Abstract
In this chapter, we construct and classify all superstring theories whose world-sheet theories have a (1, 1) superconformal algebra of gauge constraints. We discuss in detail the consistency conditions of the perturbative superstring theory, namely invariance under the mapping class group of the world-sheet and absence of dangerous tadpoles/divergences. To put things in a broader perspective, we start with a review of the mapping class group. Then we show that one-loop modular invariance implies an absence of global \(\textbf{Diff}^+\) anomalies to all loop orders. We study the conditions of modular invariance in full detail. Then we compute explicitly the path integrals for Weyl fermions on a torus \(T^2\) coupled to arbitrary flat line bundles. Finally, we work out the one-loop vacuum amplitudes for all 10d superstring models and check that they have indeed the required properties.
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Notes
- 1.
The presence of a non-zero tadpole means that the “vacuum” around which we expand is not a solution to the quantum corrected e.o.m. “Shifted away” means that we can deform a little bit the vacuum configuration and get a valid solution to the required perturbative order. If this is not possible, the theory cannot be consistently defined.
- 2.
That is, locally on the world-sheet their algebra of world-sheet gauge constraints is the (1, 1) SCFT algebra generated by the chiral currents \(T_B(z)\), \(T_F(z)\), \(\tilde{T}_B(\bar{z})\), \(\tilde{T}_F(\bar{z})\), see Chap. 2.
- 3.
We can always reduce to this case by going to the oriented double.
- 4.
There are no tadpoles on the sphere by Eq. (2.180).
- 5.
An integral bilinear form \(\langle \cdot ,\cdot \rangle \) on a lattice \(\Lambda \) is unimodular iff it induces an isomorphism (of free Abelian groups) \(\Lambda ^\vee \simeq \Lambda \). Let \(\{e_i\}\) be a set of generators of \(\Lambda \); the bilinear form \(\langle \cdot ,\cdot \rangle \) is unimodular iff \(\det \langle e_i,e_j\rangle =\pm 1\).
- 6.
Lazy readers are referred to Sect. 1.4.2 of [6].
- 7.
We us the block-matrix notation; each entry of the \(2\times 2\) “matrix” is a \(g\times g\) matrix.
- 8.
Homology is a homotopy invariant so, if \(f:\Sigma \rightarrow \Sigma \) is a diffeomorphism and \(\alpha \) a representative of a class \([\alpha ]\in H_1(\Sigma , \mathbb {Z})\), the class \([f_*\alpha ]\in H_1(\Sigma ,\mathbb {Z})\) depends on f only through its homotopy class.
- 9.
In view of the Minkowski theorem, this implies a strong result for the mapping class group:
Theorem (Minkowski [11, 12]). Let \(\Gamma \subset GL(n,\mathbb {Z})\) be a finite subgroup. For \(m\ge 3\) an integer, let \(r_m:GL(n,\mathbb {Z})\rightarrow GL(n,\mathbb {Z}/m\mathbb {Z})\) be the reduction mod m map. \(r_m\) is injective.
Corollary. \(\textsf{MCG}(\Sigma )\) contains a finite-index, torsion-less, normal subgroup.
- 10.
\(*\) stands for the free non-Abelian product of groups.
- 11.
I.e. the smallest normal subgroup of \(SL(2,\mathbb {Z})\) containing T.
- 12.
Cf. Definition 4.1.
- 13.
A closed curve is non-separating if cutting along it the surface \(\Sigma \) remains connected.
- 14.
The symbol \(\Phi \) stands for the collection of all fields we integrate over in the path integral.
- 15.
By local we mean that each operator is local with respect to all inserted operators (including itself) and also with respect to the BRST currents (left and right).
- 16.
The argument shows invariance under the pure group \(\textsf{PMCG}(\Sigma )\). To show that it is invariant under the full mapping class group \(\textsf{MCG}(\Sigma )\), we have to show that it is also invariant under the symmetric group \(\mathfrak {S}_n\) permuting the n punctures. This amounts to showing that the inserted operators have bosonic statistics, i.e. (by the 2d Spin & Statistics theorem) that their 2d spins are integral. This is an automatic consequence of BRST invariance or of left-right matching conditions.
- 17.
As before, \(\textbf{H}\) is the physical Hilbert space of the closed string.
- 18.
The Abelianization \(G^\text {ab}\) of a group G is the quotient group \(G^\text {ab}\overset{\textrm{def}}{=}G/[G,G]\), where \([G,G]\triangleleft G\) is the normal subgroup generated by the commutators.
