# Wrapped M2 brane gauge field coupling?

An M2 brane is coupled to the C3 field of SUGRA. And as you all know, a wrapped M2 brane gives us the fundamental string.

My question is, does the wrapped M2 brane (fundamental string) still couple to the C3 field to give us the strings electric charge?

## 1 Answer

In M-theory, the M2 brane couples to the $C_3$ field. The way this coupling is realized is by integrating $C_3$ over the M2.

Now if you compactify on a circle, and if the M2 wraps this circle, then it means one "leg" of the $C_3$ field it couples to is also along the circle. So in the effective 10 dimension theory you obtain by making the circle small, the M2 loses a dimension and becomes a fundamental string, and the $C_3$ becomes a 2-form in type IIA string theory. This two-form is the $B_2$ Kalb-Ramond form. And indeed you know that the fundamental string couples to the B-field in string theory.

This is the answer to your question, the fundamental string couples to the $B_2$ field, that is a dimensional reduction of the $C_3$.

Note that an M2 that does not wrap the circle becomes a D2, and this one still couples to the $C_3$ field.

• And as I understand in previous string theories, the string couples to the B field by winding around the smaller circular dimension (which is how it gets its charge). Does the same hold true for the fundamental string in M theory, or does the string couple to the B field without winding? – user164839 Aug 31 '18 at 2:27
• I don't understand your comment, the string does not couple to the B field by winding around anything. If you write down the sigma model action, the B field is the antisymmetric part of the coupling (while the symmetric traceless part is the metric, and the trace part is the dilaton). This is how the fundamental string couples to $B_2$. – Antoine Aug 31 '18 at 13:49
• Read Lubos’s answer here physics.stackexchange.com/questions/5665/… . He explains that strings get their charge and couple to the B field by winding around a circular dimension – user164839 Aug 31 '18 at 14:07
• No, I think you have misunderstood Lubos's answer. What he explains is the following. Imagine you have a string theory in 5d, and the topology of space-time is $\mathbb{R}^{1,3} \times S^1$. In 5d, the string couples to the $B_{\mu \nu}$ field as I explained before. Now assume that your energy scale is much lower than $1/R$ (with $R$ the radius of the $S^1$) and much lower than $1/\ell_s$. Then you have an effective 4d field theory description, where fields corresponding to wrapped strings couple to a gauge field (a one-form) $A_\mu = B_{5,\mu}$. – Antoine Aug 31 '18 at 14:56
• In one sentence : the stringy coupling to $B_2$ gives after compactification and low energy limit a gauge coupling in field theory for the fields corresponding to winded strings. Not the other way round :) – Antoine Aug 31 '18 at 14:57