathcal{M}_{\lambda})$ for all
$v\in V$. Moreover, the isometries $(\mathrm{id},e_{i})$ are always
orientation-reversing, and we can conclude with the statement.
$\sqcup\\!\\!\\!\\!\sqcap$

######  Corollary 3.10.

If the action of $\Phi_{\lambda}(\varphi)$ in $V$ is trivial, an isometry
$(\varphi,w)$ is fixed point-free if and only if
$w\notin\bigcup_{x\in\mathrm{Fix}(\varphi|_{\partial\mathcal{P}})}G_{x}$.

###### Proof.

By Remark 3.2, it is not hard to see that
$(\varphi,w)\in\mathrm{Adm}_{\lambda}(\mathcal{P})\ltimes
V\cong\mathrm{Isom}_{c}(\mathcal{M}_{\lambda})$ fixes a point
$(x,v)\in(\mathcal{P}\times V)/_{\sim}\cong\mathcal{M}_{\lambda}$ if, and only
if,

$(x,v)\sim(\varphi,w)\cdot(x,v)=(\varphi(x),\Phi_{\lambda}(\varphi)(v)+w),$

that is, if $\varphi(x)=x$ and
$\big{(}\Phi_{\lambda}(\varphi)(v)+w-v\big{)}\in G_{x}$. In particular, it is
enough to check this condition only on the points
$x\in\mathrm{Fix}(\varphi|_{\partial\mathcal{P}})$ and to guarantee that
$\Phi_{\lambda}(\varphi)(v)\neq w+v$ for every $v\in V$. We thus have the
following: $\sqcup\\!\\!\\!\\!\sqcap$

######  Remark 3.11.

We note that, if $\mathcal{M}_{\lambda}$ is an orientable hyperbolic surface,
the group of conformal automorphisms and orientation-preserving isometries
coincide, and we have that

$\mathrm{Aut}(\mathcal{M}_{\lambda})\cong\mathrm{Isom}^{+}(\mathcal{M}_{\lambda})>\mathrm{Isom}_{c}^{+}(\mathcal{M}_{\lambda}),$

where the last inclusion might be surjective.

##  4\. The Accola-Maclachlan surfaces

Let $P_{g}$ be the right-angled $(2g+2)$-gon and $\lambda_{g}$ the unique
lowest-rank colouring of $P_{g}$. By Example 3.4, $\mathcal{M}_{\lambda_{g}}$
a surface of genus $g$. We have the following:

######  Proposition 4.1.

The group $\text{Isom}_{c}^{+}(\mathcal{M}_{\lambda_{g}})$ of orientation-
preserving coloured isometries of $\mathcal{M}_{\lambda_{g}}$ is isomorphic to
the group $AM_{g}$ from Equation 2.

###### Proof.

Every symmetry of $P_{g}$ is admissible with respect to $\lambda_{g}$, so we
have that $\mathrm{Adm}_{\lambda_{g}}(P_{g})=\mathrm{Sym}(P_{g})\cong
D_{2g+2}$, the dihedral group of order $4g+4$. By Proposition 3.8, we have
that

$\text{Isom}_{c}(\mathcal{M}_{\lambda_{g}})\cong
D_{2g+2}\ltimes\mathbb{Z}_{2}^{2},$

a group of order $16g+16$. The group $D_{2g+2}$ is generated by two
reflections $r_{1}$, $r_{2}$, where $r_{1}$ fixes some edge of $P_{g}$ and
$r_{2}$ fixes one vertex of that edge (see Figure 2–left for an example).
These reflections act on $\mathbb{Z}_{2}^{2}$ respectively as the identity and
as the permutation of coordinates. On the other hand, $\mathbb{Z}_{2}^{2}$ is
generated by the canonical vectors $e_{1}$, $e_{2}$.

Let id be the identity element in $D_{2g+2}$ and $0$ in $\mathbb{Z}_{2}^{2}$.
Let also the product in these groups be given by $\cdot$ (which will be
omitted) and $+$, respectively; while the product $\cdot$ in
$D_{2g+2}\ltimes\mathbb{Z}_{2}^{2}$ is the one given by Proposition 3.8 and
the identity element will be noted by $1$. Put

$x=(\text{id},e_{1})\cdot(r_{1},0)=(r_{1},e_{1}),\,y=(r_{2},e_{1}),\,z=(r_{1},e_{2}),\,w=(r_{2},e_{2}).$

We have that $x^{2}=1$, $xy=(r_{1}r_{2},0)$ and
$y^{2}=(\text{id},e_{1}+e_{2})$. Moreover, it is not hard to see that
$y^{2}x=z$ and $y^{3}=w$. We also note that, since the action of $r_{1}r_{2}$
fixes $e_{1}+e_{2}$, it follows that $xy=(r_{1}r_{2},0)$ and
$y^{2}=(\text{id},e_{1}+e_{2})=y^{-2}$ commute. Similarly, since the action of
$r_{1}$ on $\mathbb{Z}_{2}^{2}$ is trivial, $x=(r_{1},e_{1})=x^{-1}$ and
$y^{2}$ commute as well, or, equivalently, $(xy^{2})^{2}=1$.

Let $a=xy$, $b=y$. By the remarks above and Corollary 3.9, these generate
$\text{Isom}_{c}^{+}(\mathcal{M}_{\lambda_{g}})$. Since $r_{1}r_{2}$ is the
rotation of $P_{g}$ by an angle of $\frac{2\pi}{2g+2}$, we also have that
$a^{2g+2}=1$. Moreover, $b^{4}=(y^{2})^{2}=1$, $[a,b^{2}]=(ab)^{2}=1$. Thus,
$\mathrm{Isom}_{c}^{+}(\mathcal{M}_{\lambda_{g}})$ is a group of order $8g+8$
with the same generators and relations of the Accola-Maclachlan group
$AM_{g}$, and it follows that these two groups are isomorphic.
$\sqcup\\!\\!\\!\\!\sqcap$

We now can prove the following:

######  Theorem 4.2.

The manifold $\mathcal{M}_{\lambda_{g}}$ is isometric to the Accola-Maclachlan
surface $X_{g}$.

###### Proof.

By Proposition 4.1, we have that
$\text{Isom}_{c}^{+}(\mathcal{M}_{\lambda_{g}})\cong AM_{g}$. It is also not
hard to see that the quotient
$\mathcal{M}_{\lambda_{g}}/_{\text{Isom}_{c}^{+}(\mathcal{M}_{\lambda_{g}})}$
is obtained by the identification of two triangles from the first barycentric
subdivision of $P_{g}$ along their sides, which gives us an orbifold with
signature $(0;2,4,2g+2)$. Since
$\text{Aut}(\mathcal{M}_{\lambda_{g}})>\text{Isom}_{c}^{+}(\mathcal{M}_{\lambda_{g}})\cong
AM_{g}$ by Remark 3.11, it follows from the non-extendability of the $AM_{g}$
group in signature $(0;2,4,2g+2)$ that
$\text{Aut}(\mathcal{M}_{\lambda_{g}})\cong AM_{g}$. By the uniqueness of the
conformal automorphism group of the Accola-Maclachlan surface on that
signature, we have that $\mathcal{M}_{\lambda_{g}}\cong X_{g}$.
$\sqcup\\!\\!\\!\\!\sqcap$

######  Corollary 4.3.

The Accola-Maclachlan surface $X_{g}$ embeds geodesically for every genus $g$.

###### Proof.

Take the right-angled Löbell polyhedron $R(2g+2)$ with colouring $\Lambda_{g}$
as described in Example 3.6. We then have that the induced colouring in one
$P_{g}$ face is precisely $\lambda_{g}$. It follows that
$X_{g}\cong\mathcal{M}_{\lambda_{g}}$ embeds geodesically in
$\mathcal{M}_{\Lambda_{g}}$. $\sqcup\\!\\!\\!\\!\sqcap$

######  Corollary 4.4.

The Accola-Maclachlan surface $X_{g}$ bounds geometrically an orientable
manifold for every genus $g$.

###### Proof.

Take the orientation-reversing involution
$(r_{1},e_{1}+e_{2})\in\mathrm{Isom}_{c}(\mathcal{M}_{\lambda_{g}})$, composed
of the orientation-reversing reflection $r_{1}$ and the orientation-reversing
translations $e_{1},e_{2}$. Since $\Phi_{\lambda_{g}}(r_{1})$ acts trivially
in $\mathbb{Z}_{2}^{2}$ and
$\mathrm{Fix}(r_{1})\cap(\partial\mathcal{P})=\\{x_{1},x_{2}\\}$ with
$G_{x_{1}}=G_{x_{2}}=[e_{1}]$, it follows from Corollary 3.10 that
$(r_{1},e_{1}+e_{2})$ is fixed point-free.

