Metric Algebroid and Poisson-Lie T-duality in DFT

Abstract

In this article we investigate the gauge invariance and duality properties of DFT based on a metric algebroid formulation given previously in Carow-Watamura et al (Metric algebroid and Dirac generating operator in Double Field Theory, 2020. https://doi.org/10.1007/JHEP10(2020)192). The derivation of the general action given in this paper does not employ the section condition. Instead, the action is determined by requiring a pre-Bianchi identity on the structure functions of the metric algebroid and also for the dilaton flux. The pre-Bianchi identity is also a sufficient condition for a generalized Lichnerowicz formula to hold. The reduction to the D-dimensional space is achieved by a dimensional reduction of the fluctuations. The result contains the theory on the group manifold, or the theory extending to the GSE, depending on the chosen background. As an explicit example we apply our formulation to the Poisson-Lie T-duality in the effective theory on a group manifold. It is formulated as a 2D-dimensional diffeomorphism including the fluctuations.

Appendices

Appendix

Spin Representation

Here, we construct a spin representation of SO(10, 10) using the basis $\Gamma _A=(\Gamma _{{\textsf{a}}},\Gamma _{{\bar{{{\textsf{a}}}}}})$. $\Gamma _A$ is defined by

$$\begin{aligned} \Gamma _A= \begin{pmatrix} \Gamma _{{\textsf{a}}}\\ \Gamma _{{\bar{{{\textsf{a}}}}}} \end{pmatrix} =\frac{1}{\sqrt{2}} \begin{pmatrix} s_{{{\textsf{a}}}b}&{}\delta _{{\textsf{a}}}{}^{b}\\ -s_{{\bar{{{\textsf{a}}}}} b}&{}\delta _{{\bar{{{\textsf{a}}}}}}{}^b \end{pmatrix} \begin{pmatrix} \gamma ^b\\ \gamma _b \end{pmatrix}~. \end{aligned}$$

(A.1)

Note that $s=\text{ diag }(-1,1,1,\cdots ,1)$ for all suffices $a, {{\textsf{a}}}, {\bar{{{\textsf{a}}}}}$, i.e.,

$$\begin{aligned} \Gamma ^{{\bar{{{\textsf{a}}}}}}=\eta ^{{\bar{{{\textsf{a}}}}}{\bar{{{\textsf{b}}}}}}\Gamma _{{\bar{{{\textsf{b}}}}}}=-s^{{\bar{{{\textsf{a}}}}}{\bar{{{\textsf{b}}}}}}\Gamma _{{\bar{{{\textsf{b}}}}}}~. \end{aligned}$$

(A.2)

The inner product of the basis is diagonalized as

$$\begin{aligned} \{\Gamma _A,\Gamma _B\}= 2\begin{pmatrix} s_{{{\textsf{a}}}{{\textsf{b}}}}&{}0\\ 0&{}-s_{{\bar{{{\textsf{a}}}}}{\bar{{{\textsf{b}}}}}} \end{pmatrix}~. \end{aligned}$$

(A.3)

The component of the generalized metric using this basis is equal to that using $\gamma _A$:

$$\begin{aligned} H_{AB}= \begin{pmatrix} s_{\textsf{ab}}&{}0\\ 0&{}s_{\bar{\textsf{a}}\bar{\textsf{b}}} \end{pmatrix}~. \end{aligned}$$

(A.4)

In this paper, we use the following spin representation for the basis $\Gamma _A$:

$$\begin{aligned} \Gamma _A=\{\Gamma _{{\textsf{a}}},\Gamma _{{\bar{{{\textsf{a}}}}}}\}=\{\Gamma _{0},\cdots ,\Gamma _9,\Gamma _{\bar{0}},\cdots ,\Gamma _{{\bar{9}}}\}\ , \end{aligned}$$

