August, 2016

Non-geometric Five-branes in Heterotic Supergravity

Shin Sasaki111shin-s(at)kitasato-u.ac.jp and Masaya
Yata222phymasa(at)nus.edu.sg

Department of Physics
Kitasato University
Sagamihara 252-0373, Japan
Department of Physics,
National University of Singapore
2, Science Drive 3, Singapore 117542, Singapore

We study T-duality chains of five-branes in heterotic supergravity where the
first order $\alpha^{\prime}$-corrections are present. By performing the
$\alpha^{\prime}$-corrected T-duality transformations of the heterotic
NS5-brane solutions, we obtain the KK5-brane and the exotic $5^{2}_{2}$-brane
solutions associated with the symmetric, the neutral and the gauge NS5-branes.
We find that the Yang-Mills gauge field in these solutions satisfies the self-
duality condition in the three- and two-dimensional transverse spaces to the
brane world-volumes. The $O(2,2)$ monodromy structures of the
$5^{2}_{2}$-brane solutions are investigated by the
$\alpha^{\prime}$-corrected generalized metric. Our analysis shows that the
symmetric $5^{2}_{2}$-brane solution, which satisfies the standard embedding
condition, is a T-fold and it exhibits the non-geometric nature. We also find
that the neutral $5^{2}_{2}$-brane solution is a T-fold at least at
$\mathcal{O}(\alpha^{\prime})$. On the other hand, the gauge $5^{2}_{2}$-brane
solution is not a T-fold but show unusual structures of space-time.

###### Contents

  1. 1 Introduction
  2. 2 Heterotic NS5-brane solutions
  3. 3 T-duality chains of five-branes and $\alpha^{\prime}$-corrected Buscher rule
    1. 3.1 Heterotic KK5-branes
    2. 3.2 Heterotic $5^{2}_{2}$-branes
  4. 4 Monodromy and T-fold
  5. 5 Conclusion and discussions
  6. A Smeared solutions for gauge type
    1. A.1 Defect gauge NS5-brane solution
    2. A.2 Smeared gauge KK5-brane solution

##  1 Introduction

Extended objects such as branes play an important role in superstring
theories. The U-duality [1], which makes non-trivial connections among
consistent superstring theories, relates various branes in each theory. It was
shown that M-theory compactified on $T^{d}$ has the U-duality symmetry group
$E_{d(d)}(\mathbb{R})$ in lower dimensions [2, 3, 4]. BPS branes in
superstring theories form lower-dimensional multiplets under the U-duality
group. For example, when we consider M-theory compactified on $T^{8}$, we have
the $E_{8(8)}(\mathbb{R})$ BPS point particle multiplet. The higher
dimensional origin of parts of these point particles is the ordinary branes
wrapped/unwrapped on cycles in $T^{8}$. Here, the ordinary branes are waves,
F-strings, D-branes, NS5-branes, Kaluza-Klein (KK) branes. However, there are
states whose higher dimensional origin does not trace back to the ordinary
branes. These states are called exotic states and their higher dimensional
origin is known as exotic branes [5, 6].

Among other things, an exotic brane in type II string theories known as the
$5^{2}_{2}$-brane, has been studied intensively [7, 8, 9, 10]. The most
tractable duality in string theory is the T-duality. Type II string theories
compactified on $T^{d}$ has T-duality symmetry group $O(d,d,\mathbb{Z})$. The
exotic $5^{2}_{2}$-brane is obtained by performing the T-duality
transformations along the transverse directions to the NS5-brane world-volume.
As its notation suggests, the $5^{2}_{2}$-brane has two isometries in the
transverse directions to the brane world-volume [3]111In this paper, we use
the notation $5^{2}_{2}$ in the sense that they are obtained by the T-duality
transformations twice on the heterotic NS5-branes.. Its tension is
proportional to $g_{s}^{-2}$ where $g_{s}$ is the string coupling constant.
Therefore the $5^{2}_{2}$-brane is a solitonic object of co-dimension two.
These co-dimension two objects in string theory, sometimes called defect
branes [11], exhibit specific properties [12, 13]. When one goes around the
center of the co-dimension two branes in the transverse two directions and
come back to the original point, the background geometry of the branes changes
according to the non-trivial monodromy. Therefore the metric and other
background fields are generically governed by multi-valued functions. In this
sense, they are called non-geometric [14]. However when the monodromy is given
by the symmetry group of the theory, which is generically the U-duality group
in string theory, then the non-geometry becomes healthy candidate of solutions
to string theory. This kind of solution is called U-fold. An important nature
of the exotic branes is that they are non-geometric objects [15]. Therefore
they are called non-geometric branes or Q-branes [16]. In particular, the
$5^{2}_{2}$-brane in type II string theories is a T-fold, whose monodromy is
given by the T-duality group $O(2,2)$.

The purpose of this paper is to study exotic branes in heterotic string
theories. Compared with type II string theories, exotic branes in type I and
heterotic string theories have been poorly understood. This is due to the non-
Abelian gauge field living in the space-time in these theories. Notably, the
Yang-Mills gauge field enters into the space-time action as the first order
$\alpha^{\prime}$-corrections. Due to the $\alpha^{\prime}$-corrections, the
Buscher rule [17] of the T-duality transformation for heterotic supergravity
is modified [18, 19, 20]. The most famous extended objects in heterotic
supergravity theories are the heterotic NS5-brane solutions. There are three
distinct NS5-brane solutions in heterotic supergravity [21, 22, 23]. We will
perform the T-duality transformations to these heterotic NS5-branes by the
$\alpha^{\prime}$-corrected Buscher rule and obtain new five-brane solutions.
We will then study the monodromy structures of these solutions.

The organization of this paper is as follows. In the next section, we
introduce the heterotic NS5-brane solutions known as the symmetric, neutral
and gauge types [21, 22]. In section 3, we introduce the isometries along the
transverse directions to the NS5-brane world-volumes and perform the T-duality
transformation by the $\alpha^{\prime}$-corrected Buscher rule. We obtain the
heterotic Kaluza-Klein five-brane (KK5-brane) solutions associated with the
three types of solutions. We then perform the second T-duality transformations
on the KK5-branes and write down the exotic $5^{2}_{2}$-brane solutions. In
section 4, we examine the monodromy structure of the heterotic
$5^{2}_{2}$-brane solutions by the $\alpha^{\prime}$-corrected generalized
metric. We show that the monodromy is given by the $O(2,2)$ T-duality group
for the symmetric and the neutral solutions while the gauge solution remains
geometric. Section 5 is devoted to conclusion and discussions. The explicit
form of the smeared gauge KK5-brane solution is found in appendix.

##  2 Heterotic NS5-brane solutions

In this section, we introduce the NS5-brane solutions in ten-dimensional
heterotic supergravity which is the low-energy effective theory of heterotic
superstring theory. Heterotic supergravity consists of the ten-dimensional
$\mathcal{N}=1$ gravity multiplet coupled with the $\mathcal{N}=1$ vector
multiplet. The relevant bosonic fields in heterotic supergravity are the
vielbein $e_{M}{}^{A}$, the dilaton $\phi$, the NS-NS $B$-field $B_{MN}$ and
the Yang-Mills gauge field $A_{M}$. Here $M,N,=0,\ldots,9$ are the curved
space indices while $A,B,\ldots=0,1,\ldots,9$ are the local Lorentz indices.
The Yang-Mills gauge field $A_{M}$ is in the adjoint representation of the
gauge group $G$ which is $SO(32)$ or $E_{8}\times E_{8}$. We employ the
convention such that the gauge field is represented by anti-hermitian matrices
and the trace is taken over the matrices of the fundamental representation.

A remarkable property of heterotic supergravity is that the Yang-Mills gauge
field contributes to the action as the first order
$\alpha^{\prime}$-correction. The famous anomaly cancellation mechanism and
the supersymmetry completion result in the Riemann curvature square term which
involves higher derivative corrections in the same order in $\alpha^{\prime}$
[24]222The authors would like to thank Tetsuji Kimura for his introducing
[24].. The ten-dimensional heterotic supergravity action for the bosonic
fields at the first order in $\alpha^{\prime}$ is given by

$\displaystyle S=$ $\displaystyle\
\frac{1}{2\kappa^{2}_{10}}\int\\!d^{10}x\sqrt{-g}e^{-2\phi}\Bigl{[}R(\omega)-\frac{1}{3}\hat{H}_{MNP}^{(3)}\hat{H}^{(3)MNP}+4\partial_{M}\phi\partial^{M}\phi$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\frac{}{}+\alpha^{\prime}\left(\text{Tr}F_{MN}F^{MN}+R_{MNAB}(\omega_{+})R^{MNAB}(\omega_{+})\right)\Bigr{]}.$
(1)

Here we employ the convention such that
$\frac{\kappa^{2}_{10}}{2g^{2}_{10}}=\alpha^{\prime}$ where $\kappa_{10}$ and
$g_{10}$ are the gravitational and the gauge coupling constants in ten
dimensions. The ten-dimensional metric $g_{MN}$ in the string frame is defined
through the vielbein as $g_{MN}=\eta_{AB}e_{M}{}^{A}e_{N}{}^{B}$ where the
metric in the local Lorentz frame is $\eta_{AB}=\text{diag}(-1,1,\ldots,1)$.
The Ricci scalar $R(\omega)$ is constructed from the spin connection
$\omega_{M}{}^{AB}$:

$\displaystyle R(\omega)=$ $\displaystyle\
e^{M}{}_{A}e^{N}{}_{B}R^{AB}{}_{MN}(\omega),$ $\displaystyle
R^{AB}{}_{MN}(\omega)=$ $\displaystyle\
\partial_{M}\omega_{N}{}^{AB}-\partial_{N}\omega_{M}{}^{AB}+\omega_{M}{}^{AD}\eta_{CD}\omega_{N}{}^{CB}-\omega_{N}{}^{AD}\eta_{CD}\omega_{M}{}^{CB}.$
(2)

