Transcript The RR charge of orientifolds Michigan Conference on Topology and Physics, Feb.

The RR charge of orientifolds
Michigan Conference on Topology and Physics,
Feb. 7, 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 string 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 string theory 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
!
X
theory of integration
over
a space of maps:
':§
S: 2d Riemannian surface
X: Spacetime endowed with geometrical
structures: Riemannian,…
exp[¡
Z
§
1
2
k d' k + ¢ ¢ ¢ ]
2
What is an orbifold?
!
':§
X
Let’s warm up with the idea of a string theory orbifold
X is smooth with ¯nite isometry group ¡
Gauge the ¡-symmetry:
Principal
G bundle
Physical
worldsheet
~
§
#
§
~
'
!
'
!
X
#
X
Spacetime
groupoid
X
= X== ¡
~ is oriented,
For orientifolds, §
!
In addition: 1 ! ¡0 ! ¡!Z2 ! 1
¡0: !(°) = +1 ¡1: !(°) = ¡1
¡0: Orientation preserving
~
On §: ¡ : Orientation reversing
1
~
§
#
Orientation
double cover
Unoriented
^
§
#
§
~
'
!
^
'
!
'
!
X
#
X== ¡0
#
X== ¡
Space
time
X is ais groupoid
More In
generally,
spacetime
an ``orbifold,’’
particular,
(c.f. Adem, Leida, Ruan, Orbifolds and Stringy Topology)
'
^
X
^ !
§
w
#
#
¼
^
!
X
'
§
w 2 H 1 (X ; Z 2 )
»
' (w) = w1 (§)
¤
There is an isomorphism
Definition: An orientifold is a string theory
defined by integration over such maps.
Orientifold Planes
For orientifold spacetimes X== ¡
a component of the fixed locus point of
2
g ¡1
is called an ``orientifold plane.’’
Worldsheet Measure
In string theory we integrate over ``worldsheets’’
For the bosonic string, space of ``worldsheets’’ is
S = f(§; ')g = Moduli(§) £ MAP(§ ! X)
R
1
§=S 2
2
k d' k ] ¢ AB
R
¤
AB = exp[2¼i §=S ' (B)]
exp[¡
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 E is a generalized cohomology
theory, then denote the
di®erential version E·
0 ! E j¡1(M; R=Z) ! E·j (M) ! -Z(M; E(pt) - R)j ! 0
j
j
·
0 ! [top:triv:] ! E (M) ! E (M) ! 0
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
E
The orientation twisting of (M ),
denoted ¿E (M )
allows us to define an ``integration map’’
RE
M
in E-theory:
E
!
E
(M
)+j
¿
j (pt)
: E
(M)
The B-field is a local geometric object,
For the oriented bosonic string its
gauge equivalence class is in
For a bosonic string orientifold
its equivalence class is in
R H·
X
·
3
H ( )
X
·
3+w
H
( )
1
·
·
AB = exp[2¼i §=S ' (¯)] 2 H (S)
¤
Integration makes sense because '¤(w) »
= w1(§)
Surprise!! For superstrings: not correct!
Superstring Orientifold B-field
Turns out that for superstring orientifolds
· 3+w (X ) ! B·3+w (X ) ! H 0 (X ; Z) £ H 1 (X ; Z2 ) ! 0
0!H
Necessary for worldsheet theory:
c.f. Talk at Singer85 (on my homepage)
That’s all for today about question (a)
G
Question b: What is RR Charge?
Type II strings have ``RR-fields’’ –
Abelian gauge fields whose fieldstrengths
are forms of fixed degree in
¤
X
¡1
R
( ; [u; u ])
deg(u) = 2
e.g. in IIB theory degree = -1:
¢
¢
¢
¡
¡
= u 1 G1 + u 2 G3 +
+
¡
u 5 G9
Naïve RR charge
In string theory there are sources of
RR fields (e.g. D-branes):
dG = j = RR current
Naively: [j] 2 H ¤
deRham
= RR charge
This notion will need to be refined…
Orientifold Planes Are Also Sources
for RR Fields
Recall that for X== ¡
2
a
component
g ¡1 of a fixed point locus of
is called an ``orientifold plane.’’