(A.5)

which we give here explicitly,

$$\begin{aligned} \Gamma _0= & {} i\sigma _1\otimes 1\otimes 1\otimes 1\otimes 1\otimes 1^{\otimes 5}\nonumber \\ \Gamma _1= & {} \sigma _2\otimes 1\otimes 1\otimes 1\otimes 1\otimes 1^{\otimes 5}\nonumber \\ \Gamma _2= & {} \sigma _3\otimes \sigma _1\otimes 1\otimes 1\otimes 1\otimes 1^{\otimes 5}\nonumber \\ \Gamma _3= & {} \sigma _3\otimes \sigma _2\otimes 1\otimes 1\otimes 1\otimes 1^{\otimes 5}\nonumber \\ \Gamma _4= & {} \sigma _3\otimes \sigma _3\otimes \sigma _1\otimes 1\otimes 1\otimes 1^{\otimes 5}\nonumber \\ \Gamma _5= & {} \sigma _3\otimes \sigma _3\otimes \sigma _2\otimes 1\otimes 1\otimes 1^{\otimes 5}\nonumber \\ \Gamma _6= & {} \sigma _3\otimes \sigma _3\otimes \sigma _3\otimes \sigma _1\otimes 1\otimes 1^{\otimes 5}\nonumber \\ \Gamma _7= & {} \sigma _3\otimes \sigma _3\otimes \sigma _3\otimes \sigma _2\otimes 1\otimes 1^{\otimes 5}\nonumber \\ \Gamma _8= & {} \sigma _3\otimes \sigma _3\otimes \sigma _3\otimes \sigma _3\otimes \sigma _1\otimes 1^{\otimes 5}\nonumber \\ \Gamma _9= & {} \sigma _3\otimes \sigma _3\otimes \sigma _3\otimes \sigma _3\otimes \sigma _2\otimes 1^{\otimes 5}\nonumber \\ \Gamma _{{\bar{0}}}= & {} (\sigma _3)^{\otimes 5}\otimes \sigma _1\otimes 1\otimes 1\otimes 1\otimes 1\nonumber \\ \Gamma _{\bar{1}}= & {} i(\sigma _3)^{\otimes 5}\otimes \sigma _2\otimes 1\otimes 1\otimes 1\otimes 1\nonumber \\ \Gamma _{\bar{2}}= & {} i(\sigma _3)^{\otimes 5}\otimes \sigma _3\otimes \sigma _1\otimes 1\otimes 1\otimes 1\nonumber \\ \Gamma _{{\bar{3}}}= & {} i(\sigma _3)^{\otimes 5}\otimes \sigma _3\otimes \sigma _2\otimes 1\otimes 1\otimes 1\nonumber \\ \Gamma _{{\bar{4}}}= & {} i(\sigma _3)^{\otimes 5}\otimes \sigma _3\otimes \sigma _3\otimes \sigma _1\otimes 1\otimes 1\nonumber \\ \Gamma _{{\bar{5}}}= & {} i(\sigma _3)^{\otimes 5}\otimes \sigma _3\otimes \sigma _3\otimes \sigma _2\otimes 1\otimes 1\nonumber \\ \Gamma _{{\bar{6}}}= & {} i(\sigma _3)^{\otimes 5}\otimes \sigma _3\otimes \sigma _3\otimes \sigma _3\otimes \sigma _1\otimes 1\nonumber \\ \Gamma _{{\bar{7}}}= & {} i(\sigma _3)^{\otimes 5}\otimes \sigma _3\otimes \sigma _3\otimes \sigma _3\otimes \sigma _2\otimes 1\nonumber \\ \Gamma _{{\bar{8}}}= & {} i(\sigma _3)^{\otimes 5}\otimes \sigma _3\otimes \sigma _3\otimes \sigma _3\otimes \sigma _3\otimes \sigma _1\nonumber \\ \Gamma _{{\bar{9}}}= & {} i(\sigma _3)^{\otimes 5}\otimes \sigma _3\otimes \sigma _3\otimes \sigma _3\otimes \sigma _3\otimes \sigma _2~, \end{aligned}$$

(A.6)

where $\sigma _1,\sigma _2$ and $\sigma _3$ are the Pauli matrices:

$$\begin{aligned} \sigma _1= \begin{pmatrix} 0&{}1\\ 1&{}0 \end{pmatrix}~,~ \sigma _2= \begin{pmatrix} 0&{}-i\\ i&{}0 \end{pmatrix}~,~ \sigma _3= \begin{pmatrix} 1&{}0\\ 0&{}-1 \end{pmatrix}~. \end{aligned}$$

(A.7)