- 19.
We use “fermion” and “spinor” interchangeably since one consequence of the analysis is that the Spin & Statistic Theorem holds in 10d, see box 5.3.
- 20.
For the full non-linear form of the self-duality constraint, see Sect. 8.5.
- 21.
Ad abundantiam we shall check absence of anomalies in IIB in Sect. 9.2 by direct computation.
- 22.
More generally, with \(\alpha \in \mathbb {R}\). Making \(\alpha \rightarrow \alpha +2\) does not modify the periodic boundary conditions of \(\boldsymbol{\lambda }\), Eq. (5.65), but changes the reference Fermi sea as \(|\alpha /2\rangle \rightarrow |\alpha /2+1\rangle \) which modifies both the energy level \(h-c/24\) and the U(1) charge J. The map \(\alpha \rightarrow \alpha +2\)—which is a non-trivial isomorphism of the operator algebra—is called spectral flow..
- 23.
For \(\lambda =1/2\) the gravitational anomaly of the U(1) current cancels; cf. Sect. 2.5.
- 24.
We shall show in Sect. 6.3 that the Jacobi’s triple product identity is the mathematical statement of bosonization of massless fermions in 2d.
- 25.
The symbols in the parenthesis specify the fermion b.c. on the torus for each spin-structure; first entry periodicity along the A-cycle, second periodicity along the B-cycle, with \(A=\) anti-periodic and \(P=\) periodic.
- 26.
See also box 1.5.
- 27.
Cf. Definition 4.1.
- 28.
Times an overall length factor; the product of all these length factors over all directions produces the \(V_{10}\) in front of Eq. (5.97).
- 29.
If the target space is Lorentzian (as contrasted to Euclidean) there is an extra overall factor i since the \(k_0^2\) term has the “wrong” sign and must be Wick rotated.
- 30.
“Standard picture” corresponds to the ghosts’ sea of OCQ; cf. Sect. 3.3. In the text we are implicity using the isomorphism between the OCQ and light-cone Hilbert spaces. More in general, the covariant chirality operator is \((-1)^{F_\text {GSO}}\equiv (-1)^{\iota \cdot \lambda }\), with \(\lambda \) the SO(10, 2) weight; cf. Eq. (3.11).
- 31.
- 32.
In particular it is required to justify restriction of the integration domain to \(F_0\).
- 33.
In the presence of zero-modes, i.e. when \(\alpha =\beta =1\) the determinants are replaced by the primed determinants with the zero eigenvalues omitted, and the amplitude is non-zero only if there are enough Fermi-field insertions to soak up all zero-modes.
- 34.
\(\textrm{Tr}^\prime _\alpha \) is the trace over the \(\alpha \) sector of the open string with the zero-modes of \(X^\mu \) omitted.
- 35.
- 36.
The analysis will show that the divergence of a planar insertion is the product of some disk amplitude times a tree-level tadpole; the tadpole will vanish if and only if both \(Z_0\), \(Z_1\) are finite.
- 37.
We mean appropriate for a single bulk R-R insertion on the disk; the left/right-pictures \(q_L,q_R\in \tfrac{1}{2}+\mathbb {Z}\) satisfy \(q_L+q_R=-2\). The most canonical solution to these conditions is as in the text.
- 38.
Alternatively we may write \(A=A^{(0)}+A^{(10)}\) where \(A^{(0)}=-i *A^{(10)}\). The equation \((d-\delta )A=0\) then yields the two equivalent conditions \(d*A^{(10)}= dA^{(0)}=0\).
- 39.
Since the zero-momentum R vacua are not BRST trivial (see Chap. 3).
- 40.
The crucial aspect here is the fact that we emphasized in Chap. 3: BRST cohomology at zero momentum is not the \(k_\mu \rightarrow 0\) limit of the non-zero momentum BRST cohomology which, in turn, is isomorphic to the light-cone Hilbert space.
- 41.
In the standard jargon of string theory, one may call (5.135) a “Chern–Simons” coupling.
- 42.
The torus does not contribute since the amplitude is finite, see Remark 5.4.
- 43.
The overall minus sign reflects (from one-loop open channel viewpoint) that R sector open string states are fermions.
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Cecotti, S. (2023). 10d Superstring Theories. In: Introduction to String Theory. Theoretical and Mathematical Physics. Springer, Cham. https://doi.org/10.1007/978-3-031-36530-0_5
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