268
12 Geometry of Moduli Spaces and Riemann Surfaces
To interpret the stub parameter, consider two local coordinates w 1 and w 2 and
rescale them by λ ∈ C with Re λ > 0:
w 1 = λ ˜
w 1 ,
w 2 = λ ˜
w 2 .
(12.55)
Then, the plumbing fixture (12.30) becomes
˜
w 1 ˜
w 2 = e
−˜ s+i ˜
θ ,
(12.56)
with
˜
s = s + 2 ln |λ|,
˜
θ = θ + i ln
λ
¯
λ
.
(12.57)
If s ∈ R + , the corresponding range of ˜
s is
˜
s ∈ [s 0 , ∞),
s 0 := 2 ln |λ|.
(12.58)
This shows that rescaling the local coordinates by a constant parameter is equivalent
to changing the stub parameter.
Note also how performing a global phase rotation in (12.57) is equivalent to
shifting the twist parameter. Working in ˆ
P g,n forces to take λ ∈ R + .
12.4 Summary
In this chapter, we have explained how to parametrize the fibre bundle P g,n , that
is, appropriate coordinates for the moduli space and the local coordinate systems.
This was realized by introducing different coordinate patches and encoding all the
information of P g,n in the transition functions. Then, this description leads to a
simple description of the tangent vectors through the Schiffer variation.
In the next chapter, we will continue the program by building the p-forms
required to describe off-shell amplitudes.
12.5 Suggested Readings
• Plumbing fixture [2, sec. 9.3].
References
1. T. Erler, Four lectures on closed string field theory. Phys. Rep. 851, 1–36 (2020). https://doi.org/
10.1016/j.physrep.2020.01.003. arXiv: 1905.06785
2. J. Polchinski, String Theory: Volume 1, An Introduction to the Bosonic String (Cambridge
University Press, Cambridge, 2005)
12 Geometry of Moduli Spaces and Riemann Surfaces
To interpret the stub parameter, consider two local coordinates w 1 and w 2 and
rescale them by λ ∈ C with Re λ > 0:
w 1 = λ ˜
w 1 ,
w 2 = λ ˜
w 2 .
(12.55)
Then, the plumbing fixture (12.30) becomes
˜
w 1 ˜
w 2 = e
−˜ s+i ˜
θ ,
(12.56)
with
˜
s = s + 2 ln |λ|,
˜
θ = θ + i ln
λ
¯
λ
.
(12.57)
If s ∈ R + , the corresponding range of ˜
s is
˜
s ∈ [s 0 , ∞),
s 0 := 2 ln |λ|.
(12.58)
This shows that rescaling the local coordinates by a constant parameter is equivalent
to changing the stub parameter.
Note also how performing a global phase rotation in (12.57) is equivalent to
shifting the twist parameter. Working in ˆ
P g,n forces to take λ ∈ R + .
12.4 Summary
In this chapter, we have explained how to parametrize the fibre bundle P g,n , that
is, appropriate coordinates for the moduli space and the local coordinate systems.
This was realized by introducing different coordinate patches and encoding all the
information of P g,n in the transition functions. Then, this description leads to a
simple description of the tangent vectors through the Schiffer variation.
In the next chapter, we will continue the program by building the p-forms
required to describe off-shell amplitudes.
12.5 Suggested Readings
• Plumbing fixture [2, sec. 9.3].
References
1. T. Erler, Four lectures on closed string field theory. Phys. Rep. 851, 1–36 (2020). https://doi.org/
10.1016/j.physrep.2020.01.003. arXiv: 1905.06785
2. J. Polchinski, String Theory: Volume 1, An Introduction to the Bosonic String (Cambridge
University Press, Cambridge, 2005)