# Complementary ADHM Instanton Sigma Model

Abbas Ali and Mohsin Ilahi
Physics Department, Aligarh Muslim University,
Aligarh, India

###### Abstract

We construct the ADHM linear sigma model that is complementary to a model
constructed by Edward Witten in 1995. Analysis of the corresponding moduli
space leads to the resolution of a nearly three decade old mystery associated
with the original construction. A number of problems are listed on which this
construction has a bearing.

The $SU(2)$ ’t Hooft instanton is a finite energy, as opposite to
perturbatively small energy, classical solutions to field theory equations.
Ever since their discovery these remain constantly fascinating. These have
been quantized [1], generalized and otherwise investigated intensely and
extensively. Their applications are wide ranging[2]. These are amenable to a
stringy generalization [3, 6, 4, 5]. The original solution depends on $5k+4$
parameters. It is well known that the most general instanton solution should
depend on $8k$ parameters for the $SU(2)$ gauge group. In 1978 Atiyah-
Drinfeld-Hitchin-Manin came up with a remarkable construction, now known as
the ADHM construction of Yang-Mills instantons, in [7]. This depends on full
set of $8k$ parameters.

On the mathematical side the ADHM construction is exceptionally brief and
incredibly linear, particularly in view of the fact that we are talking about
very non-linear system - the Yang-Mills theory. The perplexing mathematical
structure has been investigated, for example, in Refs. [8, 9, 10, 11].

On the physical side the situation is very clear and lucid. Early on Witten
pointed out that it should not be missed out that instantons are self-dual
Yang-Mills fields and hence solutions to first order partial differential
equations[12]. This is easier than the original problem of Yang-Mills
equations which happen to be second order coupled partial differential
equations. Around the same time it was clarifies in Refs.[13, 14, 15, 16] that
issue at hand concerns inverting a matrix equation with quaternion entries.

Corresponding stringy generalization was constructed by Witten in Ref.[17].
Its quantization was done in Ref. [18] (see also [19]). D-branes were included
in the ambit in Refs.[20, 21, 22].

The moduli space of this model happens to be rather mysterious. In this note
we present a complementary construction that was suggested Witten in above
reference and analyze corresponding moduli space. The long standing mystery
about the moduli space gets resolved in this process. We first collect the
notation and then construct the ADHM instanton sigma model complementary to
Witten’s construction. We then specialize to one instanton case for $SU(2)$
group. After that we cover those aspects of the complementary model that are
parallel with the original model. The comparison of the moduli space of the
resulting model with the moduli space of the original model gives surprising
and clear insight about the corresponding structure and above mentioned
resolution of the mystery. We then then summarize the additional results that
we have obtained based on this experience. We conclude with the listing of our
work under progress and future possibilities.

We begin by collecting the notation from original construction and Ref.[18].
The action for ADHM sigma model is

$\displaystyle S$ $\displaystyle=$ $\displaystyle\int
d^{2}\sigma\biggl{\\{}\partial_{-}{X}_{AY}\partial_{+}X^{AY}+i\epsilon_{A^{\prime}B^{\prime}}\epsilon_{YZ}{\psi}_{-}^{A^{\prime}Y}\partial_{+}\psi_{-}^{B^{\prime}Z}$
(1) $\displaystyle+$
$\displaystyle\partial_{-}{\phi}_{A^{\prime}Y^{\prime}}\partial_{+}\phi^{A^{\prime}Y^{\prime}}+i\epsilon_{AB}\epsilon_{Y^{\prime}Z^{\prime}}\chi_{-}^{AY^{\prime}}\partial_{+}\chi_{-}^{BZ^{\prime}}+i\lambda_{+}^{a}\partial_{-}\lambda_{+}^{a}$
$\displaystyle-$
$\displaystyle\frac{i}{2}m\lambda_{+}^{a}\Big{(}\epsilon^{AB}\frac{\partial
C^{a}_{AB^{\prime}}}{\partial
X^{BY}}\psi_{-}^{B^{\prime}Y}+\epsilon^{A^{\prime}B^{\prime}}\frac{\partial
C^{a}_{AA^{\prime}}}{\partial\phi^{B^{\prime}Y^{\prime}}}\chi_{-}^{AY^{\prime}}\Big{)}$
$\displaystyle-$
$\displaystyle\frac{1}{8}m^{2}\epsilon^{AB}\epsilon^{A^{\prime}B^{\prime}}C^{a}_{AA^{\prime}}C^{a}_{BB^{\prime}}\biggr{\\}}$

