1. The RR charge of orientifolds arXiv:0906.0795 and work in progress with Jacques Distler & Dan Freed TexPoint fonts used in EMF: AA A A A A A AA A A A A A A … Michigan Conference on Topology and Physics, Gregory Moore, Rutgers University Feb. 7, 2010

2. 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

3. Statement of the Problem That’s a complicated question a.) What is an orientifold? b.) What is RR charge? What is the RR charge of an orientifold?

4. 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.

5. Question a: What is an orientifold? Perturbative string theory is, by definition, a theory of integration over a space of maps:  : 2d Riemannian surface X : Spacetime endowed with geometrical structures: Riemannian,… ': § ! X

6. What is an orbifold? Principal  bundle Physical worldsheet Spacetime groupoid Let’s warm up with the idea of a string theory orbifold ': § ! X X i ssmoo t h w i t h¯ n i t e i some t rygroup ¡ ~ § ~ ' ! X ## § ' ! X X = X == ¡

7. Orientation double cover Unoriented Space time For orientifolds,  is oriented, In addition: ~ § ~ ' ! X ## ^ § ^ ' ! X == ¡ 0 ## § ' ! X == ¡

8. (c.f. Adem, Leida, Ruan, Orbifolds and Stringy Topology) w 2 H 1 ( X ; Z 2 ) There is an isomorphism ' ¤ ( w ) » = w 1 ( § ) More generally, spacetime is an ``orbifold,’’ X w # § ' ! X ^ § ^ ' ! ^ ¼ # Definition: An orientifold is a string theory defined by integration over such maps. I npar t i cu l ar, X i sagroupo i d

9. Orientifold Planes a component of the fixed locus point of is called an ``orientifold plane.’’ g 2 ¡ 1 F oror i en t i f o ld space t i mes X == ¡

10. Worldsheet Measure In string theory we integrate over ``worldsheets’’ B is locally a 2-form gauge potential… For the bosonic string, space of ``worldsheets’’ is

11. Differential Cohomology Theory In order to describe B we need to enter the world of differential generalized cohomology theories…

12. Variations B-field: Twisted differential cohomology RR-field: Twisted differential KR theory We will need twisted versions on groupoids Main Actors Both generalizations are nontrivial.

13. Orientation & Integration R E M : E ¿ E ( M ) + j ( M ) ! E j ( p t ) allows us to define an ``integration map’’ d eno t e d ¿ E ( M ) T h eor i en t a t i on t w i s t i ngo f E ( M ),

14. The B-field is a local geometric object, Surprise!! For superstrings: not correct! Integration makes sense because For the oriented bosonic string its gauge equivalence class is in · H 3 ( X ) For a bosonic string orientifold its equivalence class is in · H 3 + w ( X )

15. Superstring Orientifold B-field Turns out that for superstring orientifolds Necessary for worldsheet theory: c.f. Talk at Singer85 (on my homepage) That’s all for today about question (a)

16. Question b: What is RR Charge? Type II strings have ``RR-fields’’ – Abelian gauge fields whose fieldstrengths are forms of fixed degree in e.g. in IIB theory degree = -1: ­ ¤ ( X ; R [ u ; u ¡ 1 ]) d eg ( u ) = 2 G = u ¡ 1 G 1 + u ¡ 2 G 3 + ¢¢¢ + u ¡ 5 G 9

17. Naïve RR charge d G = j = RR curren t In string theory there are sources of RR fields (e.g. D-branes): N a i ve l y: [ j ] 2 H ¤ d e R h am = RR c h arge This notion will need to be refined…

18. Orientifold Planes Are Also Sources for RR Fields Our goal is to define that charge and compute it as far as possible. Computing the RP 2 amplitude shows that orientifold planes are sources of RR charge. a component of a fixed point locus of is called an ``orientifold plane.’’ g 2 ¡ 1 R eca ll t h a t f or X == ¡

19. K-theory quantization The D-brane construction implies Minasian & Moore [ j ] 2 K ( X ) S o RR curren t na t ura ll ys i t s i n · K ( X ) ( X ! X i sanon t r i v i a l genera l i za t i on )

20. KR and Orientifolds F oror i en t i f o ld srep l ace K ( X ) ! KR ( X w ) W h a t i s KR ( X w ) ? (Witten; Gukov; Bergman,Gimon,Sugimoto; Brown&Stefanski,…) Action of worldsheet parity on Chan- Paton factors