$A_+$ is defined as a generator of the Hermitian conjugate as

$$\begin{aligned} A_+\Gamma _AA_+^{-1}=\Gamma _A^\dagger ~. \end{aligned}$$

(A.8)

$A_+$ is the charge conjugate matrix in this representation given by

$$\begin{aligned} A_+=i\sigma _2\otimes \sigma _3\otimes \sigma _3\otimes \sigma _3\otimes \sigma _3\otimes \sigma _2\otimes \sigma _3\otimes \sigma _3\otimes \sigma _3\otimes \sigma _3~. \end{aligned}$$

(A.9)

To obtain the Majorana representation, we define the generator of the complex conjugate as

$$\begin{aligned} B_+\Gamma _AB_+^{-1}= & {} \Gamma _A^{*}~. \end{aligned}$$

(A.10)

The representation of $B_+$ is given by

$$\begin{aligned} B_+=\sigma _3\otimes \sigma _1\otimes \sigma _2\otimes \sigma _1\otimes \sigma _2\otimes 1\otimes \sigma _2\otimes \sigma _1\otimes \sigma _2\otimes \sigma _1~. \end{aligned}$$

(A.11)

The spinor basis $e_\alpha$ of the Majorana representation is defined by

$$\begin{aligned} B_+^{-1}e_\alpha ^*=e_{\alpha }~. \end{aligned}$$

(A.12)

A Majorana spinor $\varphi \in {\mathbb {S}}$ is denoted by

$$\begin{aligned} \varphi =\varphi ^{\alpha } e_{\alpha }~. \end{aligned}$$

(A.13)

where $\varphi ^\alpha \in {\mathbb {R}}$. $\Gamma _\chi$ is defined as

$$\begin{aligned} \{\Gamma _\chi ,\Gamma _A\}=0~. \end{aligned}$$

(A.14)

The explicit form of $\Gamma _\chi$ is given by

$$\begin{aligned} \Gamma _\chi =\sigma _3^{\otimes 5}\otimes \sigma _3^{\otimes 5}~. \end{aligned}$$

(A.15)

Finally, the spin representation of the generalized metric is given by

$$\begin{aligned} {K}\Gamma _A{K}^{-1}= & {} H_A{}^B\Gamma _B~,\end{aligned}$$

(A.16)

$$\begin{aligned} {K}= & {} 1^{\otimes 5}\otimes \sigma _3^{\otimes 5}~. \end{aligned}$$

(A.17)

1.1 A.1 Majorana representation

We list the properties under Hermitian conjugation and taking the transpose for the operators in the Majorana representation:

$$\begin{aligned} J_{\alpha \beta }= & {} e_{\alpha }^\dagger e_{\beta }~,\end{aligned}$$

(A.18)

$$\begin{aligned} \Gamma _A{}_{\alpha \beta }= & {} e_{\alpha }^\dagger \Gamma _A e_{\beta }~,\end{aligned}$$

(A.19)

$$\begin{aligned} A_+{}_{\alpha \beta }= & {} e_{\alpha }^\dagger A_+ e_{\beta }~,\end{aligned}$$

(A.20)

$$\begin{aligned} K_{\alpha \beta }= & {} e_{\alpha }^\dagger K e_{\beta }~, \end{aligned}$$

(A.21)

where $J_{\alpha \beta }$ is the metric on a Majorana spinor, i.e.,

$$\begin{aligned} e^\alpha= & {} J^{-1\alpha \beta }e_\beta ~, \end{aligned}$$

(A.22)

$$\begin{aligned} \varphi _\alpha= & {} J_{\alpha \beta }\varphi ^\beta ~. \end{aligned}$$

(A.23)

$J_{\alpha \beta },A_+{}_{\alpha \beta }$ and $K_{\alpha \beta }$ are (anti-)symmetric and (anti-)Hermitian, and $\Gamma _A$ is in the Majorana representation. Then, they are real matrices as follows,

$$\begin{aligned} J_{\alpha \beta }^{*}= & {} J_{\beta \alpha }~, \end{aligned}$$

(A.24)

$$\begin{aligned} J_{\alpha \beta }= & {} J_{\beta \alpha }~, \end{aligned}$$

(A.25)

$$\begin{aligned} \Gamma _Ae_{\alpha }= & {} {} \Gamma _{A}{}^{\beta } {}_{\alpha }e_{\beta }~, \end{aligned}$$