Computing the RP2 amplitude shows that
orientifold planes are sources of RR charge.
Our goal is to define that charge and
compute it as far as possible.
K-theory quantization
The D-brane construction implies
2
X
[j] K( )
Minasian & Moore
· X)
So RR current naturally sits in K(
( X ! X is a nontrivial generalization)
KR and Orientifolds
Action of worldsheet parity on ChanPaton factors
For orientifolds replace
K(X ) ! KR(Xw )
(Witten; Gukov; Bergman,Gimon,Sugimoto; Brown&Stefanski,…)
X
What is KR( w ) ?
X
For = X== ¡ use Fredholm model
(Atiyah, Segal, Singer )
H: Z2-graded Hilbert space with stable ¡-action
¡0: Is linear
¡1: Is anti-linear
F: Skew-adjoint odd Fredholms
X
!
F
KR( w ) = [X
]¡
· X )
Assume all goes well for KR(
w
Superstring B-field = Twisting of KR
We will interpret the B-field as a differential
twisting of differential KR theory. The twistings
live in a (2-) groupoid, just like B.
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 picture
Double-Covering
Groupoid
X
Spacetime
is a groupoid:
X: X
0
X1
Homomorphism:
X2
²w : X1 ! Z2
²w (gf ) = ²w (g) + ²w (f )
Xw;1 := ker²w
X
De¯nes w
Double cover:
Twisting KR Theory
Def: A twisting of KR(Xw ) is a quadruple ¿ = (d; L; ²a ; µ)
Degree d : X0 ! Z
L is a line bundle on X1 Z2 -grading: ²a : X1 ! Z2
Cocycle:
µg;f : ²w (f ) Lg
(
V
² V :=
V¹
²=0
²=1
Lf ! Lgf
Twistings of KR
Topological classes of twistings of KR(X )
H 0 (X ; Z) £ H 1 (X ; Z2 ) £ H 3+w (X ; Z)
d
²a
(L; µ)
Abelian group structure:
(d1 ; a1 ; h1 ) + (d2 ; a2 ; h2 )
~ a ))
= (d1 + d2 ; a1 + a2 ; h1 + h2 + ¯(a
1 2
The Orientifold B-field
So, the B-field is a geometric object
whose topological class is
2
£
£
0
1
[¯] = (d; a; h) HZ HZ
2
H w+3
Z
d=0,1 mod 2: IIB vs. IIA.
a: Related to (-1)F & Scherk-Schwarz
h: is standard
Bott
Periodicity
X
For = pt== Z
2
H 0 (X ; Z) £ H 1 (X ; Z2 ) £ H 3+w (X ; Z)
» ©
= Z Z4
We refer to these as ``universal twists’’
B = d + ¯`
2
Bott element u KR2+¯1 (pt)
»
©
h
i
Note (Z Z4 )= (2; 1) = Z8
Twisted KR Class
1. Z2 graded E ! X0 with odd skew Fredholm
with graded C`(d) action
2. On X1 we have gluing maps:
!
f : x0
x1
Ãf : ²w (f ) (Lf
E x ) ! Ex
0
3. On X2 we have a cocycle condition:
f
x0
gf
x1
g
x2
1
Where RR charge lives
2
X
·
·
·
[j] KR¯ ( w )
The RR current is
The charge is an element of
KR¯ (Xw )
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 )
2
X
¡
·
·
d
1
[A] H
( )
X
dim = n
dF = Jm
d ¤ F = Je 2
Self-dual setting:
2
d (X )
n+2¡d (X )
¤
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=3
3. Type II RR fields: GCT = K and n=10
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
E ¿ (M; R=Z) £ E ¿E (M)¡¿ ¡s (M) ! R=Z
!
degree
¿
»
E
!