21. F or X = X == ¡ use F re dh o l mmo d e l KR ( X w ) = [ X ! F ] ¡ A ssumea ll goeswe llf or · KR ( X w )

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

23. 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

24. Double-Covering Groupoid Homomorphism: Double cover: X 0 X 1 X 2 X : ² w ( g f ) = ² w ( g ) + ² w ( f ) S pace t i me X i sagroupo i d : X w ; 1 : = k er² w D e ¯ nes X w

25. Twisting KR Theory Cocycle: D e f : A t w i s t i ngo f KR ( X w ) i saqua d rup l e¿ = ( d ; L ; ² a ; µ ) L i sa l i ne b un dl eon X 1 Z 2 -gra d i ng:² a : X 1 ! Z 2 µ g ; f : ² w ( f ) L g ­ L f ! L g f ² V : = ( V ² = 0 ¹ V ² = 1 D egree d : X 0 ! Z

26. Twistings of KR Abelian group structure: T opo l og i ca l c l asseso f t w i s t i ngso f KR ( X ) ( d 1 ; a 1 ; h 1 ) + ( d 2 ; a 2 ; h 2 ) = ( d 1 + d 2 ; a 1 + a 2 ; h 1 + h 2 + ~ ¯ ( a 1 a 2 )) d ² a ( L ; µ ) H 0 ( X ; Z ) £ H 1 ( X ; Z 2 ) £ H 3 + w ( X ; Z )

27. The Orientifold B-field So, the B-field is a geometric object whose topological class is d=0,1 mod 2: IIB vs. IIA. a: Related to (-1) F & Scherk-Schwarz h: is standard [ ¯ ] = ( d ; a ; h ) 2 H 0 Z £ H 1 Z 2 £ H w + 3 Z

28. Bott Periodicity H 0 ( X ; Z ) £ H 1 ( X ; Z 2 ) £ H 3 + w ( X ; Z ) We refer to these as ``universal twists’’ » = Z © Z 4 F or X = p t == Z 2 B = d + ¯ ` B o tt e l emen t u 2 KR 2 + ¯ 1 ( p t ) N o t e ( Z © Z 4 )= h ( 2 ; 1 ) i » = Z 8

29. Twisted KR Class x 0 x 1 x 2 f g g f 1. Z 2 gra d e d E ! X 0 w i t h o dd s k ew F re dh o l m w i t h gra d e d C ` ( d ) ac t i on 2. O n X 1 we h aveg l u i ngmaps: f :x 0 ! x 1 Ã f : ² w ( f ) ( L f ­ E x 0 ) ! E x 1 3. O n X 2 we h aveacocyc l econ d i t i on:

30. Where RR charge lives The RR current is The charge is an element of Next: How do we define which element it is? [ · j ] 2 · KR · ¯ ( X w ) KR ¯ ( X w )

31. 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?

32. Generalized Maxwell Theory (A naïve model for the RR fields ) Self-dual setting:& d i m X = n [ · A ] 2 · H d ¡ 1 ( X ) d ¤ F = J e 2 ­ n + 2 ¡ d ( X ) F = ¤ F J m = J e d F = J m 2 ­ d ( X )

33. 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 Consideration of three examples: has led to a general definition ( Freed, Moore, Segal ) We need 5 pieces of data:

34. General Self-Dual Theory: Data 1. Poincare-Pontryagin self-dual mult. GCT E ¿ ( M ; R = Z ) £ E ¿ E ( M ) ¡ ¿ ¡ s ( M ) ! R = Z d egree ! ¿ F oraspace t i me X o fd i mens i onn [ · j ] 2 · E · ¿ ( X ) ¶: E ¡ s ( p t ; R = Z ) ! I 0 ( p t ; R = Z ) » = R = Z

35. 2. Families of Spacetimes · j P d i m X = P = n d i m Y = P = n + 1 d i m Z = P = n + 2

