Orientifolds & Twisted KR Theory

Background charge, over the integers

Jacques Distler

University of Texas at Austin

Why Orientifold?

Like orbifolds: procedure for generating new string theory backgrounds from old.

Has proven very useful for model building: IIB (and IIA) orientifold, “with fluxes” (KKLT et al …).

Evades the Gibbons-Maldacena-Nuñez Theorem because orientifold fixed-planes have negative tension, violating the SEC.
We think we understand moduli stabilization in this context.

Key feature: orientifolds have “background” D-brane charge. On a compact space, must be cancelled for consistency (Gauss’s Law).

Question

Is it possible to cancel the background charge rationally, but not cancel it over $ℤ$ ? I.e., could the uncancelled charge be pure torsion?

Sure. But to answer the question, need to know what the background charge is over $ℤ$ .

So far, it has only ever been computed rationally (modulo torsion). Joint work with Dan Freed and Greg Moore to rectify that.

N.b.: This is a topological question. Do not assume unbroken SUSY, or even a solution.

Subsidiary objective: clean up a bit of the bestiary of Orientifolds.

Bottom Line

Background orientifold charge, $μ$ (or $\overset{ˇ}{μ}$ ), takes values in twisted (differential) KR theory.
Only after inverting 2 does the formula for the background charge localize to the f.p.s.
Taking Chern characters, can compare with existing formulæ in the literature

$(d + u^{- 1} H \land) G = \underset{D-branes}{\underset{︸}{\overset{ˇ}{j}}} + \underset{orientifold}{\underset{︸}{\overset{ˇ}{μ}}}$
(d+ u^{-1} H\wedge) G = \underset{\text{D-branes}}{\underbrace{\check{j}}} + \underset{\text{orientifold}}{\underbrace{\check{\mu}}}

N.b.: putting the $H \land G$ term on the RHS, as is often done is morally wrong!

Orientifold

	IIB	IIA
$X$ 10-manifold	$\begin{aligned} w_{1} (X) = 0 & X oriented \\ w_{2} (X) = 0 & X spin \end{aligned}$	$w_{1}^{2} (X) + w_{2} (X) = 0 X {pin}^{-}$
$σ : X ⟲$	orientation-preserving, with lift to Spin	lift to ${Pin}^{-}$

Let $F$ be f.p.s. of $σ$ (could be all of $X$ , or could be empty).

A priori:: $σ$ orientation-preserving $\Rightarrow$ $codim (F) = even$ .; $σ$ orientation-reversing $\Rightarrow$ $codim (F) = odd$ .
In fact:: $\exists$ lift to $Spin$ or to ${Pin}^{-}$ $\Rightarrow codim (F)$ well-defined $mod (4)$ .

Accompany $σ$ by reversal of orientation on worldsheet.

Generalization

Let $Γ$ be a group of symmetries acting on $X$ (lifting to $Spin$ or ${Pin}^{-}$ ) and let

Ω : Γ \to ℤ_{2}

be a homomorphism. Can “orientifold by $Γ$ ”, where elements of $Γ$ which map nontrivially under $Ω$ are accompanied by reversal of orientation on worldsheet.

N.b.: $Γ$ can be nonabelian, even if the “orbifold subgroup,” $H = \ker Ω \subset Γ$ is abelian.

Abelian Gauge Theory

RR fields in String Theory: an exotic type of abelian gauge theory.

Recall electromagnetism.

$j_{e} \in H_{cpt}^{d - 1} (X)$ , $j_{m} \in H_{cpt}^{3} (X)$ .
Electromagnetic field trivializes these: $d F = j_{m}, d * F = j_{e}$ .
In quantum theory, need Dirac quantization condition.

$\begin{aligned} 𝒳 \\ ↓ X \\ S \end{aligned}$

RR field: replace de Rham cohomology by generalized cohomology theory, $E^{•}$ . Dirac condition: integrality of bilinear form

b (\cdot, \cdot) : E^{•} \times E^{•} \to H^{2} (S)

Self-Duality

In self-dual theory, electric current determines magnetic current (and vice versa).

j_{E} = θ (j_{M})

where $θ : E^{•} \to E^{d + 2 - •}$ is an isomorphism.