We then have that $X_{g}\cong\mathcal{M}_{\lambda_{g}}$ has a fixed point-
free, orientation-reversing involution for every $g$.555This result was also
obtained by [2, Section 2, p. 640]. By Lemma 2.1 and Corollary 4.3, we can
conclude that $X_{g}$ bounds an orientable, compact manifold for every genus.
$\sqcup\\!\\!\\!\\!\sqcap$

##  5\. The Wiman type II surfaces

Now, let $P_{g}^{\prime}$ be the $4g$-gon and $\lambda_{g}^{\prime}$ its
unique lowest-rank colouring. From Theorem 4.2, we have that

$\text{Aut}(\mathcal{M}_{\lambda_{g}^{\prime}})\cong\mathrm{Isom}_{c}^{+}(\mathcal{M}_{\lambda_{g}^{\prime}})\cong\langle
a,b\mid a^{4g}=b^{4}=(ab)^{2}=[a,b^{2}]=1\rangle,$

where $a=(r_{1}r_{2},0)$ and $b=(r_{2},e_{1})$ in
$\mathrm{Isom}_{c}(\mathcal{M}_{\lambda_{g}^{\prime}})\cong
D_{4g}\ltimes\mathbb{Z}^{2}_{2}$. For every $g\geq 2$, take
$c_{g}=a^{2g}b^{2}=\big{(}(r_{1}r_{2})^{2g},e_{1}+e_{2}\big{)}\in\text{Isom}_{c}^{+}(\mathcal{M}_{\lambda_{g}^{\prime}})$.
We have the following:

######  Theorem 5.1.

The manifold $\mathcal{M}_{\lambda_{g}^{\prime}}/_{\langle c_{g}\rangle}$ is
isometric to the Wiman type II surface $Z_{g}$.

###### Proof.

We have that $c_{g}^{2}=1$. Now, $(r_{1}r_{2})^{2g}$ is precisely the
antipodal map in $P_{g}^{\prime}$, which only fixes the interior point $0\in
P_{g}^{\prime}$. Since the action of $(r_{1}r_{2})^{2g}$ in
$\mathbb{Z}_{2}^{2}$ is trivial, it follows from Corollary 3.10 that $c_{g}$
is fixed point-free isometry of $\mathcal{M}_{\lambda_{g}^{\prime}}$.

Therefore, the orbifold
$S_{g}\cong\mathcal{M}_{\lambda_{g}^{\prime}}/_{\langle c_{g}\rangle}$ is an
orientable, connected, closed manifold and it has Euler characteristic
$\frac{4(1-g)}{2}=2-2g$, that is, is a surface of genus $g$. Also, because
both $a$ and $b$ commute with $c_{g}$, we have that the normalizer of $c_{g}$
in $\mathrm{Aut}(\mathcal{M}_{\lambda_{g}^{\prime}})$ is precisely the whole
group. By Remark 3.11 and Lemma 2.2, we have that

$\mathrm{Aut}\big{(}\mathcal{M}_{\lambda_{g}^{\prime}}/_{\langle
c_{g}\rangle}\big{)}>N_{\mathrm{Aut}(\mathcal{M}_{\lambda_{g}^{\prime}})}(c_{g})/_{\langle
c_{g}\rangle}\cong\langle a,b\mid
a^{4g}=b^{4}=(ab)^{2}=1,b^{2}=a^{2g}\rangle,$

and it follows that $a$ descends into a conformal automorphism of $S_{g}$ of
order $4g$. We thus conclude that $S_{g}$ is isometric to the Wiman type II
surface for $g\neq 3$.

We also note that $S_{g}/_{\langle
a\rangle}\cong\mathcal{M}_{\lambda_{g}^{\prime}}/_{\langle
a,c_{g}\rangle}=\mathcal{M}_{\lambda_{g}^{\prime}}/_{\langle a,b^{2}\rangle}$,
and it is not hard to see from Remark 3.2 that this is an orbifold with
signature $(0;2,4g,4g)$. It then follows that $S_{g}$ is also isometric to the
Wiman type II surface for $g=3$. $\sqcup\\!\\!\\!\\!\sqcap$

######  Remark 5.2.

We note that

$\langle a,b\mid a^{4g}=b^{4}=(ab)^{2}=1,b^{2}=a^{2g}\rangle\cong\langle
a,b^{\prime}\mid
a^{4g}=(b^{\prime})^{2}=1,(b^{\prime})^{-1}a(b^{\prime})=a^{2g-1}\rangle,$

with $b^{\prime}=ab$. By [22, Section 5.2], this group is precisely
$\mathrm{Aut}(Z_{g})$, except in $g=2$. It follows that
$\mathrm{Aut}(Z_{g})\cong\mathrm{Isom}_{c}^{+}(\mathcal{M}_{\lambda_{g}^{\prime}})/_{\langle
c_{g}\rangle}$ for every $g>2$.

######  Corollary 5.3.

The Wiman type II surface $Z_{g}$ embeds geodesically for every genus $g$.

###### Proof.

Take the Löbell polytope $R(4g)$ as described in Example 3.6. This polytope
has symmetry group $D_{4g}\rtimes\mathbb{Z}_{2}$, where $D_{4g}$ is the
symmetry group of the $4g$-gonal faces. Take also the colouring
$\Lambda_{g}^{\prime}$ described in Example 3.6. We have that the rotation $r$
of angle $\pi$ around the axis that crosses the center of the $4g$-gonal faces
preserves each colour, and thus is an admissible symmetry for
$\Lambda_{g}^{\prime}$ which acts trivially in $\mathbb{Z}_{2}^{3}$. Moreover,
it restricts to the symmetry $(r_{1}r_{2})^{2g}$ in each $4g$-gonal face.

Take, then, the isometry
$S=(r,e_{2}+e_{3})\in\mathrm{Isom}_{c}^{+}(\mathcal{M}_{\Lambda_{g}^{\prime}})$,
which is composed of the orientation-preserving rotation $r$ and two
orientation-reversing translations $e_{2}$ and $e_{3}$. We have that
$S^{2}=\mathrm{Id}$. Moreover, the fixed points of $r$ are the rotation axis
in $R(4g)$, and we have that
$\mathrm{Fix}(r)\cap(\partial\mathcal{P})=\\{x_{1},x_{2}\\}$, where
$x_{1},x_{2}$ lie in the center of the $4g$-gonal faces coloured with
$e_{1},e_{3}$ respectively. Since $G_{x_{1}}=\\{0,e_{1}\\}$ and
$G_{x_{1}}=\\{0,e_{3}\\}$, it follows from Corollary 3.10 that $S$ is a fixed
point-free, orientation-preserving isometry of
$\mathcal{M}_{\Lambda_{g}^{\prime}}$.

Now, by Example 3.6, we have that
$\big{(}P_{g}^{\prime}\times\\{0\\}\times\mathbb{Z}_{2}^{2}\big{)}/_{\sim_{\lambda_{g}^{\prime}}}$
is an embedded copy of $\mathcal{M}_{\lambda_{g}^{\prime}}$ in
$\mathcal{M}_{\Lambda_{g}^{\prime}}\cong\big{(}R(4g)\times\mathbb{Z}_{2}^{3}\big{)}/_{\sim_{\Lambda_{g}^{\prime}}}$,
obtained from the induced colouring on the $4g$-gonal face coloured with
$e_{1}$. Moreover, it is not hard to see that the isometry $S$, restricted to
$\big{(}P_{g}^{\prime}\times\\{0\\}\times\mathbb{Z}_{2}^{2}\big{)}/_{\sim}$,
acts precisely as $c_{g}$ on this copy of
$\mathcal{M}_{\lambda_{g}^{\prime}}$. It follows from Theorem 5.1 that
$Z_{g}\cong\mathcal{M}_{\lambda_{g}^{\prime}}/_{\langle c_{g}\rangle}$ embeds
geodesically in the compact, orientable, hyperbolic $3$-manifold
$\mathcal{M}_{\Lambda_{g}^{\prime}}/_{\langle S\rangle}$.
$\sqcup\\!\\!\\!\\!\sqcap$

We still need one result:

######  Corollary 5.4.

For $g\geq 3$, the Wiman type II surface $Z_{g}$ admits an orientation-
reversion, fixed point-free involution if, and only if, $g$ is odd.