with $\sigma^{\pm}=(\tau\pm\sigma)/2$ and
$\partial_{\pm}=\frac{1}{\sqrt{2}}(\partial_{0}\pm\partial_{1})$ where $\tau$
and $\sigma$ are lightcone coordinates, the worldsheet metric being
$ds^{2}=d\tau^{2}-d\sigma^{2}$. $X^{AY}$, $A=1,2$ and $Y=1,2,...,2k$ being
$4k$ bosons are part of the standard multiplet and these have right handed
superpartners $\psi_{-}^{A^{\prime}Y},A^{\prime}=1,2$. We also have another
set of $4k^{\prime}$ bosons $\phi^{A^{\prime}Y^{\prime}}$, part of the twisted
multiplet, with corresponding superpartners $\chi_{-}^{AY^{\prime}}$ with
$Y^{\prime}=1,2,...,2k^{\prime}$. The raising and lowering of $Y,Z,...$ and
$Y^{\prime},Z^{\prime},...$ indices is done with respective $Sp(k)$ and
$Sp(k^{\prime})$ tensors $\epsilon^{YZ}(\epsilon_{YZ})$,
$\epsilon^{Y^{\prime}Z^{\prime}}(\epsilon_{Y^{\prime}Z^{\prime}})$. Similarly
the $A,B,...$ and $A^{\prime},B^{\prime},...$ indices are raised (lowered)
with the help of antisymmetric tensors $\epsilon^{AB}(\epsilon_{AB})$,
$\epsilon^{A^{\prime}B^{\prime}}(\epsilon_{A^{\prime}B^{\prime}})$
respectively. The free theory has an $F\times F^{\prime}\times H\times
H^{\prime}$ symmetry that acts on $AB,YZ,A^{\prime}B^{\prime}$ and
$Y^{\prime}Z^{\prime}$ indices respectively where $F=SU(2)$,
$F^{\prime}=SU(2)$, $H=Sp(k)$ and $H^{\prime}=Sp(k^{\prime})$. This is
generally broken by the potential terms.

To begin with there is a $Z_{2}$ symmetry between the standard and the twisted
multiplet with respect to the interchange of $F$ and $F^{\prime}$ (and $H$ and
$H^{\prime}$). In the original construction this was broken because only the
$F^{\prime}$ symmetry was retained. As a result we got an action for
$k^{\prime}$ instanton number and $4k+4k^{\prime}$ target space dimensions.
This $Z_{2}$ symmetry will be broken in the present complementary construction
too because we shall retain only th $F$ symmetry and get the action for $k$
instanton number and $4k+4k^{\prime}$ target space dimensions.

At this point one issue has to be emphasized. The present complementary
construction, though straightforward, is not trivial. In particular the task
can not be assigned to a neural network nor we can use ML for this
construction.

This chiral sigma model has $(0,4)$ on-shell supersymmetry

$\displaystyle\delta_{\eta}X^{AY}$ $\displaystyle=$ $\displaystyle
i\epsilon_{A^{\prime}B^{\prime}}\eta^{AA^{\prime}}_{+}\psi^{B^{\prime}Y}_{-},\delta_{\eta}\psi^{A^{\prime}Y}_{-}=\epsilon_{AB}\eta^{AA^{\prime}}_{+}\partial_{-}X^{BY},$
$\displaystyle\delta_{\eta}\phi^{A^{\prime}Y^{\prime}}$ $\displaystyle=$
$\displaystyle
i\epsilon_{AB}\eta^{AA^{\prime}}_{+}{\chi}^{BY^{\prime}}_{-},\delta_{\eta}{\chi}^{AY^{\prime}}_{-}=\epsilon_{A^{\prime}B^{\prime}}\eta^{AA^{\prime}}_{+}\partial_{-}\phi^{B^{\prime}Y^{\prime}}$
(2)

with $\eta^{AA^{\prime}}_{+}$ being an infinitesimal anti-commuting spinor
parameter. The reality conditions on the bosons are

$X^{AY}=\epsilon^{AB}\epsilon^{YZ}\overline{X}_{BZ},~{}~{}~{}~{}~{}~{}\phi^{A^{\prime}Y^{\prime}}=\epsilon^{A^{\prime}B^{\prime}}\epsilon^{Y^{\prime}Z^{\prime}}\overline{\phi}_{B^{\prime}Z^{\prime}}$
(3)

with similar conditions for $\psi_{-}^{A^{\prime}Y}$ and
$\chi_{-}^{AY^{\prime}}$.

The supercharges $Q^{AA^{\prime}}$ satisfy

$Q^{AA^{\prime}}=\epsilon^{AB}\epsilon^{A^{\prime}B^{\prime}}{Q^{\dagger}_{BB^{\prime}}}~{}~{},~{}~{}~{}~{}\\{Q^{AA^{\prime}},Q^{BB^{\prime}}\\}=\epsilon^{AB}\epsilon^{A^{\prime}B^{\prime}}P^{+}$
(4)

with $P^{+}=P_{-}=-i\partial_{-}$. The supersymmetry transformations satisfy
the algebra

$[\delta_{\eta^{\prime}},\delta_{\eta}]=-i\epsilon_{AB}\epsilon_{A^{\prime}B^{\prime}}\eta^{AA^{\prime}}_{+}\eta{{}^{\prime}}^{BB^{\prime}}_{+}\partial_{-}.$
(5)

Before describing the complementary construction we shall make a digression to
explain the supersymmetry structure involving the Yukawa coupling term. The
Lagrangian for the left moving fermions is obtained in the following manner.
One begins with the general Yukawa type interaction

