How Does T-Duality Affect the Tension Coefficient in String Theory Actions?

[stringsandfields]

I'm working on a problem involving the following action:
$$S_1 = T_{p+1}\int{d^{p+2}\sigma (\frac{1}{4}\alpha'^2 F_{\mu\nu}F^{\mu\nu} + \frac{1}{2} \partial_\alpha X^i \partial^{\alpha} X^J \delta_{ij}+ \text{interaction terms})}$$
which represents the action of an effective free string theory in a low energy limit, thus focusing on the massless sector of the theory. This action represent a (p+1) brane in $$\mathbb{R}^{1,p} \times S^1$$ spacetime (so one dimension has been compactified into a circle). Here $F = dA$ is a 2-form on a (p+2) lorentzian manifold and $\alpha'$ is a constant (which I believe to be the Regge slope).

Then I was told that if we focus on the massless sector restricting ourself to $n=0$ modes for the momentum in the compact dimension (which is quantised), we get the following action:

$$S_{2} = -2\pi RT_{p+1} \int d^{p+1}\sigma\left(\frac{1}{4} \alpha'^2F_{\alpha\beta}F^{\alpha \beta} + \frac{1}{2} (\alpha' \partial_\alpha A_z)(\alpha' \partial^\alpha A_z)+\frac{1}{2}\partial_\alpha X^i \partial^\alpha X^j \delta_{ij}\right)$$

Now I was asked to comment on the fact on how the coefficient $T_p$ would transform under T-Duality, but I'm struggling to think about this. What I can intuitively see here is that these two actions must be the same under T-Duality so I would assume that the transformation of $T_p$ is simply given by multiplying by the circumference of the circle. However, I'm not sure because $R$ has dimensions so I'm not sure if this follows.

 

[AndyW]

I would like to first clarify the notation used in the forum post. The action $S_1$ represents the effective action of a (p+1) brane in a (p+2) dimensional spacetime, with one dimension compactified into a circle. The terms in the action involve the 2-form field $F$ and the scalar fields $X^i$. The constant $\alpha'$ is the Regge slope, which is a fundamental parameter in string theory.

Moving on to the action $S_2$, it represents the same (p+1) brane in a (p+1) dimensional spacetime, with the momentum in the compact dimension quantized to n=0 modes. The terms in this action involve the 2-form field $F$ and the scalar fields $X^i$, as well as the compactified dimension represented by the field $A_z$. The constant $R$ represents the circumference of the compactified circle.

Now, in order to understand how the coefficient $T_p$ transforms under T-duality, we need to first understand what T-duality is. T-duality is a symmetry of string theory that relates two different string theories. In this case, it relates the string theory in a (p+2) dimensional spacetime to the string theory in a (p+1) dimensional spacetime.

Under T-duality, the fields and coordinates in the two theories are related by a transformation, and the action in one theory is related to the action in the other theory by a constant factor. In this case, the two actions $S_1$ and $S_2$ are related by T-duality, and the constant factor is given by the circumference of the compactified circle, $R$. This means that the coefficient $T_p$ in the two actions is related by a factor of $R$. In other words, $T_p$ in $S_2$ is equal to $RT_p$ in $S_1$.

In conclusion, the coefficient $T_p$ in the two actions $S_1$ and $S_2$ is related by a factor of $R$ under T-duality. This shows the symmetry of T-duality in relating two different string theories.