###### Proof.

Since each of the isometries
$(r_{1},0),(r_{2},0),(\mathrm{id},e_{1}),(\mathrm{id},e_{2})\in\mathrm{Isom}_{c}(\mathcal{M}_{\lambda_{g}^{\prime}})$
commute with $c_{g}$, we have that

$\mathrm{Isom}\big{(}\mathcal{M}_{\lambda_{g}^{\prime}}/_{\langle
c_{g}\rangle}\big{)}>\mathrm{Isom}_{c}\big{(}\mathcal{M}_{\lambda_{g}^{\prime}}/_{\langle
c_{g}\rangle}\big{)}>N_{\mathrm{Isom}_{c}(\mathcal{M}_{\lambda_{g}^{\prime}})}(c_{g})/_{\langle
c_{g}\rangle}=\mathrm{Isom}_{c}(\mathcal{M}_{\lambda_{g}^{\prime}})/_{\langle
c_{g}\rangle},$

and $\mathrm{Isom}_{c}(\mathcal{M}_{\lambda_{g}^{\prime}})/_{\langle
c_{g}\rangle}$ contains
$\mathrm{Isom}_{c}^{+}(\mathcal{M}_{\lambda_{g}^{\prime}})/_{\langle
c_{g}\rangle}$ as an index–$2$ subgroup. However, by Remark 5.2, we have that
$\mathrm{Isom}^{+}\big{(}\mathcal{M}_{\lambda_{g}^{\prime}}/_{\langle
c_{g}\rangle}\big{)}\cong\mathrm{Isom}_{c}^{+}(\mathcal{M}_{\lambda_{g}^{\prime}})/_{\langle
c_{g}\rangle}$ for $g\geq 3$, from which it follows that
$\mathrm{Isom}\big{(}\mathcal{M}_{\lambda_{g}^{\prime}}/_{\langle
c_{g}\rangle}\big{)}\cong\mathrm{Isom}_{c}(\mathcal{M}_{\lambda_{g}^{\prime}})/_{\langle
c_{g}\rangle}$ since the groups have the same order.

Now, take any isometry $\varphi=(s,v)\in\mathrm{Sym(\mathcal{P})}\ltimes
V\cong\mathrm{Isom}_{c}(\mathcal{M}_{\lambda_{g}^{\prime}})$, which descents
into an isometry
$\tilde{\varphi}\in\mathrm{Isom}_{c}\big{(}\mathcal{M}_{\lambda_{g}^{\prime}}/_{\langle
c_{g}\rangle}\big{)}$. We have the following:

  1. (1)

$\tilde{\varphi}$ is orientation-reversing if $\varphi$ is orientation-
reversing;

  2. (2)

$\tilde{\varphi}$ is an involution if $\varphi^{2}\in\langle c_{g}\rangle$;

  3. (3)

$\tilde{\varphi}$ fixed point-free if $\langle\varphi,c_{g}\rangle$ acts
freely on $\mathcal{M}_{\lambda_{g}^{\prime}}$.

If $\varphi^{2}=1$, we must have that $s^{2}=1$ and
$v\in\mathrm{Fix}\big{(}\Phi_{\lambda_{g}^{\prime}}(s)\big{)}$. Up to
conjugation, we have then that $s$ must be either $r_{1}$, $r_{2}$ or
$(r_{1}r_{2})^{2g}$. If $s=r_{1}$, we have that
$\mathrm{Fix}\big{(}\Phi_{\lambda_{g}^{\prime}}(s)\big{)}=\mathbb{Z}_{2}^{2}$
and $\mathrm{Fix}(s)\cap(\partial\mathcal{P})=\\{x_{1},x_{2}\\}$ with
$G_{x_{1}}=G_{x_{2}}=[e_{1}]$. In this case, by Corollary 3.10, $(s,v)$ is
fixed point-free only if $v\in\\{e_{2},e_{1}+e_{2}\\}$. Since we want
$\varphi=(r,s)$ to be orientation-reversing, we must then have
$v=e_{1}+e_{2}$, from which it follows that $\varphi$ is a fixed point-free
orientation-reversing involution. But then we would have that $\varphi
c_{g}=(r_{1}(r_{1}r_{2})^{2g},0)$ is not fixed point-free. The other two cases
$s=r_{2}$, $s=(r_{1}r_{2})^{2g}$ are excluded analogously.

Suppose then that
$\varphi^{2}=c_{g}=\big{(}(r_{1}r_{2})^{2g},e_{1}+e_{2}\big{)}$. It follows,
in this case, that $s^{2}=(r_{1}r_{2})^{2g}$ and
$v+\Phi_{\lambda_{g}^{\prime}}(s)(v)=e_{1}+e_{2}$, which implies that
$v\notin\text{Fix}\big{(}\Phi_{\lambda_{g}^{\prime}}(s)\big{)}$. Since $s\in
D_{4g}$, we must have that $s=(r_{1}r_{2})^{\pm g}$. On the other hand, the
possible actions of $\Phi_{\lambda_{g}^{\prime}}(s)$ on $\mathbb{Z}_{2}^{2}$
are either trivial or permuting the coordinates, so we must have
$v\in\\{e_{1},e_{2}\\}$ and $\Phi_{\lambda_{g}^{\prime}}(s)$ permuting the
coordinates. For $s=(r_{1}r_{2})^{\pm g}$, this is true if, and only if, $g$
is odd. In this case, we have that $\varphi$ is orientation-reversing and
$\langle\varphi,c_{g}\rangle=\langle\varphi\rangle$ with both $\varphi$,
$\varphi^{3}=\varphi^{-1}$ acting freely, and it follows that
$\tilde{\varphi}$ is an orientation-reversing, fixed point-free involution of
$\mathcal{M}_{\lambda_{g}^{\prime}}/_{\langle c_{g}\rangle}\cong Z_{g}$.
$\sqcup\\!\\!\\!\\!\sqcap$

We can now finally prove this:

######  Corollary 5.5.

The Wiman type II surface $Y_{g}$ bounds geometrically an orientable, closed
manifold for every odd genus $g$.

###### Proof.

This follows directly from Lemma 2.1 and Corollaries 5.3, 5.4.
$\sqcup\\!\\!\\!\\!\sqcap$

##  6\. The Kulkarni Surfaces

Now, let $g\equiv 3\,(\text{mod}\,4)$ and $P_{g}$ be the $(2g+2)$-gon. Let
$\lambda_{g}^{\prime\prime}$ be the colouring on $P_{g}$ given by Figure
2–left, where each side is labelled with the vectors $e_{1}$, $e_{2}$, $e_{3}$
and $(e_{1}+e_{2}+e_{3})$ in clockwise order. Since each adjacent side has
distinct vectors, this is a proper colouring, and the surface
$\mathcal{M}_{\lambda_{g}^{\prime\prime}}$ is orientable by Proposition 3.3.
Moreover,
$\chi(\mathcal{M}_{\lambda_{g}^{\prime\prime}})=8\cdot\frac{2-2g}{4}=4-4g$,
that is, $\mathcal{M}_{\lambda_{g}^{\prime\prime}}$ is a surface of genus
$2g-1$.

Figure 2. Respective colouring of the $(2g+2)$-gon and of the Lobell
polyhedron $R(2g+2)$, as described in Section 6, where colours are represented
in binary notation.

Let $r_{1},r_{2}$ be the reflections shown in Figure 2–left, which generate
the group of symmetries $\mathrm{Sym}(P_{g})\cong D_{2g+2}$. Due to the fact
that $2g+2\equiv 0\,(\text{mod}\,8)$, we have that both these symmetries are
admissible, with induced actions on $\mathbb{Z}_{2}^{3}$ given by

$\Phi_{\lambda_{g}^{\prime\prime}}(r_{1})=\begin{pmatrix}1&1&0\\\ 0&1&0\\\
0&1&1\\\
\end{pmatrix},\,\,\Phi_{\lambda_{g}^{\prime\prime}}(r_{2})=\begin{pmatrix}0&1&1\\\
1&0&1\\\ 0&0&1\\\ \end{pmatrix},$

respectively. It follows that
$\mathrm{Adm}_{\lambda_{g}^{\prime\prime}}(P_{g})=\mathrm{Sym}(P_{g})\cong
D_{2g+2}$, and by Proposition 3.8, we have that
$\text{Isom}_{c}(\mathcal{M}_{\lambda_{g}^{\prime\prime}})\cong
D_{2g+2}\ltimes\mathbb{Z}_{2}^{3}$, with the product given by the action
$\Phi_{\lambda_{g}^{\prime\prime}}$.

Let
$d_{g}=\big{(}(r_{1}r_{2})^{g+1},e_{1}+e_{3}\big{)}\in\text{Isom}_{c}^{+}(\mathcal{M}_{\lambda_{g}^{\prime\prime}})$.
We have the following:

######  Theorem 6.1.

The surface $\mathcal{M}_{\lambda_{g}^{\prime\prime}}/_{\langle d_{g}\rangle}$
is isometric to the Kulkarni surface $Y_{g}$ for every $g\equiv
3\,(\text{mod}\,4)$.

###### Proof.

Put $x=(r_{1},e_{1})$, $y=(r_{1},e_{2})$, $z=(r_{2},e_{1})$ and
$w=(r_{2},e_{3})$. Since $xy^{2}=(r_{1},e_{3})$, $z^{3}=(r_{2},e_{2})$,
$y=z^{2}x^{-1}$ and $w=y^{2}z$, we have by Corollary 3.9 that $x,z$ generate
the orientation-preserving coloured isometry group.