$S_{\lambda}=-\frac{i}{2}\int
d^{2}\sigma\lambda^{a}_{+}G_{a\theta}{\rho}^{\theta}_{-}$ (6)

with $\rho_{-}^{\theta}$ including all of $\psi^{\prime}s$ and
$\chi^{\prime}s$. The corresponding equation of motion for $\lambda^{a}_{+}$
is

$\partial_{-}\lambda^{a}_{+}=G^{a}_{\theta}{\rho}^{\theta}_{-}.$ (7)

The closure of the algebra (5) on $\lambda^{a}_{+}$ gives

$[\delta_{\eta^{\prime}},\delta_{\eta}]\lambda^{a}_{+}=-i\epsilon_{AB}\epsilon_{A^{\prime}B^{\prime}}\eta^{AA^{\prime}}_{+}\eta{{}^{\prime}}^{BB^{\prime}}_{+}\partial_{-}\lambda^{a}_{+}.$
(8)

Using the equation of motion (7) the algebra (8) becomes

$[\delta_{\eta^{\prime}},\delta_{\eta}]\lambda^{a}_{+}=-i\epsilon_{AB}\epsilon_{A^{\prime}B^{\prime}}\eta^{AA^{\prime}}_{+}\eta{{}^{\prime}}^{BB^{\prime}}_{+}\partial_{-}G^{a}_{\theta}{\rho}^{\theta}.$
(9)

Assuming a general supersymmetry transformation for the $\lambda^{a}_{+}$ one
writes

$\delta_{\eta}\lambda^{a}_{+}=\eta^{AA^{\prime}}_{+}C^{a}_{AA^{\prime}}$ (10)

with tensor $C^{a}_{AA^{\prime}}$ being linear in $X^{\prime}s$ and
$\phi^{\prime}s$ such that

$\delta_{\eta^{\prime}}\delta_{\eta}\lambda^{a}_{+}=i\eta^{AA^{\prime}}_{+}\biggl{(}\frac{\partial{C}^{a}_{AA^{\prime}}}{\partial
X^{BY}}\epsilon_{B^{\prime}C^{\prime}}\eta^{{}^{\prime}BB^{\prime}}_{+}\psi^{C^{\prime}Y}_{-}+\frac{\partial
C^{a}_{AA^{\prime}}}{\partial\phi^{B^{\prime}Y^{\prime}}}\epsilon_{BC}\eta^{{}^{\prime}BB^{\prime}}_{+}{\chi}^{CY^{\prime}}_{-}\biggr{)}.$
(11)

From (10) we also have

$\delta_{\eta^{\prime}}\lambda^{a}_{+}=\eta^{\prime
BB^{\prime}}_{+}C^{a}_{BB^{\prime}}$ (12)

such that

$\delta_{\eta}\delta_{\eta^{\prime}}\lambda^{a}_{+}=i\eta^{\prime
BB^{\prime}}_{+}\biggl{(}\frac{\partial{C}^{a}_{BB^{\prime}}}{\partial
X^{AY}}\epsilon_{A^{\prime}C^{\prime}}\eta^{AA^{\prime}}_{+}\psi^{C^{\prime}Y}_{-}+\frac{\partial
C^{a}_{BB^{\prime}}}{\partial\phi^{A^{\prime}Y^{\prime}}}\epsilon_{AC}\eta^{AA^{\prime}}_{+}{\chi}^{CY^{\prime}}_{-}\biggr{)}.$
(13)

Combining (11) and (13) one gets

$\displaystyle[\delta_{\eta^{\prime}},\delta_{\eta}]\lambda^{a}_{+}$
$\displaystyle=$ $\displaystyle
i\eta^{AA^{\prime}}_{+}\eta{{}^{\prime}}^{BB^{\prime}}_{+}\biggl{[}\left(\epsilon_{B^{\prime}C^{\prime}}\frac{\partial{C}^{a}_{AA^{\prime}}}{\partial
X^{BY}}-\epsilon_{A^{\prime}C^{\prime}}\frac{\partial{C}^{a}_{BB^{\prime}}}{\partial
X^{AY}}\right)\psi^{C^{\prime}Y}_{-}$ (14) $\displaystyle+$
$\displaystyle\left(\epsilon_{BC}\frac{\partial
C^{a}_{AA^{\prime}}}{\partial\phi^{B^{\prime}Y^{\prime}}}-\epsilon_{AC}\frac{\partial
C^{a}_{BB^{\prime}}}{\partial\phi^{A^{\prime}Y^{\prime}}}\right){\chi}^{CY^{\prime}}_{-}\biggr{]}.$

The two terms on the RHS are independent and hence separately proportional to
$\epsilon_{AB}\epsilon_{A^{\prime}B^{\prime}}$. These should vanish if we
multiply these by $\delta^{AB}$ $\delta^{A^{\prime}B^{\prime}}$. This gives us
the relations