The spin connection is expressed by the vielbein $e_{M}{}^{A}$ and its
inverse:

$\displaystyle\omega_{M}^{~{}AB}={1\over
2}\Bigl{[}e^{AN}(\partial_{M}e_{N}^{~{}B}-\partial_{N}e_{M}^{~{}B})-e^{BN}(\partial_{M}e_{N}^{~{}A}-\partial_{N}e_{M}^{~{}A})-e^{AP}e^{BQ}(\partial_{P}e_{QC}-\partial_{Q}e_{PC})e_{M}^{~{}C}\Bigr{]}.$
(3)

The modified spin connection $\omega_{\pm M}{}^{AB}$ which enters into the
action in the $\mathcal{O}(\alpha^{\prime})$ terms is defined as

$\displaystyle\omega_{\pm
M}{}^{AB}=\omega_{M}{}^{AB}\pm\hat{H}_{M}^{(3)}{}^{AB},$ (4)

where $\hat{H}_{M}^{(3)}{}^{AB}=e^{NA}e^{PB}\hat{H}_{MNP}^{(3)}$ and the
modified $H$-flux $\hat{H}_{MNP}^{(3)}$ is defined by

$\displaystyle\hat{H}^{(3)}_{MNP}=$ $\displaystyle\
H_{MNP}^{(3)}+\alpha^{\prime}\left(\Omega_{MNP}^{\text{YM}}-\Omega_{MNP}^{\text{L}+}\right).$
(5)

Here the field strength $H_{MNP}^{(3)}$ of the $B$-field is defined by

$\displaystyle H_{MNP}^{(3)}=$ $\displaystyle\
\frac{1}{2}(\partial_{M}B_{NP}+\partial_{N}B_{PM}+\partial_{P}B_{MN}).$ (6)

The Yang-Mills and the Lorentz Chern-Simons terms which appear in (5) are
defined as follows:

$\displaystyle\Omega^{\text{YM}}_{MNP}=$ $\displaystyle\
3!\text{Tr}\left(A_{[M}\partial_{N}A_{P]}+\frac{2}{3}A_{[M}A_{N}A_{P]}\right),$
$\displaystyle\Omega^{\text{L}+}_{MNP}=$ $\displaystyle\
3!\left(\eta_{BC}\eta_{AD}\omega_{+[M}{}^{AB}\partial_{N}\omega_{+P]}{}^{CD}+\frac{2}{3}\eta_{AG}\eta_{BC}\eta_{DF}\omega_{+[M}^{AB}\omega_{+N}{}^{CD}\omega_{+P]}{}^{FG}\right).$
(7)

Here the symbol $[M_{1}M_{2}\cdots M_{n}]$ stands for the anti-symmetrization
of indices with weight $\frac{1}{n!}$. Note that the modified $H$-flux is
iteratively defined through the relations (4) and (5) order by order in
$\alpha^{\prime}$. The modified $H$-flux $\hat{H}^{(3)}$ obeys the following
Bianchi identity:

$\displaystyle d\hat{H}^{(3)}=\alpha^{\prime}\left(\text{Tr}F\wedge
F-\text{Tr}R\wedge R\right)+\mathcal{O}(\alpha^{\prime 2}),$ (8)

where $R^{AB}=\frac{1}{2!}R^{AB}{}_{MN}dx^{M}\wedge dx^{N}$ is the
$SO(1,9)$-valued curvature 2-form. The component of the Yang-Mills gauge field
strength 2-form $F=\frac{1}{2!}F_{MN}dx^{M}\wedge dx^{N}$ is

$\displaystyle
F_{MN}{}^{I}=\partial_{M}A_{N}^{I}-\partial_{N}A_{M}^{I}+f^{I}{}_{JK}A^{J}{}_{M}A^{K}{}_{N}.$
(9)

Here $I,J,K=1,\ldots,\text{dim}\mathcal{G}$ are the gauge indices and
$f^{I}{}_{JK}$ is the structure constant for the Lie algebra $\mathcal{G}$
associated with $G$. Note that the terms
$R_{ABMN}(\omega_{+})R^{ABMN}(\omega_{+})$ and $\Omega^{\text{L}+}_{MNP}$ in
the action (1) are higher derivative corrections to the ordinary second order
derivative terms.

The heterotic NS5-brane solutions satisfy the equation of motion derived from
the action (1) at $\mathcal{O}(\alpha^{\prime})$. There are three distinct
NS5-brane solutions known as the symmetric, the neutral and the gauge types in
heterotic supergravity [21, 22]. They are 1/2 BPS configurations and preserve
a half of the sixteen supercharges. They satisfy the following ansatz:

$\displaystyle\ ds^{2}=\eta_{ij}dx^{i}dx^{j}+H(r)\delta_{mn}dx^{m}dx^{n},$
$\displaystyle\
\hat{H}^{(3)}_{mnp}=-\frac{1}{2}\varepsilon_{mnpq}\partial_{q}H(r),\qquad\phi=\frac{1}{2}\log
H(r),$ $\displaystyle\ r^{2}=(x^{1})^{2}+(x^{2})^{2}+(x^{3})^{2}+(x^{4})^{2},$
(10)

where the indices $i,j=0,5,6,7,8,9$, $m,n=1,2,3,4$ represent the world-volume
and the transverse directions to the NS5-branes and $\varepsilon_{mnpq}$ is
the Levi-Civita symbol. The 1/2 BPS condition leads to the self-duality
condition for the Yang-Mills gauge field:

$\displaystyle F_{mn}=\tilde{F}_{mn},$ (11)

and other components $A_{i}$ vanish. Here the Hodge dual field strength in the
transverse four-dimensions is defined by
$\tilde{F}_{mn}=\frac{1}{2}\varepsilon_{mnpq}F^{pq}$. The harmonic function
$H(r)$ is determined by the Yang-Mills gauge field configurations of the
solutions. These heterotic NS5-branes are characterized by two charges. One is
the topological charge $k$ associated with the gauge instantons:

$\displaystyle k=-\frac{1}{32\pi^{2}}\int\mathrm{Tr}[F\wedge F],$ (12)

where the integral is defined in the transverse four-space. The other is the
charge $Q$ associated with the modified $H$-flux:

$\displaystyle
Q=-\frac{1}{2\pi^{2}\alpha^{\prime}}\int_{S^{3}}\\!\hat{H}^{(3)}.$ (13)

Here $S^{3}$ is the asymptotic three-sphere surrounding the NS5-brane. In the
following we briefly introduce the three distinct NS5-brane solutions.

#### Symmetric solution

The most tractable NS5-brane solution in heterotic theory may be the so-called
symmetric solution. The Yang-Mills gauge field which satisfies the self-
duality condition (11) is given by an instanton solution in four-dimensions.
The gauge field takes value in the $SU(2)$ subgroup of the gauge group $G$ and
the explicit solution is given by [22],

$\displaystyle A_{m}=$ $\displaystyle\
-\frac{\sigma_{mn}x^{n}}{r^{2}+n\alpha^{\prime}e^{-2\phi_{0}}},\quad
n\in\mathbb{Z},$ $\displaystyle H(r)=$ $\displaystyle\
e^{2\phi_{0}}+\frac{n\alpha^{\prime}}{r^{2}},$ (14)

where $\sigma_{mn}$ is the self-dual part of the $SO(4)$ Lorentz generator.
This is written in the following form,

$\displaystyle\sigma_{mn}=\eta^{I}_{mn}T^{I},~{}~{}~{}\eta^{I}_{mn}=\varepsilon^{I}_{~{}mn4}+\delta^{I}_{~{}m}\delta_{n4}-\delta^{I}_{~{}n}\delta_{m4},$
(15)

where $\eta^{I}_{mn}$ is the ’t Hooft symbol which satisfies the self-duality
condition in terms of the indices $m,n$:

$\displaystyle\eta^{I}_{mn}={1\over 2}\varepsilon_{mnpq}\eta^{I}_{pq}.$ (16)

The anti-hermitian matrices $T^{I}$ are defined as

$\displaystyle T^{1}=\left[\begin{array}[]{cccc}0&0&0&-1\\\ 0&0&1&0\\\
0&-1&0&0\\\ 1&0&0&0\end{array}\right],\quad
T^{2}=\left[\begin{array}[]{cccc}0&0&-1&0\\\ 0&0&0&-1\\\ 1&0&0&0\\\
0&1&0&0\end{array}\right],\quad T^{3}=\left[\begin{array}[]{cccc}0&1&0&0\\\
-1&0&0&0\\\ 0&0&0&-1\\\ 0&0&1&0\end{array}\right].$ (29)

They satisfy the $su(2)$ algebra

$\displaystyle[T^{I},T^{J}]=-2\varepsilon^{IJK}T^{K},\quad(I,J,K=1,2,3).$ (30)

The solution (14) is nothing but the BPST one-instanton in the non-singular
gauge [25]. The constant $\phi_{0}$ is the asymptotic value of the dilaton.
Compared with the general BPST instanton solution, the symmetric solution has
a fixed finite instanton size $\rho=e^{-\phi_{0}}\sqrt{n\alpha^{\prime}}$.
This solution has charges $(k,Q)=(1,n)$. A remarkable fact about the symmetric
solution is that the dilaton configuration, hence the harmonic function
$H(r)$, is obtained through the standard embedding ansatz:

$\displaystyle(A_{m})^{ab}=\omega_{+m}{}^{ab}.$ (31)

Here the $SU(2)\subset G$ indices $a,b$ of the gauge field are identified with
the indices of the $SU(2)$ subgroup of the local Lorentz group $SO(4)$ in the
transverse directions. If the relation (31) holds, the Chern-Simons terms in
the modified $H$-flux (5) cancel out. Therefore the $H$-flux only comes from
the $B$-field and the Bianchi identity (8) becomes $d\hat{H}^{(3)}=0$. From
the Bianchi identity and the relation between $H(r)$ and $\hat{H}_{mnp}^{(3)}$
in (10), the $B$-field is determined through the following condition:

$\displaystyle\partial_{m}H={1\over 2}\varepsilon_{mnpq}\partial_{n}B_{pq}.$
(32)

Although the symmetric NS5-brane solution (10) with (14) is a solution to the
equation of motion derived from the action (1) at
$\mathcal{O}(\alpha^{\prime})$, the string sigma-model analysis indicates that
the symmetric solution is an exact solution valid at all orders in
$\alpha^{\prime}$ [22]. Indeed, the symmetric solution has the
$\mathcal{N}=(4,4)$ sigma model description and it is protected against
$\alpha^{\prime}$-corrections.