Now, since $g+1\equiv 0\,(\text{mod}\,4)$, we have that
$\Phi_{\lambda_{g}^{\prime\prime}}\big{(}(r_{1}r_{2})^{g+1}\big{)}$ acts
trivially in $\mathbb{Z}_{2}^{3}$, and it follows from Corollary 3.10 that
$d_{g}$ is fixed point-free. Also, $d_{g}^{2}=1$, and since $e_{1}+e_{3}$ is
fixed by both actions $\Phi_{\lambda_{g}^{\prime\prime}}(r_{1})$,
$\Phi_{\lambda_{g}^{\prime\prime}}(r_{2})$, it follows that $d_{g}$ commutes
with $x$ and $z$ and thus with every element of
$\text{Isom}_{c}^{+}(\mathcal{M}_{\lambda_{g}^{\prime\prime}})$. Equivalently,
$N(d_{g})=\text{Isom}_{c}^{+}(\mathcal{M}_{\lambda_{g}^{\prime\prime}})$,
where the normalizer is taken inside
$\text{Isom}_{c}^{+}(\mathcal{M}_{\lambda_{g}^{\prime\prime}})$.

Put $a=xz=(r_{1}r_{2},0)$, $b=z^{3}=z^{-1}=(r_{2},e_{2})$. We have that
$a^{2g+2}=b^{4}=1$, $(ab)^{2}=x^{2}=1$ and $b^{2}ab^{2}=a^{g+2}d_{g}$. It
follows that $N(d_{g})/_{\langle d_{g}\rangle}$ is a group of order $8g+8$
with the same presentation as the Kulkarni group $K_{g}$, that is,
$N(d_{g})/_{\langle d_{g}\rangle}\cong K_{g}$. Also,

$\big{(}\mathcal{M}_{\lambda_{g}^{\prime\prime}}/_{\langle
d_{g}\rangle}\big{)}/_{\big{(}N(d_{g})/_{\langle
d_{g}\rangle}\big{)}}\cong\mathcal{M}_{\lambda_{g}^{\prime\prime}}/_{\text{Isom}_{c}^{+}(\mathcal{M}_{\lambda_{g}^{\prime\prime}})},$

which is an orbifold obtained similiarly to the one in the proof of Theorem
4.2, and thus also has signature $(0;2,4,2g+2)$. Moreover, by Remark 3.11 and
Lemma 2.2,

$\mathrm{Aut}\big{(}\mathcal{M}_{\lambda_{g}^{\prime\prime}}/_{\langle
d_{g}\rangle}\big{)}>N_{\mathrm{Aut}(\mathcal{M}_{\lambda_{g}^{\prime\prime}})}(d_{g})/_{\langle
d_{g}\rangle}>N_{\text{Isom}_{c}^{+}(\mathcal{M}_{\lambda_{g}^{\prime\prime}})}(d_{g})/_{\langle
d_{g}\rangle}=N(d_{g})/_{\langle d_{g}\rangle}\cong K_{g},$

where the second inclusion comes from the fact that
$\mathrm{Isom}_{c}^{+}(\mathcal{M}_{\lambda_{g}^{\prime\prime}})<\mathrm{Aut}(\mathcal{M}_{\lambda_{g}^{\prime\prime}})$.
We can thus conclude that $\mathcal{M}_{\lambda_{g}^{\prime\prime}}/_{\langle
d_{g}\rangle}\cong Y_{g}$, the Kulkarni surface of genus $g$.
$\sqcup\\!\\!\\!\\!\sqcap$

######  Remark 6.2.

We note that, if we would take the isometry $e=(\mathrm{id},e_{1}+e_{3})$, we
would have that $\mathcal{M}_{\lambda_{g}^{\prime\prime}}/_{\langle
e\rangle}\cong\mathcal{M}_{\lambda_{g}}\cong X_{g}$. Indeed, the isotropy
groups $G_{x}$ for the vertices $x\in P_{g}^{\prime}$ with colouring
$\lambda_{g}^{\prime\prime}$ are one of the following four subspaces of
$\mathbb{Z}_{2}^{3}$:

$[e_{1},e_{2}],[e_{2},e_{3}],[e_{3},e_{1}+e_{2}+e_{3}]=[e_{3},e_{1}+e_{2}],[e_{1}+e_{2}+e_{3},e_{1}]=[e_{1},e_{2}+e_{3}].$

Since $(e_{1}+e_{3})$ does not belong to any of these subspaces, it follows
from Corollary 3.10 that
$e\in\mathrm{Isom}_{c}^{+}(\mathcal{M}_{\lambda_{g}^{\prime\prime}})$ is fixed
point-free. Moreover, $\mathcal{M}_{\lambda_{g}^{\prime\prime}}/_{\langle
e\rangle}\cong\mathcal{M}_{(p\,\circ\lambda_{g}^{\prime\prime})}$, where
$p:\mathbb{Z}_{2}^{3}\to\mathbb{Z}_{2}^{2}$ is the projection that maps
$e_{1},e_{3}\mapsto e_{1}$ and $e_{2}\mapsto e_{2}$.We conclude with the fact
that $p\circ\lambda_{g}^{\prime\prime}=\lambda_{g}$.

######  Corollary 6.3.

The Kulkarni surface $Y_{g}$ embeds geodesically for every $g\equiv
3\,(\text{mod}\,4)$.

###### Proof.

Take the rank–$4$ colouring $\Lambda_{g}^{\prime\prime}$ of the Löbell
polyhedron $R(2g+2)$, as given in Figure 2–right, which assigns $e_{4}$ to the
$(2g+2)$-gonal faces; $e_{1}$, $e_{2}$, $e_{3}$, $(e_{1}+e_{2}+e_{3})$
cyclically in the pentagonal faces adjacent to one $(2g+2)$-gonal face; and
the same cyclical assignment rotated by two pentagonal faces in the remaining
pentagonal faces. This assignment is always possible for $2g+2\equiv
0\,(\text{mod}\,8)$, and it gives a proper colouring since every three faces
meeting at a vertex have linearly independent colours. It follows from
Propositions 3.1, 3.3 that $\mathcal{M}_{\Lambda_{g}^{\prime\prime}}$ is a
closed, orientable, hyperbolic $3$-manifold.

Now, as in the proof of Corollary 5.3, take the rotation $r^{\prime}$ of angle
$\pi$ around the axis that crosses the center of the $(2g+2)$-gonal. This
rotation fixes each colour, and thus is an admissible symmetry for
$\Lambda_{g}^{\prime\prime}$ which acts trivially in $\mathbb{Z}_{2}^{4}$.
Moreover, it restricts to the symmetry $(r_{1}r_{2})^{g+1}$ in each
$(2g+2)$-gonal face.

Take, also, the isometry
$S^{\prime}=(r^{\prime},e_{1}+e_{3})\in\mathrm{Isom}_{c}^{+}(\mathcal{M}_{\Lambda_{g}^{\prime\prime}})$.
Similarly to the proof of Corollary 5.3, we can show that $S^{\prime}$ is
fixed point-free, orientation-preserving isometry of
$\mathcal{M}_{\Lambda_{g}^{\prime\prime}}$. It’s also not hard to see that the
induced colouring in each $P_{g}$ face is $\lambda_{g}^{\prime\prime}$.
Moreover, $S^{\prime}$ acts on each $(2g+2)$-gonal face precisely as $d_{g}$
acts on $P_{g}$. Since $\mathcal{M}_{\lambda_{g}^{\prime\prime}}$ embeds
geodesically in $\mathcal{M}_{\Lambda_{g}^{\prime\prime}}$ by Proposition 3.5,
it follows from Theorem 6.1 that
$Y_{g}\cong\mathcal{M}_{\lambda_{g}^{\prime\prime}}/_{\langle d_{g}\rangle}$
embeds geodesically in the compact, orientable, hyperbolic $3$-manifold
$\mathcal{M}_{\Lambda_{g}^{\prime\prime}}/_{\langle S^{\prime}\rangle}$.
$\sqcup\\!\\!\\!\\!\sqcap$

######  Corollary 6.4.

The Kulkarni surface $Y_{g}$ bounds geometrically an orientable, closed
manifold for every $g\equiv 3\,(\text{mod}\,8)$.

###### Proof.

Similarly to the proof of Corollary 5.4, we have that $d_{g}$ commutes with
every element of $\mathrm{Isom}_{c}(\mathcal{M}_{\lambda_{g}^{\prime\prime}})$
and $\mathrm{Isom}_{c}(\mathcal{M}_{\lambda_{g}^{\prime\prime}})/_{\langle
d_{g}\rangle}<\mathrm{Isom}\big{(}\mathcal{M}_{\lambda_{g}^{\prime\prime}}/_{\langle
d_{g}\rangle}\big{)}$, where the inclusion is surjective for $g\neq 3$ by the
non-extendability of the action of the Kulkarni group $K_{g}$ on $Y_{g}$,
which implies that both groups must have the same order.