To define the quantum theory, need a quadratic refinement, $q (\cdot)$ , of the bilinear form

b (x_{1}, x_{2}) = q (x_{1} + x_{2}) - q (x_{1}) - q (x_{2}) (q (0) = 0)

The Center

A quadratic function is symmetric about its center. Let $ψ (x) = q (x) - q (- x)$ , a linear function. Since $b (\cdot, \cdot)$ is a perfect pairing, $ψ (x) = b (λ, x)$ for some $λ$ .

q (λ) = q (0) = 0

The center of $q$ is $μ$ such that $2 μ = λ$ . This determines $μ$ up to 2-torsion. Can do better, but this will suffice for today’s lecture.

Type I is an orientifold of IIB (with trivial $σ$ action). The charge group is ${KR}^{0} (X) = {KO}^{0} (X)$ . Freed and Hopkins wrote down $q (\cdot)$ and computed its center

- μ = T + 22

which is, indeed, the background charge of this orientifold. (Compare with heterotic dual.)

Objective: generalize this.

Differential Cohomology

${\overset{ˇ}{H}}^{q} (X)$ : add geometrical data to topological data in $H^{q} (X, ℤ)$ .

Examples:

${\overset{ˇ}{H}}^{0} (X) = H^{0} (X, ℤ)$
${\overset{ˇ}{H}}^{1} (X) = Maps (X, S^{1})$ : circle-valued scalar, $φ$ . $ω = d φ$ closed 1-form, de Rham representative of a class in $H^{1} (X, ℤ)$ .
${\overset{ˇ}{H}}^{2} (X) = {isom (L, \nabla)}$ : $c (L) \in H^{2} (X, ℤ)$ , the first Chern class of $L$ . $F = d A$ a de Rham representative of $c (L)$ .
etc.

Exact Sequences

More generally, the differential cohomology groups, ${\overset{ˇ}{H}}^{q} (X)$ , fit into the exact sequences

\begin{matrix} 0 \to \overset{flat}{\overset{︷}{H^{q - 1} (X, ℝ / ℤ)}} \to {\overset{ˇ}{H}}^{q} (X) \to Ω_{closed, ℤ}^{q} (X) \to 0 \\ 0 \to \underset{topologically trivial}{\underset{︸}{Ω^{q - 1} (X) / Ω_{closed, ℤ}^{q - 1} (X)}} \to {\overset{ˇ}{H}}^{q} (X) \to H^{q} (X, ℤ) \to 0 \end{matrix}

N.b.: $Ω_{closed, ℤ}^{q - 1} (X)$ is the group of gauge transformations.

Differential K-Theory

Similarly, for ${\overset{ˇ}{K}}^{q} (X)$ .

\begin{matrix} 0 \to K^{q - 1} (X, ℝ / ℤ) \to {\overset{ˇ}{K}}^{q} (X) \to \overset{currents}{\overset{︷}{Ω_{closed, ℤ}^{q} (X, R)}} \to 0 \\ 0 \to \underset{RR field strength}{\underset{︸}{Ω^{q - 1} (X, R) / Ω_{closed, ℤ}^{q - 1} (X, R)}} \to {\overset{ˇ}{K}}^{q} (X) \to K^{q} (X) \to 0 \end{matrix}

where $R = K^{•} (pt) = ℝ [u, u^{- 1}], \deg (u) = 2$ and

0 \to K^{q} (X, ℝ) / K^{q} (X) \to K^{q} (X, ℝ / ℤ) \to K_{tors}^{q + 1} (X) \to 0

K Theory

Will help to have a specific model in mind.

Usually think of $K^{0}$ as represented by formal differences of vector bundles $E_{0} ⊝ E_{1}$ . Instead, consider $ℤ_{2}$ -graded vector bundles $E = E_{0} \oplus E_{1}$ with an odd, skew-adjoint endomorphism, $T$ (requires a Hermitian metric on $E$ — a contractible choice).

Some applications will require $E$ $∞$ -dimensional and $T$ Fredholm.

${Cliff}_{n}^{\pm}$ (real Clifford algebra):: $γ_{i} γ_{j} + γ_{j} γ_{i} = \pm 2 δ_{i j}, i, j = 1, \dots, n$
$K^{\pm n}$ :: a pair, $(E, T)$ , as before, carrying a left ${Cliff}_{n}^{\pm}$ action, which graded-commutes with $T$ (the $γ_{i}$ are odd).

KR Theory

KR theory:: $σ : X ⟲$ . Consider complex, $ℤ_{2}$ -graded vector bundles, $E \to X$ , s.t. $σ$ lifts to an even, antilinear action on fibers, which commutes with $T$ and with the Clifford action. K-theory of such gadgets is called KR theory.