#### Neutral solution

The neutral solution is the one with charges $(k,Q)=(0,n)$ [22]. For this
solution, the Yang-Mills gauge field becomes trivial and the harmonic function
is given by

$\displaystyle A_{M}=$ $\displaystyle\ 0,$ $\displaystyle H(r)=$
$\displaystyle\ e^{2\phi_{0}}+\frac{n\alpha^{\prime}}{r^{2}}.$ (33)

Since the gauge field does not appear in the neutral solution, this is a
solution to type II supergravities. Indeed, this is the type II NS5-brane
solution. It is remarkable that the neutral solution (10) with (33) for
heterotic supergravity is valid at the first order in $\alpha^{\prime}$. For
this solution, we find the curvature $R_{MNAB}$ is order
$\mathcal{O}(\alpha^{\prime})$. Therefore the Lorentz Chern-Simons term in the
modified $H$-flux (5) becomes a higher order correction in $\alpha^{\prime}$
and negligible. Then the Bianchi identity again becomes $d\hat{H}^{(3)}=0$.
With the solution (33) at hand, the $B$-field is given by that of the type II
NS5-brane and its components, which are determined by (32), are the same as
the ones of the symmetric solution in the linear order in $\alpha^{\prime}$.
We note that the neutral solution has the $\mathcal{N}=(4,0)$ worldsheet sigma
model description and receives higher order $\alpha^{\prime}$-corrections in
heterotic theories [22].

#### Gauge solution

The last is the so called gauge solution which has been originally found in
[21]. The Yang-Mills gauge field is again given by the BPST instanton. As in
the case of the neutral solution, the curvature term in the modified $H$-flux
is neglected as it is a higher order in $\alpha^{\prime}$. However, the Yang-
Mills gauge field still contributes to the $H$-flux in the gauge solution and
the Bianchi identity becomes $d\hat{H}^{(3)}=\alpha^{\prime}\text{Tr}F\wedge
F$. From the Bianchi identity, the harmonic function is determined to be

$\displaystyle A_{m}=$ $\displaystyle\
-\frac{\sigma_{mn}x^{n}}{r^{2}+\rho^{2}},$ $\displaystyle H(r)=$
$\displaystyle\
e^{2\phi_{0}}+8\alpha^{\prime}\frac{r^{2}+2\rho^{2}}{(r^{2}+\rho^{2})^{2}}.$
(34)

Compared with the symmetric solution (14), the size modulus $\rho$ is not
fixed for the gauge solution and it has charges $(k,Q)=(1,8)$. Similar to the
neutral solution, the gauge solution is obtained by a perturbative series of
$\alpha^{\prime}$ and the functional forms (10), (34) are valid at the first
order in $\alpha^{\prime}$. Indeed, the gauge solution has the
$\mathcal{N}=(4,0)$ worldsheet sigma model description and is expected to
receive higher order $\alpha^{\prime}$-corrections [22]. The position of the
instanton corresponds to that of the NS5-brane. Therefore the instanton with
finite size $\rho\not=0$ resides in the core of the NS5-brane. It has been
discussed in [26], when the instanton shrink to zero size $\rho\to 0$, a gauge
multiplet on the brane world-volume becomes massless and the $SU(2)$ gauge
symmetry is enhanced. The gauge NS5-brane in the $SO(32)$ heterotic theory in
the zero size limit $\rho\to 0$ is related to the D5-brane in type I theory by
the S-duality.

We make a comment on the $B$-field for the gauge NS5-brane solution. A careful
analysis reveals that only the Yang-Mills Chern-Simons term contributes to the
modified $H$-flux and we find $H^{(3)}_{MNP}=0$ at least at
$\mathcal{O}(\alpha^{\prime})$. Therefore the $B$-field is not excited in the
gauge NS5-brane solution. Then, the components of the $B$-field take a
constant value:

$\displaystyle B_{mn}=\Theta_{mn},~{}~{}~{}(\mbox{constant}).$ (35)

##  3 T-duality chains of five-branes and $\alpha^{\prime}$-corrected Buscher
rule

In this section, we derive new five-brane solutions in heterotic supergravity
in the family of the T-duality chains. Since the gauge field enters into the
action (1) as the first order $\alpha^{\prime}$-correction, the T-duality
transformation in the heterotic supergravity should be modified from the
standard Buscher rule [17]. The first order $\alpha^{\prime}$-corrections to
the Buscher rule in heterotic supergravity have been written down in [19].
This is given by

$\displaystyle g_{\hat{M}\hat{N}}\ \longrightarrow$ $\displaystyle\
g_{\hat{M}\hat{N}}+\frac{1}{G^{\prime}_{yy}}\left(g_{yy}G^{\prime}_{y\hat{M}}G^{\prime}_{y\hat{N}}-G^{\prime}_{yy}g_{y\hat{M}}G^{\prime}_{y\hat{N}}-G^{\prime}_{yy}G^{\prime}_{y\hat{M}}g_{y\hat{N}}\right),$
$\displaystyle g_{y\hat{M}}\ \longrightarrow$ $\displaystyle\
-\frac{g_{y\hat{M}}}{G^{\prime}_{yy}}+\frac{g_{yy}G^{\prime}_{y\hat{M}}}{G^{\prime
2}_{yy}},\qquad g_{yy}\ \longrightarrow\ \frac{g_{yy}}{G^{\prime 2}_{yy}},$
$\displaystyle B_{\hat{M}\hat{N}}\ \longrightarrow$ $\displaystyle\
B_{\hat{M}\hat{N}}+\frac{1}{G^{\prime}_{yy}}(G^{\prime}_{y\hat{M}}B_{\hat{N}y}-G^{\prime}_{y\hat{N}}B_{\hat{M}y}),\qquad
B_{y\hat{M}}\ \longrightarrow\
-\frac{B_{y\hat{M}}}{G^{\prime}_{yy}}-\frac{G^{\prime}_{y\hat{M}}}{G^{\prime}_{yy}},$
$\displaystyle\phi\ \longrightarrow$ $\displaystyle\
\phi-\frac{1}{2}\log|G^{\prime}_{yy}|,\qquad A_{\hat{M}}^{I}\ \longrightarrow\
A_{\hat{M}}^{I}-\frac{G^{\prime}_{y\hat{M}}}{G^{\prime}_{yy}}A_{y}^{I},\qquad
A_{y}^{I}\ \longrightarrow\ -\frac{A_{y}^{I}}{G^{\prime}_{yy}},$ (36)

where we have decomposed the indices $M=(\hat{M},y)$. The indices $y$ and
$\hat{M},\hat{N}\ldots$ specify an isometry and non-isometry directions
respectively. In (36), we have defined the following quantity,

$\displaystyle
G^{\prime}_{MN}=g_{MN}-B_{MN}+2\alpha^{\prime}\bigl{[}\mbox{Tr}(\omega_{+M}\omega_{+N})-\mbox{Tr}(A_{M}A_{N})\bigr{]},$
(37)

where we have defined
$\mbox{Tr}\omega_{+M}\omega_{+N}=\omega_{+M}{}^{AB}\omega_{+NBA}$. We call
(36) with (37) the heterotic Buscher rule. Under the transformation (36), the
actions of $SO(32)$ and $E_{8}\times E_{8}$ heterotic supergravities
dimensionally reduced on $S^{1}$ coincide. Note that when the Yang-Mills gauge
field $A_{M}{}^{I}$ and the higher derivative corrections coming from
$\omega_{+M}{}^{AB}$ are turned off, the relation (36) reduces to the ordinary
Buscher rule for the NS-NS backgrounds in type II supergravities.

###  3.1 Heterotic KK5-branes

Before going to the exotic $5^{2}_{2}$-branes, we first write down the
heterotic KK5-brane solutions. In order to perform the T-duality
transformation for the heterotic NS5-brane solutions, we introduce a $U(1)$
isometry along the transverse direction to the brane world-volumes 333If we
perform the T-duality transformations along the world-volume direction of the
NS5-branes in the $SO(32)$ heterotic theory, we obtain the identical solutions
in the $E_{8}\times E_{8}$ theory and vice versa.. To this end, we first
compactify the $x^{4}$-direction with the radius $R_{4}$ and consider the
periodic array of the NS5-branes. Then by taking the small radius limit
$R_{4}\to 0$, we introduce the $U(1)$ isometry to the heterotic NS5-brane
solutions. The self-duality condition (11) reduces to the monopole equation
$\frac{1}{2}\varepsilon_{m^{\prime}n^{\prime}p^{\prime}}F_{n^{\prime}p^{\prime}}=\partial_{m^{\prime}}A_{4}+[A_{m^{\prime}},A_{4}]$
where $m^{\prime},n^{\prime},p^{\prime}=1,2,3$. Therefore the resulting
solutions are called the heterotic monopoles or smeared NS5-branes. The
explicit forms of the smeared NS5-brane solutions which originate from the
periodic arrays of the symmetric, the neutral and the gauge solutions have
been written down in [27, 29]. By performing the T-duality transformation of
the solutions along the isometry direction, we obtain the KK5-brane solutions
associated with the three types of the NS5-branes. In the following, we
calculate the T-duality transformations of the NS5-brane solutions and derive
the KK5-brane solutions.