As in the proof of Corollary 5.4, we have that an isometry
$\varphi=(s,v)\in\mathrm{Isom}_{c}(\mathcal{M}_{\lambda_{g}^{\prime\prime}})$
will descend to a fixed point-free, orientation-reversing involution
$\tilde{\varphi}\in\mathrm{Isom}\big{(}\mathcal{M}_{\lambda_{g}^{\prime\prime}}/_{\langle
d_{g}\rangle}\big{)}$ if, and only if, $\varphi$ is orientation-reversing,
$\varphi^{2}\in\langle d_{g}\rangle$ and $\langle\varphi,d_{g}\rangle$ acts
freely on $\mathcal{M}_{\lambda_{g}^{\prime\prime}}$. As in the proof of of
Corollary 5.4, we can show that there is no such $\varphi$ in the case
$\varphi^{2}=1$.

Take then $\varphi^{2}=d_{g}$. As in the proof of Corollary 5.4, we must have
then that $s=(r_{1}r_{2})^{\pm\frac{g+1}{2}}$ and
$v+\Phi_{\lambda_{g}^{\prime\prime}}(s)(v)=e_{1}+e_{3}$, from which it follows
that $v\notin\mathrm{Fix}\big{(}\Phi_{\lambda_{g}^{\prime\prime}}(s)\big{)}$.
If $g\equiv 7\,(\text{mod}\,8)$, however, we have that
$\Phi_{\lambda_{g}^{\prime\prime}}(s)=\mathrm{Id}$ and thus there is no such
$\varphi$ with $\varphi^{2}=d_{g}$. If $g\equiv 3\,(\text{mod}\,8)$, on the
other hand,
$\Phi_{\lambda_{g}^{\prime\prime}}(s)=\Phi_{\lambda_{g}^{\prime\prime}}(r_{1})$
and both $\varphi$, $\varphi d_{g}=\varphi^{3}=\varphi^{-1}$ are fixed point-
free for any choice of $v\in\\{e_{1},e_{2},e_{3},e_{1}+e_{2}+e_{3}\\}$, and
orientation-reversing as well.

We thus have that the Kulkarni surface
$Y_{g}\cong\mathcal{M}_{\lambda_{g}^{\prime\prime}}/_{\langle d_{g}\rangle}$
admits an orientation-reversing, fixed point-free involution if, and only if,
$g\equiv 3\,(\text{mod}\,8)$.666This result was also obtained by [2, Section
2, p. 640]. We can then conclude with Lemma 2.1 and Corollary 6.3.
$\sqcup\\!\\!\\!\\!\sqcap$

## References

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  * [17] D. Singerman, “Automorphisms of compact non-orientable Riemann surfaces”, Glasgow Mathematical Journal, 12:1 (1971), 50-59.
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# Superalgebra Doubling in $AdS_{3}$ Superstrings

Abbas Ali
Physics Department, Aligarh Muslim University, Aligarh,
India

###### Abstract

We propose a solution to the mystery of superalgebra doubling found by Giveon-
Kutasov-Seiberg in their construction of worldsheet approach to spacetime
superconformal symmetry for $AdS_{3}$ superstrings in the context of
$AdS_{3}$/$CFT_{2}$ Correspondence.

##  1 Introduction

$AdS_{3}/CFT_{2}$ is one of the two most intensely studied cases of $AdS/CFT$
correspondence. This is formulated as the $1+1$ dimensional CFT describing the
Higgs branch of the D1-D5 system on $M^{4}$= $K3$ or $T^{4}$ being dual to
Type IIB string theory on $(AdS_{3}\times S^{3})_{Q_{1}Q_{5}}\times M^{4}(Q)$
[1, 2, 3]. This naturally generalizes to $M^{4}=S^{3}\times S^{1}$. In this
regard Giveon-Kutasov-Seiberg (GKS) poineered the worldsheet approach to the
spacetime symmetry algebra for $AdS_{3}$ strings and superstrings [4]. We owe
our major insights to this approach in case of $AdS_{3}/CFT_{2}$ [5, 6].

At the moment public attention has gyrated toward the entanglement aspects of
this system but several unresolved problems are gawking at us in the core
formulation of $AdS_{3}/CFT_{2}$ correspondence. Some of these are as follows.

We know the brane configurations, namely the $D1/D5$ system, that go to the
$AdS_{3}\times S^{3}\times K3$ and $AdS_{3}\times S^{3}\times T^{4}$ geometry
in the near horizon limit. But we do not know the brane configuration that
will go to the $AdS_{3}\times S^{3}\times S^{3}\times S^{1}$ geometry in the
near horizon limit.

Similarly the CFT dual to superstrings moving on $AdS_{3}\times S^{3}\times
M^{4}$ with $M^{4}=K3$ or $T^{4}$ is the symmetric orbifold theory
$Sym^{N}(M^{4})$ but the proposed dual in case of $AdS_{3}\times S^{3}\times
S^{3}\times S^{1}$, namely $Sym^{N}(S^{3}\times S^{1})$, is known with some
conditions only after a long series of investigations [7, 8, 9, 10, 11].

In this story Gukov-Martinec-Moore-Strominger [7] explored the possibility of
a free field realization of the large $N=4$ superconformal symmetry that is
more complete than the conventionally used free field realization by Sevrin-
Troost-van Proeyen [12] so as to reach the holographic dual to superstrings
moving on $AdS_{3}\times S^{3}\times S^{3}\times S^{1}$. A non-commutativity
did not allow this to succeed. As a matter of fact corresponding complete free
field realization is already known for a long time [13, 14, 15]. To the best
of our knowledge this has not been put to good use for finding either the
brane configuration that will go to $AdS_{3}\times S^{3}\times S^{3}\times
S^{1}$ geometry in the near horizon limit or the conformal field theory dual
to superstrings moving on $AdS_{3}\times S^{3}\times S^{3}\times S^{1}$
background.

Apart from D1-D5 system the study of $AdS_{3}$ superstrings is also closely
related to the study of the three dimensional BTZ black hole [16]. While
dealing with the corresponding issue in $AdS_{3}$ the stringy generalization
of BTZ in Ref.[17] is also mostly ignored. As a result there are a number of
issues that have to be addressed in this area.

The Penrose limit of $AdS_{3}\times S^{3}\times S^{3}\times S^{1}$ has been
studied in [18, 19] but sufficient attention has not been paid to its
connection to $AdS_{3}\times S^{3}\times T^{4}$ superstrings [6].

It is well known that the superconformal symmetry relevant for the
$AdS_{3}\times S^{3}\times K3$ and $AdS_{3}\times S^{3}\times S^{3}\times
S^{1}$ superstrings are the small and large N=4 superconfromal symmetries. But
this remains largely unknown that the superconformal symmetry relevant for
superstrings moving on $AdS_{3}\times S^{3}\times T^{4}$ background is the
middle N=4 superalgebra [20].

D1-D5 system is intrinsic to the formulation of $AdS_{3}/CFT_{2}$
correspondence. In Ref.[21] Seiberg-Witten analyzed in detail the marginal
stability of this system. In absence of B-field and $\theta$-term the D1-D5
system has an instability. The branes can separate at no cost of energy. This
results in a singularity of the corresponding conformal field theory. We do
not know the structure of the corresponding singularity.

There is an additional issue concerning the analysis of the spectrum of the
chiral primary states in the D1-D5 system. The analysis of GKS found that the
chiral operators in the boundary conformal field theory have
$\Delta\leq(Q_{5}-2)/2$. The same spectrum analyzed in Ref.[22] has the bound
$\Delta\leq Q_{1}Q_{5}/2$. These two bounds do not agree with each other. This
discrepency has to be sorted out.

We have listed these problems for the sake of record. Our focus in this note
is going to be yet another problem in the context of superstrings moving on
$AdS_{3}$ backgrounds. In their study of superstrings Giveon-Kutasov-Seiberg
found a mysterious doubling of N=4 superconformal algebra. They provided
rationale for this superalgebra doubling. This explanation is conjectural.

In the rest of this note we shall first of all formulate this problem and then
discuss the solution provided by GKS. After that we shall present some details
from our construction of an ADHM sigma model [23, 24]complementary to the one
constructed by Witten in Ref. [25]. We then propose our resolution to the
problem of superalgebra doubling in $AdS_{3}$ superstrings.

##  2 The GKS Formalism

Giveon-Kutasov-Seiberg revived the old fashioned worldsheet perturbative
formalism to study propagation of strings and superstrings on $AdS_{3}\times
S^{3}\times K3$ and $AdS_{3}\times S^{3}\times T^{4}$ backgrounds. They
derived the corresponding spacetime conformal field theory, Virasoro and
current algebras. It was an establishment of the $AdS_{3}/CFT_{2}$ conjecture.
(Another derivation of the same was provided by Eberhardt-Gaberdiel-Gopakumar
in [26] from a different angle.)