$\displaystyle\frac{\partial C_{AA^{\prime}}^{a}}{\partial
X^{BY}}+\frac{\partial C_{BA^{\prime}}^{a}}{\partial X^{AY}}=0=\frac{\partial
C_{AA^{\prime}}^{a}}{\partial\phi^{B^{\prime}Y^{\prime}}}+\frac{\partial
C_{AB^{\prime}}^{a}}{\partial\phi^{A^{\prime}Y^{\prime}}}.$ (15)

The supersymmetry algebra above is satisfied if we take

$G^{a}_{\theta}\rho^{\theta}=\frac{1}{2}\biggl{[}\epsilon^{BD}\frac{\partial
C^{a}_{BB^{\prime}}}{\partial
X^{DY}}\psi^{B^{\prime}Y}_{-}+\epsilon^{B^{\prime}D^{\prime}}\frac{\partial
C^{a}_{BB^{\prime}}}{\partial\phi^{D^{\prime}Y^{\prime}}}{\chi}^{BY^{\prime}}_{-}\biggr{]}.$
(16)

This way one ends up with the Yukawa term in Eq.(1) beginning with the ansatz
(7).

The tensors $C^{a}_{AA^{\prime}}$ obey

$\sum_{a}(C^{a}_{AA^{\prime}}{C}^{a}_{BB^{\prime}}+{C}^{a}_{BA^{\prime}}{C}^{a}_{AB^{\prime}})=0.$
(17)

To obtain the ADHM equations we begin the following general form for
$C^{a}_{AA^{\prime}}$ that is linear in both $X$ and $\phi$

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

To construct the complementary branch of the ADHM sigma model we shall
maintain $F$ invariance and break $F^{\prime}$ invariance by taking $M=D=0$
and thus we have

$\displaystyle{C}^{a}_{AA^{\prime}}$ $\displaystyle=$
$\displaystyle\epsilon_{AB}N^{a}_{A^{\prime}Y}X^{BY}+\epsilon_{AB}\epsilon_{A^{\prime}B^{\prime}}E^{a}_{YY^{\prime}}X^{BY}\phi^{~{}B^{\prime}Y^{\prime}}$
(19) $\displaystyle=$ $\displaystyle X_{A}^{~{}Y}A^{a}_{A^{\prime}Y}$

where the tensor $A^{a}_{A^{\prime}Y}$ is linear in $\phi$. The condition in
Eqn.(17) this time becomes

$\sum_{a}(A^{a}_{A^{\prime}Y}A^{a}_{B^{\prime}Z}+A^{a}_{B^{\prime}Y}A^{a}_{A^{\prime}Z})=0.$
(20)

In the original construction $F^{\prime}$ invariance was maintained and $F$
invariance broken by taking $M=N=0$ and with

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

such that

$\sum_{a}(B^{a}_{AY^{\prime}}B^{a}_{BZ^{\prime}}+B^{a}_{BY^{\prime}}B^{a}_{AZ^{\prime}})=0.$
(22)

There is a duality between the two choices. This corresponds to interchange of
$X\leftrightarrow\phi$ , $\psi\leftrightarrow\chi$ and $k\leftrightarrow
k^{\prime}$ 111The complementary and original constructions can be termed
A-and B-models respectively because of the terms $A^{a}_{A^{\prime}Y}$ and
$B^{a}_{AY^{\prime}}$. This is not far fetched from the names used for
topological A-and B-models obtained by twisting N=2 superconformal algebra..
In a degenrate case the duality turns into a $Z_{2}$ symmetry.

For the original construction $C$ is homogeneous and linear in $\phi$ and the
potential quadratic. In particular the moduli space or the space of vacua is
$\mathcal{M}=R^{4k}$ ($X^{AY},A=1,2$ and $Y=1,2,..,2k^{\prime}$) that is given
by $\phi=0$ and any $X$. For the complementary branch $C$ is linear and
homogeneous in $X$ and the potential again quadratic. This time the moduli
space is given by $X=0$ and any $\phi$ and thus it is
$\mathcal{M^{\prime}}=R^{4k^{\prime}}$
($\phi^{A^{\prime}Y^{\prime}},A^{\prime}=1,2$ and
$Y^{\prime}=1,2,..,2k^{\prime}$). Both $\mathcal{M}$ and
$\mathcal{M^{\prime}}$ have hyper-Kähler structure due to (0, 4)
supersymmetry. The $\phi\leftrightarrow X$ and part of the duality is apparent
here. The $\psi\leftrightarrow\chi$ part will be apparent in a moment.

On the original branch $\mathcal{M}$ of the moduli space, that is for
$\phi=0$, the Yukawa couplings have the form
$\sum_{a}\lambda_{+}^{a}B^{a}_{AY^{\prime}}\chi^{AY^{\prime}}$ and such that
the fermionic partners of $X$, the $\psi_{-}$’s, are all massless. This is
expected from (0,4) supersymmetry.

On the complementary branch $\mathcal{M^{\prime}}$ of the moduli space we have
$X=0$ and the form of the Yukawa coupling is
$\sum_{a}\lambda_{+}^{a}A^{a}_{A^{\prime}Y}\psi^{A^{\prime}Y}$ and this time
$\chi_{-}$, the fermionic partners of $\phi$, are all massless.

