The RR charge of orientifolds

Oberwolfach June 8, 2010

Gregory Moore, Rutgers University

arXiv:0906.0795

and work… in progress with Jacques Distler & Dan Freed Outline

1. Statement of the problem & motivation 2. What is an orientifold? 3. B-field: Differential cohomology 4. RR Fields I: Twisted KR theory 5. Generalities on self-dual theories 6. RR Fields II: Quadratic form for self-duality 7. O-plane charge 8. Chern character and comparison with physics 9. Topological restrictions on the B-field 10. Precis Statement of the Problem

What is the RR charge of an orientifold?

That’s a complicated question a.) What is an orientifold? b.) What is RR charge? Motivation

• The evidence for the alleged ``landscape of vacua’’ (d=4, N=1, with moduli fixed) relies heavily on orientifold constructions.

• So we should put them on a solid mathematical foundation!

• Our question for today is a basic one, of central importance in model building.

• Puzzles related to S-duality are sharpest in orientifolds

• Nontrivial application of modern geometry & topology to physics. Question a: What is an orientifold?

Perturbative string theory is, by definition, a theory of integration over a space of maps: ϕ : Σ → X Σ: 2d Riemannian surface X: Spacetime endowed with geometrical structures: Riemannian,… 1 exp[ dϕ 2 + ] − Σ 2 k k ··· What is an ?

Let’s warm up with the idea of a string theory orbifold ϕ : Σ X → X is smooth with finite isometry Γ Gauge the Γ-symmetry: ϕ˜ Σ˜ X Principal → Spacetime Γ bundle groupoid ↓↓ϕ Σ Physical → X = X// Γ X For orientifolds, Σ˜ is oriented, ω In addition: 1 Γ0 Γ Z2 1 → → → → Γ0: ω(γ)=+1 Γ1: ω(γ)= 1 − Γ : Orientation preserving On Σ˜: 0 Γ1: Orientation reversing

ϕ˜ Σ˜ X → Orientation ↓↓ ˆ ϕˆ double cover Σ X// Γ0 → Space ↓↓ϕ Unoriented Σ X// Γ time → Orientifold Planes

For orientifold spacetimes X// Γ a component of the fixed locus point of

g Γ1 ∈ is called an ``orientifold plane.’’ More generally, spacetime is an ``orbifold,’’ In particular, is a groupoid (c.f. Adem, Leida, Ruan, andX Stringy Topology) ˆ ϕˆ Σ w → X 1 πˆ w H ( ; Z2) ↓ ϕ ↓ ∈ X Σ → X There is an isomorphism ϕ∗(w) ∼= w1(Σ)

Definition: An orientifold is a string theory defined by integration over such maps. Worldsheet Measure In string theory we integrate over ``’’ For the bosonic string, space of ``worldsheets’’ is S = (Σ, ϕ) =Moduli(Σ) MAP(Σ X) { } × → 1 2 exp[ dϕ ] B − Σ/S 2 k k ·A B =exp[2πi ϕ∗(B)] A Σ/S B is locally a 2-form gauge potential… Differential Cohomology Theory In order to describe B we need to enter the world of differential generalized cohomology theories… If is a generalized cohomology theory,E then denote the differential version ˇ E j 1 ˇj j 0 − (M,R/Z) (M) Ω (M; (pt) R) 0 → E → E → Z E ⊗ →

0 [top.triv.] ˇj (M) j (M) 0 → → E → E → Variations

We will need twisted versions on groupoids Both generalizations are nontrivial.

Main Actors

• B-field: Twisted differential cohomology

• RR-field: Twisted differential KR theory Orientation & Integration The orientation twisting of (M), E denoted τ (M) E allows us to define an ``integration map’’ in -theory: E τ (M)+j j E : E (M) (pt) M E → E Also extends to integration in differential theory Where does the B-field live? For the oriented bosonic string its Hˇ 3( ) gauge equivalence class is in X For a bosonic string orientifold Hˇ 3+w( ) its equivalence class is in X ˇ H ˇ 1 B =exp[2πi ϕ∗(β)] Hˇ (S) A Σ/S ∈

Integration makes sense because ϕ∗(w) ∼= w1(Σ) Surprise!! For superstrings: not correct! Superstring Orientifold B-field

Turns out that for superstring orientifolds

ˇ 3+w ˇ3+w 0 1 0 H ( ) ( ) H ( , Z) H ( , Z2) 0 → X → B X → X × X → Necessary for worldsheet theory: c.f. Talk at Singer85 (on my homepage) and a paper to appear soon.

That’s all for today about question (a) Question b: What is RR Charge?

