text:
\section{Introduction}
String theory is a quantum theory of gravity   with  a finite number of massless fields and a  tower of infinite number of  massive fields reflecting the stringy nature of the gravity.
An efficient way to study different phenomena in this theory is to use an effective action  which includes     only  massless fields. The effects of the massive fields appear in the action   as the  higher derivatives of the massless fields. This effective action may be found  by imposing various  symmetries/dualities in the string theory. There are various gauge symmetries in the effective actions which  are corresponding to the various massless fields, {\it e.g.,}\  the diffeomorphism symmetry corresponds to the metric and  the  gauge symmetry corresponds to the Kalb-Ramond field or $B$-field. In the bosonic string theory which has only metric, dilaton and $B$-field, they are the only local symmetries of the effective action. Imposing only these symmetries, one finds the effective action   has three couplings at order $\alpha'^0$ (two-derivative order),   has 8 couplings at order $\alpha'$ (four-derivative order)   up to field redefinitions \cite{Metsaev:1987zx},  has 60 couplings at order $\alpha'^2$ (six-derivative order) \cite{Garousi:2019cdn}  and so on. The gauge symmetries, however, can not determine the coefficients of the couplings. These parameters  may be found by S-matrix calculations \cite{Scherk:1974mc,Yoneya:1974jg},  by sigma-model calculations \cite{Callan:1985ia,Fradkin:1984pq,Fradkin:1985fq} or by imposing global symmetries of the string theory in which we are interested.

One of the global symmetries of the  string theory is   T-duality \cite{Giveon:1994fu,Alvarez:1994dn}. This   duality  like  the above gauge symmetries  may be imposed at the action level  to fix the parameters of  the   effective action   at any order of $\alpha'$. One approach for imposing this symmetry  is the   Double Field Theory (DFT) \cite{Siegel:1993xq,Siegel:1993th,Siegel:1993bj,Hull:2009mi,Aldazabal:2013sca} in which the $D$-dimensional  effective action is extended to $2D$-space. In this theory, the gauge transformations are deformed to receive $\alpha'$-corrections whereas  the T-duality symmetry is  imposed without deformation simply by writing the couplings as  $O(D,D)$ scalars \cite{Hohm:2010pp,Aldazabal:2013sca,Hohm:2014xsa,Marques:2015vua,Garousi:2018qes}.
Another  approach     is  to reduce the $D$-dimensional gauge invariant theory on a circle and impose the T-duality symmetry   by constraining   the couplings in  the $(D-1)$-dimensional spacetime  to be $Z_2$ scalars \cite{Garousi:2017fbe} where $Z_2$-group is the Buscher rules  \cite{Buscher:1987sk,Buscher:1987qj} plus their  $\alpha'$-deformations  \cite{Tseytlin:1991wr,Bergshoeff:1995cg,Kaloper:1997ux}.       Using this approach for the case that $B$-field is zero, the known gravity and dilaton  couplings in the effective actions at orders $\alpha',\alpha'^2,\alpha'^3$ have been found  in   \cite{Razaghian:2017okr,Razaghian:2018svg},  up to some overall factors. Moreover, when $B$-field is non-zero, the known couplings at order $\alpha'$ and their corresponding corrections to the Buscher rules have been found in \cite{Garousi:2019wgz} . In this paper, we are going to use this  approach to find the couplings at order $\alpha'^2$ for the case that $B$-field is non-zero. These couplings,  except its Riemann cubed couplings,  have not been  found by any other methods in string theory.

It is known  that the effective action at order $\alpha'^2$ depends on the scheme that one uses for the effective action at order $\alpha'$ \cite{Bento1990}. In the T-duality approach,  this is reflected to the T-duality transformations at order $\alpha'$. It has been observed  in \cite{Garousi:2019wgz} that the T-duality transformations at order $\alpha'$ depends on the scheme that one uses for the effective action  at order $\alpha'$. The T-duality transformation corresponding to the effective action at order $\alpha'$  which has only first time derivative \cite{Meissner:1996sa},  is given in \cite{Kaloper:1997ux}.   The T-duality transformations at order $\alpha'$ corresponding to the effective action at order $\alpha'$ in an arbitrary scheme have been found in  \cite{Garousi:2019wgz}. In this paper we are going to  find the effective action at order $\alpha'^2$ that correspond to the   effective action at order $\alpha'$ which has  minimum number of  couplings \cite{Metsaev:1987zx}.

The outline of the paper is as follows: In section 2, we write the known minimum number of couplings at orders   $\alpha'$ and $\alpha'^2$ that the gauge symmetry can fix up to field redefinitions.   In section 3, we impose the T-duality symmetry on the gauge invariant couplings to find their corresponding parameter. The calculations at order   $\alpha'$   have been already done in  \cite{Garousi:2019wgz}. That calculations produce the known couplings in the literature and the corresponding T-duality transformations. The calculations at order $\alpha'^2$ are new. We have found both the effective action and the corresponding  T-duality transformations. However, since the expressions for the T-duality transformations are very lengthy we will   write only the effective action (see \reef{S2f}). We have found that there are only 27 non-zero couplings in the effective action at order $\alpha'^2$. Two of them have  already   been found by the S-matrix calculations \cite{Metsaev:1986yb}. All other terms are new couplings that the T-duality constraint produces.  In section 4, we briefly discuss our results.

  \section{Gauge invariance constraint}\label{sec.2}

The effective action of string theory has a double expansions. One expansion is the genus expansion which includes  the  classical sphere-level and a tower of quantum effects. The other one is a stringy expansion which is an expansion in terms of higher-derivative or   $\alpha'$-expansion. The classical effective action  has the following     $\alpha'$-expansion in the string frame:
\begin{eqnarray}
\bS_{\rm eff}&=&\sum^\infty_{n=0}\alpha'^n\bS_n=\bS_0+\alpha'\bS_1+\alpha'^2 \bS_2+\cdots\ ; \quad \bS_n= -\frac{2}{\kappa^2}\int d^D x\sqrt{-G} e^{-2\Phi}\mathcal{L}_n\labell{seff}
\end{eqnarray}
The effective action must be invariant under the coordinate transformations and under the $B$-field   gauge transformations. So the metric $G_{\mu\nu}$, the  antisymmetric  $B$-field and dilaton  $\Phi$      must  appear in the Lagrangian $ \mathcal{L}_n$  trough their field strengths and their covariant derivatives. This requires the effective action at order $\alpha'^0$ to have the following couplings:
\begin{eqnarray}
\mathcal{L}_0&= &a_1R+a_2\nabla_\alpha\Phi\nabla^\alpha\Phi+a_3H_{\alpha\beta\gamma}H^{\alpha\beta\gamma}\labell{L0}
\end{eqnarray}
where $a_1,a_2,a_3$ are three   parameters which can not be fixed by the gauge invariance constraints.

At higher orders of  $\alpha'$, one has the freedom of using   field redefinitions  and the  Bianchi identities. As a result, there are no unique form for the couplings, even the number of couplings are not unique  at the higher orders of $\alpha'$. There are however, schemes in which the number of  couplings are minimum.     It has been shown in     \cite{Metsaev:1987zx} that the minimum number of couplings at order $\alpha'$ is 8. These 8 couplings can  also be written in different schemes. In one particular such  scheme, the couplings can be written as \cite{Garousi:2019cdn}
\begin{eqnarray}
 \mathcal{L}_1&=& \mathcal{L}^1_1+ \mathcal{L}^2_1\labell{L1112}
\end{eqnarray}
where $\mathcal{L}^1_1$ includes the minimum number of couplings which do not include the dilaton, {\it i.e.,}\ ,
\begin{eqnarray}
\mathcal{L}^1_1&= &  b_1 R_{\alpha \gamma \beta \delta} R^{\alpha \beta \gamma \delta}
  +b_2 H_{\alpha}{}^{\delta \epsilon} H^{\alpha \beta \gamma} H_{\beta \delta}{}^{\zeta} H_{\gamma \epsilon \zeta} \nonumber\\
  &&
  + b_3 H_{\alpha \beta}{}^{\delta} H^{\alpha \beta \gamma} H_{\gamma}{}^{\epsilon \zeta} H_{\delta \epsilon \zeta}
 +b_4 H_{\alpha}{}^{\delta \epsilon} H^{\alpha \beta \gamma} R_{\beta \delta \gamma \epsilon}\labell{L11}
 \end{eqnarray}
 and  $\mathcal{L}^2_1$ includes the other couplings  which   all include non-trivially the dilaton, {\it i.e.,}\ ,
\begin{eqnarray}
  \mathcal{L}^2_1&= & b_5 H_{\beta \gamma \delta} H^{\beta \gamma \delta} \nabla_{\alpha}\Phi \nabla^{\alpha}\Phi
 +b_6 H_{\alpha}{}^{\gamma \delta} H_{\beta \gamma \delta} \nabla^{\alpha}\Phi \nabla^{\beta}\Phi \nonumber\\&&
 +b_7 H_{\alpha}{}^{\gamma \delta} H_{\beta \gamma \delta} \nabla^{\beta}\nabla^{\alpha}\Phi
 + b_8 \nabla_{\alpha}\Phi \nabla^{\alpha}\Phi \nabla_{\beta}\Phi \nabla^{\beta}\Phi \labell{L12}
 \end{eqnarray}
 where $b_1,\cdots, b_8$ are eight parameters   which can not be fixed by the gauge invariance  constraints.