#### Symmetric KK5-brane

For the symmetric solution where the standard embedding condition is
satisfied, the ansatz and the Bianchi identity leads to the condition $\Box
e^{2\phi}=\Box H=0$. The periodic array of the symmetric NS5-brane solution is
governed by the following harmonic function $H(r)$ on $\mathbb{R}^{3}\times
S^{1}$ [29]:

$\displaystyle
H(r)=e^{2\phi_{0}}+\sum_{s}\frac{n\alpha^{\prime}}{r^{2}+(x^{4}-2\pi
R_{4}s)^{2}},\quad r^{2}=(x^{1})^{2}+(x^{2})^{2}+(x^{3})^{2}.$ (38)

Taking the compactification radius small $R_{4}\to 0$, the sum in (38) is
approximated by the integral over $s$. We call this procedure as smearing. The
result is444 The constant $C=n\alpha^{\prime}R_{4}^{-1}$ would diverge in the
limit $R_{4}\to 0$ but this is an artifact of the smearing procedure. We can
find the harmonic function (39) which finite $C$ by solving the Laplace
equation $\Box H=0$ in three dimensions.

$\displaystyle H(r)=e^{2\phi_{0}}+\frac{C}{2r},$ (39)

The corresponding solution is the smeared symmetric NS5-brane of co-dimension
three discussed in [27]. We note that the smeared symmetric NS5-brane is an
H-monopole whose quantized charge $Q$ is well-defined. On the other hand, the
Yang-Mills monopole charge for the smeared solution is not defined anymore
[29]. We now perform the T-duality transformation on the smeared symmetric
NS5-brane solution. For a solution where the standard embedding (31) is
satisfied, the heterotic Buscher rule (36) is quite simplified. This is
because the relation Tr( $\omega_{+M}\omega_{+N})=\mbox{Tr}(A_{M}A_{N})$ holds
and the last term in (37) vanishes.

We perform the T-duality transformation by utilizing the heterotic Buscher
rule (36). Then we obtain the symmetric KK5-brane solution:

$\displaystyle ds^{2}=$ $\displaystyle\
\eta_{ij}dx^{i}dx^{j}+H\left[(dx^{1})^{2}+(dx^{2})^{2}+(dx^{3})^{2}\right]+H^{-1}\left[dx^{4}+\beta_{m^{\prime}}dx^{m^{\prime}}\right]^{2},$
$\displaystyle\phi=$ $\displaystyle\ 0,\qquad
B_{MN}=0,\qquad(m^{\prime}=1,2,3),$ (40)

where the harmonic function $H$ is given in (39) and $\beta_{m^{\prime}}$ is
determined by the following relation:

$\displaystyle\partial_{m^{\prime}}H=\varepsilon_{m^{\prime}n^{\prime}p^{\prime}}\partial_{n^{\prime}}\beta_{p^{\prime}}.$
(41)

The dilaton and the NS-NS $B$-field vanish but the Yang-Mills gauge field
remains non-trivial. The symmetric KK5-brane solution is not a purely
geometric solution but the geometry is dressed up with the Yang-Mills gauge
fields. The gauge field $A_{M}{}^{I}$ is obtained as

$\displaystyle A_{1}$ $\displaystyle={1\over
2H^{2}}\bigl{[}-\beta_{1}\partial_{1}HT^{1}-(H\partial_{3}H+\beta_{1}\partial_{2}H)T^{2}+(H\partial_{2}H-\beta_{1}\partial_{3}H)T^{3}\bigr{]},$
$\displaystyle A_{2}$ $\displaystyle={1\over
2H^{2}}\bigl{[}(H\partial_{3}H-\beta_{2}\partial_{1}H)T^{1}-\beta_{2}\partial_{2}HT^{2}-(H\partial_{1}H+\beta_{2}\partial_{3}H)T^{3}\bigr{]},$
$\displaystyle A_{3}$ $\displaystyle={1\over
2H^{2}}\bigl{[}-(H\partial_{2}H+\beta_{3}\partial_{1}H)T^{1}+(H\partial_{1}H-\beta_{3}\partial_{2}H)T^{2}-\beta_{3}\partial_{3}HT^{3}\bigr{]},$
$\displaystyle A_{4}$ $\displaystyle={1\over
2H^{2}}\bigl{[}-\partial_{1}HT^{1}-\partial_{2}HT^{2}-\partial_{3}HT^{3}\bigr{]},~{}~{}~{}~{}~{}A_{i}=0.$
(42)

We find that the Yang-Mills gauge field configuration (42) satisfies the anti-
monopole (Bogomol’nyi) equation

$\displaystyle B_{i}=-D_{i}\phi.$ (43)

This equation is nothing but the anti-self-duality condition
$F_{mn}=-\tilde{F}^{mn}$ in disguise555 The flip of the sign in the Hodge
dualized field strength $\tilde{F}_{mn}$ comes from the choice of the freedom
for the overall sign in the heterotic Buscher rule (36). If we choose another
sign in front of $A_{y}^{I}$ in the right hand side of (36), the gauge field
satisfies the self-duality condition instead of the anti-self-duality
condition. . Since the field configuration (42) satisfies the equation
$F_{mn}=-\tilde{F}_{mn}$, this belongs to the general class of spherically
symmetric solutions discussed in [28].

Meanwhile, by using the relation (41), the modified spin connection
$\omega_{+}$ associated with the geometry (40) is calculated to be

$\displaystyle\omega_{+1}$ $\displaystyle={1\over
2H^{2}}\bigl{[}-\beta_{1}\partial_{1}HT^{1}-(H\partial_{3}H+\beta_{1}\partial_{2}H)T^{2}+(H\partial_{2}H-\beta_{1}\partial_{3}H)T^{3}\bigr{]},$
$\displaystyle\omega_{+2}$ $\displaystyle={1\over
2H^{2}}\bigl{[}(H\partial_{3}H-\beta_{2}\partial_{1}H)T^{1}-\beta_{2}\partial_{2}HT^{2}-(H\partial_{1}H+\beta_{2}\partial_{3}H)T^{3}\bigr{]},$
$\displaystyle\omega_{+3}$ $\displaystyle={1\over
2H^{2}}\bigl{[}-(H\partial_{2}H+\beta_{3}\partial_{1}H)T^{1}+(H\partial_{1}H-\beta_{3}\partial_{2}H)T^{2}-\beta_{3}\partial_{3}HT^{3}\bigr{]},$
$\displaystyle\omega_{+4}$ $\displaystyle={1\over
2H^{2}}\bigl{[}-\partial_{1}HT^{1}-\partial_{2}HT^{2}-\partial_{3}HT^{3}\bigr{]},~{}~{}~{}~{}~{}\omega_{+i}=0.$
(44)

From the results (42) and (44), we find that the standard embedding condition
(31) still holds for the symmetric KK5-brane solution.

#### Neutral KK5-brane

We next study the neutral KK5-brane solution. For the neutral NS5-brane
solution, we have the trivial Yang-Mills gauge field $A_{M}{}^{I}=0$. Again,
the smearing procedure is applicable to the neutral solution. The smeared
neutral NS5-brane solution is governed by the harmonic function (39). As we
have claimed in the previous section, the modified spin connection
$\omega_{+}$ is in the $\mathcal{O}(\alpha^{\prime})$ for the neutral
solution. Therefore it is negligible in $G^{\prime}_{MN}$ in the heterotic
Buscher rule (37). Applying the heterotic Buscher rule, we find that the
metric, the $B$-field and the dilaton for the neutral KK5-brane solution are
given by (40) and the gauge field remains vanishing $A_{M}{}^{I}=0$. Therefore
the neutral KK5-brane is a purely geometric Taub-NUT solution at least at the
leading order in $\alpha^{\prime}$. Indeed, this is nothing but the KK5-brane
solution in type II theories.