In the original model if the number $n$ of components of $\lambda_{+}^{a}$ is
larger than $4k^{\prime}$, the number of components of
$\chi_{-}^{AY^{\prime}}$, then generically all of the $\chi_{-}$’s get mass
and $N=n-4k^{\prime}$ components of $\lambda_{+}$ are massless.

In the complementary model if the number $n$ of the left moving fermions
$\lambda^{a}_{+}$ is larger than $4k$, that is, the number of
$\psi^{A^{\prime}Y}_{-}$ ($A^{\prime}=1,2,Y=1,2,...,4k$) then all components
of $\psi_{-}$ get masses. By supersymmetry this happens when all components of
$X$ get masses. The $N^{\prime}=n-4k$ components of $\lambda^{a}_{+}$ are
massless.

We now work out the details for the complementary branch in parallel with the
original construction. Let $v^{a}_{i}$, $i=1,2,...,N^{\prime}$ be a basis for
massless components of $\lambda^{a}_{+}$. These are solutions to the equation

$\sum_{a}v^{a}_{i}A^{a}_{A^{\prime}Y}=0.$ (23)

We shall choose $v^{a}_{i}$ to be orthonormal, that is,

$\sum_{a}v^{a}_{i}v^{a}_{j}=\delta_{ij}.$ (24)

The tensor $A^{a}_{A^{\prime}Y}$ and hence $v^{a}_{i}$ are $\phi$-dependent.
There is an $SO(N)$ transformation on the index $i$. We shall take it as a
gauge invariance of the low energy theory. Setting

$\lambda^{a}_{+}=\sum_{i=1}^{N^{\prime}}v^{a}_{i}\lambda_{+i}$ (25)

amounts to putting the massive modes of $\lambda^{a}_{+}$ to zero. Here
$\lambda_{+i}$ are the massless left moving modes. Using
$\partial_{-}(v^{a}_{j}\lambda_{+j})=\partial_{-}v^{a}_{j}\lambda_{+j}+v^{a}_{j}\partial_{-}\lambda_{+j}$
and $\partial_{-}v^{a}_{j}=\frac{\partial
v^{a}_{j}}{\partial\phi^{A^{\prime}Y^{\prime}}}\partial_{-}\phi^{A^{\prime}Y^{\prime}}$
the kinetic energy term for massless $\lambda_{+i}$ becomes

$\displaystyle\frac{i}{2}\int
d^{2}\sigma\sum_{i,j,a}(v^{a}_{i}\lambda_{+i})\partial_{-}(v^{a}_{j}\lambda_{+j})$
(26) $\displaystyle=$ $\displaystyle\frac{i}{2}\int
d^{2}\sigma\sum_{i,j}\\{\lambda_{+i}[\delta_{ij}\partial_{-}+A_{ijA^{\prime}Y^{\prime}}\partial_{-}\phi^{A^{\prime}Y^{\prime}}]\lambda_{+j}\\}$

with

$A_{ijA^{\prime}Y^{\prime}}=\sum_{a}v^{a}_{i}\frac{\partial
v^{a}_{j}}{\partial\phi^{A^{\prime}Y^{\prime}}}$ (27)

which is the standard fermion coupling to spacetime gauge field
$A_{ijA^{\prime}Y^{\prime}}$. These gauge fields should be compatible with the
hyper-Kähler structure on the moduli space $\mathcal{M}={R}^{4k^{\prime}}$.
The covariant derivative for the original branch was
$\delta_{ij}\partial_{-}+A_{ijAY}\partial_{-}X^{AY}$ and the instanton

$A_{ijAY}=\sum_{a}v^{a}_{i}\frac{\partial v^{a}_{j}}{\partial X^{AY}}.$ (28)

Clearly the $v_{i}^{a}$’s of (27) and (28) are different.

The original construction was ADHM sigma model for an $SO(N)$ instanton with
instanton number $k^{\prime}$ given by the tensor $B^{a}_{AY^{\prime}}$
obeying Eqn.(22). It had a degeneracy implying that components of $\phi$ are
massive for all $X$. This will be apparent from the expression for potential
later on.

In the present construction we have got a complementary ADHM sigma model for
an $SO(N^{\prime})$ instanton with instanton number $k$ given by the tensor
$A^{a}_{A^{\prime}Y}$ obeying Eqn.(20). The degeneracy condition too is
complementary and it says that all the components of $X$ are massive for all
$\phi$. The structure of the potential for the complementary branch makes it
clear.

Both the formulas (27) and (28) for ADHM instanton gauge field are standard
and complementary to each other in the sense that pervades the two
constructions.