At order $\alpha'^2$,   the minimum number of couplings is 60. In one particular minimal scheme in which there is no  $R,\,R_{\mu\nu},\,\nabla_\mu H^{\mu\alpha\beta}$, $ \nabla_\mu\nabla^\mu\Phi$, the couplings are \cite{Garousi:2019cdn}
\begin{eqnarray}
\mathcal{L}_2&=&\mathcal{L}_2^1+\mathcal{L}_2^2 \labell{L2122}
\end{eqnarray}
where $\mathcal{L}_2^1$ has the minimum number of couplings in which the dilaton does not appear, {\it i.e.,}\ ,
\begin{eqnarray}
\mathcal{L}_2^1&=& c_1R_{\alpha}{}^{\epsilon}{}_{\gamma}{}^{\zeta} R^{\alpha \beta \gamma \delta} R_{\beta \zeta \delta \epsilon} + c_2 R_{\alpha \beta}{}^{\epsilon \zeta} R^{\alpha \beta \gamma \delta} R_{\gamma \epsilon \delta \zeta}+c_3 H_{\alpha}{}^{\delta \epsilon} H^{\alpha \beta \gamma} H_{\beta \delta}{}^{\zeta} H_{\gamma}{}^{\iota \kappa} H_{\epsilon \iota}{}^{\mu} H_{\zeta \kappa \mu}\nonumber\\&& + c_4 H_{\alpha \beta}{}^{\delta} H^{\alpha \beta \gamma} H_{\gamma}{}^{\epsilon \zeta} H_{\delta}{}^{\iota \kappa} H_{\epsilon \zeta}{}^{\mu} H_{\iota \kappa \mu} + c_{5} H_{\alpha \beta}{}^{\delta} H^{\alpha \beta \gamma} H_{\gamma}{}^{\epsilon \zeta} H_{\delta \epsilon}{}^{\iota} H_{\zeta}{}^{\kappa \mu} H_{\iota \kappa \mu}\nonumber\\&& + c_{6} H_{\alpha}{}^{\delta \epsilon} H^{\alpha \beta \gamma} H_{\beta}{}^{\zeta \iota} H_{\delta \zeta}{}^{\kappa} R_{\gamma \epsilon \iota \kappa}  + c_{7} H_{\alpha}{}^{\delta \epsilon} H^{\alpha \beta \gamma} R_{\beta}{}^{\zeta}{}_{\delta}{}^{\iota} R_{\gamma \zeta \epsilon \iota} + c_{8} H_{\alpha \beta}{}^{\delta} H^{\alpha \beta \gamma} H_{\epsilon \zeta}{}^{\kappa} H^{\epsilon \zeta \iota} R_{\gamma \iota \delta \kappa}\nonumber\\&& + c_{9} H^{\alpha \beta \gamma} H^{\delta \epsilon \zeta} R_{\alpha \beta \delta}{}^{\iota} R_{\gamma \iota \epsilon \zeta}  + c_{10} H_{\alpha}{}^{\delta \epsilon} H^{\alpha \beta \gamma} R_{\beta}{}^{\zeta}{}_{\delta}{}^{\iota} R_{\gamma \iota \epsilon \zeta} + c_{11} H_{\alpha}{}^{\delta \epsilon} H^{\alpha \beta \gamma} R_{\beta}{}^{\zeta}{}_{\gamma}{}^{\iota} R_{\delta \zeta \epsilon \iota}\nonumber\\&& + c_{12} H_{\alpha \beta}{}^{\delta} H^{\alpha \beta \gamma} R_{\gamma}{}^{\epsilon \zeta \iota} R_{\delta \zeta \epsilon \iota}  + c_{13} H_{\alpha \beta}{}^{\delta} H^{\alpha \beta \gamma} H_{\gamma}{}^{\epsilon \zeta} H_{\epsilon}{}^{\iota \kappa} R_{\delta \iota \zeta \kappa} + c_{14} H_{\alpha}{}^{\delta \epsilon} H^{\alpha \beta \gamma} H_{\beta \delta}{}^{\zeta} H_{\gamma}{}^{\iota \kappa} R_{\epsilon \iota \zeta \kappa}\nonumber\\&& + c_{15} H_{\alpha \beta}{}^{\delta} H^{\alpha \beta \gamma} H_{\gamma}{}^{\epsilon \zeta} H_{\delta}{}^{\iota \kappa} R_{\epsilon \iota \zeta \kappa}  + c_{16} H_{\alpha}{}^{\delta \epsilon} H^{\alpha \beta \gamma} \nabla_{\iota}H_{\delta \epsilon \zeta} \nabla^{\iota}H_{\beta \gamma}{}^{\zeta}\nonumber\\&& + c_{17} H_{\alpha}{}^{\delta \epsilon} H^{\alpha \beta \gamma} \nabla_{\zeta}H_{\gamma \epsilon \iota} \nabla^{\iota}H_{\beta \delta}{}^{\zeta} + c_{18} H_{\alpha}{}^{\delta \epsilon} H^{\alpha \beta \gamma} \nabla_{\iota}H_{\gamma \epsilon \zeta} \nabla^{\iota}H_{\beta \delta}{}^{\zeta}\nonumber\\&& + c_{19} H_{\alpha \beta}{}^{\delta} H^{\alpha \beta \gamma} \nabla_{\zeta}H_{\delta \epsilon \iota} \nabla^{\iota}H_{\gamma}{}^{\epsilon \zeta} + c_{20} H_{\alpha \beta}{}^{\delta} H^{\alpha \beta \gamma} \nabla_{\iota}H_{\delta \epsilon \zeta} \nabla^{\iota}H_{\gamma}{}^{\epsilon \zeta}\labell{L21}
\end{eqnarray}
 and $\mathcal{L}_2^2$ has  the other couplings which all include  derivatives of the dilaton, {\it i.e.,}\ ,
\begin{eqnarray}
\mathcal{L}_2^2&\!\!\!\!\!=\!\!\!\!\!& c_{21} H_{\beta}{}^{\epsilon \zeta} H^{\beta \gamma \delta} H_{\gamma \epsilon}{}^{\iota} H_{\delta \zeta \iota} \nabla_{\alpha}\Phi \nabla^{\alpha}\Phi + c_{22} R_{\beta \gamma \delta \epsilon} R^{\beta \gamma \delta \epsilon} \nabla_{\alpha}\Phi \nabla^{\alpha}\Phi + c_{23} H_{\beta}{}^{\epsilon \zeta} H^{\beta \gamma \delta} R_{\gamma \epsilon \delta \zeta} \nabla_{\alpha}\Phi \nabla^{\alpha}\Phi\nonumber\\&& + c_{24} H_{\alpha}{}^{\gamma \delta} H_{\beta}{}^{\epsilon \zeta} H_{\gamma \epsilon}{}^{\iota} H_{\delta \zeta \iota} \nabla^{\alpha}\Phi \nabla^{\beta}\Phi + c_{25} R_{\alpha}{}^{\gamma \delta \epsilon} R_{\beta \delta \gamma \epsilon} \nabla^{\alpha}\Phi \nabla^{\beta}\Phi \nonumber\\&&+ c_{26} H_{\alpha}{}^{\gamma \delta} H_{\beta}{}^{\epsilon \zeta} R_{\gamma \epsilon \delta \zeta} \nabla^{\alpha}\Phi \nabla^{\beta}\Phi + c_{27} H_{\gamma \delta \epsilon} H^{\gamma \delta \epsilon} \nabla^{\alpha}\Phi \nabla_{\beta}\nabla_{\alpha}\Phi \nabla^{\beta}\Phi \nonumber\\&&+ c_{28} H_{\alpha}{}^{\gamma \delta} H_{\beta}{}^{\epsilon \zeta} H_{\gamma \epsilon}{}^{\iota} H_{\delta \zeta \iota} \nabla^{\beta}\nabla^{\alpha}\Phi + c_{29} H_{\alpha}{}^{\gamma \delta} H_{\beta}{}^{\epsilon \zeta} H_{\gamma \delta}{}^{\iota} H_{\epsilon \zeta \iota} \nabla^{\beta}\nabla^{\alpha}\Phi \nonumber\\&&+ c_{30} H_{\alpha}{}^{\gamma \delta} H_{\beta \gamma}{}^{\epsilon} H_{\delta}{}^{\zeta \iota} H_{\epsilon \zeta \iota} \nabla^{\beta}\nabla^{\alpha}\Phi + c_{31} H_{\gamma \delta}{}^{\zeta} H^{\gamma \delta \epsilon} R_{\alpha \epsilon \beta \zeta} \nabla^{\beta}\nabla^{\alpha}\Phi + c_{32} R_{\alpha}{}^{\gamma \delta \epsilon} R_{\beta \delta \gamma \epsilon} \nabla^{\beta}\nabla^{\alpha}\Phi \nonumber\\&&+ c_{33} H_{\alpha}{}^{\gamma \delta} H_{\gamma}{}^{\epsilon \zeta} R_{\beta \epsilon \delta \zeta} \nabla^{\beta}\nabla^{\alpha}\Phi + c_{34} H_{\alpha}{}^{\gamma \delta} H_{\beta}{}^{\epsilon \zeta} R_{\gamma \epsilon \delta \zeta} \nabla^{\beta}\nabla^{\alpha}\Phi + c_{35} H_{\alpha}{}^{\delta \epsilon} \nabla^{\alpha}\Phi \nabla^{\beta}\Phi \nabla_{\gamma}H_{\beta \delta \epsilon} \nabla^{\gamma}\Phi \nonumber\\&&+ c_{36} \nabla_{\alpha}\Phi \nabla^{\alpha}\Phi \nabla_{\beta}\Phi \nabla^{\beta}\Phi \nabla_{\gamma}\Phi \nabla^{\gamma}\Phi + c_{37} \nabla_{\alpha}\Phi \nabla^{\alpha}\Phi \nabla^{\beta}\Phi \nabla_{\gamma}\nabla_{\beta}\Phi \nabla^{\gamma}\Phi \nonumber\\&&+ c_{38} H_{\beta}{}^{\delta \epsilon} H_{\gamma \delta \epsilon} \nabla^{\alpha}\Phi \nabla^{\beta}\Phi \nabla^{\gamma}\nabla_{\alpha}\Phi + c_{39} H_{\beta}{}^{\delta \epsilon} H_{\gamma \delta \epsilon} \nabla^{\beta}\nabla^{\alpha}\Phi \nabla^{\gamma}\nabla_{\alpha}\Phi\nonumber\\&& + c_{40} \nabla^{\alpha}\Phi \nabla^{\beta}\Phi \nabla_{\gamma}\nabla_{\beta}\Phi \nabla^{\gamma}\nabla_{\alpha}\Phi + c_{41} \nabla^{\beta}\nabla^{\alpha}\Phi \nabla_{\gamma}\nabla_{\beta}\Phi \nabla^{\gamma}\nabla_{\alpha}\Phi \nonumber\\&&+ c_{42} H_{\beta}{}^{\delta \epsilon} H_{\gamma \delta \epsilon} \nabla_{\alpha}\Phi \nabla^{\alpha}\Phi \nabla^{\gamma}\nabla^{\beta}\Phi + c_{43} H_{\alpha}{}^{\delta \epsilon} \nabla^{\alpha}\Phi \nabla_{\gamma}H_{\beta \delta \epsilon} \nabla^{\gamma}\nabla^{\beta}\Phi \nonumber\\&&+ c_{44} \nabla_{\alpha}\Phi \nabla^{\alpha}\Phi \nabla_{\gamma}\nabla_{\beta}\Phi \nabla^{\gamma}\nabla^{\beta}\Phi + c_{45} H_{\alpha \gamma}{}^{\epsilon} H_{\beta \delta \epsilon} \nabla^{\alpha}\Phi \nabla^{\beta}\Phi \nabla^{\delta}\nabla^{\gamma}\Phi \nonumber\\&&+ c_{46} R_{\alpha \gamma \beta \delta} \nabla^{\alpha}\Phi \nabla^{\beta}\Phi \nabla^{\delta}\nabla^{\gamma}\Phi + c_{47} H_{\alpha \gamma}{}^{\epsilon} H_{\beta \delta \epsilon} \nabla^{\beta}\nabla^{\alpha}\Phi \nabla^{\delta}\nabla^{\gamma}\Phi + c_{48} R_{\alpha \gamma \beta \delta} \nabla^{\beta}\nabla^{\alpha}\Phi \nabla^{\delta}\nabla^{\gamma}\Phi \nonumber\\&&+ c_{49} H_{\beta}{}^{\delta \epsilon} \nabla^{\alpha}\Phi \nabla^{\gamma}\nabla^{\beta}\Phi \nabla_{\epsilon}H_{\alpha \gamma \delta} + c_{50} H^{\gamma \delta \epsilon} \nabla^{\alpha}\Phi \nabla^{\beta}\nabla_{\alpha}\Phi \nabla_{\epsilon}H_{\beta \gamma \delta} \nonumber\\&&+ c_{51} \nabla^{\alpha}\Phi \nabla^{\beta}\Phi \nabla_{\delta}H_{\beta \gamma \epsilon} \nabla^{\epsilon}H_{\alpha}{}^{\gamma \delta} + c_{52} \nabla^{\beta}\nabla^{\alpha}\Phi \nabla_{\delta}H_{\beta \gamma \epsilon} \nabla^{\epsilon}H_{\alpha}{}^{\gamma \delta} \nonumber\\&&+ c_{53} \nabla^{\alpha}\Phi \nabla^{\beta}\Phi \nabla_{\epsilon}H_{\beta \gamma \delta} \nabla^{\epsilon}H_{\alpha}{}^{\gamma \delta} + c_{54} \nabla^{\beta}\nabla^{\alpha}\Phi \nabla_{\epsilon}H_{\beta \gamma \delta} \nabla^{\epsilon}H_{\alpha}{}^{\gamma \delta} \nonumber\\&&+ c_{55} \nabla_{\alpha}\Phi \nabla^{\alpha}\Phi \nabla_{\epsilon}H_{\beta \gamma \delta} \nabla^{\epsilon}H^{\beta \gamma \delta} + c_{56} H_{\alpha}{}^{\beta \gamma} R_{\gamma \zeta \delta \epsilon} \nabla^{\alpha}\Phi \nabla^{\zeta}H_{\beta}{}^{\delta \epsilon} \nonumber\\&&+ c_{57} H_{\beta \gamma}{}^{\epsilon} H^{\beta \gamma \delta} H_{\delta}{}^{\zeta \iota} \nabla^{\alpha}\Phi \nabla_{\iota}H_{\alpha \epsilon \zeta} + c_{58} H_{\alpha}{}^{\beta \gamma} H_{\delta \epsilon}{}^{\iota} H^{\delta \epsilon \zeta} \nabla^{\alpha}\Phi \nabla_{\iota}H_{\beta \gamma \zeta} \nonumber\\&&+ c_{59} H_{\alpha}{}^{\beta \gamma} H_{\beta}{}^{\delta \epsilon} H_{\delta}{}^{\zeta \iota} \nabla^{\alpha}\Phi \nabla_{\iota}H_{\gamma \epsilon \zeta} + c_{60} H_{\alpha}{}^{\beta \gamma} H_{\beta}{}^{\delta \epsilon} H_{\gamma}{}^{\zeta \iota} \nabla^{\alpha}\Phi \nabla_{\iota}H_{\delta \epsilon \zeta}\labell{AA}
\end{eqnarray}
 where $c_1,\cdots, c_{60}$ are 60  parameters   which can not be fixed by the gauge invariance  constraint.

Up to this point, the above couplings are valid for any higher derivative  theory which includes metric, $B$-field and dilaton. In the string theory, however, the parameters in \reef{L0}, \reef{L1112} and \reef{L2122} may be fixed by imposing some other specific constraints which are  valid only in the string theory. For example, one may construct the appropriate S-matrix elements with the above couplings and then compare them with the $\alpha'$-expansion of the corresponding sphere-level S-matrix elements in the  string theory to fix the parameters. This method has been used in  \cite{Metsaev:1987zx} to find the parameters in   \reef{L0}, \reef{L1112}. The parameters $c_1,c_2$ in  \reef{L2122} have been also found by the S-matrix method in \cite{Metsaev:1986yb}. The S-matrix  method for fixing all parameters in \reef{L2122}, however,  requires one to calculate six-point function in string theory in full details which has not been done yet.

Instead of comparing the S-matrix elements of above couplings with the corresponding S-matrix elements in the string theory, one may impose some other symmetries of the string theory   to fix the parameters in \reef{L0}, \reef{L1112} and \reef{L2122} . The bosonic string theory has the global T-duality symmetry as well as the gauge  symmetries that have been used to find the couplings  in \reef{L0}, \reef{L1112} and \reef{L2122}. It has been shown in \cite{Garousi:2019wgz} that the T-duality symmetry can fix correctly  the couplings in \reef{L0}, \reef{L1112} up to   overall factors at each order of $\alpha'$.   In the next section, we show that imposing the T-duality   on the couplings in \reef{L2122} can also  fix all 60 parameters in terms of the overall factor at order $\alpha'$.

