Type I Strings, Orientifolds, and Chan--Paton Factors

So far the superstring theories have been oriented. Closed strings have distinguishable left- and right-moving sectors, and open strings have oriented boundaries. There is another important operation: worldsheet parity. It reverses the orientation of the string worldsheet and exchanges left and right movers.

Projecting by worldsheet parity produces unoriented strings. Applied to Type IIB, and supplemented by open strings, this gives Type I string theory. The open strings carry endpoint labels called Chan—Paton factors, and their massless modes become spacetime gauge fields.

This page explains the basic projections and the origin of the Type I gauge group.

Type 0 projections

Before discussing Type I, it is useful to separate two different uses of GSO projections. Type II theories use a chiral GSO projection that removes the NS tachyon and produces spacetime supersymmetry. There are also non-supersymmetric closed-string theories called Type 0 theories.

Their sectors can be organized schematically as follows:

$$\begin{array}{ccl} \text{Type 0B} &:& (\mathrm{NS}+,\mathrm{NS}+),\ (\mathrm{NS}-,\mathrm{NS}-),\ (\mathrm{R}+,\mathrm{R}+),\ (\mathrm{R}-,\mathrm{R}-),\\ \text{Type 0A} &:& (\mathrm{NS}+,\mathrm{NS}+),\ (\mathrm{NS}-,\mathrm{NS}-),\ (\mathrm{R}+,\mathrm{R}-),\ (\mathrm{R}-,\mathrm{R}+). \end{array}$$

The $\mathrm{NS}-$ sector contains a tachyon. Therefore Type 0 theories are not supersymmetric and are perturbatively unstable around flat space. They are nevertheless useful as a clean example of how different modular-invariant projections produce different closed-string theories.

Type 0 projections keep the NS tachyon and double the R—R sector. Type 0A and Type 0B differ in how the two Ramond chiralities are paired.

Worldsheet parity

For a closed string, worldsheet parity $\Omega$ reverses the spatial coordinate on the string:

$$\Omega: \sigma\mapsto 2\pi-\sigma.$$

It exchanges left and right movers:

$$\Omega\alpha_n^\mu\Omega^{-1}=\tilde\alpha_n^\mu, \qquad \Omega\tilde\alpha_n^\mu\Omega^{-1}=\alpha_n^\mu,$$

and similarly for the fermions,

$$\Omega\psi_r^\mu\Omega^{-1}=\tilde\psi_r^\mu, \qquad \Omega\tilde\psi_r^\mu\Omega^{-1}=\psi_r^\mu,$$

up to signs that depend on the spin structure and detailed convention.

Worldsheet parity reverses the orientation of the closed string and exchanges left-moving and right-moving oscillators.

A closed-string state survives the unoriented projection only if it is invariant under $\Omega$.

Type I as an orientifold of Type IIB

Type IIB is special because its left and right sectors have the same chirality. This makes it possible to quotient by $\Omega$. The resulting closed unoriented theory keeps the $\Omega$-even states and removes the $\Omega$-odd states.

In the NS—NS sector, the projection keeps

$$G_{\mu\nu}, \qquad \Phi,$$

and removes the antisymmetric tensor

$$B_{\mu\nu}.$$

In the R—R sector, the projection keeps the two-form potential $C_2$ and removes the R—R scalar $C_0$ and the self-dual four-form $C_4$ in the simplest Type I projection. The closed-string sector is therefore ten-dimensional $\mathcal N=1$ supergravity.

But the orientifold projection also produces tadpoles. Consistency requires adding open strings ending on D9-branes. Tadpole cancellation selects the celebrated gauge group

$$SO(32).$$

Type I string theory is obtained from Type IIB by worldsheet parity, together with open strings. The perturbatively consistent ten-dimensional theory has $SO(32)$ gauge symmetry.

Thus Type I theory contains both closed and open strings. Its low-energy limit is ten-dimensional $\mathcal N=1$ supergravity coupled to $SO(32)$ super Yang—Mills theory.

Chan—Paton labels

An oriented open string can carry labels at its two endpoints:

$$i,j=1,\ldots,N.$$

A state is then written as

$$|\Psi;ij\rangle.$$

Equivalently, one attaches a matrix $\lambda$ to the vertex operator:

$$V\quad \longrightarrow \quad V_\lambda=\lambda_{ij}V.$$

The massless vector states are

$$\alpha_{-1}^\mu|0;k;ij\rangle$$

in the bosonic string, or

$$\psi_{-1/2}^\mu|0;k;ij\rangle$$

in the NS superstring. Their Chan—Paton matrices become spacetime gauge fields,

$$A_\mu(x)=A_\mu^a(x)\lambda^a.$$

Chan—Paton labels live at the endpoints of open strings. The corresponding matrices become nonabelian gauge indices in spacetime.

For oriented open strings, the natural gauge group is $U(N)$: the Chan—Paton matrices span the algebra of $N\times N$ Hermitian matrices.

Orientation reversal and $SO/Sp$ projections

For unoriented open strings, worldsheet parity also acts on the Chan—Paton labels:

$$\Omega: |\Psi;ij\rangle \longmapsto \pm |\Psi;ji\rangle.$$

More generally, one may include a unitary matrix $\gamma_\Omega$ acting on Chan—Paton space:

$$\lambda \longmapsto \pm\gamma_\Omega\lambda^T\gamma_\Omega^{-1}.$$

The sign depends on the worldsheet oscillator part of the state. The massless vector projection gives either an orthogonal or symplectic Lie algebra:

$$\mathfrak{so}(N): \quad \lambda=-\lambda^T,$$

or

$$\mathfrak{usp}(N): \quad \lambda=-J\lambda^TJ^{-1},$$

where $J$ is the symplectic form. In ten-dimensional Type I string theory, the consistent choice is $SO(32)$.