Type II strings have ``RR-fields’’ – Abelian gauge fields whose fieldstrengths are forms of fixed degree in 1 Ω∗( , R[u, u− ]) deg(u)=2 X e.g. in IIB theory degree = -1:

1 2 5 G = u− G1 + u− G3 + + u− G9 ··· Naïve RR charge

In string theory there are sources of RR fields:

dG = j = RR current

Naively: [j] H∗ =RRcharge ∈ deRham This notion will need to be refined… Sources for RR Fields Worldsheet computations show there are two sources of RR charge: • D- • Orientifold planes Recall that for X// Γ a component of a fixed point locus of

g Γ1 is called an ``orientifold plane.’’ ∈ Our goal is to define precisely the orientifold plane charge and compute it as far as possible. K-theory quantization The D- construction implies

[j] K( ) Minasian & Moore ∈ X So RR current naturally sits in Kˇ ( ) X ( X is a nontrivial generalization) → X KR and Orientifolds

Action of worldsheet parity on Chan- Paton factors For orientifolds replace K( ) KR( w) X → X (Witten; Gukov; Hori; Bergman, Gimon, Horava; Bergman,Gimon,Sugimoto; Brown&Stefanski,…)

What is KR( w)? X For = X// Γ use Fredholm model X (Atiyah, Segal, Singer )

: Z2-graded Hilbert space with stable Γ-action H Γ0:Islinear Γ1:Isanti-linear : Skew-adjoint odd Fredholms F Γ KR( w):=[X ] X → F This fits well with `` condensation.’’ We need twisted KR-theory… Following Witten and Bouwknegt & Mathai, we will interpret the B-field as a (differential) twisting of (differential) KR theory.

It is nontrivial that this is compatible with what we found from the worldsheet viewpoint.

As a bonus: This point of view nicely organizes the zoo of K-theories associated with various kinds of orientifolds found in the physics literature. Twistings

• We will consider a special class of twistings with geometrical significance.

• We will consider the degree to be a twisting, and we will twist by a ``graded gerbe.’’

• We now describe a simple geometric model Double-Covering Groupoid Spacetime is a groupoid: X

: X0 X1 X2 X Homomorphism: ²w : X1 Z2 → ²w(gf)=²w(g)+²w(f) Double cover: Xw,1 := ker²w

Defines w X Twisting KR Theory

Def: A twisting of KR( w)isaquadrupleτ =(d, L, ²a, θ) X Degree d : X0 Z →

L is a line bundle on X1 Z2-grading: ²a : X1 Z2 → ² (f) Cocycle: θg,f : w Lg Lf Lgf ⊗ →

² V²=0 V := ¯ (V²=1 Twistings of KR

Topological classes of twistings of KR( w) X 0 1 3+w H ( ; Z) H ( ; Z2) H ( ; Z) X × X × X

d ²a (L, θ)

Abelian group structure:

(d1,a1,h1)+(d2,a2,h2)

=(d1 + d2,a1 + a2,h1 + h2 + β˜(a1a2)) The Orientifold B-field

So, the B-field is a geometric object whose topological class is [β]=(d, a, h) H0 H1 Hw+3 ∈ Z × Z2 × Z d=0,1 mod 2: IIB vs. IIA.

a: Related to (-1)F & Scherk-Schwarz

h: is standard Bott Periodicity For = ℘ := pt// Z2 X 0 1 3+w H (℘; Z) H (℘; Z2) H (℘; Z) = Z Z4 × × ∼ ⊕ We refer to these as ``universal twists’’

B = d + β` Bott element u KR2+β1 (pt) ∈ Note (Z Z4)/ (2, 1) = Z8 ⊕ h i ∼ So, there are 8 distinct universal B-fields Twisted KR Class

1. Z2 graded E X0 with odd skew Fredholm → with graded C`(d)action

2. On X1 we have gluing maps:

²w (f) f : x0 x1 ψf : (Lf Ex ) Ex → ⊗ 0 → 1 3. On X2 we have a cocycle condition:

x1 f g x0 gf x2 Where RR charge lives

The RR current is βˇ [ˇj] KRˇ ( w) ∈ X The charge is an element of

β KR ( w) X Next: How do we define which element it is? The RR field is self-dual

The key to defining and computing the background charge is the fact that the RR field is a self-dual theory.