 \section{T-duality invariance constraint}\label{sec.3}

 The T-duality  constraint on the $D$-dimensional effective action $\bS_{\rm eff}$,  in the most simple form, is to reduce the theory on a circle with $U(1)$ isometry to find the $(D-1)$-dimensional effective action $S_{\rm eff}(\psi)$ where $\psi$ represents all massless fields in the $(D-1)$-dimensional base space. Then one has to transform it under the T-duality transformations to produce  $S_{\rm eff}(\psi')$ where $\psi'$ represents the T-duality transformations of the massless fields in the base space. The T-duality invariance constraint is then
 \begin{eqnarray}
 S_{\rm eff}(\psi)-S_{\rm eff}(\psi')&=&\int d^{D-1}x \sqrt{-\bg}\nabla_a(e^{-2\bphi}J^a)\labell{TS}
 \end{eqnarray}
where $\bg_{ab}$, $\bphi$ are the metric and dilaton in the    base space, and $J^a$ is an arbitrary covariant vector made of the $(D-1)$-dimensional fields. It has the following $\alpha'$-expansion:
\begin{eqnarray}
J^a&=&\sum^\infty_{n=0}\alpha'^nJ^a_n
\end{eqnarray}
 where $J^a_n$ is an arbitrary covariant vector at order $\alpha'^n$.

  To have a background with $U(1)$ isometry,  it is convenient to use the following background for  the metric, $B$-field and dilaton:
  \begin{eqnarray}
G_{\mu\nu}=\left(\matrix{\bg_{ab}+e^{\varphi}g_{a }g_{b }& e^{\varphi}g_{a }&\cr e^{\varphi}g_{b }&e^{\varphi}&}\right),\, B_{\mu\nu}= \left(\matrix{\bb_{ab}+\frac{1}{2}b_{a }g_{b }- \frac{1}{2}b_{b }g_{a }&b_{a }\cr - b_{b }&0&}\right),\,  \Phi=\bar{\phi}+\varphi/4\labell{reduc}\end{eqnarray}
where $ \bb_{ab}$  is  the   B-field in the base space, and $g_{a},\, b_{b}$ are two vectors  in this space. Inverse of the above $D$-dimensional metric is
\begin{eqnarray}
G^{\mu\nu}=\left(\matrix{\bg^{ab} &  -g^{a }&\cr -g^{b }&e^{-\varphi}+g_{c}g^{c}&}\right)\labell{inver}
\end{eqnarray}
where $\bg^{ab}$ is the inverse of the base  metric which raises the index of the   vectors.

The T-duality transformations at the leading order of $\alpha'$ on the $(D-1)$-dimensional fields   are given by the  Buscher rules \cite{Buscher:1987sk,Buscher:1987qj}. In the above parametrisation, they  become the following linear transformations:
\begin{eqnarray}
\varphi'= -\varphi
\,\,\,,\,\,g'_{\mu }= b_{\mu }\,\,\,,\,\, b'_{\mu }= g_{\mu } \,\,\,,\,\,\bg_{\alpha\beta}'=\bg_{\alpha\beta} \,\,\,,\,\,\bb_{\alpha\beta}'=\bb_{\alpha\beta} \,\,\,,\,\,  \bar{\phi}'= \bar{\phi}\labell{T2}
\end{eqnarray}
They form a $Z_2$-group, {\it i.e.,}\  $ (x')'= x$ where $x$ is any field in the base space. At higher orders of $\alpha'$, the above transformations receive higher derivative corrections, {\it i.e.,}\
\begin{eqnarray}
\psi'&=&\sum^\infty_{n=0}\alpha'^n\psi'_n
\end{eqnarray}
where $\psi'_0$ is the Buscher rules \reef{T2}, $\psi'_1$ contains corrections to the Buscher rules at order $\alpha'$ and so on. The deformed transformations must satisfy the $Z_2$-group.

\subsection{T-duality constraint at orders $\alpha'^0,\, \alpha'$}

To impose the constraint \reef{TS} on the diffeomorphism invariant couplings  \reef{L2122}, we first review how imposing this constraint  on the couplings  at orders $\alpha'^0$ and $\alpha'$ can  fix their parameters \cite{Garousi:2019wgz}. The constraint \reef{TS} at order $\alpha'^0$ is
\begin{eqnarray}
S_0(\psi)-S_0(\psi'_0)&=& \int d^{D-1}x \sqrt{-\bg}\nabla_a(e^{-2\bphi}J_0^a)\labell{TS0}
\end{eqnarray}
where $J^a_0$ is an arbitrary vector at the leading order of $\alpha'$, and $\psi'_0$ is the Buscher rules \reef{T2}. Reduction of different scalar terms in $\bS_0$ are the following \cite{Garousi:2019wgz}:
\begin{eqnarray}
e^{-2\Phi}\sqrt{-G}&=&e^{-2\bphi}\sqrt{-\bg}\nonumber\\
a_1R&=&a_1(\bR-\nabla^a\nabla_a\vp-\frac{1}{2}\nabla_a\vp \nabla^a\vp -\frac{1}{4}e^{\vp}V^2 )\labell{R}\\
a_2\nabla_{\mu}\Phi\nabla^{\mu}\Phi&=&a_2(\nabla_a\bphi\nabla^a \bphi+\frac{1}{2}\nabla_a\bphi\nabla^a\vp+\frac{1}{16}\nabla_a\vp\nabla^a\vp)\nonumber\\
a_3H^2&=&a_3(\bH_{abc}\bH^{abc}+3e^{-\vp}W^2) \nonumber
\end{eqnarray}
 where $V_{ab}$ is field strength of the $U(1)$ gauge field $g_{a}$, {\it i.e.,}\  $V_{ab}=\nabla_{a}g_{b}-\nabla_{b}g_{a}$, and $W_{\mu\nu}$ is field strength of the $U(1)$ gauge field $b_{a}$, {\it i.e.,}\  $W_{ab}=\nabla_{a}b_{\nu}-\nabla_{b}b_{a}$. The    three-form $\bH$ is defined as $\bH_{abc}=\tilde{H}_{abc}-g_{a}W_{bc}-g_{c}W_{ab}-g_{b}W_{ca}$ where the three-form  $\tilde{H}$ is field strength of the two-form $\bb_{ab}+\frac{1}{2}b_{a}g_{b}-\frac{1}{2}b_b g_a $ in \reef{reduc}.  The three-form $\bH$ is invariant under the Buscher rules and is not the field strength of a two-form. It satisfies the following Bianchi identity  \cite{Kaloper:1997ux}:
 \begin{eqnarray}
\nabla_{[a}\bH_{bcd]}&=&-\frac{3}{2}V_{[ab}W_{cd]}\labell{Bian}
 \end{eqnarray}
 which is invariant under the Buscher rules \reef{T2}.

The transformations of different terms in \reef{R} under the Buscher rules \reef{T2} can easily be found. Then the T-duality constraint \reef{TS0} fixes the parameters $a_1,a_2,a_3$   in the $D$-dimensional action \cite{Garousi:2019wgz}, {\it i.e.,}\
 \begin{eqnarray}
\bS_0&=& -\frac{2a_1}{\kappa^2}\int d^Dx e^{-2\Phi}\sqrt{-G}\,  \left(  R + 4\nabla_{a}\Phi \nabla^{a}\Phi-\frac{1}{12}H^2\right)\,.\labell{S0bf}
\end{eqnarray}
which is the standard effective action at order $\alpha'^0$, up to an overall factor. The overall factor must be $a_1=1$ to be the effective action of string theory. The constraint \reef{TS0} fixes also the form of vector $J_0^a$ in which we are not interested.

The constraint \reef{TS} at order $\alpha'$ is
\begin{eqnarray}
S_0(\psi)+\alpha'S_1(\psi)-S_0(\psi'_0+\alpha'\psi'_1)-\alpha'S_1(\psi'_0)&=& \int d^{D-1}x \sqrt{-\bg}\nabla_a[e^{-2\bphi}(J_0^a+\alpha'J_1^a)]\labell{TS1}
\end{eqnarray}
where $J^a_1$ is an arbitrary vector at   order of $\alpha'$, and $\psi'_0+\alpha'\psi'_1$ is the Buscher rule plus its  deformation at order $\alpha'$, {\it i.e.,}\
\begin{eqnarray}
&&\varphi'= -\varphi+\alpha'\Delta\vp^{(1)}
\,\,\,,\,\,g'_{a }= b_{a }+\alpha'e^{\vp/2}\Delta g^{(1)}_a\,\,\,,\,\, b'_{a }= g_{a }+\alpha'e^{-\vp/2}\Delta b^{(1)}_a \,\,\,,\,\,\nonumber\\
&&\bg_{ab}'=\bg_{ab}+\alpha'\Delta \bg^{(1)}_{ab} \,\,\,,\,\,\bH_{abc}'=\bH_{abc}+\alpha'\Delta\bH^{(1)}_{abc} \,\,\,,\,\,  \bar{\phi}'= \bar{\phi}+\alpha'\Delta\bphi^{(1)}\labell{T22}
\end{eqnarray}
where $\Delta \vp^{(1)}, \cdots,\Delta\bphi^{(1)}$ contains some  contractions of $\nabla\vp,\nabla\bphi, e^{\vp/2}V, e^{-\vp/2}W,\bH,\bR$ and their covariant derivatives at order $\alpha'$. Since the constraint \reef{TS1} should have terms up to order $\alpha'$, in expanding $S_0(\psi'_0+\alpha'\psi'_1)$, one should keep the terms up to order $\alpha'$, {\it i.e.,}\
\begin{eqnarray}
S_0(\psi'_0+\alpha'\psi'_1)&=&S_0(\psi'_0)+\alpha'\delta S_0^{(1)}+\cdots
\end{eqnarray}
Using this expansion and the constraint \reef{TS0}, one can simplify the constraint \reef{TS1} to the following constraint which is only at order $\alpha'$:
 \begin{eqnarray}
 S_1(\psi)-S_1(\psi'_0)-\delta S_0^{(1)}&=& \int d^{D-1}x \sqrt{-\bg}\nabla_a[e^{-2\bphi} J_1^a]\labell{TS11}
\end{eqnarray}
It has been shown in \cite{Garousi:2019wgz} that the above constraint can fix the parameters in the effective action \reef{L1112} as well as the parameters in the corrections to the Buscher rules up to an overall factor. The result is that $\cL_1^2$ is zero and all terms in  $\cL_1^1$ are non-zero, {\it i.e.,}\
  \begin{eqnarray}
 \bS_1&=&\frac{-2b_1 }{\kappa^2}\alpha'\int d^{D}x e^{-2\Phi}\sqrt{-G}\Big(   R_{\alpha \beta \gamma \delta} R^{\alpha \beta \gamma \delta} -\frac{1}{2}H_{\alpha}{}^{\delta \epsilon} H^{\alpha \beta \gamma} R_{\beta  \gamma \delta\epsilon}\nonumber\\
&&\qquad\qquad\qquad\qquad\qquad\quad+\frac{1}{24}H_{\epsilon\delta \zeta}H^{\epsilon}{}_{\alpha}{}^{\beta}H^{\delta}{}_{\beta}{}^{\gamma}H^{\zeta}{}_{\gamma}{}^{\alpha}-\frac{1}{8}H_{\alpha \beta}{}^{\delta} H^{\alpha \beta \gamma} H_{\gamma}{}^{\epsilon \zeta} H_{\delta \epsilon \zeta}\Big)\labell{S1bf}
\end{eqnarray}
   Up to the overall factor $b_1$, the above couplings are   the standard effective action of the   string theory   which has been found in \cite{Metsaev:1987zx} by the S-matrix calculations. For the bosonic string theory $b_1=1/4$,   for the heterotic theory  $b_1=1/8$ and for the  superstring theory  $b_1=0$.

The corrections to the Buscher rules corresponding to the above action are the following \cite{Garousi:2019wgz}:
\begin{eqnarray}
\Delta \bar{g}_{ab}^{(1)}&=&2b_1\Big(e^\vp V_a {}^c V_{bc}-e^{-\vp}W_a {}^c W_{bc}\Big) \nonumber\\
\Delta\bphi^{(1)}&=&\frac{b_1}{2}\Big(e^\vp V^2- e^{-\vp}W^2\Big) \nonumber\\
\Delta\vp^{(1)}&=&2b_1\Big(\nabla_a\vp\nabla^a\vp+e^\vp V^2+e^{-\vp}W^2\Big) \nonumber\\
\Delta g_{a}^{(1)}&=&b_1\Big(2e^{-\vp/2}\nabla^bW_{ab}+e^{\vp/2}\bH_{abc} V^{bc}-4e^{-\vp/2}\nabla^b\bphi W_{ab}\Big)\nonumber\\
\Delta b_{a}^{(1)}&=&-b_1\Big(2e^{\vp/2}\nabla^b V_{ab}+e^{-\vp/2}\bH_{abc} W^{b}-4e^{\vp/2}\nabla^b\bphi V_{ab}\Big)\nonumber\\
\Delta\bH_{abc}^{(1)}&=&12b_1 \nabla_{[a}(W_b{}^d V_{cd]})-3e^{\vp/2} V_{[ab}\Delta g^{(1)}_{c]}-3e^{-\vp/2} W_{[ab}\Delta b _{c]}^{(1)}\labell{dbH1}
\end{eqnarray}
 Replacing these corrections into \reef{T22}, one finds the corresponding T-duality transformations at order $\alpha'$ satisfy the $Z_2$-group as well as the Bianchi identity \reef{Bian} \cite{Garousi:2019wgz}. The constraint \reef{TS11} fixes also the vector $J_1^a$ in which we are not interested.