In the original model the two instantons are equivalent iff two
$B^{a}_{AY^{\prime}}$ tensors can be mapped onto each other by the action of
$SO(n)\times Sp(k^{\prime})$ that comes from the linear action of this group
$a$ and $Y^{\prime}$ indices of $\lambda^{a}_{+}$ and
$\phi^{A^{\prime}Y^{\prime}}$ respectively. For the complementary construction
two instantons are equivalent of the corresponding $A^{a}_{A^{\prime}Y}$
tensors are related by the transformation of the group $SO(n)\times Sp(k)$
acting on the indices $a$ and $Y$ of $\lambda^{a}_{+}$ and $X^{AY}$
respectively. We believe that the duality between the original and
complementary branches is the same that becomes more apparent after Douglas’
generalization to D-branes and that was pointed out in [19, 21].

This is the construction of the complementary ADHM instanton sigma model in
its generality. Now we shall specialize to the case for SU(2) group with
$k=k^{\prime}=1$. These are the complementary cases of $SU(2)$ instantons with
instanton number one in the two, original and complementary, models. The
$SU(2)$ instanton can be embedded in any larger group and the natural gauge
groups in the two models are $SO(N^{\prime})$ and $SO(N)$ respectively. We
shall take $N^{\prime}=N=4$ and use embedding of $SU(2)$ in $SO(4)$ given by
$SO(4)=SU(2)\times SU(2)$. If we identify the one instanton with one of the
$SU(2)$’s then the other $SU(2)$ will be a trivial global symmetry group of
the solution. The global $SU(2)$ for original and complementary constructions
will be called $K$ and $K^{\prime}$ respectively. We already have the global
symmetry group $F^{\prime}=SU(2)$ for original and $F=SU(2)$ for complementary
model.

Because of the rotation groups $SO(4)$ of $R^{4}$ for the original model and
$SO(4)^{\prime}$ of $R^{{}^{\prime}4}$ of complementary model we also have
global groups $SO(4)=SU(2)_{L}\times SU(2)_{R}$ for orginal model and
$SO(4)^{\prime}=SU(2)^{{}^{\prime}}_{L}\times SU(2)^{{}^{\prime}}_{R}$ for the
complementary model. Thus the total symmetry group for complementary model is
$F\times K^{\prime}\times SU(2)^{\prime}_{L}\times SU(2)^{\prime}_{R}\cong
SU(2)^{4}$ and for orginal model it is $F^{\prime}\times K\times
SU(2)_{L}\times SU(2)_{R}\cong SU(2)^{4}$.

For the $SU(2)$ instanton with intanton number there are four $X$’s and four
$\phi$’s and a global symmetry group $F\times F^{\prime}\times H\times
H^{\prime}$ as identified in the beginning. In the original construction it
was identified with $F^{\prime}\times K\times SU(2)_{L}\times SU(2)_{R}$. For
the the complementary construction we shall identify it with $F\times
K^{\prime}\times SU(2)^{\prime}_{L}\times SU(2)^{\prime}_{R}$.

To realize above global symmetry groups for the original model in the
Lagrangian action of $F\times F^{\prime}\times H\times H^{\prime}$ was needed
on left moving fermions $\lambda^{a}_{+}$. To get $N=4$ with $N=n-4k^{\prime}$
with $k^{\prime}=1$ the needed value of $n$ was 8 and so is it with the
complementary construction due to $N^{\prime}=4$ and $N^{\prime}=n-4k$ with
$k=1$.

Experimentation lead to the realization that $\lambda_{+}$’s of the original
model must be in
$(\frac{1}{2},0,0,\frac{1}{2})\oplus(0,0,\frac{1}{2},\frac{1}{2})$. The
$\lambda_{+}^{AY^{\prime}},A,Y=1,2$ and
$\lambda_{+}^{YY^{\prime}},Y,Y^{\prime}=1,2$ were chosen with the reality
conditions

$\displaystyle\lambda^{AY^{\prime}}_{+}=\epsilon^{AB}\epsilon^{Y^{\prime}Z^{\prime}}\overline{\lambda}_{+BZ^{\prime}},~{}$
$\displaystyle\lambda^{YY^{\prime}}_{+}=\epsilon^{YZ}\epsilon^{Y^{\prime}Z^{\prime}}\overline{\lambda}_{+ZZ^{\prime}}.$
(29)

In complementary construction the suitable representation turns out to be
$(0,\frac{1}{2},\frac{1}{2},0)\oplus(0,0,\frac{1}{2},\frac{1}{2})$ that is
$\lambda_{+}^{A^{\prime}Y}$ and $\lambda_{+}^{YY^{\prime}}$ with reality
conditions

$\displaystyle\lambda^{A^{\prime}Y}_{+}=\epsilon^{A^{\prime}B^{\prime}}\epsilon^{YZ}\overline{\lambda}_{+B^{\prime}Z},~{}$
$\displaystyle\lambda^{Y^{\prime}Y}_{+}=\epsilon^{YZ}\epsilon^{Y^{\prime}Z^{\prime}}\overline{\lambda}_{+ZZ^{\prime}}.$
(30)

The Yukawa coupling tensor ${C}^{a}_{BB^{\prime}}$ has two pieces

$\displaystyle{C}^{Y^{\prime}Y}_{~{}~{}~{}~{}BB^{\prime}}=X_{B}^{~{}Y}\phi^{~{}Y^{\prime}}_{B^{\prime}},~{}$
$\displaystyle{C}^{A^{\prime}Y}_{~{}~{}~{}~{}BB^{\prime}}=\frac{\omega}{\sqrt{2}}\delta^{A^{\prime}}_{~{}B^{\prime}}X^{~{}Y}_{B}.$
(31)

Here $\omega$ is like $\rho$ of the original construction-the size of the
complementary instanton. These expressions are complementary to the
expressions for the original model.