The bosonic string on $AdS_{3}$ times some compact manifold has the Lagrangian

$\mathcal{L}=\frac{2l^{2}}{l_{s}^{2}}(\partial\phi\bar{\partial}\phi+e^{2\phi}\bar{\partial}\gamma\partial\gamma)$
(1)

where $l_{s}$ is the fundamental string length and $l$ is the radius of the
curvature of the hyperspace used to define $AdS_{3}$ metric. A $B$-field of
the form

$B=l^{2}e^{2\phi}d\gamma\wedge d\bar{\gamma}$ (2)

has been incorporated and the $AdS_{3}$ metric is

$ds^{2}=l^{2}(d\phi^{2}+e^{2\phi}\partial\gamma\partial\bar{\gamma}).$ (3)

The theory has affine $SL(2,R)\times SL(2,R)$ Lie algebra symmetry at level
$k$

$J^{a}(z)J^{b}(z)=\frac{k\eta^{AB}/2}{(z-w)^{2}}+\frac{i\eta_{CD}\epsilon^{ABC}J^{D}(w)}{z-w}+...$
(4)

with $\eta^{AB}$ being the $SL(2,R)$ metric with signature $(+,+,-)$ and
$\epsilon_{ABC}$ are the structure constants of $SL(2,R)$. Also the level $k$
is related to $l$ and $l_{s}$ by

$l^{2}=l^{2}_{s}k.$ (5)

The central charge of the model is

$c=\frac{3k}{k-2}.$ (6)

The current algebra has a Wakimoto free field representation

$\displaystyle J^{-}$ $\displaystyle=$
$\displaystyle\beta,~{}~{}~{}J^{3}=\beta\gamma+\frac{\alpha_{+}}{2}\partial\phi,$
$\displaystyle J^{+}$ $\displaystyle=$
$\displaystyle\beta\gamma^{2}+\alpha_{+}\gamma\partial\phi+k\partial\gamma$
(7)

with

$\alpha_{+}^{2}=2k-4.$ (8)

The relevant vertex operators are

$V_{jm\bar{m}}=\gamma^{j+m}\bar{\gamma}^{j+\bar{m}}exp\left(\frac{2j}{{\alpha_{+}}}\phi\right)$
(9)

with scaling dimension of $V_{jm\bar{m}}$ being

$h=-\frac{j(j+1)}{k-2}.$ (10)

The unitarity condition puts a condition on the quantum numbers

$-1<j<\frac{k}{2}-1$ (11)

as analyzed in [27] and [28]. The presence of above $SL(2,R)$ Lie algebra
gives rise to three conserved charges

$\displaystyle L_{0}=-\oint dzJ^{3}(z),~{}L_{1}$ $\displaystyle=$
$\displaystyle-\oint dzJ^{+}(z),~{}L_{-1}=-\oint dzJ^{-}(z)$ (12)

which is generalized to full Virasoro algebra along the lines of Ref. [29] in
the following way. The full Virasoro algebra can be introduced using affine
Kac-Moody algebra in the following manner. If the string background is
$AdS_{3}\times\mathcal{N}$ with $\mathcal{N}$ having a worldsheet affine Lie
algebra $\hat{G}$ for a compact group $G$ with currents $K^{a}(z)$ satisfying

$K^{a}(z)K^{b}(z)=\frac{k^{\prime}\delta^{ab}/2}{(z-w)^{2}}+\frac{if^{ab}_{~{}c}k^{\prime}(w)}{(z-w)}+...,~{}a,b,c=1,...,\text{dimG}$
(13)

then the spacetime theory can be endowed with an affine Kac-Moody algebra with
the generators

$T^{a}_{n}=\oint dzK^{a}(z)\gamma^{n}(z)$ (14)

and the algebra closes with the spacetime level

$k_{s-t}\equiv\tilde{k}=pk^{\prime}$ (15)

with the integer

$p\equiv\oint dz\frac{\partial_{z}\gamma}{\gamma}.$ (16)

The Virasoro generators are defined as

$L_{n}=-\oint
dz[(1-n^{2})J^{3}\gamma^{n}+\frac{n(n-1)}{2}J^{-}\gamma^{n+1}+\frac{n(n+1)}{2}J^{+}\gamma^{n-1}].$
(17)

The corresponding spacetime Virasoro algebra has the central charge

$c_{s-t}=6kp.$ (18)

To discuss fermionic strings GKS introduced fermions $\psi^{A}(z)$ and
$\chi^{a}(z)$ and ‘bosonic’ currents $j^{a}(z)$ and $k^{a}(z)$ with

$\displaystyle
J^{A}=j^{A}-\frac{i}{k}\epsilon^{A}_{~{}BC}\psi^{B}\psi^{C},~{}K^{a}=k^{a}-\frac{i}{k^{\prime}}f^{a}_{~{}bc}\chi^{b}\chi^{c}$
(19)

with two point functions

$\displaystyle\langle\psi^{A}(z)\psi^{B}(w)\rangle=\frac{k\eta^{AB}/2}{z-w},~{}\langle\chi^{a}(z)\chi^{b}(w)\rangle=\frac{k^{\prime}\delta^{ab}/2}{z-w}.$
(20)

With $A,B=1,2,3~{}\text{and}~{}a,b=1,..,\text{dimG}$.

The worldsheet supercurrent $G(z)$ is proposed to be

$G(z)=\frac{2}{k}(\eta_{AB}\psi^{A}j^{B}-\frac{i}{3k}\epsilon_{ABC}\psi^{A}\psi^{B}\psi^{C})+\frac{2}{k^{\prime}}(\chi^{a}k^{a}-\frac{i}{3k^{\prime}}f_{abc}\chi^{a}\chi^{b}\chi^{c})+G_{rest}.$
(21)

When we take up the discussion of superstrings on $AdS_{3}\times S^{3}\times
T^{4}$ then the worldsheet theory has a three sphere or $SU(2)$ WZW model. The
$AdS_{3}$ currents are $(\psi^{A},J^{A})$ and $SU(2)$ currents are
$(\chi^{a},K^{a})$ and the central charge on the worldsheet is

$c=\left[\frac{3(k+2)}{k}+\frac{3}{2}\right]+\left[\frac{3(k^{\prime}+2)}{k^{\prime}}+\frac{3}{2}\right].$
(22)

The energy-momentum tensor $T(z)$ and the supercurrent $G(z)$ in this case are

$T(z)=\frac{1}{k}(j^{A}j_{A}-\psi^{A}\partial\psi_{A})+\frac{1}{k}(k^{a}k_{a}-\chi^{a}\partial\chi_{a})+\frac{1}{2}(\partial
Y^{i}\partial Y_{i}-\lambda^{i}\partial\lambda_{i})$ (23)

and

$G(z)=\frac{2}{k}(\psi^{A}j_{A}-\frac{i}{3k}\epsilon_{ABC}\psi^{A}\psi^{B}\psi^{C})+\frac{2}{k}(\chi^{a}k^{a}-\frac{i}{3k}\epsilon_{abc}\chi^{a}\chi^{b}\chi^{c})+\lambda^{i}\partial\lambda_{i}.$
(24)

Where $Y^{i}(z),~{}i=1,2,3,4$ are the four free bosons corresponding to
$T^{4}$.

##  3 Superalgebra Doubling

With above GKS formalism we would now like to demand spacetime supersymmetry
in the theory. It is well known that to have N=1 spacetime supersymmetry the
worldsheet superconformal symmetry must be enhanced to N=2.

The N=1 superconformal algebra has only two generators: the energy-momentum
tensor $T(z)$ and super-tensor $G(z)$. Both of these have been constructed
above. N=2 superconformal symmetry has four generators-the energy-momentum
tensor $T(z)$, two super-tensors $G^{\pm}(z)$ and a $U(1)$ current $J(z)$.

GKS devised an ingenious method to get N=2 superconformal symmetry on
worldsheet to get spactime supersymmetry. In particular they abandoned the
usual approach of identifying the $U(1)$ currents. The reason is that this
usual approach, discussed in their Appendix B, suffers from the problem of
superalgebra doubling. We now spell out this problem. The N=2 $U(1)$ current
is written as sum of parts corresponding to isometries

$\frac{SL(2,R)}{U(1)}\times\frac{SU(2)}{U(1)}\times U(1)^{2}\times U(1)^{4}$
(25) $J(z)=J^{(1)}+J^{(2)}+J^{(3)}+J^{(4)}$ (26)

with

$\displaystyle J^{(1)}(z)$ $\displaystyle=$
$\displaystyle\frac{2}{k}(J^{3}+i\psi^{1}\psi^{2}),~{}J^{(2)}(z)=\frac{2}{k}(i\chi^{1}\chi^{2}-K^{3})$
$\displaystyle J^{(3)}(z)$ $\displaystyle=$
$\displaystyle\frac{2}{k}\psi^{3}\chi^{3},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}J^{(4)}(z)=i\lambda^{1}\lambda^{2}+i\lambda^{3}\lambda^{4}.$
(27)

Here $J^{3}$ and $K^{3}$ are total $SL(2,R)$ and $SU(2)$ currents
respectively. To construct spacetime supercharges $J(z)$ is bosonised as