 \subsection{T-duality constraint at order $  \alpha'^2$}

 We now study in details the constraint \reef{TS} at order $\alpha'^2$ to fix the 60 parameters in \reef{L2122}. This constraint at order $\alpha'^2$ is
 \begin{eqnarray}
&&S_0(\psi)+\alpha'S_1(\psi)+\alpha'^2S_2(\psi)-S_0(\psi'_0+\alpha'\psi'_1+\alpha'^2\psi_2')-\alpha'S_1(\psi'_0+\alpha'\psi_1')-\alpha'^2S_2(\psi_0')\nonumber\\
 &&\qquad\qquad\qquad\qquad\qquad\qquad= \int d^{D-1}x \sqrt{-\bg}\nabla_a[e^{-2\bphi}(J_0^a+\alpha'J_1^a+\alpha'^2 J_2^a)]\labell{TS2}
\end{eqnarray}
where $J^a_2$ is an arbitrary vector at   order  $\alpha'^2$,  $ \alpha'\psi'_1$ represents  the   corrections to the Buscher rules at order $\alpha'$, {\it e.g.,}\  \reef{dbH1},  and  $\alpha'^2\psi_2'$ represents the corrections to the Buscher rules at order $\alpha'^2$, {\it i.e.,}\
\begin{eqnarray}
&&\varphi'= -\varphi+\alpha'\Delta\vp^{(1)}+\frac{1}{2}\alpha'^2\Delta\vp^{(2)}
\,\,\,,\,\,g'_{a }= b_{a }+\alpha'e^{\vp/2}\Delta g^{(1)}_a+\frac{1}{2}\alpha'^2e^{\vp/2}\Delta g^{(2)}_a \nonumber\\
&& b'_{a }= g_{a }+\alpha'e^{-\vp/2}\Delta b^{(1)}_a +\frac{1}{2}\alpha'^2e^{-\vp/2}\Delta b^{(2)}_a\,\,\,,\,\,\bg_{ab}'=\bg_{ab}+\alpha'\Delta \bg^{(1)}_{ab}+\frac{1}{2}\alpha'^2\Delta \bg^{(2)}_{ab}\nonumber\\ &&\bH_{abc}'=\bH_{abc}+\alpha'\Delta\bH^{(1)}_{abc}+\frac{1}{2}\alpha'^2\Delta\bH^{(2)}_{abc} \,\,\,,\,\,  \bar{\phi}'= \bar{\phi}+\alpha'\Delta\bphi^{(1)}+\frac{1}{2}\alpha'^2\Delta\bphi^{(2)}\labell{T221}
\end{eqnarray}
where $\Delta \vp^{(2)}, \cdots,\Delta\bphi^{(2)}$ contains all  contractions of $\nabla\vp,\nabla\bphi, e^{\vp/2}V, e^{-\vp/2}W,\bH,\bR$ and their covariant derivatives at order $\alpha'^2$ with  unknown coefficients. Since the constraint \reef{TS2} should have terms up to order $\alpha'^2$, in expanding $S_0(\psi'_0+\alpha'\psi'_1+\alpha'^2\psi'_2)$ and $\alpha'S_1(\psi'_0+\alpha'\psi'_1 )$, one should keep the terms up to order $\alpha'^2$, {\it i.e.,}\
\begin{eqnarray}
S_0(\psi'_0+\alpha'\psi'_1+\alpha'^2\psi'_2)&=&S_0(\psi'_0)+\alpha'\delta S_0^{(1)}+\alpha'^2\delta S_0^{(2)} +\cdots\nonumber\\
\alpha'S_1(\psi'_0+\alpha'\psi'_1 )&=&\alpha'S_1(\psi'_0)+\alpha'^2\delta S_1^{(1)}+\cdots
\end{eqnarray}
Using these expansions and the constraints \reef{TS0} and \reef{TS11}, one can simplify the constraint \reef{TS2} to the following constraint which is only at order $\alpha'^2$:
 \begin{eqnarray}
 S_2(\psi)-S_2(\psi'_0)-\delta S_0^{(2)}-\delta S_1^{(1)}&=& \int d^{D-1}x \sqrt{-\bg}\nabla_a[e^{-2\bphi} J_2^a]\labell{TS21}
\end{eqnarray}
The speculation is that this constraint as well as the constraint that the T-duality transformations should satisfy the $Z_2$-group and the anomalous Bianchi identity \reef{Bian} may fix all parameters in the diffiomorphism invariant couplings \reef{L2122}.

The T-duality transformations \reef{T221} should satisfy the $Z_2$-group. This produces the following constraint between the corrections at orders $\alpha'$ and $\alpha'^2$:
\begin{eqnarray}
-\Delta\vp^{(2)}(\psi)+\Delta\vp^{(2)}(\psi_0')+2\delta\Delta\vp^{(1)}(\psi_0')&=&0\nonumber\\
\Delta b_a^{(2)}(\psi)+\Delta g_a^{(2)}(\psi'_0)+2\delta\Delta g_a^{(1)}(\psi_0')&=&0\nonumber\\
\Delta g_a^{(2)}(\psi)+\Delta b_a^{(2)}(\psi'_0)+2\delta\Delta b_a^{(1)}(\psi_0')&=&0\nonumber\\
\Delta \bg_{ab}^{(2)}(\psi)+\Delta \bg_{ab}^{(2)}(\psi'_0)+2\delta\Delta \bg_{ab}^{(1)}(\psi_0')&=&0\nonumber\\
\Delta \bH_{abc}^{(2)}(\psi)+\Delta \bH_{abc}^{(2)}(\psi'_0)+2\delta\Delta \bH_{abc}^{(1)}(\psi_0')&=&0\nonumber\\
\Delta\bphi^{(2)}(\psi)+\Delta\bphi^{(2)}(\psi_0')+2\delta\Delta\bphi^{(1)}(\psi_0')&=&0\labell{Z2}
\end{eqnarray}
where we have used the $Z_2$-constraint at order $\alpha'$ which are \cite{Garousi:2019wgz}
\begin{eqnarray}
-\Delta\vp^{(1)}(\psi)+\Delta\vp^{(1)}(\psi_0') &=&0\nonumber\\
\Delta b_a^{(1)}(\psi)+\Delta g_a^{(1)}(\psi'_0) &=&0\nonumber\\
\Delta g_a^{(1)}(\psi)+\Delta b_a^{(1)}(\psi'_0)&=&0\nonumber\\
\Delta \bg_{ab}^{(1)}(\psi)+\Delta \bg_{ab}^{(1)}(\psi'_0)&=&0\nonumber\\
\Delta \bH_{abc}^{(1)}(\psi)+\Delta \bH_{abc}^{(1)}(\psi'_0) &=&0\nonumber\\
\Delta\bphi^{(1)}(\psi)+\Delta\bphi^{(1)}(\psi_0') &=&0\labell{Z22}
\end{eqnarray}
The perturbations $\delta\Delta\vp^{(1)}(\psi_0'), \cdots,\delta\Delta\bphi^{(1)}(\psi_0') $ in \reef{Z2} are defined as
\begin{eqnarray}
\Delta\vp^{(1)}(\psi_0'+\alpha'\psi_1')&=&\Delta\vp^{(1)}(\psi_0')+\alpha'\delta\Delta\vp^{(1)}(\psi_0')+\cdots\nonumber\\
e^{\vp/2}(\psi_0'+\alpha'\psi_1')\Delta g_a^{(1)}(\psi_0'+\alpha'\psi_1')&=&e^{-\vp/2}\Delta g_a^{(1)}(\psi_0' )+\alpha'e^{-\vp/2}\delta\Delta g_a^{(1)}(\psi_0')+\cdots\nonumber\\
e^{-\vp/2}(\psi_0'+\alpha'\psi_1')\Delta b_a^{(1)}(\psi_0'+\alpha'\psi_1')&=&e^{\vp/2}\Delta b_a^{(1)}(\psi_0' )+\alpha'e^{\vp/2}\delta\Delta b_a^{(1)}(\psi_0')+\cdots\nonumber\\
\Delta\bg_{ab}^{(1)}(\psi_0'+\alpha'\psi_1')&=&\Delta\bg_{ab}^{(1)}(\psi_0' )+\alpha'\delta\Delta \bg_{ab}^{(1)}(\psi_0')+\cdots\nonumber\\
\Delta \bH_{abc}^{(1)}(\psi_0'+\alpha'\psi_1')&=&\Delta \bH_{abc}^{(1)}(\psi_0')+\alpha'\delta\Delta \bH_{abc}^{(1)}(\psi_0')+\cdots\nonumber\\
\Delta\bphi^{(1)}(\psi_0'+\alpha'\psi_1')&=&\Delta\bphi^{(1)}(\psi_0' )+\alpha'\delta\Delta\bphi^{(1)}(\psi_0')+\cdots
\end{eqnarray}
where dots represent the perturbations at higher orders of $\alpha'$ which do not appear in our calculations.

The Bianchi identity \reef{Bian} in   terms of 3-form $H$ and 1-forms $g$, $b$  is $dH=-(3/2)dg\wedge db$. The T-dual fields should satisfy this identity as well, {\it i.e.,}\
\begin{eqnarray}
d(\bH+\alpha'\Delta\bH^{(1)}+\frac{1}{2}\alpha'^2\Delta\bH^{(2)}+\cdots)&=&-\frac{3}{2}d(b+\alpha'e^{\vp/2}\Delta g^{(1)}+\frac{1}{2}\alpha'^2e^{\vp/2}\Delta g^{(2)}+\cdots)\nonumber\\
&&\wedge d( g+\alpha'e^{-\vp/2}\Delta b^{(1)}  +\frac{1}{2}\alpha'^2e^{-\vp/2}\Delta b^{(2)} +\cdots)\labell{aa}
\end{eqnarray}
This relation at order $\alpha'^0$ gives the Bianchi identity \reef{Bian}. At order $\alpha'$ it gives the following relation between the corrections to the Buscher rules at order $\alpha'$:
\begin{eqnarray}
\Delta\bH^{(1)}&=&\tilde H^{(1)}-\frac{3}{2}\Big[db\wedge (e^{-\vp/2}\Delta b^{(1)})+(e^{\vp/2}\Delta g^{(1)})\wedge dg\Big]
\end{eqnarray}
where $\tilde H^{(1)}$ is a $U(1)\times U(1)$ invariant closed 3-form, {\it i.e.,}\  $d \tilde H^{(1)}=0$, at order $\alpha'$ which is odd under parity. The corrections \reef{dbH1} satisfy this relation. At order $\alpha'^2$, the Bianchi identity \reef{aa} produces the following relation between the corrections at orders $\alpha'$ and $\alpha'^2$:
\begin{eqnarray}
\Delta\bH^{(2)}&=&\tilde H^{(2)}-\frac{3}{2}\Big[db\wedge (e^{-\vp/2}\Delta b^{(2)})+(e^{\vp/2}\Delta g^{(2)})\wedge dg\nonumber\\
&&+(e^{\vp/2}\Delta g^{(1)})\wedge d(e^{-\vp/2}\Delta b^{(1)})+d(e^{\vp/2}\Delta g^{(1)})\wedge (e^{-\vp/2}\Delta b^{(1)})\Big]\labell{HH2}
\end{eqnarray}
where $\tilde H^{(2)}$ is a   closed 3-form, {\it i.e.,}\  $d\tilde H^{(2)}=0$,   which  contains all  contractions of $\nabla\vp$, $\nabla\bphi$,  $e^{\vp/2}V, e^{-\vp/2}W$, $\bH,\bR$ and their covariant derivatives at order $\alpha'^2$ with  unknown coefficients.

Therefore, the second order corrections $\Delta\vp^{(2)},\, \Delta g_a^{(2)},\, \Delta b_a^{(2)}, \,\Delta \bg_{ab}^{(2)}$ and $\Delta\bphi^{(2)}$ should be  all  contractions of $\nabla\vp,\nabla\bphi, e^{\vp/2}V, e^{-\vp/2}W,\bH,\bR$ and their covariant derivatives at order $\alpha'^2$ with  unknown coefficients. The correction $\Delta \bH_{abc}^{(2)}$ is then can be calculated from \reef{HH2}. All corrections should satisfy the $Z_2$-relations \reef{Z2}. They produce some algebraic equations between the parameters of the corrections at order $\alpha'^2$ and the parameter $b_1$ in the corrections at order $\alpha'$, {\it i.e.,}\  \reef{dbH1}. These parameters and the 60 parameters in the action \reef{L2122} should satisfy the constraint \reef{TS21} as well.