¡
0
s
R
Z
R
Z
¶:
(pt; = )
I (pt; = ) = R=Z
For a spacetime X of dimension n
2
E
X
·
·
¿· ( )
[j]
2. Families of Spacetimes
·j
X
P
dim = = n
dim Y =P = n + 1
Z
P
dim = = n + 2
P
3. Isomorphism of
Electric
& Magnetic
Currents
E
X
!
E
·¿ (X )¡¿ +2¡s (X )
µ0 : ·¿ ( )
µ1 : E·¿ (Y ) ! E·¿ (Y )¡¿ +1¡s (Y )
µ2 : E·¿ (Z ) ! E·¿ (Z )¡¿ ¡s (Z )
4. Symmetric pairing of currents:
RE
b0 (·j1 ; ·j2 ) = ·¶ X µ0 (·j1 )·j2 2 I·2 (P )
RE
b1 (·j1 ; ·j2 ) = ·¶ Y µ1 (·j1 )·j2 2 I·1 (P )
RE
b2 (·j1 ; ·j2 ) = ·¶ Z µ2 (·j1 )·j2 2 I·0 (P )
5. Quadratic Refinement
¡
¡
·
·
·
qi (j1 + j2 ) qi (j1 ) qi (·j2 ) + qi (0) = bi (·j1 ; ·j2 )
line bundles
q0 (·j) 2 I·2 (P ) = Z2 -graded
P
over
with connection
2
P
P
·
·
1
q1 (j) I ( ) = M ap( ; R=Z)
q2 (·j) 2 I·0 (P ) = M ap(P ; Z)
Formulating the Theory
Using these data one can
formulate a self dual theory.
The topological data of q2 and µ2
in (n + 2) dimensions
determines the remaining ones
Physical Interpretation: Holography
Generalizes the well-known example of the
holographic duality between 3d abelian ChernSimons 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
· 2 E·¿ (Y ): Chern-Simons gauge ¯eld.
[A]
· 2 M ap(P ; R=Z): Chern-Simons action.
q1 (A)
X
Y
Edge modes = self-dual gauge ¯eld
A·jX = ·j
Chern-Simons wavefunction= Self-dual partition function
ª(A·jX ) = Z(·j)
Definition of the Background
Charge
Identify automorphisms of ·j with ® 2 E ¿ ¡2 (X ; R=Z)
Identify these with global gauge transformations
Gauge group acts on CS wavefunction
(® ¢ ª)(·j) = e2¼iq1 (·j+®· t·) ª(·j)
® ! q (·
®t·) is linear
1
q1 ( ®
· t·) = ¶
¹ 2 E ¿ (X )
RE
X
S1
µ(¹)®
``background charge’’
X
Computing Background Charge
A simple argument shows that
twice the charge is computed
by
R
¡
¡
q1 (y) q1 ( y) =
y = ®t
Heuristically:
q1 (y) =
1 (y
2
E
X
¡
µ0 ( 2¹)®
¡ ¹)2 + const:
Self-Duality for Type II RR Field
2
Z
Z
P
Now j K( ) and dim = = 12
It turns out that R
q2 (j) =
KO
Z
¹jj 2 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
2
Z
Z
P
¯
Now j KR ( ) and dim = = 12
We want to makeRsense of a formula like
q2 (j) =
But:
KO
Z
¹jj 2 Z
Z
¹
¹jj 2 KR¯+¯
( )
Need to interpret this in KO
The real lift
Lemma: There exists maps
< : Twist (M ) ! Twist (M)
KR
w
KO
½ : KR¯ (Mw ) ! KO <(¯) (M)
So that under complexification:
! ¡ ¹
½(j)
u
d jj
<(¯) ! ¯ + ¯¹ ¡ d¿ (u)
Twisted Spin Structure
In order to integrate in KO,
½(j) 2 KO<(¯) (M)
Must be an appropriately twisted density
For simplicity now take M = M== Z2
R
KOZ
2
M
: KO
¿KO
Z2
Z2
(M )+j
! KOj (pt)
Z2
Twisted Spin Structure- II