#### Gauge KK5-brane

For the gauge NS5-brane solution, the periodic harmonic function (39) does not
work as a solution since $\Box e^{2\phi}=0$ is not satisfied for the gauge
solution. The ansatz (10) and the Bianchi identity indicates $\Box
H(r)=\frac{\alpha^{\prime}}{2}\mathrm{Tr}[\varepsilon^{mnpq}F_{mn}F_{pq}]$. In
order to find a solution of co-dimension three associated with the gauge
solution, we perform the singular gauge transformation of the solution (34).
Then the gauge field becomes

$\displaystyle
A_{m}=\frac{-2\rho^{2}\bar{\sigma}_{mn}x^{n}}{r^{2}(r^{2}+\rho^{2})},\qquad
r^{2}=(x^{1})^{2}+(x^{2})^{2}+(x^{3})^{2}+(x^{4})^{2},$ (45)

where $\bar{\sigma}_{mn}$ is the antisymmetric and anti-self-dual matrix and
is written as

$\displaystyle\bar{\sigma}_{mn}=\overline{\eta}^{I}_{mn}T^{I},~{}~{}~{}\bar{\eta}^{I}_{mn}=\varepsilon^{I}_{~{}mn4}-\delta^{I}_{~{}m}\delta_{n4}+\delta^{I}_{~{}n}\delta_{m4}.$
(46)

This is the BPST instanton in the singular gauge. The solution is obtained by
the famous ’t Hooft ansatz:

$\displaystyle A_{m}=\bar{\sigma}_{mn}\partial_{n}\log f.$ (47)

The function $f$ satisfies $\frac{1}{f}\Box f=0$ and given by
$f(r)=1+\sum_{i=1}^{k}\frac{\rho^{2}}{(x-x_{i})^{2}}$ for the $k$-instanton
solution. Here $x_{i}$ are the positions of the instantons. Note that the
solution (45) corresponds to $k=1$. Using this fact, we can perform the
smearing procedure for $f$ along the line of obtaining (39). This periodic
array of the instantons is just the calorons of the Harrington-Shepard type
[30]. The harmonic function $H$ for the smeared gauge NS5-brane is determined
by the Laplace equation whose source is given by the $R_{4}\to 0$ limit of the
calorons, namely, the smeared instantons. After the smearing, we find
$f=1+\frac{\tilde{\rho}}{r}$ where $r^{2}=(x^{1})^{2}+(x^{2})^{2}+(x^{3})^{2}$
and $\tilde{\rho}=\frac{\rho^{2}}{2R_{4}}$ is a rescaled size modulus. Then
the smeared gauge NS5-brane solution is found to be

$\displaystyle ds^{2}$
$\displaystyle=\eta_{ij}dx^{i}dx^{j}+e^{2\phi}\delta_{mn}dx^{m}dx^{n},$
$\displaystyle A_{m}$
$\displaystyle=\bar{\sigma}_{mn}{-\tilde{\rho}x^{\nu}\over
r^{2}(r+\tilde{\rho})},~{}~{}~{}~{}~{}e^{2\phi}=e^{2\phi_{0}}-2\alpha^{\prime}{\tilde{\rho}^{2}\over
r^{2}(r+\tilde{\rho})^{2}},~{}~{}~{}~{}~{}\hat{H}_{mnp}^{(3)}=-{1\over
2}\varepsilon_{mnpq}\partial_{q}e^{2\phi}.$ (48)

As in the case of the gauge NS5-brane solution, we find that the $B$-fields in
the smeared gauge NS5-brane solution becomes trivial. This property is also
found in the heterotic monopole solution [29]. We also note that the smeared
gauge NS5-brane does not exhibit the H-monopole property. Indeed, we find the
modified $H$-flux behaves like $\hat{H}^{(3)}_{mn4}\sim
4\alpha^{\prime}\tilde{\rho}\epsilon_{mnq4}\frac{x^{q}}{r^{6}}\ (r\to\infty)$
and the charge $Q$ vanishes. This is in contrast to the symmetric and the
neutral solutions. We also comment that the solution (48) based on the smeared
instanton does not have a finite monopole charge. This is again in contrast to
the monopole solution in [29].

Since the $B$-field is trivial in the gauge solution, we can set these
components as

$\displaystyle
B_{14}=\Theta_{1},~{}~{}~{}B_{24}=\Theta_{2},~{}~{}~{}B_{34}=\Theta_{3},$ (49)

where $\Theta_{i}$ are constants. The other components do not appear in the
solution and they can be set to zero. Now we perform the T-duality
transformation of the smeared gauge NS5-brane solution along the
$x^{4}$-direction. After calculations, we find the following gauge KK5-brane
solution:

$\displaystyle ds^{2}$ $\displaystyle=\eta_{ij}dx^{i}dx^{j}+{\cal
H}\delta_{m^{\prime}n^{\prime}}dx^{m^{\prime}}dx^{n^{\prime}}+e^{-4\phi_{0}}{\cal
H}(dx^{4}+\Theta_{m^{\prime}}dx^{m^{\prime}})^{2},$ $\displaystyle\phi$
$\displaystyle={1\over 2}\log e^{-4\phi_{0}}{\cal H},~{}~{}~{}~{}~{}{\cal
H}=e^{2\phi_{0}}-2\alpha^{\prime}{\tilde{\rho}^{2}\over
r^{2}(r+\tilde{\rho})^{2}},$ $\displaystyle A_{1}$
$\displaystyle={\tilde{\rho}\over
2r^{2}(r+\tilde{\rho})}\Bigl{(}(x^{3}T^{2}-x^{2}T^{3})+e^{-2\phi_{0}}\Theta_{1}(x^{1}T^{1}+x^{2}T^{2}+x^{3}T^{3})\Bigr{)},$
$\displaystyle A_{2}$ $\displaystyle={\tilde{\rho}\over
2r^{2}(r+\tilde{\rho})}\Bigl{(}(x^{1}T^{3}-x^{3}T^{1})+e^{-2\phi_{0}}\Theta_{2}(x^{1}T^{1}+x^{2}T^{2}+x^{3}T^{3})\Bigr{)},$
$\displaystyle A_{3}$ $\displaystyle={\tilde{\rho}\over
2r^{2}(r+\tilde{\rho})}\Bigl{(}(x^{2}T^{1}-x^{1}T^{2})+e^{-2\phi_{0}}\Theta_{3}(x^{1}T^{1}+x^{2}T^{2}+x^{3}T^{3})\Bigr{)},$
$\displaystyle A_{4}$ $\displaystyle=e^{-2\phi_{0}}{\rho^{2}\over
2r^{2}(\rho^{2}+r)}(x^{1}T^{1}+x^{2}T^{2}+x^{3}T^{3}),\quad A_{i}=0.$ (50)

The solution seems complicated but one finds that the Yang-Mills gauge field
satisfies the anti-monopole equation (43). Notably, the function $\mathcal{H}$
becomes negative at a finite value of $r$. The solution is ill-defined near
the center of the brane. This property is similar to the symmetric and neutral
NS5-brane solutions with $n<0$ whose physical interpretation is unclear [29].

###  3.2 Heterotic $5^{2}_{2}$-branes

Now we are in a position where the second T-duality transformation is
performed on the KK5-brane solutions. We introduce another isometry along the
transverse direction $x^{3}$ to the KK5-brane world-volumes. By performing the
second T-duality transformation along the $x^{3}$-direction, we obtain the
exotic $5^{2}_{2}$-brane solutions associated with the symmetric, the neutral
and the gauge solutions.

#### Symmetric $5^{2}_{2}$-brane

The harmonic function $H(r)$ in $\mathbb{R}^{2}\times T^{2}$ is obtained by
the periodic array of the KK5-brane. This is given by

$\displaystyle H(r)=$ $\displaystyle\
e^{2\phi_{0}}+\frac{1}{2}\sum_{s}\frac{n\alpha^{\prime}R_{3}^{-1}}{\sqrt{r^{2}+(x^{3}-2\pi
R_{3}s)^{2}}},$ $\displaystyle r^{2}=$ $\displaystyle\
(x^{1})^{2}+(x^{2})^{2}.$ (51)

Taking the compactification radius small $R_{3}\to 0$, the sum in (51) is
approximated by the integral over $s$ with the cutoff scale $\Lambda$. The
result is [15]

$\displaystyle H(r)=h_{0}-\frac{\sigma}{2}\log\frac{r}{\mu},$ (52)

where $\sigma=\frac{R_{3}R_{4}}{2\pi\alpha^{\prime}}$ and $\mu$ is a constant
which specifies the region where the solution is valid [31]. The constant
$h_{0}$ diverges in the large cutoff scale limit $\Lambda\to\infty$. Since the
harmonic function only depends on $x^{1},x^{2}$, the NS-NS $B$-field does so.
The only non-zero component of the $B$-field $B_{34}=\omega$ is determined by
the following relations:

$\displaystyle\partial_{1}H=\partial_{2}\omega,\qquad\partial_{2}H=-\partial_{1}\omega,$
(53)

The explicit form of $\omega$ is found to be

$\displaystyle\omega=-\sigma\tan^{-1}\left(\frac{x^{2}}{x^{1}}\right).$ (54)

If we employ the coordinate system $x^{1}=r\cos\theta,x^{2}=r\sin\theta$ where
$\theta$ is the angular coordinate in the $x^{1}x^{2}$-plane, then $\omega$ is
proportional to $\theta$. This result leads to the fact that the space-time
metric, the dilaton and the NS-NS $B$-field in the solution are not single-
valued anymore. The logarithmic behaviour of the harmonic function is
characteristic to co-dimension two objects. We call the solution (40) on which
the harmonic function is replaced by (52) the smeared symmetric KK5-brane
solution. Indeed, the heterotic vortex and the domain wall are discussed in
[32, 33] where the harmonic function behaves as logarithmic and linear
functions.