(A.26)

$$\begin{aligned} \Gamma _{A\alpha \beta }= & {} \Gamma _{A}^{*} {}_{\alpha \beta }~, \end{aligned}$$

(A.27)

$$\begin{aligned} A_{+\alpha \beta }^{*}= & {} -A_{+\beta \alpha }~, \end{aligned}$$

(A.28)

$$\begin{aligned} A_{+\alpha \beta }= & {} - A_{+\beta \alpha }~, \end{aligned}$$

(A.29)

$$\begin{aligned} K_{\alpha \beta }^{*}= & {} K_{\beta \alpha }~, \end{aligned}$$

(A.30)

$$\begin{aligned} {K}_{\alpha \beta }= & {} {K}_{\beta \alpha }~. \end{aligned}$$

(A.31)

1.2 A.2 Vacuum

The vacuum of the spin space ${\mathbb {S}}$ is defined by

$$\begin{aligned} B_+^{-1}\vert 0\rangle ^*= & {} \vert 0\rangle ~,\nonumber \\ \gamma _a\vert 0\rangle= & {} 0~. \end{aligned}$$

(A.32)

$\gamma _a$ is a suffix of standard O(D, D). On the other hand, $K\vert 0\rangle$ is a dual vacuum and satisfies

$$\begin{aligned} \gamma ^aK\vert 0\rangle =0~. \end{aligned}$$

(A.33)

1.3 A.3 A-product

An O(D, D) transformation of a spinor is given by

$$\begin{aligned} \varphi \mapsto \exp \Big (\frac{1}{4}\Lambda _{AB}\Gamma ^{AB}\Big )\varphi ~, \end{aligned}$$

(A.34)

where $\Lambda _{AB}=-\Lambda _{BA}$. We define the A-product $(\cdot ,\cdot )_{A}$ as an O(D, D) invariant product of any two spinors $\varphi _1,\varphi _2$:

$$\begin{aligned} (\varphi _1,\varphi _2)_A=\varphi _1^\dagger A_+\varphi _2~. \end{aligned}$$

(A.35)

We show the O(D, D) invariance of the A-product as follows,

$$\begin{aligned} (\varphi _1,\varphi _2)_A\mapsto & {} \Big (\exp \Big (\frac{1}{4}\Lambda _{AB}\Gamma ^{AB}\Big )\varphi _1,\exp \Big (\frac{1}{4}\Lambda _{A'B'}\Gamma ^{A'B'}\Big )\varphi _2\Big )_A\nonumber \\= & {} \varphi _1^\dagger \exp \Big (\frac{1}{4}\Lambda _{AB}\Gamma ^{AB}\Big )^\dagger A_+\exp \Big (\frac{1}{4}\Lambda _{A'B'}\Gamma ^{A'B'}\Big )\varphi _2\nonumber \\= & {} \varphi _1^\dagger A_+\exp \Big (-\frac{1}{4}\Lambda _{AB}\Gamma ^{AB}\Big ) \exp \Big (\frac{1}{4}\Lambda _{A'B'}\Gamma ^{A'B'}\Big )\varphi _2\nonumber \\= & {} \varphi _1^\dagger A_+\varphi _2\nonumber \\ {}= & {} (\varphi _1,\varphi _2)_A~. \end{aligned}$$

(A.36)

The A-product of two Majorana spinors $\varphi _1,\varphi _2$ is real:

$$\begin{aligned} (\varphi _1,\varphi _2)_A=\varphi _1^{\alpha }e_{\alpha }^\dagger A_+\varphi _2^{\beta }e_{\beta }=\varphi _1^{\alpha } A_{+\alpha \beta }\varphi _2^{\beta }\in {\mathbb {R}}~. \end{aligned}$$

(A.37)

As we see from (Eq. 4.42), $(\vert 0\rangle ,K\vert 0\rangle )_A$ includes the measure. The representation of $(\vert 0\rangle ,K\vert 0\rangle )_A$ is given by

$$\begin{aligned} (\vert 0\rangle ,K\vert 0\rangle )_A=\langle 0 \vert A_+K\vert 0\rangle ~. \end{aligned}$$

(A.38)