N.b., if $σ$ acts trivially, this is just an antilinear involution for the fibers, i.e. a real structure $\Rightarrow KO theory$ .

Equivariant K-theory, $K_{G}^{•} (X)$ :: $G$ acts on $X$ , lifts to an even linear action on $E$ , which commutes with $T$ and with the Clifford action.
Hybrid, ${}^{Ω}K_{Γ}^{•}$ :: $Ω : Γ \to ℤ_{2}$; lift of $γ \in Γ$ is even-linear or -antilinear, depending on whether $Ω (γ)$ is nontrivial.

Twisted KR Theory

Freed-Hopkins-Teleman model for twisting K-theory (and its variants).

Isomorphism classes of twistings of KR given by the central extension

0 \to H_{ℤ_{2}}^{3} (X, \tilde{ℤ}) \to [{Twist}_{KR} (X)] \to H_{ℤ_{2}}^{1} (X, ℤ_{2}) \to 0

$H_{ℤ_{2}}^{3} (X, \tilde{ℤ})$ are B-fields; $H_{ℤ_{2}}^{1} (X, ℤ_{2})$ is a local version¹ of $(- 1)^{F_{L}}$ .

Under tensor product of twisted bundles, twistings add with the law

(η_{1}, κ_{1}) + (η_{2}, κ_{2}) = (η_{1} + η_{2}, κ_{1} + κ_{2} + \tilde{β} (η_{1} \cup η_{2}))

where $(η_{i}, κ_{i}) \in H_{ℤ_{2}}^{1} (X, ℤ_{2}) \times H_{ℤ_{2}}^{3} (X, \tilde{ℤ})$ and $\tilde{β}$ is the Bockstein $\tilde{β} : H_{ℤ_{2}}^{2} (X, ℤ_{2}) \to H_{ℤ_{2}}^{3} (X, \tilde{ℤ})$ associated to $0 \to \tilde{ℤ} \overset{\times 2}{\to} \tilde{ℤ} \to ℤ_{2} \to 0$

There are “universal” twistings pulled back from $H_{ℤ_{2}}^{3} (pt, \tilde{ℤ}) ⋊ H_{ℤ_{2}}^{1} (pt, ℤ_{2})$ . Depending on $codim (F) mod 4$ , only 2 of 4 are consistent, and lead to ${O p}^{\pm}$ .
↩

Integration

$\begin{aligned} 𝒳 \\ ↓ X \\ S \end{aligned}$

Integration in equivariant KO-theory given by Dirac Index (for families).

\int_{X} : {KO}_{ℤ_{2}}^{n} (𝒳) \to {KO}_{ℤ_{2}}^{n - \dim X} (S)

I’ll take a shortcut: consider 12-manifold (2-parameter family). So we map ${KO}_{ℤ_{2}}^{0} \to {KO}_{ℤ_{2}}^{- 12} (pt) = {KO}^{- 4} (pt) \otimes RO (ℤ_{2}) = ℤ \oplus ϵ ℤ$

Quadratic Refinement

Let $x \in {KR}^{0 + τ} (X)$ (IIB) or $x \in {KR}^{1 + τ} (X)$ (IIA). Can lift $x \bar{x}$ to ${KO}_{ℤ_{2}}^{0} (X)$ . Then we integrate over a 12-manifold, and pick off the coefficient of $ϵ$ in the result.

q (x) = {(\int_{𝒳} x \bar{x}) ∣}_{ϵ}

The center (or, rather, twice the center) can be computed from

ψ (x) = {(\int_{𝒳} ({(x \bar{x})}_{+} - {(π x \bar{π x})}_{+})) ∣}_{ϵ}

Localization

After inverting 2, the formula for $ψ (x)$ localizes to the f.p.s, $F$

ψ (x) = \int_{F}^{KO [1 / 2]} \frac{2 ψ_{2} (x ∣_{F})}{Δ (ν)}

where $Δ (ν)$ is the spinor bundle of the normal bundle, and $ψ_{2} (V) = {Sym}^{2} V ⊝ \land^{2} V$ is the Adams operation.