Since the heterotic KK5-brane solution satisfies the standard embedding
condition, this again simplify the heterotic Buscher rule (36). By performing
the second T-duality transformation along the $x^{3}$-direction, we obtain the
following symmetric $5^{2}_{2}$-brane solution:

$\displaystyle ds^{2}=$ $\displaystyle\
\eta_{ij}dx^{i}dx^{j}+H\left[(dx^{1})^{2}+(dx^{2})^{2}\right]+\frac{H}{K}\left[(dx^{3})^{2}+(dx^{4})^{2}\right],$
$\displaystyle\phi=$ $\displaystyle\ \frac{1}{2}\log\frac{H}{K},\quad
B_{34}=-\frac{\omega}{K},\quad K=H^{2}+\omega^{2}.$ (55)

The metric, the dilaton and the NS-NS $B$-field in (55) are the same with the
ones in type II theory. However, in heterotic theory, the Yang-Mills gauge
field remains non-trivial. The gauge field $A_{M}{}^{I}$ is calculated as

$\displaystyle A_{1}$ $\displaystyle={1\over
2H}(\partial_{2}H)T^{3},~{}~{}~{}~{}~{}A_{2}=-{1\over
2H}(\partial_{1}H)T^{3},$ $\displaystyle A_{3}$ $\displaystyle={1\over
2HK}\Bigl{(}(H\partial_{2}H+\omega\partial_{1}H)T^{1}-(H\partial_{1}H-\omega\partial_{2}H)T^{2}\Bigr{)},$
$\displaystyle A_{4}$ $\displaystyle=-{1\over
2HK}\Bigl{(}(H\partial_{1}H-\omega\partial_{2}H)T^{1}+(H\partial_{2}H+\omega\partial_{1}H)T^{2}\Bigr{)},$
$\displaystyle A_{i}$ $\displaystyle=0.$ (56)

We find that (56) satisfies the vortex-like equation in two dimensions:

$\displaystyle D_{1}\varphi-
iD_{2}\varphi=0,\qquad[\varphi,\bar{\varphi}]=B_{3},$ (57)

where $\varphi=\frac{1}{\sqrt{2}}(A_{3}+iA_{4})$ is a complexified adjoint
scalar field. The vortex-like equation (57) is obtained by dimensionally
reducing the self-duality condition (11) to two dimensions. We note that since
the Yang-Mills gauge field contains the explicit angular coordinate in
$\omega$, $A_{m}$ is not a single-valued function also in the symmetric
$5^{2}_{2}$-brane solution.

The modified spin connection $\omega_{+}$ associated with the solution (55) is
calculated as

$\displaystyle\omega_{+1}=\frac{1}{2H}(\partial_{2}H)T^{3}+K^{-1}\left(H\partial_{2}H+\omega\partial_{1}H\right)N^{34},\quad\omega_{+2}=-\frac{1}{2H}(\partial_{1}H)T^{3}-K^{-1}\left(H\partial_{1}H-\omega\partial_{2}H\right)N^{34},$
$\displaystyle\omega_{+3}=-\frac{1}{2H}K^{-\frac{3}{2}}\left(\omega^{2}\partial_{2}H-H^{2}\partial_{2}H-2\omega
H\partial_{1}H\right)T^{1}+\frac{1}{2H}K^{-\frac{3}{2}}\left(\omega^{2}\partial_{1}H-H^{2}\partial_{1}H+2\omega
H\partial_{2}H\right)T^{2},$ $\displaystyle\omega_{+4}=\
\frac{1}{2H}K^{-\frac{3}{2}}\left(\omega^{2}\partial_{1}H-H^{2}\partial_{1}H+2\omega
H\partial_{2}H\right)T^{1}+\frac{1}{2H}K^{-\frac{3}{2}}\left(\omega^{2}\partial_{2}H-H^{2}\partial_{2}H-2\omega
H\partial_{1}H\right)T^{2}.$ (58)

Here $(N^{ab})^{AB}=\delta^{aA}\delta^{bB}-\delta^{bA}\delta^{aB}$ is the
generator of the $SO(4)$ Lorentz group. More explicitly, they are given by

$\displaystyle(N^{12})=\left[\begin{array}[]{cccc}0&1&0&0\\\ -1&0&0&0\\\
0&0&0&0\\\
0&0&0&0\end{array}\right],\quad(N^{13})=\left[\begin{array}[]{cccc}0&0&1&0\\\
0&0&0&0\\\ -1&0&0&0\\\
0&0&0&0\end{array}\right],\quad(N^{14})=\left[\begin{array}[]{cccc}0&0&0&1\\\
0&0&0&0\\\ 0&0&0&0\\\ -1&0&0&0\end{array}\right],$ (71)
$\displaystyle(N^{23})=\left[\begin{array}[]{cccc}0&0&0&0\\\ 0&0&1&0\\\
0&0&0&0\\\
-1&0&0&0\end{array}\right],\quad(N^{24})=\left[\begin{array}[]{cccc}0&0&0&0\\\
0&0&0&1\\\ 0&0&0&0\\\
0&-1&0&0\end{array}\right],\quad(N^{34})=\left[\begin{array}[]{cccc}0&0&0&0\\\
0&0&0&0\\\ 0&0&0&1\\\ 0&0&-1&0\end{array}\right].$ (84)

At first sight, the standard embedding condition (31) does not hold for the
symmetric $5^{2}_{2}$-brane solution. However we find that the condition (31)
is satisfied up to the gauge transformation. To see this, we calculate the
gauge (and the $SO(4)$) invariant quantity $\text{Tr}F_{MN}F^{MN}$ and
$R_{MNAB}(\omega_{+})R^{MNBA}(\omega_{+})$ and find

$\displaystyle\text{Tr}F_{MN}F^{MN}=R_{MNAB}(\omega_{+})R^{MNBA}(\omega_{+}).$
(85)

This result indicates that $A_{m}$ and $\omega_{+}$ are identified up to a
gauge transformation.

When the standard embedding condition is satisfied, all the
$\alpha^{\prime}$-corrections are expected to be canceled in the T-duality
transformations [34]. We therefore conclude that the heterotic Buscher rule
(36) is exact for solutions of the symmetric type. The family of solutions
related by the T-duality transformations seem not to suffer from the
$\alpha^{\prime}$-corrections. We note that the multi-valuedness which appears
in the gauge invariant quantity $\text{Tr}F_{MN}F^{MN}$ in the action is
canceled by the standard embedding condition. This is similar to the situation
discussed in the multi-monopole solutions [27] where divergences in
$\mathrm{Tr}F_{MN}F^{MN}$ are canceled in the term $R_{MNAB}R^{MNBA}$.

#### Neutral $5^{2}_{2}$-brane

For the neutral KK5-brane, we again find that the heterotic Buscher rule is
simplified due to the same reason discussed in the previous subsection. After
some calculations, the metric, the dilaton and the NS-NS $B$-field are given
by (55). This is a conceivable result since the neutral solution does not
involve gauge field anymore and it is a solution in type II theory. We stress
that although the symmetric solution (55) is an exact solution, the neutral
solution is a perturbative solution in the $\alpha^{\prime}$ expansion.

#### Gauge $5^{2}_{2}$-brane

We introduce another isometry along the same way of obtaining the Harrington-
Shepard calorons for the smeared gauge NS5-brane. The calculation is the same
with the harmonic function (52) in $\mathbb{R}^{2}\times T^{2}$. One obtains
the function
$f=\tilde{h}_{0}-\frac{\tilde{\sigma}}{2}\log\left(\frac{r}{\mu}\right)+\mathcal{O}(r/\Lambda)$
in the ’t Hooft ansatz (47). Here $r^{2}=(x^{1})^{2}+(x^{2})^{2}$ and
$\tilde{h}_{0}=1+\frac{\tilde{\rho}}{\pi R_{3}}\log[4\pi R_{3}\Lambda/\mu]$,
$\tilde{\sigma}=\frac{2\tilde{\rho}}{\pi R_{3}}$ are constants. For the
smeared gauge KK5-brane of co-dimension two, only the one component of the
$B$-field is non-trivial. As discussed before, this is just a constant and we
choose this $B_{34}=\Theta$. The explicit form of the smeared gauge KK5-brane
solution is found in appendix. Now we perform the second T-duality
transformation of the gauge solution. For the gauge KK5-brane, again the
$\omega_{+}^{2}$ term in $G^{\prime}_{MN}$ in the heterotic Buscher rule is
negligible as it is a higher order in $\alpha^{\prime}$. However, the Yang-
Mills gauge field remains non-trivial and contributes to the Buscher rule.
After calculations, we find the following gauge $5^{2}_{2}$-brane solution:

$\displaystyle ds^{2}$ $\displaystyle=\eta_{ij}dx^{i}dx^{j}+{\cal
I}[(dx^{1})^{2}+(dx^{2})^{2}]+{{\cal I}\over
e^{4\phi_{0}}+\Theta^{2}}[(dx^{3})^{2}+(dx^{4})^{2}],$ $\displaystyle B_{34}$
$\displaystyle=-{\Theta\over
e^{4\phi_{0}}+\Theta^{2}},~{}~{}~{}~{}~{}\phi_{=}{1\over 2}\log({{\cal I}\over
e^{4\phi_{0}}+\Theta^{2}}),~{}~{}~{}~{}~{}{\cal
I}=e^{2\phi_{0}}-{\alpha^{\prime}\tilde{\sigma}^{2}\over
2r^{2}\bigl{(}\tilde{h}_{0}-{\tilde{\sigma}\over 2}\log(r/\mu)\bigr{)}^{2}},$
$\displaystyle A_{1}$ $\displaystyle={-\tilde{\sigma}x^{2}\over
4r^{2}\bigl{(}\tilde{h}_{0}-{\tilde{\sigma}\over
2}\log(r/\mu)\bigr{)}}T^{3},~{}~{}~{}~{}~{}A_{2}={\tilde{\sigma}x^{1}\over
4r^{2}\bigl{(}\tilde{h}_{0}-{\tilde{\sigma}\over 2}\log(r/\mu)\bigr{)}}T^{3},$
$\displaystyle A_{3}$ $\displaystyle={\tilde{\sigma}\over
4r^{2}\bigl{(}\tilde{h}_{0}-{\tilde{\sigma}\over
2}\log(r/\mu)\bigr{)}(e^{4\phi_{0}}+\Theta^{2})}\Bigl{(}e^{2\phi_{0}}(x^{1}T^{2}-x^{2}T^{1})+\Theta(x^{1}T^{1}+x^{2}T^{2})\Bigr{)},$
$\displaystyle A_{4}$ $\displaystyle={\tilde{\sigma}\over
4r^{2}\bigl{(}\tilde{h}_{0}-{\tilde{\sigma}\over
2}\log(r/\mu)\bigr{)}(e^{4\phi_{0}}+\Theta^{2})}\Bigl{(}e^{2\phi_{0}}(x^{1}T^{1}+x^{2}T^{2})+\Theta(x^{1}T^{2}-x^{2}T^{1})\Bigr{)},$
$\displaystyle A_{i}$ $\displaystyle=0.$ (86)

The gauge field satisfies the vortex-like equation (57) which is a reminiscent
of the self-duality equation (11). In the symmetric and the neutral
$5^{2}_{2}$-brane solutions, there was the function $\omega$ which contains
explicit angular coordinate on the $x^{1}x^{2}$-plane. In the gauge
$5^{2}_{2}$-brane solution, the $B$-field is governed by a parameter $\Theta$
instead of $\omega$. However, since $\Theta$ is a constant parameter, the
gauge $5^{2}_{2}$-brane solution is completely determined by single-valued
functions and it is a geometric solution. The function $\mathcal{I}$ becomes
negative at a finite value of $r$. This is the same situation in the case of
the gauge KK5-brane and, unfortunately, its physical meaning is still obscure.
We will make a comment on this property in section 5. A summary of the
smearing and the T-duality relations for the heterotic five-branes is found in
Figure 1 .