Since $A_+$ and K are constant matrices, $\vert 0\rangle$ has to be a half density.

$$\begin{aligned} \vert 0\rangle = \sqrt{dXe^{-2d}\sqrt{\det \eta _{NM}}}\vert 0\rangle ~, \end{aligned}$$

(A.39)

where $\vert 0\rangle$ is a constant spinor and satisfies

$$\begin{aligned} \langle 0 \vert A_+K\vert 0\rangle =c_0~. \end{aligned}$$

(A.40)

1.4 A.4 AK-product

We can define the AK-product $(-,-)_{AK}$ as the $O(1,D-1)\times O(D-1,1)$ invariant product of any two spinors $\varphi _1,\varphi _2$:

$$\begin{aligned} (\varphi _1,\varphi _2)_{AK}=\varphi _1^\dagger A_+ K\varphi _2~. \end{aligned}$$

(A.41)

We can see its $O(1,D-1)\times O(D-1,1)$ invariance as:

$$\begin{aligned} (\varphi _1,\varphi _2)_{AK}\mapsto & {} \Big (\exp \Big (\frac{1}{4}\Lambda _{AB}\Gamma ^{AB}\Big )\varphi _1,\exp \Big (\frac{1}{4}\Lambda _{A'B'}\Gamma ^{A'B'}\Big )\varphi _2\Big )_{AK}\nonumber \\= & {} \varphi _1^\dagger \exp \Big (\frac{1}{4}\Lambda _{AB}\Gamma ^{AB}\Big )^\dagger A_+ K\exp \Big (\frac{1}{4}\Lambda _{A'B'}\Gamma ^{A'B'}\Big )\varphi _2\nonumber \\= & {} \varphi _1^\dagger A_+\exp \Big (-\frac{1}{4}\Lambda _{AB}\Gamma ^{AB}\Big )K \exp \Big (\frac{1}{4}\Lambda _{A'B'}\Gamma ^{A'B'}\Big )\varphi _2\nonumber \\= & {} \varphi _1^\dagger A_+ K\exp \Big (-\frac{1}{4}\Lambda _{AB}\Gamma ^{CD}H_{C}{}^AH_D{}^B\Big ) \exp \Big (\frac{1}{4}\Lambda _{A'B'}\Gamma ^{A'B'}\Big )\varphi _2\nonumber \\= & {} \varphi _1^\dagger A_+ K\exp \Big (-\frac{1}{4}\Lambda _{AB}\Gamma ^{AB}\Big ) \exp \Big (\frac{1}{4}\Lambda _{A'B'}\Gamma ^{A'B'}\Big )\varphi _2\nonumber \\= & {} \varphi _1^\dagger A_+ K\varphi _2\nonumber \\= & {} (\varphi _1,\varphi _2)_{AK}~, \end{aligned}$$

(A.42)

where $\Lambda _{AB}$ satisfies

$$\begin{aligned} \Lambda _{AB}=-\Lambda _{BA}~,~\Lambda _{AB}=H_A{}^{A'}H_B{}^{B'}\Lambda _{A'B'}~. \end{aligned}$$

(A.43)

As an A-product, the AK-product is real.

$$\begin{aligned} (\varphi _1,\varphi _2)_{AK}=\varphi _1^{\alpha }e_{\alpha }^\dagger A_+ K\varphi _2^{\beta }e_{\beta }=\varphi _1^{\alpha } A_{+\alpha \gamma }S^{\gamma \delta } K_{\delta \beta }\varphi _2^{\beta } \end{aligned}$$

(A.44)

$$\begin{aligned} (\varphi _1,\varphi _2)_{AK}\in {\mathbb {R}}~. \end{aligned}$$

(A.45)

B. Reduction of $S_{RR}$ to DFT$_{sec}$

In this section, we show that for the case of a flat background $S_{RR}$ reduces to DFT$_{sec}$. For this end we consider the case where the metric and the structure function $\phi '$ are given by

$$\begin{aligned} \eta _{MN}= \begin{pmatrix} 0&{}\delta ^m{}_n\\ \delta _m{}^n&{}0 \end{pmatrix}~,~\phi '_{LM}{}^N=0~. \end{aligned}$$

(B.1)