In $KO [1 / 2]$ , $ψ_{2}$ has an inverse, $ψ_{1 / 2}$ ,¹ with which we can write

ψ (x) = 2 \int_{F}^{KO} ψ_{2} (\frac{x ∣_{F}}{ψ_{1 / 2} (Δ (ν))})

Use splitting principle and the fact that $ψ_{2}$ is a ring homomorphism. $ψ_{2} (L) = L^{2}$ . Let $L = 1 + x$ , with $x$ nilpotent. (Note: only powers of 2 in the denominators!)

$ψ_{1 / 2} (L) = (1 + x)^{1 / 2} = 1 + \frac{1}{2} x - \frac{1}{8} x^{2} + \frac{1}{16} x^{3} - \dots$
\psi_{1/2}(L) = (1+x)^{1/2} = 1 + \tfrac{1}{2} x - \tfrac{1}{8} x^2 + \tfrac{1}{16} x^3 - \dots
↩

Cannibalistic Class

The Bott Cannibalistic Class is the $KO$ analogue of the Wu class

\int_{M}^{KO [1 / 2]} ψ_{2} (y) = \int_{M}^{KO [1 / 2]} y \cup ρ (M)

It can be written as (at least, for $M$ even-dim and spin)

ρ (M) = \prod_{i = 1}^{\dim (M) / 2} (l_{i}^{1 / 4} \oplus l_{i}^{- 1 / 4}) = ψ_{1 / 2} (Δ (M))

where we used the splitting principle

TM \otimes ℂ = \oplus_{i = 1}^{\dim (M) / 2} (l_{i} \oplus l_{i}^{- 1})

The Background Charge (after inverting 2)

Using this, we can finally write ( $i : F ↪ X$ )

μ = - i_{*} μ_{F}, μ_{F} = ψ_{1 / 2} (\frac{Δ (F)}{Δ (ν)})

Note that, even though, in the derivation I presented, I assumed that $F$ was spin, the ratio, $Δ (F) / Δ (ν)$ , makes sense even when neither the numerator nor the denominator makes sense separately (say, because $F$ is only ${Spin}_{ℂ}$ ).

To compare with the existing formulæ we take Chern characters. The coupling to the RR “connection” is

\begin{aligned} \int_{X} C \land Ch (μ) \sqrt{\hat{A} (X)} & = \int_{F} i^{*} C \land \frac{Ch (μ_{F}) i^{*} \sqrt{\hat{A} (X)}}{\hat{A} (ν)} \\ = \int_{F} i^{*} C \land Ch (μ_{F}) \sqrt{\frac{\hat{A} (F)}{\hat{A} (ν)}} \end{aligned}

Scruca-Serone

It’s now a very pretty little computation¹, using the splitting principle, to check that this is

- 2^{5 - codim (F)} \int_{F} i^{*} C \land \sqrt{\frac{L (R_{F} / 4)}{L (R_{ν} / 4)}}

which agrees with the standard formulæ that you find in the physics literature.

You’ll need the characteristic polynomials $\hat{A} (R) = \prod \frac{x_{i} / 2}{\sinh (x_{i} / 2)}$ and $L (R) = \prod \frac{x_{i}}{\tanh (x_{i})}$ .
↩

So What?

After inverting 2, I presented you with a pretty nice, computable formula for the background charge

μ = - i_{*} (ψ_{1 / 2} (\frac{Δ (F)}{Δ (ν)}))

Though we’ve lost 2-torsion, there are still interesting examples (with 3-torsion, etc) where one can compute this explicitly.

Can be generalized to cases with non-torsion flux ( $H \neq 0$ rationally). “RR flux” is not a separate issue. From our point of view:

You prescribe $H$ (which determines what twisted KR theory we need).
You compute $\overset{ˇ}{μ}$ .
You prescribe $\overset{ˇ}{j}$ .
If $μ + j$ is trivial in the relevant twisted KR group, then you can solve for the RR field. Multiple solutions ⇔ “choices of RR flux”
If it’s nontrivial, then no solution for $G$ (over $ℤ$ !). The background is inconsistent.

Destructive String Theory

As I alluded, these consideration may, in fact, rule out some previously proposed string backgrounds.

If not, it certainly should be considered as a check on newly proposed ones.

There are a number of variations on the basic orientifold construction ( $O p^{+}, {\tilde{O p}}^{-}, {\tilde{O p}}^{+}, \dots$ ) to which we should extend our results

The rubric of twisted (differential) KR-theory seems to be a powerful organizing principle for this bestiary of orientifolds. Perhaps you’ll find some of the ideas to be useful in other contexts as well.

Orientifolds

LMS Durham Symposium, August 23, 2007.