Strings with variable tension from membranes

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 $x_{1} (z)$ and $x_{2} (z)$ . Then, a membrane with constant tension $| d x | \land | d log z |$ , 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) \geq |\int_{x_{2} (z)}^{x_{1} (z)} d x \land d log z| = |x_{1} \frac{d z}{z} - x_{2} \frac{d z}{z}| .

(6.2.1)

Denoting $λ_{i} (z) = x_{i} d z ∕ z$ , we ﬁnd that

(tension at z) \geq | λ (z) | where λ (z) = λ_{1} (z) - λ_{2} (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

ℝ^{3, 1} \times X \times ℝ^{3} .

(6.2.3)

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

ℝ^{3, 1} \times Σ \times {0}

(6.2.4)

where $Σ \subset X$ is the curve, and $0$ is the origin of the additional $ℝ^{3}$ . 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

ℝ^{0, 1} \times disc \times {0}

(6.2.5)

where $ℝ^{0, 1} \subset ℝ^{3, 1}$ is the worldline of a particle in the four-dimensional spacetime, and the $disc \subset X$ 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 $ℝ^{3, 1} \times C$ . This six-dimensional theory is known as the 6d $𝒩 = (2, 0)$ theory of type $S U (2)$ .

6.2.2 Example: pure $S U (2)$ theory

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

Σ : Λ^{2} z + \frac{Λ^{2}}{z} = x^{2} - 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 $\pm x (z)$ . 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 $d z ∕ z$ , and say that two sheets have coordinates $\pm λ = x (z) d z ∕ z$ . 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 $2 λ$ , and the mass is given by

M \geq | 2 \int_{A} λ | = | 2 a | .

(6.2.7)

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

M = | 2 a | .

(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 $S U (2)$ . 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 $a_{D} (u)$ 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 | \int_{z_{-}}^{z_{+}} λ | = | \int_{B} λ | = | a_{D} | .

(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 $2 λ$ becomes zero.

Figure 6.5: Dyon as a string and as a suspended membrane. Note that it automatically has the charge

a_{D} + 2 a

, not

a_{D} + a

We can also connect the two branch points $z_{\pm}$ 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 = | 2 a + a_{D} | .

(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 $| 2 n a + a_{D} |$ for integral $n$ . We also see there is no way to connect the branch points to have dyons with mass $| (2 n + 1) a + a_{D} |$ , which is compatible with the ﬁeld theory analysis in Sec. 1.3.

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6.2 Strings with variable tension from membranes

6.2.1 General idea

6.2.2 Example: pure SU(2) theory

6.2.2 Example: pure $S U (2)$ theory