Then, the vielbein $E_A{}^N$ is given by

$$\begin{aligned} E_A{}^N= \begin{pmatrix} e^{-T}{}^a{}_m&{}0\\ 0&{}e_a{}^m \end{pmatrix} \begin{pmatrix} \delta ^m{}_n&{}0\\ -B_{mn}&{}\delta _m{}^n \end{pmatrix}~. \end{aligned}$$

(B.2)

For the purpose to define the B-transformation later, we separate $E_A{}^N$ into a GL(D) part $E^{(e)}_A{}^M$ and a B-field part $E^{(B)}_M{}^N$ as

$$\begin{aligned} E_A{}^M= & {} E_A^{(e)}{}^NE_N^{(B)}{}^M~,\end{aligned}$$

(B.3)

$$\begin{aligned} E^{(e)}_A{}^N= & {} \begin{pmatrix} e^{-T}{}^a{}_m&{}0\\ 0&{}e_a{}^m \end{pmatrix}~,\end{aligned}$$

(B.4)

$$\begin{aligned} E^{(B)}_N{}^M= & {} \begin{pmatrix} \delta ^m{}_n&{}0\\ -B_{mn}&{}\delta _m{}^n \end{pmatrix}~. \end{aligned}$$

(B.5)

The spin operators ${{\textsf{S}}}_{E^{(e)}},{{\textsf{S}}}_{E^{(B)}}$ related to $E^{(e)}$ and $E^{(B)}$, respectively, can be defined as

$$\begin{aligned} {{\textsf{S}}}_{E^{(e)}}\gamma _A{{\textsf{S}}}_{E^{(e)}}^{-1}= & {} E^{(e)}_A{}^N\delta _N{}^B\gamma _B~,\end{aligned}$$

(B.6)

$$\begin{aligned} {{\textsf{S}}}_{E^{(B)}}\gamma _A{{\textsf{S}}}_{E^{(B)}}^{-1}= & {} \delta _A{}^ME^{(B)}_M{}^N\delta _N{}^B\gamma _B~. \end{aligned}$$

(B.7)

The concrete form of ${{\textsf{S}}}_{E^{(e)}}$ and ${{\textsf{S}}}_{E^{(B)}}$ are

$$\begin{aligned} {{\textsf{S}}}_{E^{(e)}}= & {} \exp (-\frac{1}{2}\lambda _a{}^b\gamma ^a{}_b)~,~(e^{\lambda })_a{}^b=e_a{}^m\delta _m{}^b~,\end{aligned}$$

(B.8)

$$\begin{aligned} {{\textsf{S}}}_{E^{(B)}}= & {} e^{\frac{1}{4}B_{mn}\delta _a{}^m\delta _b{}^n\gamma ^{ab}}~, \end{aligned}$$

(B.9)

Using these spin operators, we define the spin operator ${{\textsf{S}}}_E$ related to $E_A{}^N$ as

$$\begin{aligned} {{\textsf{S}}}_E= & {} {{\textsf{S}}}_{E^{(B)}}{{\textsf{S}}}_{E^{(e)}}~,\end{aligned}$$

(B.10)

$$\begin{aligned} {{\textsf{S}}}_E\gamma _A{{\textsf{S}}}_E^{-1}= & {} E_A{}^N\delta _N{}^B\gamma _B~. \end{aligned}$$

(B.11)

With this ${{\textsf{S}}}_E$, the corresponding Dirac generating operator denoted by $\partial _N:=E_N^{-1}{}^A\partial _A$ is

(B.12)

is a Dirac operator in the local coordinate basis. To see the relation between $S_{RR}$ and DFT$_{sec}$, we rewrite the action of the R–R sector $S_{RR}$ by :

(B.13)

where $\breve{\chi }$ is defined by

$$\begin{aligned} \breve{\chi }=e^{-d}{{\textsf{S}}}_E\chi ~. \end{aligned}$$

(B.14)

The coefficient of the R–R flux is defined by

(B.15)

This defines the RR field $\breve{F}^{RR}$ from the spinor $\breve{\chi }$. It is the equivalent to the relation used in DFT$_{sec}$ [63]. Using $\breve{F}^{RR}$, the action of the R–R sector becomes