To use the constraint \reef{TS21} one needs to reduce the couplings in \reef{L2122}. The reduction of each term at order $\alpha'^2$ is a very lengthy expression. However, the final result for the reduction of each term must be an invariant term under the  $U(1)\times U(1)$ gauge transformations. Using this fact as a constraint, the calculations of the reduction of $\bS_2$ can be simplified greatly. The couplings in \reef{L2122} have only Riemann curvature, $H$, $\nabla H$, $\nabla\Phi$ and $\nabla\nabla\Phi$. So we need to reduce these terms and then contract them with the metric \reef{inver}. In the  reduction of these terms, there are many terms which contains gauge field  $g_a$ without its field strength. These terms must be cancelled at the end of the day for the scalar couplings. Hence, to simplify the calculation we drop those terms in the reduction of  $R_{\mu\nu\alpha\beta}$, $H_{\mu\nu\alpha}$, $\nabla_{\mu} H_{\nu\alpha\beta}$, $\nabla_{\mu}\Phi$, $\nabla_{\mu}\nabla_{\nu}\Phi$  and $G^{\mu\nu}$ which have the gauge field $g_a$.  Using this simplification, the reduction of Riemann curvature becomes\footnote{We have used the package ''xAct" \cite{Nutma:2013zea} for performing the calculations in this paper.}
\begin{eqnarray}
R_{abcd}&=&\bR_{abcd}+\frac{1}{4}e^\vp (V_{ad}V_{bc}-V_{ac}V_{bd}-2V_{ab}V_{cd})\nonumber\\
R_{abcy}&=&\frac{1}{4}e^\vp(V_{bc}\nabla_a\vp-V_{ac}\nabla_b\vp-2V_{ab}\nabla_c\vp-2\nabla_cV_{ab})\nonumber\\
R_{aycy}&=&\frac{1}{4}e^\vp(e^\vp V_a{}^bV_{cb}-\nabla_a\vp\nabla_c\vp-2\nabla_c\nabla_a\vp)\labell{rR}
\end{eqnarray}
All other components  are either zero or related to the above terms by the Riemann symmetries.  The reduction of different components of $\nabla\nabla\Phi$ and  $\nabla\Phi$ become
\begin{eqnarray}
\nabla_a\nabla_b\Phi&=&\nabla_a\nabla_b\bphi+\frac{1}{4}\nabla_a\nabla_b\vp\nonumber\\
\nabla_a\nabla_y\Phi&=&-\frac{1}{2}e^\vp(V_{ab}\nabla^b\bphi+\frac{1}{4}V_{ab}\nabla^b\vp)\nonumber\\
\nabla_y\nabla_y\Phi&=&\frac{1}{2}e^\vp(\nabla_a\vp\nabla^a\bphi+\frac{1}{4}\nabla_a\vp\nabla^a\vp)\nonumber\\
 \nabla_a\Phi&=&\nabla_a\bphi+\frac{1}{4}\nabla_a\vp\,\,\,;\,\,\,
\nabla_y\Phi\,=\,0\labell{rP}
\end{eqnarray}
The reduction of different components of  $\nabla H$ and  $H$ become
\begin{eqnarray}
\nabla_aH_{bcd}&=&\frac{1}{2}(V_{ad}W_{bc}-V_{ac}W_{bd}+V_{ab}W_{cd}+2\nabla_a\bH_{bcd})\nonumber\\
\nabla_aH_{bcy}&=&\frac{1}{2}(-e^\vp\bH_{bcd}V_{a}{}^d-W_{bc}\nabla_a\vp+2\nabla_a W_{bc})\nonumber\\
\nabla_y H_{bcd}&=&\frac{1}{2}e^\vp(\bH_{bda}V_c{}^a-\bH_{cda}V_b{}^a-\bH_{bca}V_d{}^a)+\frac{1}{2}(W_{bd}\nabla_c\vp-W_{cd}\nabla_b\vp-W_{bc}\nabla_d\vp)\nonumber\\
\nabla_yH_{bcy}&=&\frac{1}{2}e^\vp(\bH_{bca}\nabla^a\vp-V_c{}^aW_{ba}+V_b{}^aW_{ca})\labell{rDH}\\
 H_{abc}&=&\bH_{abc}\,\,\,;\,\,\, H_{aby}\,\,=\,\,W_{ab}\nonumber
 \end{eqnarray}
 The covariant derivatives on the right-hand side of \reef{rR}, \reef{rP} and \reef{rDH} are $(D-1)$-dimensional, and the indices are raised by the inverse metric $\bg^{ab}$. The reduction of inverse of the $D$-dimensional metric in this case also becomes
 \begin{eqnarray}
G^{\mu\nu}=\left(\matrix{\bg^{ab} & 0&\cr 0&e^{-\varphi}&}\right)\labell{inver1}
\end{eqnarray}
Using above reductions, one can calculate the reduction of different scalar terms in \reef{L2122}.

Then using the constraint \reef{TS21}, one finds some   equations involving the 60 parameters in \reef{L2122}, the arbitrary parameters of $J_2^a$ and  the parameters of  $\Delta\vp^{(2)},\, \Delta g_a^{(2)},\, \Delta b_a^{(2)}, \,\Delta \bg_{ab}^{(2)}$, $\Delta\bphi^{(2)}$ and $\tilde{H}^{(2)}$. The parameters  of $\Delta\vp^{(2)},\, \Delta g_a^{(2)},\, \Delta b_a^{(2)}, \,\Delta \bg_{ab}^{(2)}$, $\Delta\bphi^{(2)}$ and $\tilde{H}^{(2)}$ should also satisfy the constraints \reef{Z2} and \reef{HH2}. To solve these  constraints, one has to write the couplings in them in terms of independent couplings. Then   coefficients  of the independent couplings which involve the above parameters should be zero.  To perform this last steps, one has to impose  the following Bianchi identities into the constraints as well:
\begin{eqnarray}
 \bR_{a[bcd]}&=&0,\nonumber\\
 \nabla_{[a}\bR_{bc]de}&=&0\nonumber \\
{[}\nabla,\nabla {]}\cO &=& \bR\cO\nonumber\\
\nabla_{[a}V_{bc]}&=&0\nonumber\\
\nabla_{[a}W_{bc]}&=&0\nonumber\\
\nabla_{[a}\bH_{bcd]}+\frac{3}{2}V_{[ab}W_{cd]}&=&0\labell{iden}
\end{eqnarray}
To impose the last identity above, we     contract it with tensors  $\nabla\vp$, $\nabla\bphi$,  $e^{\vp/2}V, e^{-\vp/2}W$, $\bH,\bR$  and their derivatives with arbitrary parameters and then add them to the constraints.  To impose the first three identities above,   we  use the locally inertial frame in which these   identities are    automatically satisfied.
In the locally inertial frame, the metric $\bg_{ab}$ takes its canonical form and its first derivatives are all vanish, {\it i.e.,}\ ,
\[
\bg_{ab}=\eta_{ab},\qquad \partial_a \bg_{bc}=0
\]
The second and higher derivatives of metric, however, are non-zero. In this coordinate,   by rewriting the covariant derivative in terms of partial derivatives, one finds the first three identities in \reef{iden}  are satisfied. To satisfy  the Bianchi identities $dV=0=dW$ as well, in the couplings which involve derivatives of $V$ and $W$, we   rewrite  them    in terms of  their gauge fields, {\it i.e.,}\   $V_{ab}=\partial_a g_{b}-\partial_b g_{a} $ and  $W_{ab}=\partial_a b_{b}-\partial_b b_{a} $.

After using the above steps to write the couplings in the constraints \reef{TS21}, \reef{Z2} and \reef{HH2} in terms of  independent couplings in the local frame, one can set their coefficients to zero to produce some algebraic equations involving only the   parameters. Interestingly, these algebraic equations    fix all the 60 parameters in \reef{L2122} in terms of $b_1$, the overall factor at order $\alpha'$ which should be $b_1=1/4$ for the bosonic string theory. All 20 parameters in \reef{L21} are non-zero and only 7 parameters in \reef{AA} are non-zero. They are
\begin{eqnarray}
\bS_2&=&\frac{-2b_1 }{\kappa^2}\alpha'^2\int d^{D}x e^{-2\Phi}\sqrt{-G}\Big(  - \frac{1}{12} H_{\alpha}{}^{\delta \epsilon} H^{\alpha \beta
\gamma} H_{\beta \delta}{}^{\zeta} H_{\gamma}{}^{\iota \kappa}
H_{\epsilon \iota}{}^{\mu} H_{\zeta \kappa \mu}\nonumber\\&& +
\frac{1}{30} H_{\alpha \beta}{}^{\delta} H^{\alpha \beta
\gamma} H_{\gamma}{}^{\epsilon \zeta} H_{\delta}{}^{\iota
\kappa} H_{\epsilon \zeta}{}^{\mu} H_{\iota \kappa \mu} +
\frac{3}{10} H_{\alpha \beta}{}^{\delta} H^{\alpha \beta
\gamma} H_{\gamma}{}^{\epsilon \zeta} H_{\delta
\epsilon}{}^{\iota} H_{\zeta}{}^{\kappa \mu} H_{\iota \kappa
\mu} \nonumber\\&&+ \frac{13}{20} H_{\alpha}{}^{\epsilon \zeta}
H_{\beta}{}^{\iota \kappa} H_{\gamma \epsilon \zeta} H_{\delta
\iota \kappa} R^{\alpha \beta \gamma \delta} + \frac{2}{5}
H_{\alpha}{}^{\epsilon \zeta} H_{\beta \epsilon}{}^{\iota}
H_{\gamma \zeta}{}^{\kappa} H_{\delta \iota \kappa} R^{\alpha
\beta \gamma \delta}\nonumber\\&& + \frac{18}{5} H_{\alpha
\gamma}{}^{\epsilon} H_{\beta}{}^{\zeta \iota} H_{\delta
\zeta}{}^{\kappa} H_{\epsilon \iota \kappa} R^{\alpha \beta
\gamma \delta} -  \frac{43}{5} H_{\alpha \gamma}{}^{\epsilon}
H_{\beta}{}^{\zeta \iota} H_{\delta \epsilon}{}^{\kappa}
H_{\zeta \iota \kappa} R^{\alpha \beta \gamma \delta} \nonumber\\&&-
\frac{16}{5} H_{\alpha \gamma}{}^{\epsilon} H_{\beta
\delta}{}^{\zeta} H_{\epsilon}{}^{\iota \kappa} H_{\zeta \iota
\kappa} R^{\alpha \beta \gamma \delta} - 2 H_{\beta
\epsilon}{}^{\iota} H_{\delta \zeta \iota}
R_{\alpha}{}^{\epsilon}{}_{\gamma}{}^{\zeta} R^{\alpha \beta
\gamma \delta} - 2 H_{\beta \delta}{}^{\iota} H_{\epsilon
\zeta \iota} R_{\alpha}{}^{\epsilon}{}_{\gamma}{}^{\zeta}
R^{\alpha \beta \gamma \delta}\nonumber\\&& -  \frac{4}{3}
R_{\alpha}{}^{\epsilon}{}_{\gamma}{}^{\zeta} R^{\alpha \beta
\gamma \delta} R_{\beta \zeta \delta \epsilon} + \frac{4}{3}
R_{\alpha \beta}{}^{\epsilon \zeta} R^{\alpha \beta \gamma
\delta} R_{\gamma \epsilon \delta \zeta} + 3
H_{\beta}{}^{\zeta \iota} H_{\epsilon \zeta \iota} R^{\alpha
\beta \gamma \delta} R_{\gamma}{}^{\epsilon}{}_{\alpha \delta}
\nonumber\\&&+ 2 H_{\beta \epsilon}{}^{\iota} H_{\delta \zeta \iota}
R^{\alpha \beta \gamma \delta}
R_{\gamma}{}^{\epsilon}{}_{\alpha}{}^{\zeta} + 2 H_{\alpha
\beta \epsilon} H_{\delta \zeta \iota} R^{\alpha \beta \gamma
\delta} R_{\gamma}{}^{\epsilon \zeta \iota} + \frac{13}{10}
H_{\alpha}{}^{\gamma \delta} H_{\beta \gamma}{}^{\epsilon}
H_{\delta}{}^{\zeta \iota} H_{\epsilon \zeta \iota}
\nabla^{\beta}\nabla^{\alpha}\Phi\nonumber\\&& + \frac{13}{5}
H_{\gamma}{}^{\epsilon \zeta} H_{\delta \epsilon \zeta}
R_{\alpha}{}^{\gamma}{}_{\beta}{}^{\delta}
\nabla^{\beta}\nabla^{\alpha}\Phi -  \frac{52}{5} H_{\beta
\delta}{}^{\zeta} H_{\gamma \epsilon \zeta}
R_{\alpha}{}^{\gamma \delta \epsilon}
\nabla^{\beta}\nabla^{\alpha}\Phi\nonumber\\&& -  \frac{26}{5} H_{\alpha
\gamma \epsilon} H_{\beta \delta \zeta} R^{\gamma \delta
\epsilon \zeta} \nabla^{\beta}\nabla^{\alpha}\Phi +
\frac{13}{5} \nabla^{\beta}\nabla^{\alpha}\Phi
\nabla_{\epsilon}H_{\beta \gamma \delta}
\nabla^{\epsilon}H_{\alpha}{}^{\gamma \delta} \nonumber\\&&+ \frac{13}{10}
H_{\beta \gamma}{}^{\epsilon} H^{\beta \gamma \delta}
H_{\delta}{}^{\zeta \iota} \nabla^{\alpha}\Phi
\nabla_{\iota}H_{\alpha \epsilon \zeta}  -  \frac{13}{20}
H_{\alpha}{}^{\beta \gamma} H_{\delta \epsilon}{}^{\iota}
H^{\delta \epsilon \zeta} \nabla^{\alpha}\Phi
\nabla_{\iota}H_{\beta \gamma \zeta} \nonumber\\&&+ \frac{1}{20}
H_{\alpha}{}^{\delta \epsilon} H^{\alpha \beta \gamma}
\nabla_{\iota}H_{\delta \epsilon \zeta}
\nabla^{\iota}H_{\beta \gamma}{}^{\zeta} + \frac{1}{5}
H_{\alpha}{}^{\delta \epsilon} H^{\alpha \beta \gamma}
\nabla_{\zeta}H_{\gamma \epsilon \iota}
\nabla^{\iota}H_{\beta \delta}{}^{\zeta}\nonumber\\&& -  \frac{6}{5}
H_{\alpha}{}^{\delta \epsilon} H^{\alpha \beta \gamma}
\nabla_{\iota}H_{\gamma \epsilon \zeta}
\nabla^{\iota}H_{\beta \delta}{}^{\zeta} -  \frac{6}{5}
H_{\alpha \beta}{}^{\delta} H^{\alpha \beta \gamma}
\nabla_{\zeta}H_{\delta \epsilon \iota}
\nabla^{\iota}H_{\gamma}{}^{\epsilon \zeta}\nonumber\\&&\qquad\qquad\qquad\qquad\qquad\qquad\qquad  + \frac{17}{10}
H_{\alpha \beta}{}^{\delta} H^{\alpha \beta \gamma}
\nabla_{\iota}H_{\delta \epsilon \zeta}
\nabla^{\iota}H_{\gamma}{}^{\epsilon \zeta}\Big)\labell{S2f}
\end{eqnarray}
Note that the 60 parameters are fixed when the Bianchi constraint \reef{HH2} is imposed as well as the T-duality constraint \reef{TS21} and \reef{Z2}. If  one only uses the constraint   \reef{TS21} and \reef{Z2}, then 12 parameters of \reef{L2122} remain arbitrary. It is the constraint  \reef{HH2} which fixes these 12 parameters as well.