De¯nition: A twisted spin structure on M is
»
<
M
¡
M
· : (¯) = ¿KO (T
dim )
Note: A spin structure on M allows us to integrate
» isomorphism ¡
in KO. It is an
0 = ¿KO (T M
Existence of tss
dim M )
Topological
conditions on B
Orientifold Quadratic Refinement
R
KO
Z
¡12
2
·½(j) KO
(pt)
¡12
KO (pt)
Z
Z2
»
¡4 (pt)
KO
=
2
¶:
KO¡4 (pt)
©
Z
Z")
(
! I 0 (pt) »
=Z
£ R
¤
De¯nition: q2 (j) := ¶ ZKO ·½(j)
"
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) =
RE
X
¡
µ0 ( 2¹)®
© R
ª
R
¶ KOZ2 [½(y)) ¡ ½(¡y)] = ¶ KR µ(¡2¹)y
Y
"
Y
Localize wrt S = f(1 ¡ ")n g ½ R(Z2 )
Atiyah-Segal localization theorem
Background charge with 2 inverted
localizes on the O-planes.
K-theoretic O-plane charge
2
¹ = i¤ (¤) KR¯ [ 1 ](X)
2
!
i : F , X º= Normal bundle
ª(¤) = 2d
``Adams Operator”
C(F )
Euler(º)
ª : KR¯ [ 1 ](Y ) ! S ¡1 KOZRe(¯) (Y )
2
2
C(F )
KR-theoretic Wu class generalizing
Bott’s cannibilistic class
Special case: Type I String
FreedX& Hopkins (2001)
Type I theory:
= X== Z2
With Z2 acting trivially and ¯ = 0
2¹ = ¡¥(X)
¥(F ) :
R
KO
F
KO-theoretic
Wu class
R
Ã2 (x) =
KO
F
¥(F )x
¡¹ = T X + 22 + F ilt(¸ 8)
The physicists’ formula
Taking Chern characters we get the physicist’s
(Morales-Scrucca-Serone) formula for the charge in
de Rham cohomology:
q
q 0
L (T F )
p¡4
^
¡ A(T X)ch(¹) = §2 ¶¤
L0 (º)
0
L (V ) =
Q
xi =4
i tanh(xi =4)
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
X
w1 ( ) = dw
X
w2 ( ) =
d(d+1) w 2
2
+ aw
[¯] = (d; a; h)
This general result unifies scattered older
observations in special cases.
Examples
Zero B-field
If [b]=0 then we must have IIB theory on
X which is orientable and spin.
Op-planes
X = Rp+1 £ Rr == Z
2
p+r =9
w1 (X ) = rw
w2 (X ) =
Compute:
d = rmod2
(
0
a=
w
r(r¡1) w 2
2
r = 0; 3 mod 4
r = 1; 2 mod 4
Pinvolutions
X
= X== Z2
Deck transformation ¾ on X lifts to P in¡ bundle.
r = cod. mod 4 of orientifold planes
Older Classification
(Bergman, Gimon, Sugimoto, 2001)
Orientifold Précis : NSNS
Spacetime
1. X : 10-dimensional Riemannian orbifold with dilaton.
2. Orientifold double cover Xw , w 2 H 1 (X ; Z2 ).
X
·
3. B: Di®erential twisting of KR( w )
4. Twisted spin structure:
»
<
X
¡
X
· : (¯) = ¿KO (T
dim )
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
¹ = i¤ (¤) ª(¤) =
inverting two, and
4. Well-defined spacetime fermions and
couplings to RR fields.
5. Possibly, new NSNS solitons.
2d C(F )
Euler(º)
Conclusion
The main future direction is in applications
• Destructive String Theory?
•Tadpole constraints (Gauss law)
•Spacetime anomaly cancellation
• S-Duality Puzzles