(B.16)

where we used

$$\begin{aligned} {{\textsf{S}}}_{E^{(e)}}^{-1}\vert 0\rangle =(\det g_{mn})^\frac{1}{4}\vert 0\rangle ~. \end{aligned}$$

(B.17)

To compare with the action in DFT$_{sec}$, we determine the constant $\beta _{RR}$ as

$$\begin{aligned} \beta _{RR}=-\frac{1}{2c_0}~, \end{aligned}$$

(B.18)

which yields the action of R–R sector in a flat space as:

$$\begin{aligned} S_{RR}=-\frac{1}{4}\int dX\sum _{p}\sqrt{\det G_{ll'}}\frac{1}{p!}\breve{F}^{RR}_{m_1\cdots m_p}\breve{F}^{RR}_{n_1\cdots n_p}G^{m_1n_1}\cdots G^{m_pn_p}~. \end{aligned}$$

(B.19)

Thus, the action of the R–R sector (7.3) reduced to a flat background is consistent with the results given in the literature for DFT$_{sec}$.

C. GSE and DFT

The Generalized Supergravity Equations (GSE) are defined by

$$\begin{aligned} R^{(e)}+4\nabla ^m\partial _m\phi -4\vert \partial \phi \vert {}^2-\frac{1}{2}\vert {}H\vert {}^2-4(I^mI_m+U^mU_m+2U^m\partial _m\phi -\nabla _mU^m)= & {} 0~,\nonumber \\ R^{(e)}_{mn}-\frac{1}{4}H_{mpq}H_n{}^{pq}+2\nabla _m\partial _n\phi +\nabla _mU_n+\nabla _nU_m= & {} 0~,\nonumber \\ -\frac{1}{2}\nabla ^kH_{kmn}+\partial _k\phi H^k{}_{mn}+U^kH_{kmn}+\nabla _mI_n-\nabla _nI_m= & {} 0~. \nonumber \\ \end{aligned}$$

(C.1)

Here $R^{(e)}$ is the Ricci scalar given by the D-dimensional vielbein and $I=I^m\partial _m$ is a constant Killing vector which satisfies

$$\begin{aligned} L_IG=0~,~L_IB=0~,~L_I\phi =0~. \end{aligned}$$

(C.2)

$U_m$ is defined by

$$\begin{aligned} U_m=I^nB_{nm}~. \end{aligned}$$

(C.3)

It was shown in [29] that the GSE can be derived from the DFT$_{sec}$ by taking an ansatz

$$\begin{aligned} H_{MN}= \begin{pmatrix} G^{-1}&{}-G^{-1}B\\ BG^{-1}&{}G-BG^{-1}B \end{pmatrix}~,~ d=\phi -\frac{1}{4}\log (\det G)+I^m{{\tilde{x}}}_m~. \end{aligned}$$

(C.4)

On the other hand, we would like to use a different ansatz, to obtain the Poisson-Lie T-duality in which the sign of $B_0$ equals to that of $\Pi$ and ${\bar{\Pi }}$ as in section 6.1 Since a change of the ansatz $(B,I)\rightarrow (-B,-I)$ respects the GSE (C.1), we use here the ansatz

$$\begin{aligned} H_{MN}= \begin{pmatrix} G^{-1}&{}G^{-1}B\\ -BG^{-1}&{}G-BG^{-1}B \end{pmatrix}~, \end{aligned}$$

(C.5)

$$\begin{aligned} d=\phi -\frac{1}{4}\log (\det G)-I^m{{\tilde{x}}}_m~. \end{aligned}$$

(C.6)

Moreover, when $I^m$ is not constant, we use the following redefinition of $F_A$:

$$\begin{aligned} d= & {} \phi -\frac{1}{4}\log (\det G)~,\nonumber \\ F_M= & {} 2\partial _Md-2I_M~, \end{aligned}$$

(C.7)

where $I_M=(I^m,I_m)$ as in the modified DFT.

Thus, in this paper, we denote the DFT action $\mathcal{I}(0,8c_0^{-1})$ using the ansätze (C.5), (C.7) which derives the GSE of $S_{DFT_{sec}}^{mod}[E_A{}^M,d,I^m]$. The resulting action coincides with the modified DFT action defined in [30], and thus is giving the missing algebraic background of their modification for the Drinfel’d double case.