Abstract
In the first part of this chapter we compute the one-loop partition function of the closed fermionic string. We will do this in light cone gauge. The possibility to assign to the world-sheet fermions periodic or anti-periodic boundary conditions leads to the concept of spin structures. The requirement of modular invariance is then shown to result in the GSO projection. We also generalize some of the results of Chap. 6 to the case of fermions. We then consider open superstrings, i.e. we extend the formalism of conformal field theories with boundaries to include free fermionic fields. This gives rise to D-branes in superstring theories. We also discuss non-oriented superstrings, which result form performing a quotient of the type IIB superstring by the world-sheet parity transformation. We show that one-loop diagrams are divergent unless D-branes are present in the model. This defines the type I superstring, whose construction we discuss in some detail.
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Notes
- 1.
Even if this condition is not satisfied, one might still have spinors. But this requires the existence of a suitable complex line bundle to which they couple. This leads to the notion of Spin\({}_{\mathbb{C}}\)-structures.
- 2.
For a real fermion the only choices for g, hare ± 1. For complex fermions they can be arbitrary phases and if we consider several fermions they can be orthogonal matrices; see also the appendix to this chapter.
- 3.
For instance, if one computes the partition function of a fermionic oscillator in the path integral formulation, one obtains the correct result only if one uses anti-periodic boundary conditions for the fermion along the compact Euclidean time (temperature) direction.
- 4.
This will be derived in the appendix to this chapter.
- 5.
There is a subtlety here related to the fact that e.g. the second Stiefel-Whitney class w 2of \(\mathbb{R}{\mathbb{P}}_{2}\)does not vanish. For non-orientable manifolds there is a second construction of spinors for which the obstruction is \({w}_{2} + {w}_{1} \cup {w}_{1}\)which vanishes on any two-dimensional manifold.
- 6.
We choose the (0, 1)-directions as the light-cone directions.
- 7.
Note that this linear combination is different from the one introduced in Eq. (8.14), which did not mix left- and right moving zero modes.
- 8.
We will say more about the effective world-volume action of D-branes in Chap. 16.
- 9.
The calculation is similar to the one in the appendix of Chap. 6; we have to use the propagator for C p + 1as derived from the gauge fixed bulk action \({ 1 \over 2{\kappa }_{10}^{2}} \int olimits olimits {d}^{10}x\,{ 1 \over 2(p+1)!} {C}_{{\mu }_{1}\ldots {\mu }_{p+1}}\square \,{C}^{{\mu }_{1}\ldots {\mu }_{p+1}}\).
- 10.
The general rule is that if the exchange particle has even spin, equal sign charges attract and opposite charges repel. If the spin is odd, the situation is reversed. Exchange of an antisymmetric tensor particle leads to repulsion between equal charges.
- 11.
We have already discussed aspects of non-oriented strings in Chaps. 4, 6 and 8, but so far did not construct a complete model.
- 12.
While in the case considered here only D9 branes and O9 planes exhibit this problem, in the compactified theory also lower dimensional D-branes and orientifold planes will lead to potential inconsistencies whenever their transverse space is compact.
- 13.
While for this interaction the two endpoints of the string which join have to meet on a brane, there is another interaction between open strings: \((\mathit{ij}) + (\mathit{kl}) \rightarrow (\mathit{il}) + (\mathit{kj})\). Here the two strings touch at interior points where they split and join while respecting the orientation. This can happen anywhere in the bulk. This local interaction allows for the emission of closed strings from D-branes: two interior points of an (ij) string touch, split and joint, leaving an (ij) open string plus a closed string. The reverse process is the absorption of a closed string. In the CFT description these processes are described by various correlation functions of open and closed string vertex operators.
- 14.
Different relative signs for the massless NS and R states correspond to O9-planes whose tension has the opposite sign as the R-R charge.
- 15.
This is most easily seen by representing \(\Omega = {e}^{i\pi ({N}_{\alpha }+{N}_{b})}\)where N αand N b are the number operators for the bosonic and fermionic oscillators, respectively.
- 16.
It is straightforward to show that if we change the overall sign in (9.76) we need the lower sign in (9.71) and \(\lambda = \gamma {\lambda }^{\mathrm{T}}{\gamma }^{-1}\). In this case we can choose \(\gamma = \begin{array}{cc} 0 & Vdash \\ - Vdash & 0\end{array}\)and we also find 496 states in the adjoint of SO(32).
- 17.
In other words, we do not sum over the two orientations as we did in the orientable case.
- 18.
Duality symmetries will be discussed in Chap. 18.
- 19.
This representation of the theta-function has an immediate generalization to higher genus Riemann surfaces Σ g :
$${\vartheta}\left [{ a \atop b} \right ](z\vert \Omega ) ={ \sum olimits }_{n\in \mathbb{Z}}\exp \left [i\pi {(n + a)}^{T}\Omega (n + a) + 2\pi i{(n + a)}^{T}(z + b)\right ]\,.$$(9.99)\(a,b,z \in {\mathbb{R}}^{g}\), \(n \in {\mathbb{Z}}^{g}\)and Ωis the period matrix. \(a\)and \(b\)denote the spin-structure or, more generally, encode the periodicity conditions around the aand bcycles of the Riemann surface.
- 20.
Here a bar means complex conjugate, not left-movers.
- 21.
In terms of the coordinates \({\xi }^{1},{\xi }^{2}\)the boundary conditions (9.119) and (9.125) were given in (9.12). \({\xi }^{1} \rightarrow {\xi }^{1} + 1\)corresponds to \(({\sigma }^{0},{\sigma }^{1}) \rightarrow ({\sigma }^{0},{\sigma }^{1} + 2\pi )\)and \({\xi }^{2} \rightarrow {\xi }^{2} + 1\)to \(({\sigma }^{0},{\sigma }^{1})) \rightarrow ({\sigma }^{0} + 2\pi {\tau }_{2},{\sigma }^{1} + 2\pi {\tau }_{1})\). The complex coordinates on the torus are \(z = {\sigma }^{1} + i{\sigma }^{0} = 2\pi ({\xi }^{1} + \tau {\xi }^{2})\).
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© 2012 Springer-Verlag Berlin Heidelberg
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Blumenhagen, R., Lüst, D., Theisen, S. (2012). Superstrings. In: Basic Concepts of String Theory. Theoretical and Mathematical Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-29497-6_9
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DOI: https://doi.org/10.1007/978-3-642-29497-6_9
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