$J(z)=i\partial(H+H_{1}+H_{2}+H_{3})$ (28)

with

$\displaystyle i\partial H$ $\displaystyle=$
$\displaystyle\frac{2}{k}(-i\psi^{1}\psi^{2}+i\chi^{1}\chi^{2}+J^{3}-K^{3}),~{}i\partial
H_{1}=\frac{2}{k}\psi^{3}\chi^{3},$ $\displaystyle\partial H_{2}$
$\displaystyle=$ $\displaystyle\lambda^{1}\lambda^{2},~{}\partial
H_{3}=\lambda^{3}\lambda^{4}.$ (29)

The spacetime charges are

$\displaystyle Q_{\alpha}^{\pm}=\oint
dzS^{\pm}_{\alpha},~{}\bar{Q}_{\alpha}^{\pm}=\oint dz\bar{S}^{\pm}_{\alpha}$
(30)

with

$\displaystyle
S_{\alpha}^{\pm}=e^{\pm\frac{i}{2}(H_{2}+H_{3})},~{}S_{\bar{\alpha}}^{\pm}=e^{\pm\frac{i}{2}(H_{2}-H_{3})}$
(31)

Now the spacetime superalgebra is

$\displaystyle\\{Q_{\alpha}^{+},Q_{\beta}^{-}\\}$ $\displaystyle=$
$\displaystyle\delta_{\alpha\beta}(J^{3}-K^{3}),~{}\\{\bar{Q}_{\alpha}^{+},\bar{Q}_{\beta}^{-}\\}=\delta_{\bar{\alpha}\bar{\beta}}(J^{3}+K^{3})$
$\displaystyle\\{Q_{\alpha}^{-},\bar{Q}_{\beta}^{+}\\}$ $\displaystyle=$
$\displaystyle\\{Q_{\alpha}^{+},\bar{Q}_{\bar{\beta}}^{-}\\}=\gamma^{i}_{\alpha\bar{\beta}}P_{i}.$
(32)

First of these equations is the Ramond sector N=4 superalgebra. With that
interpretation the second of the above equations give another copy of the
Ramond sector superalgebra.

For a giveN theory there should be only one (small) N=4 superconformal
algebra. Here we have two copies. Why do we have two copies of the Ramond
sector (small) N=4 superconformal algebra is the problem we are trying to
resolve in this note.

##  4 The GKS Resolution

One immediate idea to interpret this superalgebra doubling is to propose that
the second equation of (3) is the copy of superalgebra corresponding to the
other chirality. GKS reject this possibility on the basis of two rather stark
arguments. Firstly the left and right moving generators commute or anti-
commute, that is, have zero commutators or anti-commutators with the other
chirality. From third of Eqn.(3) it is clear that the superalgebra of two
copies of the superalgebra have non-zero anti-commutators. Secondly the
worldsheet and spacetime chiralities in string theory are related. Thus a
single worldsheet chirality giving rise to both chiralities in the spacetime
theory is not feasible. Moreover the other chirality of worldsheet too will
give rise to both chiralities on the spacetime side.

The GKS solution to the Ramond superalgebra doubling is to propose that the
two superalgebras are different Ramond twists [30] of the NS generators

$\displaystyle T^{3}_{n}(\eta)$ $\displaystyle=$ $\displaystyle
T^{3}_{n}(0)-\frac{\eta
kp}{2}\delta_{n,0},~{}T^{\pm}_{n\pm\eta}=T^{\pm}_{n}(0),$ $\displaystyle
Q^{1}_{n+\frac{\eta}{2}}(\eta)$ $\displaystyle=$ $\displaystyle
Q^{1}_{\eta}(0),~{}~{}Q^{2}_{n-\frac{\eta}{2}}(\eta)=Q^{2}_{n}(0),$
$\displaystyle L_{n}(\eta)$ $\displaystyle=$ $\displaystyle L_{n}(0)-\eta
T^{3}_{n}(0)+\eta^{2}\frac{kp}{4}\delta_{n,0}.$ (33)

Now since both vacua cannot be described at the same time they proposed the
following procedure to restrict to single sector of the vacuum. If $J(z)$
contains the combination $J^{3}-K^{3}$ then the physical states are defined by
restricting to the cohomology of operators $Q_{\alpha}^{\pm}$:

$\displaystyle
Q_{\alpha}^{\pm}|phy\rangle=0,~{}|phy\rangle\sim|phy\rangle+Q|anything\rangle.$
(34)

Because of (3) such states have $J^{3}-K^{3}=0$. After that the four remaining
supercharges $\bar{Q}_{\bar{\alpha}}^{\pm}$ together with $J^{3}+K^{3}$ form
the left moving small N=4 superconformal algebra

$\displaystyle\left[{L}_{m},{L}_{n}\right]$ $\displaystyle=$
$\displaystyle(m-n){L}_{m+n}+\frac{{c}}{12}(m^{3}-m)\delta_{{m+n},0},$
$\displaystyle\left[{L}_{m},{G}_{r}^{a}\right]$ $\displaystyle=$
$\displaystyle\left(\frac{m}{2}-r\right){G}^{a}_{m+r},~{}~{}~{}\left[{L}_{m},{J}^{i}_{n}\right]=-n{J}^{i}_{m+n},$
$\displaystyle\\{{G}^{a}_{r},{G}^{b}_{s}\\}$ $\displaystyle=$ $\displaystyle
2\delta^{ab}{L}_{r+s}-4(r-s)\alpha^{+i}_{ab}{J}^{i}_{r+s}+\frac{{c}}{3}\left(r^{2}-\frac{1}{4}\right)\delta^{ab}\delta_{{r+s},0},$
$\displaystyle\left[{J}^{i}_{m},{G}^{a}_{r}\right]$ $\displaystyle=$
$\displaystyle\alpha^{+i}_{ab}{G}^{b}_{m+r},~{}~{}~{}~{}\left[{J}^{i}_{m},{J}^{j}_{n}\right]=\epsilon^{ijk}{J}^{k}_{m+n}-m\frac{{k}}{2}\delta^{ij}\delta_{m+n,0},$
$\displaystyle\alpha^{+i}_{ab}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\delta^{i}_{a}\delta^{0}_{b}-\delta^{i}_{b}\delta^{0}_{a}\right)+\frac{1}{2}\epsilon^{iab},~{}~{}~{}{c}=6{k}.$
(35)

When the current $J(z)$ contains $J^{3}+K^{3}$ the roles of $Q$ and $\bar{Q}$
as well as $J^{3}-K^{3}$ and $J^{3}+K^{3}$ are reversed.

By this procedure the states of the other chirality are killed. Moreover, GKS
argue, that this procedure ensures $P^{i}=0$ using the unitarity argument. The
projecting out of the states of other chirality by Eqn.(34) the worldsheet and
spactime chirality correlation is restored. They also compare the projection
(34) to the topological twist [31, 32, 33] of N=2 theory whose physical states
are in one to one correspondence with the chiral ring of the original theory.

This resolution of the superalgebra doubling is admittedly conjectural and
kills half of the spectrum in an ad hoc manner.

##  5 Our Resolution

To present our resolution to the problem of doubling of Ramond superalgebra we
need some background from Refs.[23, 24, 25] (see also Ref. [34]). In [25] an
ADHM instanton sigma model was constructed whose action is

$S_{W}=S^{kin}+S^{int}$ (36)

with

$\displaystyle S^{kin}$ $\displaystyle=$ $\displaystyle\int
d^{2}\sigma(\epsilon_{AB}\epsilon_{YZ}\partial_{-}X^{AY}\partial_{+}X^{BZ}+i\epsilon_{A^{\prime}B^{\prime}}\epsilon_{YZ}\psi_{-}^{A^{\prime}Y}\partial_{+}\psi_{-}^{B^{\prime}Z}$
(37) $\displaystyle+$
$\displaystyle\epsilon_{A^{\prime}B^{\prime}}\epsilon_{Y^{\prime}Z^{\prime}}\partial_{-}\phi^{A^{\prime}Y^{\prime}}\partial_{+}\phi^{B^{\prime}Z^{\prime}}+i\epsilon_{AB}\epsilon_{Y^{\prime}Z^{\prime}}\chi_{-}^{AY^{\prime}}\partial_{+}\chi_{-}^{BZ^{\prime}}$
$\displaystyle+$ $\displaystyle i\lambda_{+}^{a}\partial_{-}\lambda_{+}^{a})$

and

$\displaystyle S^{int}$ $\displaystyle=$ $\displaystyle-\frac{i}{2}m\int
d^{2}\sigma\lambda^{a}_{+}[(\epsilon^{BD}\frac{\partial
C^{a}_{BB^{\prime}}}{\partial
X^{DY}}\psi^{B^{\prime}Y}_{-}+\epsilon^{B^{\prime}D^{\prime}}\frac{\partial
C^{a}_{BB^{\prime}}}{\partial\phi^{D^{\prime}Y^{\prime}}}{\chi}^{BY^{\prime}}_{-})$
(38) $\displaystyle-$
$\displaystyle\frac{im}{4}\epsilon^{AB}\epsilon^{A^{\prime}B^{\prime}}C^{a}_{AA^{\prime}}C^{a}_{BB^{\prime}})]$