The algebraic equations also fix  some of the parameters in the T-duality transformations and the parameters of total derivative terms at order $\alpha'^2$  in terms of $b_1$, and leave many of them to be arbitrary. Some of the  arbitrary parameters in the T-duality transformations may be removed by the Bianchi identities and  some of them are related to the coordinate transformations at order $\alpha'^2$. Even when all the arbitrary parameters are set to zero, there are still too may terms in the T-duality transformations at order $\alpha'^2$, so we do not   write them explicitly. On the other hand, those corrections are only needed if one would like to extend the above couplings to the order $\alpha'^3$ in the bosonic theory in which we are not interested in this paper. The important part of the calculations is that there are 60 relations between the  60 parameters in \reef{L2122} and the parameter $b_1$, {\it i.e.,}\  the T-duality constraint fixes  all 60 parameters at order $\alpha'^2$ in terms of the overall factor of the couplings at order $\alpha'$!  This ends our illustration of the fact that the T-duality constraint on the effective action can fix uniquely  the effective action of bosonic string  theory at order $\alpha'^2$.

\section{Discussion}

In this paper, we have shown that  imposing the gauge symmetries and the T-duality symmetry on the effective action of string theory for metric, $B$-field and dilaton at order $\alpha'^2$, can fix the effective action, {\it i.e.,}\  \reef{S2f}, up to an overall factor which is the overall factor of the effective action at order $\alpha'$. This is extension of the similar calculation at order $\alpha'$  done in \cite{Garousi:2019wgz} which fixes the effective action at order $\alpha'$ up to the  overall factor $b_1$, {\it i.e.,}\  \reef{S1bf}. In fact, the gauge symmetries require to have 60 couplings at order $\alpha'^2$ with unfixed coefficients \cite{Garousi:2019cdn}, and the T-duality symmetry which is imposed  on the reduction of the effective action   on a circle,  fixes these 60 parameters.

In the base space, we have done the calculations   in  the local frame in which the first derivatives of the base metric is zero. After solving the constraints, we have  imposed the solution for the parameters in the constraints \reef{TS21}, \reef{Z2}, \reef{HH2}  and found that they are satisfied even when the first derivative of metric is non-zero. It is as expected, because the constraints are some   covariant  identities. If they satisfy in one particular frame like the local inertial frame, they would  satisfy in all other frames as well.

Most of the couplings in \reef{S2f} are new couplings which have not been found in the literature by other methods in string theory. When $B$-field is zero, the   couplings \reef{S2f}  reduce to two Riemann cubed terms that their coefficients, after using the cyclic symmetry of the Riemann curvature,  become exactly the same as the coefficients  that have been found in  \cite{Metsaev:1986yb} by the S-matrix method. These couplings are invariant under the field redefinitions. However, the couplings which have $B$-field are not invariant under the field redefinitions. When $B$-field is non-zero, one may check the couplings involving four fields with the corresponding four-point S-matrix elements in bosonic  string theory. To check this comparison,    one has  to use a field redefinition that change the Riemann squared terms in \reef{S1bf} to the Gauss-Bonnet  combination in which the propagators do not receive $\alpha'$-correction.  That field redefinitions  would then change the form of the couplings in \reef{S2f}. The resulting couplings then may be checked with the corresponding S-matrix elements. We leave the details of this calculation for the future works.

We have found that 7  dilaton couplings in \reef{AA} are non-zero. On the other hand, it is known that  the couplings at order $\alpha'^2$ depends on the effective action at order $\alpha'$  \cite{Bento1990}. We have used the minimal action \reef{S1bf} and the corresponding T-duality transformations \reef{dbH1}. Using another  scheme for the couplings at order $\alpha'$, some of  the parameters in \reef{S2f} may be changed. It would be interesting to check if there is a scheme for the couplings at order $\alpha'$ for which all the dilaton couplings in \reef{AA} become zero.

We have found the effective action \reef{S2f} by imposing only the symmetries of string theory, {\it i.e.,}\  the $B$-field gauge invariance, diffeomorphism  and T-duality invariances. As a result, the effective action   \reef{S2f} is background independent. However, the total derivative terms are   ignored
 in imposing the T-duality constraint. Hence the effective action \reef{S2f} is valid for all backgrounds that have no boundary. It would be interesting to take into account in details  the total derivative terms to find the boundary terms as well as the bulk terms for the general case that the background has    boundary.

We have done the calculations in the curved   base space to make sure that the constraints  \reef{TS21}, \reef{Z2}, \reef{HH2} are satisfied in full details. We have performed the calculations in flat base space as well and found exactly the same parameters for \reef{L2122} as in \reef{S2f}. In the  T-duality calculations at order $\alpha'$    \cite{Garousi:2019wgz} which have correctly reproduced the effective action at order $\alpha'$, it is also assumed that the base space is flat. Hence, for the calculations at the higher orders of $\alpha'$ which would be very lengthy calculations, one may safely assume the base space is flat. The most simple calculations at   order $\alpha'^3$  is for superstring theory in which the T-duality transformations have no deformation at orders $\alpha'$ and $\alpha'^2$.  It would be interesting to perform this calculations at order $\alpha'^3$ in the superstring to find the $B$-field couplings which are not known in the literature.

 If one extends the calculations in the  bosonic theory to the   order  $\alpha'^3$, one would find a   set of couplings   which are proportional to  $b_1$ and another   set of couplings that their overall factor is arbitrary. The comparison with the four-point S-matrix elements  dictates that this factor should be $\zeta(3)$.  At order $\alpha'^4$, again one should find a   set of couplings which are proportional to $b_1$, a set of couplings proportial to $\zeta(3)$ and some other   sets of couplings that their overall factor may be fixed by the corresponding S-matrix elements. Continuing these logic, one would find   sets of couplings at each order of $\alpha'$ which are proportional to $b_1$. Hence, one expects the T-duality constraint produces a   set of couplings at each order of $\alpha'$ that are proportional to $b_1$.  They form a complete set of couplings which would be  invariant   under the T-duality transformations at all orders of $\alpha'$. That T-dual set of  couplings may have de Sitter solution  \cite{Hohm:2019jgu}. It would be interesting to find this T-dual set.

 In this paper, while we have deformed the T-duality transformations, we have assumed the gauge transformations are the standard diffeomorphisms and $B$-field gauge transformations which are the correct transformations in the bosonic and superstring theories. In the superstring theory $b_1=0$, hence,  the   couplings \reef{S2f} are zero in the superstring theory as expected.
On the other hand, the 60 parameters in \reef{S2f} do not dependent on the dimension of spacetime. That does not indicate   the   result \reef{S2f} is valid also for the heterotic theory for $b_1=1/8$. The reason is that in the heterotic theory the $B$-field gauge transformation is deformed at order $\alpha'$ which is resulted from the Green-Schwarz anomaly cancellation mechanism  \cite{Green:1984sg}. To produce the heterotic result, one has  to add to the couplings \reef{L2122}  the  fixed couplings at order $\alpha'^2$ which are resulted from the deformed  gauge transformations, {\it i.e.,}\  $-\frac{\alpha'^2}{12}\Omega_{\mu\nu\alpha}\Omega^{\mu\nu\alpha}$ where $\Omega$ is the three-form Chern-Simons which can be written in terms of spin connection,
\begin{eqnarray}
\Omega_{\mu\nu\alpha}&=&\omega_{[\mu i}{}^j\prt_\nu\omega_{\alpha]j}{}^i+\frac{2}{3}\omega_{[\mu i}{}^j\omega_{\nu j}{}^k\omega_{\alpha]k}{}^i\,\,;\,\,\,\omega_{\mu i}{}^j=\prt_\mu e_\beta{}^j e^\beta{}_i-\Gamma_{\mu\beta}{}^\gamma e_\gamma{}^j e^\beta{}_i
\end{eqnarray}
 where $e_\mu{}^ie_\nu{}^j\eta_{ij}=G_{\mu\nu}$.
 Adding this term, the corresponding T-duality transformations and the 60 parameters in \reef{L2122} may be fixed in the heterotic theory. We didn't perform this calculations, however, one expects all parameters in \reef{L2122} to be zero.   Similar calculation at order $\alpha'$ has been done in \cite{Garousi:2019wgz}. The T-duality structure of the couplings in the heterotic theory  has been studied in \cite{Bedoya:2014pma,Coimbra:2014qaa,Lee:2015kba,Baron:2017dvb,Baron:2018lve} in the DFT formalism.

 There is  another deformation of $B$-field gauge transformations which correspond to the Chiral string theory \cite{Hohm:2013jaa}. The deformation at order $\alpha'$ is the same as the deformation in the heterotic theory in which spin connection is replaced by  the Christoffel connection \cite{Hohm:2014eba}. The low energy effective action of this theory at the leading order,   is given by the T-duality invariant action \reef{S0bf} and  at the order $\alpha'$, it is given by the T-duality invariant coupling   $H^{\mu\nu\alpha}\bf{\Omega}_{\mu\nu\alpha}$ \cite{Hohm:2015doa}\footnote{The parameter $b_1$, in the  parity invariant    action \reef{S1bf}, corresponding to the Chiral string is $b_1=0$.} where the three-form Chern-Simons $\bf{\Omega}_{\mu\nu\alpha}$  is resulted from the deformed gauge transformation, {\it i.e.,}\
\begin{eqnarray}
\bf{\Omega}_{\mu\nu\alpha}&=&\Gamma_{[\mu\beta}{}^\gamma\prt_\nu\Gamma_{\alpha]\gamma}{}^\beta+\frac{2}{3}\Gamma_{[\mu\beta}{}^\gamma\Gamma_{\nu\gamma}{}^\lambda\Gamma_{\alpha]\lambda}{}^\beta
\end{eqnarray}
To find the effective action at order $\alpha'^2$, one has  to add to the couplings \reef{L2122}  the  fixed couplings at order $\alpha'^2$, {\it i.e.,}\  $-\frac{\alpha'^2}{12}\bf{\Omega}_{\mu\nu\alpha}\bf{\Omega}^{\mu\nu\alpha}$.  Adding this term, the corresponding T-duality transformations and the 60 parameters in \reef{L2122} may be fixed in the Chiral string  theory by the T-duality constraint method. It would be interesting to perform this calculation to find the $\alpha'^2$-order terms. This theory has been studied in  the DFT formalism in \cite{Hohm:2015mka,Lescano:2016grn}.

In general, both the diffeomorphisms and the $B$-field gauge transformations  may receive higher derivative deformations in a general gauge invariant higher-derivative theory. One may impose these gauge transformations and the deformed T-duality transformations to study the effective action of a higher-derivative theory which is invariant under the gauge transformations and under  the T-duality transformations.  The effective action at the leading order of $\alpha'$ is given by \reef{S0bf}. At order $\alpha'$, the parity invariant part of the effective action would be more general than the action \reef{S1bf}. It would be interesting to find this effective action.

 \vskip .3 cm
{\bf Acknowledgments}:   This work is supported by Ferdowsi University of Mashhad under grant  1/50251(1398/04/31).

redpajama_set_name: RedPajamaArXiv


text:
package common;

public class HttpHeadKey {
	public final static String Host = "Host";
	public final static String Refer = "Refer";
}

redpajama_set_name: RedPajamaGithub


Autistic people have trouble making facial expressions appropriate to the circumstances.
Two new parent questionnaires could help clinicians identify autism in babies and in members of racial or ethnic minority groups.
Communication problems have always been considered a core feature of autism. Yet there are substantial and wide-ranging differences in how people with autism communicate.
Many girls hide their autism, sometimes evading diagnosis well into adulthood. These efforts can help women on the spectrum socially and professionally, but they can also do serious harm.
The way children view both social and nonsocial situations may be determined at least in part by their genes.
Autism and schizophrenia share a long and tangled history. Comparing the social features of the two conditions could lead to better treatments and a deeper understanding of each.