Figure 1:  The smearing and the T-duality relations for the heterotic five-
branes of various co-dimensions.

##  4 Monodromy and T-fold

In this section, we study the monodromy of the heterotic $5^{2}_{2}$-brane
solutions. The T-duality symmetry in heterotic string theories compactified on
$T^{d}$ with Wilson lines have been studied in detail [35, 36]. The Wilson
line fields break the $SO(32)$ or $E_{8}\times E_{8}$ gauge group down to a
Cartan subgroup of the gauge group. The off-diagonal parts of the Yang-Mills
gauge field are Higgsed and becomes massive. In the lower-dimensions, there
are $U(1)$ gauge fields which originate from the Kaluza-Klein reduction of the
metric, the NS-NS $B$-field and the Yang-Mills gauge field that correspond to
the Cartan subgroup. In this case, the T-duality group has been determined to
be $O(d,d+\text{dim}G,\mathbb{R})$ where $\text{dim}G$ is the number of the
$U(1)$ sector associated with the Cartan subgroup [37].

On the other hand, when the Wilson line fields are absent, the non-Abelian
gauge group is not broken and the T-duality group
$O(d,d+\text{dim}G,\mathbb{R})$ reduces to $O(d,d,\mathbb{R})$ [38]. Their
study relies on the S-matrix analysis of strings and the result is true in all
orders in $\alpha^{\prime}$. In order to clarify the $O(d,d)$ covariance of
field configurations, it is convenient to consider the generalized metric
$\mathcal{H}$. The generalized metric $\mathcal{H}$ in type II supergravity is
a $2d\times 2d$ matrix and defined through the metric and the $B$-field.
Although, the T-duality group $O(d,d)$ itself does not change for all orders
in $\alpha^{\prime}$, the generalized metric receives
$\alpha^{\prime}$-corrections. This is analogous to the
$\alpha^{\prime}$-corrected Buscher rule of the T-duality transformation.
Remarkably, in addition to the metric and the $B$-field, the Yang-Mills gauge
field plays an important role in heterotic theories. The generalized metric in
heterotic theories is determined by utilizing the heterotic supergravity
action (1) compactified on $T^{d}$ [38]. This is given by

$\displaystyle\mathcal{H}=\left[\begin{array}[]{cc}G^{-1}&-G^{-1}B\\\
BG^{-1}&G-BG^{-1}B\end{array}\right],$ (89)

where $G$ is defined by

$\displaystyle
G_{\mu\nu}=g_{\mu\nu}+2\alpha^{\prime}\bigl{[}\mbox{Tr}(\omega_{+\mu}\omega_{+\nu})-\mbox{Tr}(A_{\mu}A_{\nu})\bigr{]},$
(90)

Here $\mu,\nu$ are the isometry directions. The generalized metric (89) takes
the same form in type II supergravities but the second term in (90) is
characteristic to heterotic theories. The spin connection term in (90) has
been introduced as it enters into the action (1) in the same way as the Yang-
Mills gauge field [38]. In the following, we investigate the monodromy
structures of the heterotic $5^{2}_{2}$-branes by using the generalized metric
(90).

#### Symmetric $5^{2}_{2}$-brane

For the symmetric $5^{2}_{2}$-brane, since the standard embedding condition is
satisfied, we can choose a gauge where the second term in (90) is canceled. We
find that the generalized metric is the same with the one in type II theory.
For the solution (55), this is given by

$\displaystyle\mathcal{H}(\theta)=\left[\begin{array}[]{cccc}H^{-1}K&0&0&H^{-1}\omega\\\
0&H^{-1}K&-H^{-1}\omega&0\\\ 0&-H^{-1}\omega&HK^{-1}+(HK)^{-1}\omega^{2}&0\\\
H^{-1}\omega&0&0&HK^{-1}+(HK)^{-1}\omega^{2}\end{array}\right].$ (95)

When we go around the center of the $5^{2}_{2}$-brane and come back to the
original point, namely if the angular position changes as $\theta=0\to 2\pi$,
then the generalized metric is evaluated as

$\displaystyle\mathcal{H}(2\pi)=O^{t}\mathcal{H}(0)O,$ (96)

where

$\displaystyle O=\left[\begin{array}[]{cc}-i\tau_{2}&0\\\
2\pi\sigma\mathbf{1}_{2}&-i\tau_{2}\end{array}\right]\in O(2,2).$ (99)

This implies that the monodromy is given by the $O(2,2)$ T-duality
transformation. Therefore, although the symmetric $5^{2}_{2}$-brane solution
is non-geometric, it is a T-fold and a consistent solution to heterotic string
theories.

#### Neutral $5^{2}_{2}$-brane

For the neutral $5^{2}_{2}$-brane, the bulk gauge field is trivial and we can
always choose the gauge where $A_{M}=0$. Again, the modified spin connection
is $\mathcal{O}(\alpha^{\prime})$ and it does not contribute to (90) and to
the generalized metric. Then the generalized metric is given by (95) and its
monodromy structure is the same with the symmetric case. Therefore we find
that the neutral $5^{2}_{2}$-brane is a T-fold at least at
$\mathcal{O}(\alpha^{\prime})$ in heterotic theories.

#### Gauge $5^{2}_{2}$-brane

For the gauge $5^{2}_{2}$-brane, the situation is different. The gauge field
contributes to the generalized metric through (90) at
$\mathcal{O}(\alpha^{\prime})$. However, since all the fields in the gauge
solution do not depend on the angle $\theta$ in the two-dimensional base
space, they do not inherit multi-valuedness of the geometry. Therefore the
monodromy becomes trivial. This can be seen by evaluating the generalized
metric for example in $\Theta=0$ gauge. In this gauge, we have

$\displaystyle\mathcal{H}=\left[\begin{array}[]{cc}G^{-1}&0\\\
0&G\end{array}\right],\qquad G=e^{-2\phi_{0}}\mathbf{1}_{2}.$ (102)

This implies $\mathcal{H}(2\pi)=\mathcal{H}(0)$. Therefore we concludes that
the gauge $5^{2}_{2}$-brane does not exhibit non-geometric nature.

##  5 Conclusion and discussions

In this paper we studied the T-duality chains of five-branes in heterotic
supergravity. A specific feature of heterotic supergravity is the Yang-Mills
gauge sector which appears in the linear order in the
$\alpha^{\prime}$-corrections. There are also higher derivative corrections of
the curvature square term in the same order in $\alpha^{\prime}$. The three
different half BPS five-brane solutions in this $\alpha^{\prime}$ order are
known. They are the symmetric, the neutral and the gauge NS5-brane solutions.
These are distinguished by the topological charge of instantons of Yang-Mills
gauge field and the charge associated with the modified $H$-flux.

We introduced the $U(1)$ isometry along a transverse direction to the
NS5-brane world-volume and explicitly performed the T-duality transformation
of these solutions. Due to the $\alpha^{\prime}$-corrections in heterotic
supergravity, the Buscher rule is modified by the corrections. For the
symmetric solution, where the standard embedding condition is satisfied, the
$\alpha^{\prime}$-corrections in the modified Buscher rule cancel out. The
resulting metric, the $B$-field and the dilaton are nothing but the ones for
the KK5-brane solution in type II theory. We demonstrated that the Yang-Mills
gauge field satisfies the standard embedding condition again and it is given
by the solution to the monopole equation in three dimensions. For the neutral
solution, we find that the T-dualized solution is given by the KK5-brane in
type II theory at least at $\mathcal{O}(\alpha^{\prime})$. The solution is
given by the purely geometric Taub-NUT metric. For the gauge solution, the
geometry is ill-defined near the brane core after the smearing procedure. This
property is carried over to the T-dualized solution. For the gauge KK5-brane
solution, the $B$-field is not excited which is the same with the KK5-brane in
type II theory. However, the geometry is well-defined only at the asymptotic
region.

We then introduce another $U(1)$ isometry to the KK5-brane solutions and
perform the second T-duality transformation. The resulting solutions are the
exotic $5^{2}_{2}$-branes in heterotic theory. For the symmetric solution, the
metric, $B$-field and the dilation are given by that of the $5^{2}_{2}$-brane
in type II theory. The gauge field satisfies the vortex-like equation in two
dimensions. We found that the standard embedding condition is satisfied up to
a gauge transformation. For the neutral solution, we found that the neutral
$5^{2}_{2}$-brane is the same with the one in type II theory. They exhibit a
non-geometric nature due to the multi-valuedness of the $B$-field. For the
gauge $5^{2}_{2}$-brane solution, we found that the fields do not show the
multi-valuedness and they remain geometric.