In both the constructions the number of components of $\lambda_{+}$ is
$2\times 2+2\times 2=8$. With the definitions of the original constructions

$X^{2}=\epsilon_{AB}\epsilon_{YZ}X^{AY}X^{BZ},~{}~{}\phi^{2}=\epsilon_{A^{\prime}B^{\prime}}\epsilon_{Y^{\prime}Z^{\prime}}\phi^{A^{\prime}Y^{\prime}}\phi^{B^{\prime}Z^{\prime}}.$
(32)

The potentials for the original and complementary models become

$V_{W}=\frac{1}{8}(X^{2}+\rho^{2})\phi^{2},~{}~{}~{}V_{C}=\frac{1}{8}(\phi^{2}+\omega^{2})X^{2}$
(33)

respectively.

To get the formula for instanton guage field of the complementary construction
in the familiar form the fermionic field ansatz is

$\displaystyle\lambda^{Y^{\prime}Y}=\frac{\omega\zeta^{~{}Y^{\prime}Y}_{+}}{\sqrt{\omega^{2}+\phi^{2}}},~{}$
$\displaystyle\lambda^{A^{\prime}Y}=-\frac{\sqrt{2}\phi^{A^{\prime}}_{~{}~{}Y^{\prime}}\zeta_{+}^{~{}Y^{\prime}Y}}{\sqrt{\omega^{2}+\phi^{2}}}.$
(34)

The kinetic energy for left moving fermions, then, becomes

$\displaystyle\lambda_{+Y^{\prime}Y}\partial_{-}\lambda^{Y^{\prime}Y}_{+}+\lambda_{+A^{\prime}Y}\partial_{-}\lambda^{A^{\prime}Y}_{+}=\zeta_{+Y^{\prime}Y}\partial_{-}\zeta^{~{}Y^{\prime}Y}_{+}-\zeta_{+Y^{\prime}Y}\times$
$\displaystyle\frac{\frac{1}{2}\epsilon_{A^{\prime}B^{\prime}}(\phi^{A^{\prime}Y^{\prime}}\partial_{-}\phi^{B^{\prime}Z^{\prime}}+\phi^{A^{\prime}Z^{\prime}}\partial_{-}\phi^{B^{\prime}Y^{\prime}})}{\phi^{2}+\omega^{2}}\zeta^{Y}_{+Z^{\prime}}$
(35)

which has the standard instanton expression.

This concludes the discussion of the special case of one instanton for $SU(2)$
gauge group for the complementary model. As expected the expressions are
complementary to the expressions for the original construction.

We shall now describe the main outcome of our construction by taking up the
discussion of the moduli spaces. We shall talk about the moduli space of the
original model, moduli space of the our complementary model and finally about
the resolution of the mystery of the moduli space of the original model as
pointed out by Witten in his paper on the original construction. We find that
the nearly three decades old mystery has a very simple resolution.

Let us recall the structure of the moduli space of the original model. From
the expression for the potential $V_{W}$ Eqn. (33) we see that $\phi$ has an
$X$ dependent mass term. We also get the moduli space for $\phi=0$ and any
$X$. In the small instanton regime we have $\rho\rightarrow 0$ the potential
becomes proportional to $\phi^{2}X^{2}$. In this case a new branch of the
moduli space opens for $X=0$ and any $\phi$. This is the mystery of the moduli
space of the original ADHM instanton sigma model constructed by Witten.

Witten revisted the problem of the moduli space in Ref. [23]. The mystery
persisted but some new insides were gained. In particular by comparing with
the Callan-Harvey-Strominger (CHS) instanton sigma model we realize in the
small instanton limit the two branches of the moduli space are infinitely far
away.

Let us take up the discussion of the moduli space of the complementary model.
Initially it is parallel to the discussion for the original model. From the
expression for the potential $V_{C}$ in Eqn.(33) we see that $X$ has a $\phi$
dependent mass term. Apart from that there is moduli space for $X=0$ and any
$\phi$. This is complementary to the structure in the original model, as
expected.

Even the supplementary analysis is in parallel with [23]. The metric obtained
after including terms of order $\alpha^{\prime}$ [3, 4], for the complementary
case, is given by

$ds^{2}=(d\phi)^{2}\Bigl{(}e^{2\phi_{0}}+8\alpha^{\prime}\frac{\phi^{2}+2\omega^{2}}{(\phi^{2}+\omega^{2})^{2}}+...\Bigr{)}.$
(36)

As in the original case the $X$ and $\phi$ branches are infinitely far away in
the small instanton limit.