How to formulate self-duality? Generalized Maxwell Theory

(A naïve model for the RR fields ) ˇ ˇ d 1 dim = n [A] H − ( ) X ∈ X d dF = Jm Ω ( ) ∈ X n+2 d d F = Je Ω − ( ) ∗ ∈ X

Self-dual setting: F = F & Jm = Je ∗ Consideration of three examples:

1. Self-dual scalar: n=2 and d=2

2. M-theory 5-brane: n=6 and d=4

3. Type II RR fields: n=10 & GCT = K has led to a general definition (Freed, Moore, Segal)

We need 5 pieces of data: General Self-Dual Theory: Data

1. Poincare-Pontryagin self-dual mult. GCT τ τ ( ) τ s ( , R/Z) E M − − ( ) R/Z E M ×E M →

s 0 degree τ ι : − (pt; R/Z) I (pt; R/Z) = R/Z → E → ∼ For a spacetime of dimension n X [ˇj] ˇτˇ( ) ∈ E X 2. Families of Spacetimes

dim / = n X P ˇj dim / = n +1 Y P dim / = n +2 Z P

P 3. Isomorphism of Electric & Magnetic Currents τ τ( ) τ+2 s θ0 : ˇ ( ) ˇ X − − ( ) E X → E X τ τ( ) τ+1 s θ1 : ˇ ( ) ˇ Y − − ( ) E Y → E Y ˇτ ˇτ( ) τ s θ2 : ( ) Z − − ( ) E Z → E Z 4. Symmetric pairing of currents:

2 b0(ˇj1, ˇj2)=ˇι E θ0(ˇj1)ˇj2 Iˇ ( ) X ∈ P ˇ ˇ ˇ ˇ ˇ1 b1(j1, j2)=ˇι E θR1(j1)j2 I ( ) Y ∈ P 0 b2(ˇj1, ˇj2)=ˇι ERθ2(ˇj1)ˇj2 Iˇ ( ) Z ∈ P R 5. Quadratic Refinement qi(ˇj1 + ˇj2) qi(ˇj1) qi(ˇj2)+qi(0) = bi(ˇj1, ˇj2) − − q (ˇj) Iˇ2( )= Z2-graded line bundles 0 over with connection ∈ P P ˇ ˇ1 q1(j) I ( )=Map( , R/Z) ∈ P P ˇ ˇ0 q2(j) I ( )=Map( , Z) ∈ P P Formulating the theory

Using these data one can formulate a self dual theory.

The topological data of q2 and θ2 in (n +2)dimensions

determines q1,q0 Physical Interpretation: Holography

Generalizes the well-known example of the holographic duality between 3d abelian Chern- Simons theory and 2d RCFT.

See my Jan. 2009 AMS talk on my homepage for this point of view, which grows out of the work of Witten and Hopkins & Singer, and is based on my work with Belov and Freed & Segal Holographic Formulation [Aˇ] ˇτ ( ): Chern-Simons gauge field. ∈ E Y ˇ q1(A) Map( , R/Z): Chern-Simons action. ∈ P

Edge modes = self-dual gauge field X Y Aˇ = ˇj |X Chern-Simons wavefunction= Self-dual partition function

Ψ(Aˇ )=Z(ˇj) |X Defining the Background Charge - I

ˇ τ 2 Identify automorphisms of j with α − ( , R/Z) ∈ E X Identify these with global gauge transformations Automorphisms act on CS wavefunction

ˇ ˇ (α Ψ)(ˇj)=e2πiq1(j+αˇt)Ψ(ˇj) · S1 Global gauge transformations:

2πiα (α Ψ)(ˇj)=e ·QΨ(ˇj) · X Defining the Background Charge - II

2πiα 2πiq1(ˇj+αˇtˇ) e ·QΨ(ˇj)=e Ψ

q1(ˇj + αˇtˇ)= θ(j)α + q1(αˇtˇ)

α q1(αˇtˇ) is linear → Poincare duality: q1(αˇtˇ):=ι E θ(μ)α X μ τ ( ) ``backgroundR charge’’ ∈ E X Computing Background Charge

A simple argument shows that twice the charge is computed by q1(y) q1( y)= E θ0( 2μ)α − − − y = αt X Heuristically: 1 2 q1(y)= (y μ) + const. 2 − Self-Duality for Type II RR Field Now [ˇj] K( ) and dim / =12 ∈ Z Z P It turns out that KO ¯ q2(j)= jj Z Z ∈ correctly reproduces many known facts in string theory and M-theory

Witten 99, Moore & Witten 99, Diaconescu, Moore & Witten, 2000, Freed & Hopkins, 2000, Freed 2001 Self-duality for Orientifold RR field

