In string theory, can strings have zero string tension? Energy = length * string tension
Is it possible for a string to have zero tension, or is it possible that its tension decreases while maintaining length or even growing in length?
 A: It depends what exactly you mean by tension. As Luboš Motl describes here, the Nambu-Goto string action is:
$$ S_{NG} = -\frac{T_0}{c}\int d\mathcal A \tag{1} $$
where the constant $T_0$ has the units of a tension, and this is what we call the string tension. This is a fundamental constant so it cannot change at all, let alone become zero.
However in string theory we can have open strings, and obviously the effective tension at the ends of the string must be zero otherwise the tension would pull in the ends and the string would contract away to nothing. If $v$ is the perpendicular velocity of the string then the effective tension is:
$$ T_{eff} = T_0 \sqrt{1 - \frac{v^2}{c^2}} \tag{2} $$
The ends of an open string have $v = c$, and substituting this into equation (2) gives us:
$$ T_{eff} = 0 $$
So in this sense the tension in a string can be zero at the ends of an open string.
A: It seems relevant to mention that there exist in the literature theories for tensionless strings and for null strings. E.g. in the Hamiltonian formulation of the bosonic string, the Virasoro constraints are then changed into $$\begin{align*}0~\approx~\chi_1&:=~P\cdot X^{\prime} ,\cr0~\approx~\chi_2&:=~P^2+T_0^2(X^{\prime})^2~ \longrightarrow ~ P^2\qquad\text{for}\qquad T_0~\equiv~\frac{1}{2\pi\hbar c\alpha^{\prime}}~\to~ 0.\end{align*}$$
References: 


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*A. Schild, Classical null strings, Phys.Rev. D16 (1977) 1722.

*A. Bredthauer, U. Lindström, J. Persson & L. Wulff, Type IIB tensionless superstrings in a pp-wave background, arXiv:hep-th/0401159.

*U. Schrieber, The string coffee table blog, 2004.
(See also references within Refs. 1-3.)
