On the effective action of confining strings

Abstract

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).

Figures (7)

• Figure 1: The 2-loop contribution to the partition function at $O(T^{-1})$.
• Figure 2: The 3-loop contribution to the partition function. Diagrams (1) and (3) are the two contributions to $\langle S_4^2 \rangle$ and diagram (2) is a single vertex diagram appearing in $\langle S_6 \rangle$.
• Figure 3: The 2-point fermion diagrams: (1) $\Delta \Pi_1$, (2) $\Delta \Pi_2$. The fermionic propagator is marked by a dashed line. The numbers indicate the propagator indices; e.g. in (2) there are two contributions coming from $G_{12}$ and $G_{21}$. External solid lines mark the incoming scalars $X^{i}$ and $X^j$, with momenta $k$ and $-k$ respectively.
• Figure 4: The 2-point scalar diagram $\Delta\Pi_3$. Both massive and massless scalars are marked by solid lines.
• Figure 5: The 4-point fermion diagrams: (1)$M^4_{F1}$, (2)$M^4_{F2}$, (3)$M^4_{F3}$, (4)$M^4_{F4}$, (5+6)$M^4_{F5}$+$M^4_{F6}$, (7)$M^4_{F7}$, (8)$M^4_{F8}$, (9)$M^4_{F9}$, (10)$M^4_{F10}$, (11)$M^4_{F11}$.