We next studied the monodromies of the three different $5^{2}_{2}$-branes. We
calculated the generalized metric for these solutions. We found that the
symmetric and the neutral $5^{2}_{2}$-branes have a monodromy given by the
$O(2,2)$ T-duality transformation. Therefore they are T-folds. On the other
hand, the gauge solution does not show the nature of a T-fold.

Following the general discussion in [34], the symmetric solution seems to be
exact in terms of $\alpha^{\prime}$. On the other hand, for the neutral and
the gauge $5^{2}_{2}$-brane solutions, they generically receive
$\alpha^{\prime}$-corrections. The result is summarized in Table 1.

Type | Geometry | Valid order in $\alpha^{\prime}$ | Notes
---|---|---|---
Symmetric | T-fold |  $\alpha^{\prime}$-exact | standard embedding
Neutral | T-fold | $\mathcal{O}(\alpha^{\prime})$ | Type II solution
Gauge | geometric | $\mathcal{O}(\alpha^{\prime})$ | ill-defined near the center
Table 1: Properties of the heterotic $5^{2}_{2}$-branes.

A few comments are in order about the solutions. We introduced the $U(1)$
isometry to the gauge NS5-brane by the smearing procedure of the instantons.
The resulting gauge field is just the Harrington-Shepard calorons in the small
radius limit. The dilaton and the metric at $\mathcal{O}(\alpha^{\prime})$ are
determined through the Bianchi identity
$d\hat{H}^{(3)}=\alpha^{\prime}\mathrm{Tr}F\wedge F$ where the right-hand side
is given by the topological charge density for the smeared caloron. The
resulting geometry is ill-defined near the center of the brane. We stress that
there is another co-dimension three solution based on the BPS monopole of ’t
Hooft-Polyakov type instead of the smeared caloron [27, 29]. In the gauge
NS5-brane solution of co-dimension three based on the BPS monopole type, the
metric and the $B$-field behave well-defined near the core of the five-brane.
The gauge KK5-and $5^{2}_{2}$-branes of BPS monopole type would show better
physical interpretation of T-dualized solutions. A related property of the
solutions is the logarithmic behaviour of the $5^{2}_{2}$-branes of all types.
This is characteristic to the co-dimension two objects and found also in the
solution in type II theory [15]. Similar to the gauge solutions of the smeared
caloron type, the $5^{2}_{2}$-branes discussed in section 3.2 seem to be ill-
defined at asymptotic region. However this does not indicate any inconsistency
of the solutions but the general property of co-dimension two objects.
Analogous to the D7-brane in type IIB string theory, the exotic
$5^{2}_{2}$-brane is not well-defined as the stand-alone object. We need other
co-existing branes in order to write down asymptotically flat globally well-
defined solutions 666An example is the $SL(2,\mathbf{Z})$ multiplet of
7-branes in type IIB string theory [39].. Indeed, the scale $\mu$ in (52)
specifies the “cutoff” point where the effect of the next duality branes is
not negligible [31]. We believe that globally well-defined $5^{2}_{2}$-brane
solutions exist even in heterotic theories.

We note that the most tractable way to study the non-geometric nature of
string theory solutions is the double field theory construction of
supergravity [40, 41]. There are several studies about double field theory
formulation of heterotic supergravity [42] and the inclusion of
$\alpha^{\prime}$-corrections [43]. Although the spin connection term in the
generalized metric (90) is a conjectural one, the supersymmetry transformation
law of the heterotic supergravity and the double field theory analysis
strongly suggest that this is true [38]. Comprehensive studies are presented
in the generalized geometry with $\alpha^{\prime}$-corrections [45]. The
heterotic $5^{2}_{2}$-brane is expected to be a source of the non-geometric
flux or mixed-symmetric tensor and they are in the T-duality multiplets in
lower-dimensions [46]. It is also interesting to study the world-volume
effective action for the non-geometric branes [47, 48] in heterotic theory. We
will come back to these issues in future studies.

### Acknowledgments

The authors would like to thank T. Kimura and S. Mizoguchi for useful
discussions and comments. The work of S. S. is supported in part by Kitasato
University Research Grant for Young Researchers. The work of M. Y. is
supported by NUS Tier 1 FRC Grant R-144-000-316-112.

##  Appendix A Smeared solutions for gauge type

In this appendix, we introduce the explicit solutions of the smeared gauge
KK5-brane. It is convenient first to introduce the defect gauge NS5-brane
solution before we write down the smeared KK5-brane solution. The defect gauge
NS5-brane solution is obtained by the smearing procedure along the
$x^{4}$-direction to the $x^{3}$-smeared gauge NS5-brane. The resulting
solution is a brane of co-dimension two. By performing the T-duality
transformation along the $x^{3}$-direction, we obtain the $x^{4}$-smeared
gauge KK5-brane.

###  A.1 Defect gauge NS5-brane solution

The defect gauge NS5-brane is the co-dimension two gauge NS5-brane and it is
obtained by smearing the two directions of the gauge NS5-brane solution (34)
along the same way to obtain the smeared gauge NS5-brane solution (48). The
result is

$\displaystyle ds^{2}$ $\displaystyle=\eta_{ij}dx^{i}dx^{j}+{\cal
I}\delta_{mn}dx^{m}dx^{n},~{}~{}~{}~{}~{}{\cal
I}=e^{2\phi_{0}}-{\alpha^{\prime}\tilde{\sigma}^{2}\over
2r^{2}\bigl{(}\tilde{h}_{0}-{\tilde{\sigma}\over 2}\log(r/\mu)\bigr{)}^{2}},$
$\displaystyle A_{m}$
$\displaystyle=-\bar{\sigma}_{mn}{\tilde{\sigma}x^{n}\over
4r^{2}\bigl{(}\tilde{h}_{0}-{\tilde{\sigma}\over
2}\log(r/\mu)\bigr{)}},~{}~{}~{}~{}\phi={1\over 2}\log{\cal
I},~{}~{}~{}\hat{H}^{(3)}_{mnp}=-{1\over 2}\varepsilon_{mnpq}\partial_{q}{\cal
I},$ (103)

where $r^{2}=(x^{1})^{2}+(x^{2})^{2}$. In the gauge solution, since the non-
zero components of the modified $H$-flux $\hat{H}_{mnp}$ come from the Yang-
Mills Chern-Simons term, the $B$-field is taken to be a constant. For the
defect NS5-brane solution, the relevant non-zero component of the $B$-field is
$B_{34}=\Theta$. When we perform the heterotic T-duality transformation along
the $x^{4}$-direction for the defect NS5-brane solution, we can obtain the
smeared gauge KK5-brane solution shown below.

###  A.2 Smeared gauge KK5-brane solution

The smeared gauge KK5-brane solution is obtained by smearing $x^{3}$-direction
in the gauge KK5-brane solution (86). The explicit form is as follows:

$\displaystyle ds^{2}$ $\displaystyle=\eta_{ij}dx^{i}dx^{j}+{\cal
I}\delta_{m^{\prime}n^{\prime}}dx^{m^{\prime}}dx^{n^{\prime}}+{\cal
I}e^{-4\phi_{0}}[dx^{4}+\Theta dx^{3}]^{2},$ $\displaystyle B_{MN}$
$\displaystyle=0,~{}~{}~{}~{}~{}\phi={1\over 2}\log(e^{-2\phi_{0}}{\cal
I}),~{}~{}~{}~{}~{}{\cal
I}=e^{2\phi_{0}}-{\alpha^{\prime}\tilde{\sigma}^{2}\over
2r^{2}\bigl{(}\tilde{h}_{0}-{\tilde{\sigma}\over 2}\log(r/\mu)\bigr{)}^{2}},$
$\displaystyle A_{1}$ $\displaystyle={-\tilde{\sigma}x^{2}\over
4r^{2}\bigl{(}\tilde{h}_{0}-{\tilde{\sigma}\over
2}\log(r/\mu)\bigr{)}^{2}}T^{3},~{}~{}~{}~{}~{}A_{2}={\tilde{\sigma}x^{1}\over
4r^{2}\bigl{(}\tilde{h}_{0}-{\tilde{\sigma}\over
2}\log(r/\mu)\bigr{)}^{2}}T^{3},$ $\displaystyle A_{3}$
$\displaystyle={\tilde{\sigma}\over
4r^{2}\bigl{(}\tilde{h}_{0}-{\tilde{\sigma}\over
2}\log(r/\mu)\bigr{)}^{2}}\Bigl{(}(x^{2}T^{1}-x^{1}T^{2})+e^{-2\phi_{0}}\Theta(x^{1}T^{1}+x^{2}T^{2})\Bigr{)},$
$\displaystyle A_{4}$ $\displaystyle=e^{-2\phi_{0}}{\tilde{\sigma}\over
4r^{2}\bigl{(}\tilde{h}_{0}-{\tilde{\sigma}\over
2}\log(r/\mu)\bigr{)}^{2}}(x^{1}T^{1}+x^{2}T^{2}).$ (104)

As we mentioned above, the solution is obtained by taking the heterotic
T-duality transformation along the $x^{4}$-direction on the defect gauge
NS5-brane solution (103). When we take the heterotic T-duality transformation
with the $x^{3}$-direction instead of the $x^{4}$-direction on (103), we find
the other type of smeared gauge KK5-brane solution. The solution is different
with the sign in front of $\Theta$ in (104), but the physical meanings are the
same for both of the solutions. On the other hands, if we take the heterotic
T-duality along the $x^{3}$-direction for the smeared gauge KK5-brane, we
obtain the gauge $5^{2}_{2}$-brane solution as we see in (86).

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