6.2 Strings with variable tension from membranes

6.2.1 General idea

Figure 6.2: How the variable-tension string arises from higher dimensions.

One might say strings with variable tension is slightly weird. One way to realize this variation of the tension in a natural manner is to consider that the extra-dimensional space C which have two dimensions is further embedded in a four-dimensional ambient space X, and there are two sheets of Σ covering C separated in the additional directions of X. We then furthermore suppose that there is a membrane extending along two spatial directions plus one temporal direction, which can have ends on the sheets of Σ. The situation is depicted in Fig. 6.2. Let z be the coordinate of C, and X has complex coordinates (z,x). Then two sheets of Σ define two functions x1(z) and x2(z). Then, a membrane with constant tension |dx||d log z| , suspended between two sheets, can be regarded as a string with variable string whose tension at a given value of z is given by

(tension at z) x2(z)x1(z)dx d log z = x1dz z x2dz z . (6.2.1)

Denoting λi(z) = xidzz, we find that

(tension at z) |λ(z)|whereλ(z) = λ1(z) λ2(z). (6.2.2)

In M-theory, there are indeed higher-dimensional objects with these properties. We consider an eleven dimensional spacetime of the form

3, 1 × X × 3. (6.2.3)

M-theory has six-dimensional objects called M5-branes. We put one M5-brane on

3, 1 × Σ ×{0} (6.2.4)

where Σ X is the curve, and 0 is the origin of the additional 3. This gives a four-dimensional theory. M-theory also has three-dimensional objects called M2-branes, which can have ends on M5-branes. We can take one M2-brane on

0, 1 ×disc ×{0} (6.2.5)

where 0, 1 3, 1 is the worldline of a particle in the four-dimensional spacetime, and the disc X has its boundary on Σ as depicted in Fig. 6.2. For more details on this point, the reader should start from the original paper [52].

It is also useful to regard the intermediate situation when we regard the system as a six-dimensional one on 3, 1 × C. This six-dimensional theory is known as the 6d 𝒩=(2, 0) theory of type SU(2).

6.2.2 Example: pure SU(2) theory

Let us apply this higher-dimensional idea to the curve (4.3.1) of the pure SU(2) theory concretely. For easy reference we reproduce the curve here:

Σ : Λ2z + Λ2 z = x2 u. (6.2.6)

We consider Σ to be embedded in a four-dimensional space X. Given a point z on C, we find two x coordinates by solving the quadratic equation above, as depicted on the left hand side of Fig. 6.3. Let the solutions be ± x(z). As the point z moves on C, they form two sheets of the curve Σ, see the right hand side of Fig. 6.3. The coordinate x always appears as a way to describe the one-form on C giving the tension, so it is convenient to multiply them always by dzz, and say that two sheets have coordinates ± λ = x(z)dzz. We use this convention from now on.

Figure 6.3: W-boson as a string and as a suspended membrane

We can now consider a ring-shaped membrane suspended between the two sheets over the A cycle, see Fig. 6.3. Note that the tension as a string on C is 2λ, and the mass is given by

M |2Aλ| = |2a|. (6.2.7)

We can minimize the tension by solving (6.1.6), which give rise to a configuration with the mass

M = |2a|. (6.2.8)

Note that this has the correct mass to be a W-boson, which has electric charge n = 2 in our normalization, which is for the triplets of SU(2). It is also to be noted that there is no way to have a membrane whose mass is given by

M = |a|, (6.2.9)

because there is simply no way to suspend the membrane to have just one ends over the A-cycle. Therefore, this higher-dimensional reasoning has more explanatory power than just regarding the curve Σ as an auxiliary object producing the holomorphic functions a(u) and aD(u) with the correct monodromy properties. This procedure knows that there is no dynamical particle with electric charge n = 1 in this system.

Figure 6.4: Monopole as a string and as a suspended membrane

Next, we can consider a disc-shaped membrane suspended between the sheets of Σ so that they have endpoints over the branch points z+, z of C, see Fig. 6.4. By a similar reasoning as above, the mass of this membrane is

M = 2|zz+λ| = |Bλ| = |aD|. (6.2.10)

This is a correct mass formula for the monopole, whose magnetic charge is m = 1. In terms of a variable-tension string on C, it is to be noted that this corresponds to an open string, ending at the points where the tension 2λ becomes zero.

Figure 6.5: Dyon as a string and as a suspended membrane. Note that it automatically has the charge aD + 2a, not aD + a.

We can also connect the two branch points z± by going around the phase direction of z, as shown in Fig. 6.5. As shown there, the membrane is topologically the sum of the two configurations considered so far, and we find that the mass of this configuration is

M = |2a + aD|. (6.2.11)

This is the correct mass formula for the dyon, with the electric charge n and the magnetic m given by (n,m) = (2, 1). By going around n times when we connect the branch points, we see that there are dyons with mass |2na + aD| for integral n. We also see there is no way to connect the branch points to have dyons with mass |(2n + 1)a + aD|, which is compatible with the field theory analysis in Sec. 1.3.