Correct Type IIA supergravity action

[Ygor]

Hi,

I'm trying to figure out what the correct form is of the Type IIA supergravity action in the String frame. The peculiar thing is that I'm finding different versions in the literature. Let me state two different ones.

In Polchinski, the action is given by

$$S = \frac{1}{2\kappa_{10}} \int_X e^{-2\phi} (R \ast \mathbf{1} + 4 \phi \wedge \ast \phi -\frac{1}{2} H \wedge \ast H) - \frac{1}{2}(F_2 \wedge \ast F_2 + F_4 \wedge \ast F_4) + - \frac{1}{2} B_2 \wedge dC_3 \wedge dC_3,$$
where
$$F_4 = dC_3 - A_1 \wedge H$$

Then in articles by Louis and Micu if find

$$\begin{equation} S = \int \,e^{-2\hat\phi} \big( \frac12 \hat R ^\ast\! {\bf 1} + 2 d \hat\phi \awedge d \hat\phi - \frac12 \hat H_3 \awedge \hat H_3 \big) - \frac12 \, \big( \hat F_2 \awedge \hat F_2 + \hat F_4 \awedge \hat F_4 \big) - \frac12\Big[ \hat B_2 \wg d\hat C_3 \wg d\hat C_3 - (\hat B_2)^2 \wg d\hat C_3 \wg d\hat A_1 + \frac13 (\hat B_2)^3\wg d\hat A_1 \wg d \hat A_1\Big] \notag , \end{equation} where \begin{equation} \hat F_4 = d\hat C_3 - d\hat A_1 \wg \hat B_2\ , \qquad \hat F_2 = d\hat A_1\ , \qquad \hat H_3 = d \hat B_2 \notag \end{equation}$$

Now my question is, which one is correct? (In particular, note the difference in F_4). Moreover, sometimes i see these actions with a minus sign in front of the Ricci scalar. Does anyone know where that comes from?

Note also, that according to Louis and Micu the action can be rewritten by redefining $$F_4$$ as $$\hat C_3\to \hat C_3 + \hat A_1\wedge\hat B_2$$ so that it becomes

$$\begin{equation} S = \int \, e^{-2\hat\phi} \left( \frac12 \hat R ^\ast\! {\bf 1} + 2 d \hat\phi \awedge d \hat\phi - \frac14 \hat H_3\awedge \hat H_3 \right) - \frac12 \, \left(\hat F_2 \awedge \hat F_2 + \hat F_4 \awedge \hat F_4 \right) + \frac12 \hat H_3 \wedge \hat C_3 \wedge d \hat C_3 \notag , \end{equation}$$
where $$\hat F_4 = d \hat C_3 - \hat A_1 \wedge\hat H_3$$. It thus appears that the F_4 defintion as given in Polchinski belongs is associated to a different Chern Simmons term?

I hope anyone can help.

Thanks,

Ygor

 

[javierR]

Hi Ygor, your post was over a month ago, so I hope this isn't pointless...

They are equivalent actions, just different ways of writing them.
(1) the sign in from of the Ricci scalar term *may* be due to different spacetime signature conventions (-++...+) vs. (+--...-)...I don't know what conventions they have.
(2) The key to the equivalence of the actions is
(A) the "local field redefinition" $$C\rightarrow C+A\wedge B$$ is allowed since the field you're redefining,C, only appears with derivatives $$dC$$ in the initial Louis/Micu action; and
(B) $$\int B\wedge dC\wedge dC = -\int dB\wedge C\wedge dC$$ by integration by parts and assuming $$\int d[...]=0$$, a typical assumption in field theories for non-compact dimensions.

Look ok?

 

[Entropia]

Hi Ygor,

Thank you for reaching out and sharing your confusion about the correct form of the Type IIA supergravity action. This is a common issue in theoretical physics, as different authors may use different conventions or notations.

After reviewing the two versions you provided, it seems that both actions are equivalent and can be obtained from each other through a field redefinition. The difference in the definition of F_4 is due to the fact that in the first version, F_4 is defined as a field strength, while in the second version it is defined as a gauge field. This difference can be resolved by using the Bianchi identity F_4 = dC_3 - A_1 \wedge H, which is satisfied by both definitions.

Regarding the minus sign in front of the Ricci scalar, it is a matter of convention and can be absorbed into the overall normalization of the action. Some authors choose to include it, while others do not.

In summary, both versions of the Type IIA supergravity action are correct and equivalent. The difference in notation and conventions can be resolved through field redefinitions and the inclusion of the minus sign can be absorbed into the overall normalization of the action.

I hope this helps clarify your confusion. Feel free to reach out with any further questions.