It has $4k$ bosons $X^{AY}$, $A=1,2,$ $Y=1,2,...,2k$ and $4k^{\prime}$ bosons
$\phi^{A^{\prime}Y^{\prime}}$, $A^{\prime}=1,2,$
$Y^{\prime}=1,2,...,2k^{\prime}$ and corresponding right-handed superpartners
$\psi_{-}^{A^{\prime}Y}$ for $X^{AY}$, and $\chi_{-}^{AY^{\prime}}$ for
$\phi^{A^{\prime}Y^{\prime}}$. There are also left-handed fermions
$\lambda_{+}^{a}$, $a=1,2,...,n$. The indices are raised and lowered by taking
the help of antisymmetric tensors $\epsilon^{AB}(\epsilon_{AB})$,
$\epsilon^{A^{\prime}B^{\prime}}(\epsilon_{A^{\prime}B^{\prime}})$,
$\epsilon^{YZ}(\epsilon_{YZ})$ and
$\epsilon^{Y^{\prime}Z^{\prime}}(\epsilon_{Y^{\prime}Z^{\prime}})$ belonging
to $SU(2)$, $SU(2)^{\prime}$, $Sp(2k)$ and $Sp(2k^{\prime})$ respectively. The
conditions on $C^{a}_{AA^{\prime}}$ are

$\displaystyle\frac{\partial C_{AA^{\prime}}^{a}}{\partial
X^{BY}}+\frac{\partial C_{BA^{\prime}}^{a}}{\partial X^{AY}}=0=\frac{\partial
C_{AA^{\prime}}^{a}}{\partial\phi^{B^{\prime}Y^{\prime}}}+\frac{\partial
C_{AB^{\prime}}^{a}}{\partial\phi^{A^{\prime}Y^{\prime}}},$ (39)

$\sum_{a}(C^{a}_{AA^{\prime}}{C}^{a}_{BB^{\prime}}+{C}^{a}_{BA^{\prime}}{C}^{a}_{AB^{\prime}})=0.$
(40)

In Ref.[23] we constructed a Complementary ADHM instanton sigma model with the
action

$S_{C}=\hat{S}^{kin}+\hat{S}^{int}$ (41)

with

$\displaystyle\hat{S}^{kin}$ $\displaystyle=$ $\displaystyle\int
d^{2}\sigma(\epsilon_{AB}\epsilon_{YZ}\partial_{-}X^{AY}\partial_{+}X^{BZ}+i\epsilon_{A^{\prime}B^{\prime}}\epsilon_{YZ}\psi_{-}^{A^{\prime}Y}\partial_{+}\psi_{-}^{B^{\prime}Z}$
(42) $\displaystyle+$
$\displaystyle\epsilon_{A^{\prime}B^{\prime}}\epsilon_{Y^{\prime}Z^{\prime}}\partial_{-}\phi^{A^{\prime}B^{\prime}}\partial_{+}\phi^{B^{\prime}Z^{\prime}}+i\epsilon_{AB}\epsilon_{Y^{\prime}Z^{\prime}}\chi_{-}^{AY^{\prime}}\partial_{+}\chi_{-}^{BZ^{\prime}}$
$\displaystyle+$ $\displaystyle
i\hat{\lambda}_{+}^{a^{\prime}}\partial_{-}\hat{\lambda}_{+}^{a^{\prime}})$

and

$\displaystyle\hat{S}^{int}$ $\displaystyle=$ $\displaystyle-\frac{i}{2}m\int
d^{2}\sigma\hat{\lambda}^{a^{\prime}}_{+}[(\epsilon^{BD}\frac{\partial\hat{C}^{a^{\prime}}_{BB^{\prime}}}{\partial
X^{DY}}\psi^{B^{\prime}Y}_{-}+\epsilon^{B^{\prime}D^{\prime}}\frac{\partial\hat{C}^{a^{\prime}}_{BB^{\prime}}}{\partial\phi^{D^{\prime}Y^{\prime}}}{\chi}^{BY^{\prime}}_{-})$
(43) $\displaystyle-$
$\displaystyle\frac{im}{4}\epsilon^{AB}\epsilon^{A^{\prime}B^{\prime}}\hat{C}^{a^{\prime}}_{AA^{\prime}}\hat{C}^{a^{\prime}}_{BB^{\prime}})]$

with the conditions

$\displaystyle\frac{\partial\hat{C}_{AA^{\prime}}^{a^{\prime}}}{\partial
X^{BY}}+\frac{\partial\hat{C}_{BA^{\prime}}^{a^{\prime}}}{\partial
X^{AY}}=0=\frac{\partial\hat{C}_{AA^{\prime}}^{a^{\prime}}}{\partial\phi^{B^{\prime}Y^{\prime}}}+\frac{\partial\hat{C}_{AB^{\prime}}^{a^{\prime}}}{\partial\phi^{A^{\prime}Y^{\prime}}},$
(44)

$\sum_{a^{\prime}}(\hat{C}^{a^{\prime}}_{AA^{\prime}}{\hat{C}}^{a^{\prime}}_{BB^{\prime}}+{\hat{C}}^{a^{\prime}}_{BA^{\prime}}{\hat{C}}^{a^{\prime}}_{AB^{\prime}})=0$
(45)

The tensors $C^{a}_{AA^{\prime}}$ and $\hat{C}^{a^{\prime}}_{AA^{\prime}}$ are
linear in both $X$ and $\phi$ and their general expression are

$C^{a}_{AA^{\prime}}=M^{a}_{AA^{\prime}}+\epsilon_{AB}N^{a}_{A^{\prime}Y}X^{BY}+\epsilon_{A^{\prime}B^{\prime}}D^{a}_{AY^{\prime}}\phi^{~{}B^{\prime}Y^{\prime}}+\epsilon_{AB}\epsilon_{A^{\prime}B^{\prime}}E^{a}_{YY^{\prime}}X^{BY}\phi^{~{}B^{\prime}Y^{\prime}}$
(46)

and

$\hat{C}^{a^{\prime}}_{AA^{\prime}}=\hat{M}^{a^{\prime}}_{AA^{\prime}}+\epsilon_{AB}\hat{N}^{a^{\prime}}_{A^{\prime}Y}X^{BY}+\epsilon_{A^{\prime}B^{\prime}}\hat{D}^{a^{\prime}}_{AY^{\prime}}\phi^{~{}B^{\prime}Y^{\prime}}+\epsilon_{AB}\epsilon_{A^{\prime}B^{\prime}}\hat{E}^{a^{\prime}}_{YY^{\prime}}X^{BY}\phi^{~{}B^{\prime}Y^{\prime}}.$
(47)

For the one instanton cases the potentials in action (36) and 41 become

$V_{W}=\frac{m^{2}}{8}(X^{2}+\rho^{2})\phi^{2}~{}\text{and}~{}V_{C}=\frac{m^{2}}{8}(\phi^{2}+\omega^{2})X^{2}.$
(48)

These two models are independent of each other and there is a duality between
them given by the following interchanges
$X\leftrightarrow\phi,~{}~{}k\leftrightarrow
k^{\prime},~{}~{}\psi\leftrightarrow\chi$.

We assert that the two copies of the Ramond superalgebra that Giveon-Kutasov-
Seiberg get in their construction are the two distinct and independent
branches of the theory just like the two branches of the moduli space of the
ADHM instanton sigma model. Thus these should be treated separately. In
particular there is no need to kill the Hilbert space of physical states of
one branch or the other.

##  6 Discussion

Above resolution comes with several questions. For example why did we not
discover the other branch of the moduli space so far? Here we would like
recall that as long back as 1988 Sevrin, Troost and van Proeyen [12] told us
that the small N=4 superconformal algebra is contained in large N=4
superconformal algebra in two different ways. They also pointed out that there
is a duality between these two embeddings. Thus the doubling of small N=4
superconformal algebra is known for a long time.

Another question concerns whether the lessons learned from ADHM instanton
sigma models can be applied to $AdS_{3}$ superstrings or not. Inhis regard it
is a long held belief that the ADHM instanton sigma model with (small) N=4
supersymmetry and flow to a superconformal limit in the infrared [35].

Finally it is clear from our construction of the Complete ADHM instanton sigma
model [24] that the duality between the original and Complementary models
become a $Z_{2}$ symmetry in case of the Complete model. This $Z_{2}$ symmetry
plays a role in getting the complete free field realizations of the large N=4
superconformal symmetry [36] that is relevant for $AdS_{3}\times S^{3}\times
S^{3}\times S^{1}$ superstrings [7].

Acknowledgements: These investigations were made possible by the academic
environment created by my students Dr. P.P. Abdul Salih, Dr. Shafeeq Rahman
Thottoli and Dr. Mohsin Ilahi. I thank Prof. D.P. Jatkar and the Harish-
Chandra Research Institute, Prayagraj, where part of this work was done, for
hospitality.

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