Worldsheet parity transposes Chan—Paton matrices. Depending on the orientifold action, the surviving massless vectors generate an $SO$ or $Sp$ gauge algebra.

This is the string-theoretic origin of nonabelian gauge symmetry: gauge indices are not put into the worldsheet action as local fields; they arise as labels on open-string endpoints.

Color ordering on the disk

Disk amplitudes know about Chan—Paton matrices through the cyclic order of boundary insertions. For oriented open strings, an $n$-point amplitude decomposes as

$$\mathcal A_n^{\mathrm{open}} = g_o^{n-2} \sum_{\sigma\in S_n/\mathbb Z_n} \mathrm{Tr}\left( \lambda^{a_{\sigma(1)}}\lambda^{a_{\sigma(2)}}\cdots \lambda^{a_{\sigma(n)}} \right) A_n\bigl(\sigma(1),\ldots,\sigma(n)\bigr),$$

where $A_n$ is a color-ordered partial amplitude. The trace records the order in which vertex operators appear around the disk boundary.

Open-string disk amplitudes produce cyclic traces of Chan—Paton matrices. This is the worldsheet origin of color ordering.

For unoriented strings, one must also include the reversed ordering. This is why the projection to $SO$ or $Sp$ gauge groups is naturally tied to worldsheet parity.

Gauge-boson vertices with Chan—Paton factors

The NS open-string gauge-boson vertex in the $-1$ picture becomes

$$V_A^{(-1)}(y) = g_o\,c(y)\,\lambda^a\, \epsilon_\mu\psi^\mu(y)e^{-\phi(y)}e^{ik\cdot X(y)}.$$

In the zero picture,

$$V_A^{(0)}(y) \propto g_o\,c(y)\,\lambda^a\, \epsilon_\mu \left( i\partial_yX^\mu+2\alpha'k_\nu\psi^\nu\psi^\mu \right)e^{ik\cdot X(y)}.$$

The Chan—Paton matrix is carried along as an external label. Its algebra determines the spacetime gauge group, while the worldsheet correlator determines the kinematic partial amplitude.

The Type I low-energy theory

The massless Type I spectrum is the field content of ten-dimensional $\mathcal N=1$ supergravity coupled to $SO(32)$ super Yang—Mills:

$$\begin{array}{ccl} \text{closed sector} &:& G_{\mu\nu},\ \Phi,\ C_{\mu\nu},\ \text{gravitino},\ \text{dilatino},\\ \text{open sector} &:& A_\mu^a,\ \lambda^a_{\mathrm{gaugino}},\qquad a=1,\ldots,496. \end{array}$$

The number $496$ is

$$\dim SO(32)=\frac{32\cdot31}{2}=496.$$

The same number appears in anomaly cancellation. This is one of the first signs that string consistency is much more restrictive than ordinary quantum field theory.

Exercises

Exercise 1

Show that the adjoint dimension of $SO(32)$ is $496$.

Solution

The Lie algebra $\mathfrak{so}(N)$ consists of antisymmetric real $N\times N$ matrices. The number of independent entries is the number of unordered pairs $(i,j)$ with $i<j$:

$$\dim SO(N)=\frac{N(N-1)}{2}.$$

For $N=32$,

$$\dim SO(32)=\frac{32\cdot31}{2}=16\cdot31=496.$$

Exercise 2

For an oriented open-string four-point amplitude, write the Chan—Paton factor associated with the cyclic ordering $(1,2,3,4)$.

Solution

The cyclic ordering $(1,2,3,4)$ contributes the trace

$$\mathrm{Tr}(\lambda^{a_1}\lambda^{a_2}\lambda^{a_3}\lambda^{a_4}).$$

Because the disk boundary is cyclic, this is equivalent to cyclic rotations such as

$$\mathrm{Tr}(\lambda^{a_2}\lambda^{a_3}\lambda^{a_4}\lambda^{a_1}).$$

The reversed ordering is distinct for oriented open strings but is paired by $\Omega$ in unoriented theories.

Exercise 3

Explain why the antisymmetric tensor $B_{\mu\nu}$ is projected out by the simplest Type I worldsheet-parity projection.

Solution

The NS—NS massless polarization decomposes into symmetric traceless, antisymmetric, and trace parts:

$$\zeta_{\mu\nu} = \zeta_{(\mu\nu)}^{\mathrm{traceless}} +\zeta_{[\mu\nu]} +\text{trace}.$$

Worldsheet parity exchanges left and right movers, so it transposes the two polarization indices:

$$\zeta_{\mu\nu}\longmapsto \zeta_{\nu\mu}.$$

The symmetric graviton and trace dilaton are even. The antisymmetric part changes sign, so it is odd under $\Omega$ and is projected out.

Exercise 4

Why do Chan—Paton factors produce nonabelian gauge symmetry rather than merely many copies of a $U(1)$ photon?

Solution

The open-string vertex operators carry matrices $\lambda^a$. Disk amplitudes contain ordered products and traces of these matrices:

$$\mathrm{Tr}(\lambda^{a_1}\lambda^{a_2}\cdots\lambda^{a_n}).$$

The noncommutativity of matrix multiplication produces the structure constants of a nonabelian Lie algebra. Equivalently, the massless vector field is matrix-valued,

$$A_\mu=A_\mu^a\lambda^a,$$

and its low-energy field strength contains

$$F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu+[A_\mu,A_\nu].$$

The commutator term is the hallmark of nonabelian gauge theory.