# Two perturbative expansions in String Theory

In my String Theory course we learned about the two perturbative expansions of the theory: the one in $$g_s$$ and the one in $$\alpha'$$, but I can't picture the difference between the two of them. I was told that the $$g_s$$ expansion is an expansion in different topologies (sphere + torus + bitorus + ...) while the $$\alpha'$$ expansion is an expansion in couplings (quadratic + cubic + quartic + ...) but they both seem like the same loop expansion, because to consider a torus (order 1 in $$g_s$$) one needs to have at least a cubic interaction term (order 1 in $$\alpha'$$).

• First is the expansion in the dimensionless string coupling $$g_s=\langle e^\phi\rangle$$ while performing string perturbation theory. This is an expansion in the genus of the Riemann surface, i.e. the worldsheet of interacting strings, suppressing worldsheets of higher genus. This sum over topologies has a direct analogue in ordinary Yang-Mills theory: the expansion in terms of the 't Hooft coupling $$g_\mathrm{YM}^2 N$$. In summary, the $$g_s$$ series governs interactions and quantum effects.

• The expansion in the dimensionful inverse string tension $$\alpha'\equiv[L]^2$$ (or more precisely, a dimensionless quantity formed by pairing it with a momentum/energy/inverse length) corresponds to expanding in "stringy effects" away from the point particle limit, encoding the fact that a string is an extended object.

For example, take the non-linear sigma model arising from the low-energy effective action of the string action. For any scattering process, we can analyse the resulting interacting worldsheet CFT through an expansion around the free theory in the dimensionless parameter $$\alpha'/r^2$$, where $$r$$ is the e.g. radius of curvature of the target spacetime (which defines a characteristic length scale).

The most important effect that it governs is the geometry probed by the string: it differs from what a 0D particle "sees" and so the $$\alpha'$$-series describes stringy geometry. One consequence of this is that while probing gravitational effects in regions of high curvature, perturbation theory in $$\alpha'$$ breaks down.

In a general background, it is necessary to perform both expansions at the same time. Often one does not compute the sums explicitly since we implicitly employ non-renormalisation theorems and symmetries rather than brute force - but naturally, such tools are not at our disposal in general backgrounds.

As an afterword, it is a consequence of the AdS/CFT correspondence that the two expansions are related in certain backgrounds: see e.g. https://arxiv.org/abs/1703.05217.

• I understand what you mean in the first point, and I think I understand your example of the non-linear sigma model in the low-energy case, but I don't think I get what you mean by "stringy effects". Does it describe how "not-pointlike" our loop worldsheet expansion looks like? – Mauro Giliberti Jun 9 at 10:39
• @MauroGiliberti well, not specifically the worldsheet series, but yes, the expansion in $\alpha'$ encodes the fact that a string is an extended object rather than a point particle – Nihar Karve Jun 9 at 10:44
• Before they become technical, Brian Greene's notes have a nice description of the two expansions: arxiv.org/abs/hep-th/9702155 – Connor Behan Jun 9 at 11:22
• @NiharKarve can you elaborate on your last comment (that according to AdS/CFT they are related in certain backgrounds) – Wakabaloola Jun 9 at 21:29
• @NiharKarve i’m also not sure i understand why you say it is necessary to perform both expansions at the same time in a general background – Wakabaloola Jun 9 at 21:34