  We study the low-energy effective action on confining strings (in the
fundamental representation) in SU(N) gauge theories in D space-time dimensions.
We write this action in terms of the physical transverse fluctuations of the
string. We show that for any D, the four-derivative terms in the effective
action must exactly match the ones in the Nambu-Goto action, generalizing a
result of Luscher and Weisz for D=3. We then analyze the six-derivative terms,
and we show that some of these terms are constrained. For D=3 this uniquely
determines the effective action for closed strings to this order, while for D>3
one term is not uniquely determined by our considerations. This implies that
for D=3 the energy levels of a closed string of length L agree with the
Nambu-Goto result at least up to order 1/L^5. For any D we find that the
partition function of a long string on a torus is unaffected by the free
coefficient, so it is always equal to the Nambu-Goto partition function up to
six-derivative order. For a closed string of length L, this means that for D>3
its energy can, in principle, deviate from the Nambu-Goto result at order
1/L^5, but such deviations must always cancel in the computation of the
partition function. Next, we compute the effective action up to six-derivative
order for the special case of confining strings in weakly-curved holographic
backgrounds, at one-loop order (leading order in the curvature). Our
computation is general, and applies in particular to backgrounds like the
Witten background, the Maldacena-Nunez background, and the Klebanov-Strassler
background. We show that this effective action obeys all of the constraints we
derive, and in fact it precisely agrees with the Nambu-Goto action (the single
allowed deviation does not appear).