36. 4. Symmetric pairing of currents: 3. Isomorphism of Electric & Magnetic Currents µ 2 : · E ¿ ( Z ) ! · E ¿ ( Z ) ¡ ¿ ¡ s ( Z ) b 2 ( · j 1 ; · j 2 ) = · ¶ R E Z µ 2 ( · j 1 ) · j 2 2 · I 0 ( P ) µ 0 : · E ¿ ( X ) ! · E ¿ ( X ) ¡ ¿ + 2 ¡ s ( X ) b 0 ( · j 1 ; · j 2 ) = · ¶ R E X µ 0 ( · j 1 ) · j 2 2 · I 2 ( P ) µ 1 : · E ¿ ( Y ) ! · E ¿ ( Y ) ¡ ¿ + 1 ¡ s ( Y ) b 1 ( · j 1 ; · j 2 ) = · ¶ R E Y µ 1 ( · j 1 ) · j 2 2 · I 1 ( P )

37. 5. Quadratic Refinement q 2 ( · j ) 2 · I 0 ( P ) = M ap ( P ; Z ) q 1 ( · j ) 2 · I 1 ( P ) = M ap ( P ; R = Z ) q 0 ( · j ) 2 · I 2 ( P ) = Z 2 -gra d e dl i ne b un dl es over P w i t h connec t i on q i ( · j 1 + · j 2 ) ¡ q i ( · j 1 ) ¡ q i ( · j 2 ) + q i ( 0 ) = b i ( · j 1 ; · j 2 )

38. Formulating the Theory Using these data one can formulate a self dual theory. T h e t opo l og i ca ld a t ao f q 2 an dµ 2 i n ( n + 2 ) d i mens i ons d e t erm i nes t h erema i n i ngones

39. 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

40. Holographic Formulation Y X [ · A ] 2 · E ¿ ( Y ) : C h ern- S i monsgauge ¯ e ld. q 1 ( · A ) 2 M ap ( P ; R = Z ) : C h ern- S i monsac t i on. · A j X = · j E d gemo d es = se lf - d ua l gauge ¯ e ld C h ern- S i monswave f unc t i on = S e lf - d ua l par t i t i on f unc t i on ª ( · A j X ) = Z ( · j )

41. Definition of the Background Charge Gauge group acts on CS wavefunction X S 1 ``background charge’’ I d en t i f yau t omorp h i smso f · j w i t h ® 2 E ¿ ¡ 2 ( X ; R = Z ) I d en t i f y t h esew i t h g l o b a l gauge t rans f orma t i ons ( ® ¢ ª )( · j ) = e 2 ¼ i q 1 ( · j + · ® · t ) ª ( · j ) ® ! q 1 ( · ® · t ) i s l i near q 1 ( · ® · t ) = ¶ R E X µ ( ¹ ) ® ¹ 2 E ¿ ( X )

42. A simple argument shows that twice the charge is computed by Heuristically: Computing Background Charge q 1 ( y ) = 1 2 ( y ¡ ¹ ) 2 + cons t : q 1 ( y ) ¡ q 1 ( ¡ y ) = R E X µ 0 ( ¡ 2 ¹ ) ® y = ® t

43. Self-Duality for Type II RR Field N ow j 2 K ( Z ) an dd i m Z = P = 12 q 2 ( j ) = R KO Z ¹ jj 2 Z It turns out that 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

44. Self-duality for Orientifold RR field Need to interpret this in KO N ow j 2 KR ¯ ( Z ) an dd i m Z = P = 12 We want to make sense of a formula like ¹ jj 2 KR ¹ ¯ + ¯ ( Z ) But: q 2 ( j ) = R KO Z ¹ jj 2 Z

45. The real lift Lemma: There exists maps So that under complexification: < : T w i s t KR ( M w ) ! T w i s t KO ( M ) ½: KR ¯ ( M w ) ! KO < ( ¯ ) ( M ) < ( ¯ ) ! ¯ + ¹ ¯ ¡ d ¿ ( u ) ½ ( j ) ! u ¡ d ¹ jj

46. Twisted Spin Structure Must be an appropriately twisted density In order to integrate in KO, ½ ( j ) 2 KO < ( ¯ ) ( M ) F ors i mp l i c i t ynow t a k e M = M == Z 2 R KO Z 2 M : KO ¿ KO Z 2 ( M ) + j Z 2 ! KO j Z 2 ( p t )

47. Twisted Spin Structure- II Note: A spin structure on M allows us to integrate in KO. It is an isomorphism ·: < ( ¯ ) » = ¿ KO ( T M ¡ d i m M ) D e ¯ n i t i on: A t w i s t e d sp i ns t ruc t ureon M i s Existence of tss Topological conditions on B 0 » = ¿ KO ( TM ¡ d i m M )