Now j KRβ( ) and dim / =12 ∈ Z Z P We want to make sense of a formula like KO ¯ q2(j)= jj Z Z ∈ But ¯jj KRβ¯+β ( ), not in KO ∈ Z And, we need a KO density! The real lift

Lemma: There exists maps

:TwistKR( w) TwistKO( ) < M → M β (β) ρ : KR ( w) KO< ( ) M → M So that under complexification: d ρ(j) u− ¯jj → (β) β + β¯ dτ(u) < → − Twisted Spin Structure - I

In order to integrate in KO,

(β) ρ(j) KO< ( ) ∈ M Must be an appropriately twisted density

For simplicity now take = M//Z2 M

KO τKO (M)+j Z2 : KO Z2 KOj (pt) M Z2 → Z2 R Twisted Spin Structure- II

Definition: A twisted spin structure on is M

κ : (β) = τKO(T dim ) < ∼ M − M Note: A spin structure on M allows us to integrate in KO. It is an isomorphism

0 = τKO(TM dim M) ∼ − Existence of tss Topological conditions on B Orientifold Quadratic Refinement

KO 12 κρ(j) KO− (pt) Z2 Z ∈ 12 4 KO− (pt) = KO− (pt) (Z Zε) Z2 ∼ ⊗ ⊕ 4 0 ι : KO− (pt) I (pt) = Z → ∼ KO Definition: q2(j):= ι κρ(j) ε Z At this point we have defined the

``background RR charge’’

of an orientifold spacetime.

How about computing it? Localization of the charge on X// Z2

q1(y) q1( y)= E θ0( 2μ)α − − X − KO KR ι Z2 [ρ(y)) ρ( y)] = ι θ( 2μ)y Y − − ε Y − © R ª n R Localize wrt S = (1 ε) R(Z2) { − } ⊂ Atiyah-Segal localization theorem Background charge with 2 inverted localizes on the O-planes. K-theoretic O-plane charge

μ = i (Λ) KRβ[ 1 ](X) ∗ ∈ 2 i : F , X ν=Normalbundle → d C(F ) Ψ(Λ)=2 Euler(ν)

β 1 1 Re(β) ``Adams Operator” Ψ : KR [ ](Y ) S− KO (Y ) 2 → Z2 C(F ) KR-theoretic Wu class generalizing Bott’s cannibilistic class Special case: Type I String Freed & Hopkins (2001)

Type I theory: = X// Z2 X With Z2 acting trivially and β =0 2μ = Ξ(X) − Ξ(F ):KO-theoretic Wu class KO KO F ψ2(x)= F Ξ(F )x Rμ = TX +22+R Filt( 8) − ≥ The physicists’ formula

Taking Chern characters we get the physicist’s (Morales-Scrucca-Serone) formula for the charge in de Rham cohomology: L˜(TF) Aˆ(TX)ch(μ)= 2kι − ± ∗s L˜(ν) q L˜(V ):= xi/4 k =dimF 5 i tanh(xi/4) − Q Topological Restrictions on the B-field One corollary of the existence of a twisted spin structure is a constraint relating the topological class of the B-field to the topology of X

d(d+1) 2 w1( )=dw w2( )= w + aw X X 2 [β]=(d, a, h)

This general result unifies scattered older observations in special cases. Examples

Zero B-field If [β]=0 then we must have IIB theory on X which is orientable and spin. Op-planes p+1 r = R R // Z2 p + r =9 X × r(r 1) 2 Compute: w1( )=rw w2( )= − w X X 2 0 r =0, 3mod4 d = rmod2 a = (wr=1, 2mod4 Pinvolutions

= X// Z2 X Deck transformation σ on X lifts to Pin− bundle.

r = cod. mod 4 of orientifold planes Older Classification

(Bergman, Gimon, Sugimoto, 2001) Orientifold Précis : NSNS Spacetime

1. : 10-dimensional Riemannian orbifold with . X 1 2. Orientifold double cover w, w H ( , Z2). X ∈ X 3. B:Differential twisting of KRˇ ( w) X 4. Twisted spin structure:

κ : (β) = τKO(T dim ) < ∼ X − X Orientifold Précis: Consequences

1. Well-defined worldsheet measure. 2. K-theoretic definition of the RR charge of an orientifold spacetime. 3. RR charge localizes on O-planes after inverting two, and d μ = i (Λ) Ψ(Λ)= 2 C(F ) ∗ Euler(ν) 4. Well-defined spacetime fermions and couplings to RR fields. 5. Possibly, new NSNS solitons. Conclusion The main future direction is in applications

• Destructive String Theory?

•Tadpole constraints (Gauss law)

•Spacetime cancellation

• S-Duality Puzzles