The moduli spaces of original and complementary models develop a second branch
in the small instanton limit. The metric 36 and its counter part for the
original model develop a semi-infinite throat. This itself is a puzzle. This
was already sorted out in Ref.[18] where it was explained how to go from the
ADHM moduli space to the CHS moduli space.

We now present the resolution to the nearly three decade old mystery of the
moduli space of the original model. In the small instanton limit the the
potentials of the two models coincide and the appearance of the new branch of
the moduli space is the appearance of the moduli space of the other model.

Now the questin of the infinite distance between two branches in the Callan-
Harvey-Strominger analysis too has a natural explanation. It simply means that
the two models are independent of each other. In case of the finite instanton
there is a duality between the two models and that in particular means that
the two models are independent of each other. In the small instanton limit
this duality collapses to a $Z_{2}$ symmetry and hence the appearance of the
second branch of the moduli space. This is consistent with the well known fact
that in the small instanton limit the theory gets an enhanced symmetry[24]. We
got a glimpse of this duality long back in Ref.[21].

We shall now turn our focus to those issues that lead us to this exploration
and what have we learned about those issues from our indulgence in the present
topic. We came to this problem to get some insight about the issues related to
D1-D5 system and $AdS_{3}$ superstrings. In the $AdS_{3}/CFT_{2}$ version of
the famous Maldacena conjecture, the AdS/CFT Correspondence, the field theory
dual to strings moving on $AdS_{3}\times S^{3}\times M^{4}$ with
$M^{4}=K3,T^{4}$ and $S^{3}\times S^{1}$ has small, middle and large N=4
superconfromal symmetry, respectively [25, 26, 27, 28, 29, 30, 31]. In this
case we have the long standing problem of doubling of the Ramond superalgebra
mentioned in the Appendix B of the Ref.[31]. The complementary ADHM instanton
sigma model constructed in this note suggest a way to solves this quarter of a
century old problem. The doubling of Ramond superalgebra in this case is due
to the other branch of the theory [32]. The two algebras are simply related to
each other by the very same $Z_{2}$ duality that we have uncovered here. The
two branches are independent of each other. The same $Z_{2}$ symmetry is at
the core of the route to get to the proper $N=4$ superconformal field that
theory for $AdS_{3}\times S^{3}\times S^{3}\times S^{1}$ superstrings. In
Ref.[33] this two decade old problem [34] is resolved.

We have also analyzed the quantization of the complementary branch in Ref.[35]
along the lines adopted in Ref.[18] for the original ADHM instanton sigma
model. We have found that the mechanism for cancellation between
$\Gamma_{Div}(b)$ and $\Gamma_{Div}(f)$ in the two cases, original and
complementary, are different from each other. It was also suggested in the
original paper to construct a model that contains both of the symmetries $F$
and $F^{\prime}$. We have done that too in Ref. [36].

We next take up the discussion of those aspects of the complementary model
that are in parallel with the original model. To obtain stringy corrections in
both models renormalization group flow to the supposedly $(0,4)$ conformal
field theory in the infrared has to be considered. This involves integrating
out the massive fields rather than making these zero. The resulting theory may
or may not be conformally invariant. In the latter eventuality we have to
consider the additional flow of the massless fields.

In integrating out the massive fields the expansion parameter in the original
model wass $1/(X^{2}+\rho^{2})$ and in the complementary model it is
$1/(\phi^{2}+\omega^{2})$. In the original model the perturbation theory is
accurate for all $X$ for large $\rho$ while for small $\rho$ it is accurate
for large $X$ only. For the complementary model the perturbation theory is
accurate for all $\phi$ in case of large $\omega$ while for small $\omega$ it
is accurate only for large $\phi$.

The complementary construction also has something to say about the case when
the sigma model is formulated on curved two manifold. In spite of being super-
renormalizable the theory has a one-loop logarithmic divergence proportional
to worldsheet curvature $R$ when it is minimally coupled to world-sheet
metric. To avoid this one should add an extra term $X^{2}$. This spoils the
$Z_{2}$ symmetry that we get for vanishing instanton (see later). This is in
parallel with the original model.

We now conclude with a very brief discussion of future possibilities. We would
also like to mention that it would be interesting to discuss both of these
sigma models for general groups as a sigma model construction corresponding to
[13] generalization of ADHM construction. In [37] Witten’s ADHM instanton
sigma model was formulated in projective superspace. The complementary model
too can be cast in that formalism. In [38] we have formulated the
complementary ADHM instanton sigma model in harmonic superspace following
Galperin-Sokatchev [39].

We would like to conclude with the following remark. It is well known that
$N=4$ theories do not flow under renormalization group. The models considered
in this note too have the same property [18]. In view of that the journey from
ADHM (or even ’t Hooft) instanton to conformal field theory has a problem and
we do not have any insight about it.

Acknowledgements: This work was carried out as part of the Ph.D. Thesis of
Mohsin Ilahi. We are thankful to P.P Abdul Salih and Shafeeq Rahman Thottoli
for discussions.

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