One of the pleasures of cataloging manuscripts is learning about authors and texts that are relatively little known. One such Syriac author is Athanasios (Abū Ġalib) of Ǧayḥān (Ceyhan). Two fifteenth-century manuscripts, CFMM 417 and 418, which I have recently cataloged, each contain different texts attributed to him. Barsoum surveys his life and work briefly in Scattered Pearls (pp. 441-442), and prior to that Vosté wrote an article on him; more recently Vööbus and Carmen Fotescu Tauwinkl have further reported on him. (See the bibliography below; I have not seen all of these resources.) According to Barsoum, he died in 1177 at over 80 years old. As far as I know, none of his work has been published.
The place name associated with this author is the Turkish Ceyhan. The Syriac spelling of the place in the Gazetteer has gyḥʾn, but in both of these manuscripts it is gyḥn. The former is probably an imitation of the Arabic-script spelling, while the form without ālap in the manuscripts still indicates ā in the second syllable by means of an assumed zqāpā.
Now for the CFMM texts.
An untitled monastic selection. These two pages make up the whole of this short text. As you can see, it follows something from Isaac of Nineveh, and it precedes Ps.-Evagrius, On the Perfect and the Just (CPG 2465 = Hom. 14 of the Liber Graduum). The manuscript is dated March, 1785 AG (= 1474 CE).
Excerpts "from his teaching". Here are the first and last pages of the text. This longer text follows Isaac of Nineveh's Letter on how Satan Takes Pains to Remove the Diligent from Silence (ff. 223v-235v, Eggartā ʿal hāy d-aykannā metparras Sāṭānā la-mbaṭṭālu la-ḥpiṭē men šelyā) and precedes some Profitable Sayings attributed to Isaac. This manuscript — written by more than one scribe, but at about the same time, it seems — is dated on f. 277v with the year 1482, but the 14- is to be read 17-, so we have 1782 AG (= 1470/1 CE; cf. Vööbus, Handschriftliche Überlieferung der Mēmrē-Dichtung des Jaʿqōb von Serūg, III 97).
Tauwinkl, Carmen Fotescu, "Abū Ghālib, an Unknown West Syrian Spiritual Author of the XIIth Century", Parole de l'Orient 36 (2010): 277-284.
Tauwinkl, Carmen Fotescu, "A Spiritual Author in 12th Century Upper Mesopotamia: Abū Ghālib and his Treatise on Monastic Life", Pages 75-93 in The Syriac Renaissance. Edited by Teule, Herman G.B. and Tauwinkl, Carmen Fotescu and ter Haar Romeny, Robert Bas and van Ginkel, Jan. Eastern Christian Studies 9. Leuven / Paris / Walpole, MA: Peeters, 2010.
Vööbus, Arthur, History of Asceticism in the Syrian Orient: A Contribution to the History of Culture in the Near East, III, CSCO 500, Subs. 81. Louvain: Secrétariat du CorpusSCO, 1988, pp. 407-410.
Vööbus, Arthur, "Important Discoveries for the History of Syrian Mysticism: New Manuscript Sources for Athanasius Abû Ghalîb", Journal of Near Eastern Studies 35:4 (1976): 269-270.
Vosté, Jacques Marie, "Athanasios Aboughaleb, évêque de Gihân en Cilicie, écrivain ascétique du XIIe siècle", Revue de l'Orient chrétien III, 6  (1927-1928): 432-438. Available here.
Athanasius is indeed one of the little known monastic authors. The fragments you came across most likely come from his extensive monastic treatise the oldest copy of which was in the Catholic Church of Mar Tuma in Mosul (and now apparently elsewhere).


Marcus Aurelius is considered to one of the great Emperors of Rome who was not only a skilled military leader but also a great philosopher in the Stoic tradition.
He was born in AD 121 and both of his parents came from wealthy backgrounds. His father died when Marcus was three years old and by the time he was six he had gained the attention of the Emperor Hadrian who oversaw his education. Hadrian ensured that Marcus was taught by some of the greatest scholars in Rome who educated him in literature, drama, geometry, Greek oratory, Greek and Latin. Marcus later abandoned most of those subjects in favour of philosophy, with the work of the Greek philosopher Epictetus being a major influence on his thinking.
In 138, while still a young man, the Emperor Antoninus Pius adopted Marcus, and in 161 he himself became Emperor. Marcus insisted he would only take up the Emperorship if Lucius Verus were also installed. Marcus's insistence on Lucius joining him as Emperor was a military one. At that time Rome was fighting wars on multiple fronts and Marcus wanted someone he could trust to marshal the troops who he knew would not at some point lead a revolt against him. Lucius and Marcus were loyal to each other up until Lucius's death in 169.
After Lucius's death, Marcus was the sole Emperor and due to the incessant wars in the provinces he was unable to spend much time indulging his philosophical pursuits. He did manage to found four Chairs of Philosophy in Athens, one for each of the main philosophical schools of thought, Aristotelian, Epicurean, Platonic and Stoic.
Marcus Aurelius died on 17 March 180 in the city of Vindobona, which was situated where Vienna is today. History remembers him as the last of the 'Five Good Emperors' of the Nervan-Antonian dynasty. Marcus' son Commodus replaced his father as Emperor and although he reigned over a relatively stable period in Roman history, in terms of war and peace, his personal behavior and antics were not in the spirit of those Emperors that came immediately before him. Commodus was eventually murdered in a plot that involved his mistress Marcia, thus bringing to an end the highly regarded Nervan-Antonian dynasty.


text:
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London's longest serving courier appoints new CEO - Read more
They've been working on this for months. Luckily, we've been working on this since 1865.
Absolutely might sound like a new name on the courier block. But actually we've been around in this industry for over 150 years.
Our original holding company, G. Thompson Ltd, was founded in 1865 by George Thompson. A station master at St Pancras station, he identified a business opportunity to work as a freight agent for the privately owned railway companies. Aided by five of his seven sons, George grew the business from its headquarters in Hatton Garden, employing over thirty people.
The company transported a wide range of people and goods across the capital, using carriages pulled by Welsh Cobs, cared for in the company stables.
Been there, delivered that since 1865
George Thompson passed away in 1906, having built the business into one of the most well-known and respected transportation companies of its time. G. Thompson Ltd has continued to lead the way in new developments in the courier industry, working under brand names such as Mach 1 Couriers, Pink Express, Destinations Green and A-Z Couriers, to name just a few.
The company is now in the hands of the fifth generation of the Thompson family. We no longer deliver by horse and cart, but much about the core values of the company remain the same to this day.
Some familiar names
Our current Managing Director, Jeremy Thompson launched Mach 1 Couriers Ltd as a subsidiary brand in 1984, and it began carving a name for itself in the London dispatch scene. The company grew organically, joining up with the APC Network for overnight courier deliveries, while also acquiring a number of other courier companies including well-known brands such as The Warwick Group and Reuter Brooks.
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redpajama_set_name: RedPajamaCommonCrawl


text:
Svenska Missionsförbundets förlagsrörelse omfattande Svenska Missionsförbundets förlag och Gummessons bokförlag, grammofonskivetiketterna Sirius, Solist och Signatur.

Svenska Missionsförbundets förlag startade 1895 i samband med utgivningen av Svenska Missionsförbundets sångbok. 1896 utgavs Sånger för söndagsskolan. Övrig produktion begränsade sig till material som hade en direkt inriktning på verksamheten i församlingarna. 1905 gavs första förlagskatalogen ut. Där lanserades bland annat missionssparbössan. Missionsförbundet blev medlem i Nya bokförläggareföreningen 1910. Nya bokförläggareföreningen upphörde 1912 och Missionsförbundet anslöt sig då till Svenska Förläggareföreningen. 1905 anslöt Missionsförbundet till Sällskapet de religiösa bokförlagen (Kristna bokförläggareföreningen). Wilhelm Sjöholm var sedan 1904 redaktör för Missionsförbundets förlag. Sjöholm tog initiativ till Missionsförbundets julkalender Ansgarius 1906, som ett år senare blev årsboken Ansgarius. Tre verk kom ut under Sjöholms redaktörskap, som dokumenterade missionen: Dagbräckning i Kongo (1916), 25 år i Kina (1916) samt På obanade stigar, tjugofem år i Ost-Turkestan (1917).

En rad fristående förlag stod i nära kontakt med Missionsförbundets verksamhet. P.P. Waldenströms skrifter gavs ut på A L Normans förlag. P.P. Waldenström var under en tid delägare i Normans förlag. Waldenström ägde även Pietistens förlagsexpedition. Missionsföreståndare Jakob Ekman publicerade sina böcker på eget förlag E J Ekmans förlagsexpedition, till vilken även ett tryckeri hörde. Johan Bernhard Gauffin, utgivare av söndagsskoltidningen Barnavännen producerade på J B Gauffins förlag i Uppsala böcker och material för söndagsskolan. Dagens bokförlag ägdes på 1920-talet av Axel Andersson. 1913 förvärvade Missionsförbundets förlag utgivningsrätten utgivningsrätten för Waldenströms skrifter. 1913 anställdes bokförlagsföreståndaren Josef Roth. 1920 förvärvades J B Gauffins förlag som specialiserat sig på söndagsskolelitteratur. Tidigare hade detta förlag till Missionsförbundet överlämnat Barnavännen och en jultidning för söndagsskolbarn Kring granen och brasan. 1935 övertogs Dagens bokförlag. Företagsmässigt var förlagsverksamheten i Svenska Missionsförbundet inte ett självständigt bolag utan ingick som en integrerad del i samfundets övriga verksamhet under missionsföreståndarens ledning. Gummessons bokförlag kom att under 1950-talet etableras som förlagsnamn, för att under 1960-talet vara namn för hela Missionsförbundets förlagsverksamhet, inklusive utgivande av skivetiketterna Solist, Sirius och Signatur. Jan-Erik Wikström var chef för Gummessons bokförlag 1963–1976. 1982 gick Gummessons bokförlag samman med Svenska kyrkans förlagsrörelse och bildade förlaget Verbum.

Källor
Kussak, Åke, Författaren som predikant; ett frikyrkosamfunds litterära verksamhet 1910-1939, Gummessons 1982
Hallingberg, Gunnar; Tidningar, böcker, förlag och moderna medier, s 305-322 ur Andreasson, Hans; Liv och rörelse, Verbum, 2007

Noter

Ej längre existerande svenska bokförlag

redpajama_set_name: RedPajamaWikipedia


text:
Tomohiko Taniguchi
Special Advisor to the Cabinet of Prime Minister Shinzo Abe
TANIGUCHI, Tomohiko, as Councillor, Cabinet Secretariat, 2013-14, and as Special Advisor to the Cabinet of Shinzo Abe since April 2014, has been involved in PM Abe's strategic communications. He is also Professor, Keio University Graduate School of System Design and Management, reading international relations. After spending 20 years with Nikkei Business, Japan's foremost business magazine, Mr. Taniguchi joined the Foreign Ministry in 2005 as Deputy Press Secretary. Until he left the ministry three years later he wrote policy speeches for Foreign Ministers and Prime Ministers. While at the magazine he spent a spell in London, 1997-2000, as the magazine's first European correspondent. In 1999 the Foreign Press Association in London elected him President, the first from Asia. Mr. Taniguchi has spent sabbaticals at Princeton University as Fulbright Visiting Fellow, at the Shanghai Institute for International Studies, and at the Brookings Institution as CNAPS Fellow. LL.B. from the University of Tokyo, he has authored or coauthored more than 10 books on modern history and political economy.
Leadership: Thinking the Unthinkable (Panel Session)
Panelists engage in what must be done to confront "unthinkables."