48. Orientifold Quadratic Refinement R KO Z ·½ ( j ) 2 KO ¡ 12 Z 2 ( p t ) D e ¯ n i t i on:q 2 ( j ) : = £ ¶ R KO Z ·½ ( j ) ¤ " ¶: KO ¡ 4 ( p t ) ! I 0 ( p t ) » = Z KO ¡ 12 Z 2 ( p t ) » = KO ¡ 4 ( p t ) ­ ( Z © Z " )

49. How about computing it? At this point we have defined the ``background RR charge’’ of an orientifold spacetime.

50. Background charge with 2 inverted localizes on the O-planes. L oca l i zewr t S = f ( 1 ¡ " ) n g ½ R ( Z 2 ) Atiyah-Segal localization theorem q 1 ( y ) ¡ q 1 ( ¡ y ) = R E X µ 0 ( ¡ 2 ¹ ) ® © ¶ R KO Z 2 Y [ ½ ( y )) ¡ ½ ( ¡ y )] ª " = ¶ R KR Y µ ( ¡ 2 ¹ ) y L oca l i za t i ono f t h ec h argeon X == Z 2

51. K-theoretic O-plane charge ª ( ¤ ) = 2 d C ( F ) E u l er ( º ) ``Adams Operator” º = N orma lb un dl e C ( F ) KR-theoretic Wu class generalizing Bott’s cannibilistic class ¹ = i ¤ ( ¤ ) 2 KR ¯ [ 1 2 ]( X ) i : F, ! X ª : KR ¯ [ 1 2 ]( Y ) ! S ¡ 1 KO R e ( ¯ ) Z 2 ( Y )

52. KO-theoretic Wu class Special case: Type I String Freed & Hopkins (2001) T ype I t h eory: X = X == Z 2 W i t h Z 2 ac t i ng t r i v i a ll yan d¯ = 0 2 ¹ = ¡ ¥ ( X ) R KO F Ã 2 ( x ) = R KO F ¥ ( F ) x ¡ ¹ = TX + 22 + F i l t ( ¸ 8 ) ¥ ( F ) :

53. The physicists’ formula Taking Chern characters we get the physicist’s (Morales-Scrucca-Serone) formula for the charge in de Rham cohomology:

54. 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 Topological Restrictions on the B-field This general result unifies scattered older observations in special cases. w 2 ( X ) = d ( d + 1 ) 2 w 2 + aw w 1 ( X ) = d w [ ¯ ] = ( d ; a ; h )

55. Examples If [  ]=0 then we must have IIB theory on X which is orientable and spin. Zero B-field Op-planes Compute: d = rmo d 2 X = R p + 1 £ R r == Z 2 w 1 ( X ) = rw w 2 ( X ) = r ( r ¡ 1 ) 2 w 2

56. Pinvolutions X = X == Z 2 D ec k t rans f orma t i on¾on X l i f t s t o P i n ¡ b un dl e. r = co d. mo d 4 o f or i en t i f o ld p l anes

57. Older Classification (Bergman, Gimon, Sugimoto, 2001)

58. Orientifold Précis : NSNS Spacetime 2. O r i en t i f o ldd ou bl ecover X w, w 2 H 1 ( X ; Z 2 ). 3. B : D i ® eren t i a l t w i s t i ngo f · KR ( X w ) 4. T w i s t e d sp i ns t ruc t ure: ·: < ( ¯ ) » = ¿ KO ( T X ¡ d i m X ) 1. X : 10 - d i mens i ona l R i emann i anor b i f o ld w i t hd i l a t on.

59. Orientifold Précis: Consequences ¹ = i ¤ ( ¤ ) ª ( ¤ ) = 2 d C ( F ) E u l er ( º ) 1. Well-defined worldsheet measure. 2. K-theoretic definition of the RR charge of an orientifold spacetime. 4. Well-defined spacetime fermions and couplings to RR fields. 5. Possibly, new NSNS solitons. 3. RR charge localizes on O-planes after inverting two, and

60. Conclusion The main future direction is in applications Destructive String Theory? Tadpole constraints (Gauss law) Spacetime anomaly cancellation S-Duality Puzzles