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Well, I knew this would be a show with its ups and downs, but after five incredibly implausible but entertaining and cute installments, man, did we ever hit a turkey. "The Mystery of the Ghostwriters' Cruise" is terrible. Characters don't know things about their own pasts they should definitely remember, other characters have utterly astonishing technical skills that border on the supernatural, and other characters are just plain annoying. The direction and editing are unbelievably clumsy, too. In order to keep conveying a sense of mystery and keep everybody a suspect, the camera lingers on everybody way too long. It's a very annoying hour of television.
A couple of interesting cast notes, though. David Wayne plays the famous mystery writer John Addams, who is retiring and taking a cruise, but somebody plans to kill him, and TV's first Captain America, Reb Brown, is one of about six people set up as suspects. Les Lannom, who had been so entertaining as Lester Hodges in several episodes of Harry O, gets to play the ship's entertainment director, who is pretty much the only man on the ship who doesn't seem to want to kill Addams. Sadly, he's so incredibly creepy and pushy and touchy in that seventies way that he's more troubling than a potential murderer.
Also, the wannabe killer misspells "you're" as "your" in the first threatening note. I thought that Nancy would say something about that. I'm not sure what prison sentence awaits the would-be assassin, but because of that note, I hope they threw away the key.
A couple of months ago, I checked out The Aristocats from the library to show our son. Before the film, there was an ad for other Disney selections and our son hooted. "I want to see that cowboy movie," he yelled. Well, if we must, I said.
I don't know how I've never seen this movie, but I guess I never did. Between HBO showing all sorts of live-action Disney movies and the public library having summers of films, I thought I must have seen this and forgotten, but I didn't recognize a frame of it. I guess I must've seen the sequel!
For more than an hour, I figured I'd write something brief and possibly dismissive about this silly movie. It's cute, but it didn't raise much more than a chuckle. However, that wouldn't be entirely true. Honesty compels me to report that John McGiver delivers a line about how stupid Theodore and Amos are that, a full minute later, had me gasping for air, I laughed so hard. I mean, you miss a minute of a movie from laughing, you can't call it a bad movie.
McGiver's just a small piece of a terrific cast. I'll always make time for a seventies Disney live-action film because they're full of great character actors. Everybody seems to think of this as a vehicle for Don Knotts and Tim Conway, but they're actually providing supporting roles to a story led by Bill Bixby as a hapless gambler suddenly burdened by three orphans. He thinks that a marriage of convenience to a stagecoach driver played by Susan Clark might give the kids a home as well as a chance to nip out and play some poker, but things get complicated when the children, who own a deed to a mine everybody thinks is worthless, unearth a giant gold nugget valued at more than $87,000. Suddenly everybody wants to be part of these kids' lives. Harry Morgan tries to keep order as the town's sheriff, judge, and barber, with supporting roles for McGiver, David Wayne, and Slim Pickens.
But Conway and Knotts do walk away with the proceedings in one perfectly-timed slapstick scene after another. They play criminals so incompetent that the sheriff just lets them wander around freely, because bad guys who can't afford the bullets to "throw lead" don't present much of a danger to the public. I can imagine that, in lesser hands, stopping a movie's narrative for a full five minutes to watch two characters steal a ladder might be an indulgence, but darned if our son didn't spend every second of them chuckling and giggling. This is perfectly judged comedy for seven year-olds. It ends with a chase and everybody getting dunked in the river, inevitably, but our kid whooped that this was the greatest "chase montage" he's ever seen, and the "boat fire truck" that Bixby and Pickens find themselves on in the end was his favorite part of the movie.
I'm not entirely sure I need to watch the sequel. Or Million Dollar Duck, if there's an ad for that hiding on some other DVD at the library. Fingers crossed.
In pretty sharp contrast to the most recent episode of The Avengers that we watched, which featured an element that your typical secret agent show had never seen before, here's an episode of The Twilight Zone with a twist that I can't imagine any adult in 1959 finding at all surprising. I picked this one for its guest appearance by David Wayne as a bored hypochondriac, but the real fun is seeing an actor called Thomas Gomez, who plays the Devil, under the name "Mr. Cadwallader," with a mischievous sense of fun with all the dialogue that he's been given.
Again, we had to pause to explain the narrative a little bit. Newly immortal, David Wayne's character becomes a thrill-seeker arranging accidents and then collecting thousands in insurance money. Sometimes, we watch the story unfold with a little smile before remembering that there's a six year-old in the audience. "So… this probably doesn't make any sense at all to you, does it?" Still, he understood enough to see what a mess Wayne had got himself into with the devil, and facepalmed admirably when the inevitable problem came crashing down, which is delicious for younger viewers if obvious to their parents. The sentence the court passes on this immortal character for the crime of murdering his wife is, of course, life imprisonment.
Part two of this story is indeed better than part one, as I remembered. That's despite a completely ridiculous subplot about Batman and Robin's supposed death. They apparently chose to fake their deaths in the nuclear reactor to put the Mad Hatter off guard, but word got out and kind of got away from them. Within hours, President Johnson, Premier Khrushchev, and the Queen of England are all flying to Gotham City, Aunt Harriet is organizing a funeral committee of four thousand women, and the same two bits of stock footage of crowds that were employed in the Batman movie get used again.
But never mind that, this has one of the all-time great Batfights, which looks like it happened because the stunt team convinced the producers that they could stage a really great one on a water tower. There's absolutely no practical reason in the plot for the Mad Hatter's hideout, at the defunct Green Derby restaurant, to have a water tower, and the best that the writer can come up with is that the Hatter plans to zap Batman and Robin with his mesmerizer as they climb the ladder. But the wind carries his hat off, so it's down to fisticuffs, and you know what? It looks terrific. That is a great, great fight scene.
So in a scene that is made more glorious by David Wayne's extremely mannered and fussy body language and fey delivery, the Mad Hatter basks in his infamy and immediately sets about plotting the next details of his plan. Then moll-of-the-week Polly, played by Jean Hale, wonders whether they're being disrespectful to Batman's memory and Mad Hatter patiently corrects her. Since Batman and Robin were made into celebrities because of criminals, then the best way to honor their memory is to be crooked. "It's the LEAST you can do!" What a delightful, funny scene!
I am almost certain that I remember that part two of this story has a couple of very entertaining moments. I hope I'm right, because part one is incredibly dumb and boring.
In the previous story, writer Charles Hoffman gave us a Batcomputer that spits out spaghetti. This time, Hoffman again thinks that computers are great opportunities for comedy. We learn that the computer is tired of crimefighting, and so Batman has to activate a compute-harder switch to make it come up with some possible suggestions for Mad Hatter's next move. That's about the level of thought put into this story.
Last time, Mad Hatter wanted Batman's cowl. His big plan this time is to irradiate it and turn it pink, so Batman has to go to a nuclear plant to have it decontaminated and he can intercept it. We also learn that Batman has several other cowls in the wash – bad timing, Alfred – and that new cowls are on order, so, you know, he could have just thrown the pink one away and wait for the dryer cycle to finish.
Well, Riddler, you have lost your crown as "most unnerving Bat-villain." The Mad Hatter succeeded in freaking the almighty blazes out of our son tonight. Not content with dumping super-fast hardening plaster over Batman last time, he and his men had a big fight with our heroes in a room with a deathtrap machine on a conveyor belt and Daniel did not enjoy it at all.
This really struck me as unusual until I thought about it. See, the Batfight is always Daniel's favorite part of the episode. He jumps and swings and laughs and enjoys them tremendously. But this time out, they really sold this machine as something full of spikes and blades and the Mad Hatter couldn't wait to get Batman on it and… this gets really outre… stretch him and shrink him and turn him into a hat.
"He's flipped his lid," said one henchman to the other.
So now, with this silly machine firmly established as a really ugly threat, the Batfight in this room – which, incidentally, goes on forever – had Daniel on the edge of his seat, teeth clenched, security blanket in hand, and then Robin got knocked onto the darn thing and the world ended.
"I HATE the Mad Hatter! He's worse than the Riddler!" Daniel bellowed. And he must be. The Riddler only made Daniel cry once in four episodes. The Hatter is two for two!
Oh, and how did Batman get out of that plaster "shroud" in which he was encased last time? He held his breath until he was chipped free. Seriously.
Oh, dear! I'm afraid that Daniel was upset by the cliffhanger to this episode, the first appearance of one of Batman's sillier foes, the Mad Hatter. Everything was going swell until the very end. Previous villains unnerved him simply by virtue of their laughs or appearance, but David Wayne's very mannered and prissy antics didn't frighten him.
I think he was charmed on the one hand by the Mad Hatter's moxie – just stepping up behind people and snatching their hats from their heads, before zapping them with his mesmerizing zap-eyes in his own hat – and bored on the other, because his scheme involves taking revenge on the twelve jurors who last convicted him, and Batman, who was a witness for the prosecution, and he is four, and doesn't know what a trial is. Nor has he seen or been read a version of Alice's Adventures in Wonderland yet, so he asked "Why is he mad?" a couple of times.
Distracted, Batman then got shoved against a wall and Mad Hatter yanked a chain, and Batman got drenched by super-quick hardening plaster. Daniel lost it. "NO! WHY DID HE DO THAT?!" And there were tears. It took a couple of readings of a Mo Willems book before a bath, and he got calmed down.
This episode's lineage is a very strange one. Wikipedia notes that it's a mash-up of three different comic book adventures, but unlike the episodes we've looked at so far, none of them were available as reprints for the producers to easily pick up in late 1965, and the first of them hadn't been seen since its original appearance in 1949. I wonder where they got copies? Maybe somebody looked through DC/National's archive after all?
On the acting front, David Wayne had been in dozens of films in the 1950s, the best known of which are probably Adam's Rib and How to Marry a Millionaire, before spending about five years on the New York stage and making very sporadic TV and film appearances. This is kind of the career arc for many of the Bat-villains, as the major studios started deliberately scaling back the number of pictures they released each year in the sixties, and former leading men, now in their forties, found themselves less in demand and with lots of TV options available. His sidekick Lisa is played by Diane McBain, in the first of her two supporting roles as different characters in the show.


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13 Asians among 19 dead in Egypt balloon crash
Four Japanese citizens and nine tourists from Hong Kong were among the fatalities in a hot air balloon crash in Egypt.
The hot air balloon crashed to Earth after catching fire and exploding over the ancient city of Luxor on Tuesday, killing at least 19 tourists, sources said.
The balloon, which was carrying 20 tourists and a pilot, was landing after a flight over the west bank of Luxor, one of Egypt's most renowned archaeological sites, when a landing cable got caught around a helium tube and a fire erupted, according to an investigator.
The balloon shot up in the air then plunged some 300 meters into a sugar cane field.
At least 18 bodies were found among the charred remnants of the balloon. An Egyptian official identified the tourists from Hong Kong, Japan, Britain, France and Hungary.
According to AP, nine of the killed tourists were from Hong Kong and four from Japan. The French foreign ministry confirmed two of its nationals had died while the British Foreign Office only said that "a small number of British nationals" were involved in the tragic accident.
Three survivors – two British tourists and the Egyptian pilots – were taken to a local hospital. One of the Britons later died, bringing the total number of casualties to 19.
Luxor Governor Ezzat Saad imposed an immediate ban on all hot air balloon flights in the province as authorities ordered an investigation into the fatal crash, reports AFP.
The crash, one of the world's deadliest ballooning accidents, comes amid widespread anger over safety standards in Egypt's transportation.
Tags: egypt, Egypt balloon crash, hot air balloon crash, Luxor, tragic balloon crash in Egypt, world's deadliest ballooning accidents
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Demus Whisper
Digital Memory Music
Digital Memory Music label boss Demus Whisper Started as a promoter in the early 90's in NYC to such clubs as Palladium, USA, Tunnel, Limelight, Roxy... Got into Djing in 1995 playing Progressive House leading into Hard Trance, Hard Techno and UK Hard House & Nu Nrg.. Took a break for a few years and got back into djing in 2005. Some places he has played in NYC are Tunnel, Limelight / Avalon, Sound Factory, Katra, The Green Room, Chelsea Brewing Company, The Grand, Leopard Lounge, The East Side Company Bar, Azza, Bar 13, Le Souk, China 1, Karma, Aji Lounge, National Underground, T heater Bar, Luca Lounge, Studio B just to name a few.He has also played all over the US and overseas in London twice. Demus released his first single with a collaboration with GRon for a heavy techno workout called "Breaking My Concentration" in January 2013. Expect more singles and remixes from Demus in 2013 on Digital Memory Music and Keoki Records.


By far one of the most common aesthetic procedures performed in Dallas Fort Worth, liposuction is a surgical procedure that removes fat. Because it seems a simple concept to remove fat, and there are very few people who don't have areas of fat that they would like removed, it is unfortunately an area that is ripe for hype and marketing tactics that overplay its simplicity and results. Because the body needs just the right layers of fat underneath the skin and there is far less control by the cosmetic surgeon than most aesthetic procedures, this is perhaps where you want the surgeon with the best understanding of the procedure and has the best judgment in precisely how much and where to remove fat. Too much fat removed and the skin sags. Fat removed from the main area without attention to surrounding areas leaves unevenness. Fat removed from superficial areas can leave ridging and dimpling that is very difficult to correct. Lack of understanding of where fat decreases or increases with age can produce acceptable immediate results, but poor results in the long term.
Perhaps the biggest advance that has contributed to the safety of liposuction in Dallas is the understanding that placing the right fluid in content and amount within the fatty tissue, allows safer removal of fat (less blood) and the replacement of fluid that is essential to the body when fat is removed. There are many marketing forms of liposuction including "laser liposuction", "external liposuction", and "ultrasonic liposuction". Suffice it to say that the surgeons at DPSI of Dallas, because of the volume of cases they do and their prominence in the field of plastic surgery, have the opportunity to study all of these techniques and incorporate those that are truly useful and safe, and eliminate those that are empty marketing promotional techniques.
Liposuction can frequently be done safely in combination with other cosmetic plastic surgery procedures such as breast augmentation, tummy tuck, rhinoplasty, and others. You want advice and recommendations from a surgeon that is experienced and knowledgeable in all fields of cosmetic surgery and understands the physiological implications on the body.
The surgeon and patient are partners in achieving the best possible long term results. A patient does not have to be at his or her ideal weight, but needs to have a nutritional and activity lifestyle that allows the necessary postoperative maintenance of results (see section on nutrition). Uncontrolled weight gain after the surgery can overcome many of the benefits produced by the liposuction surgery.