A-field and B-field from Freed-Witten anomaly

Transcription

1 A-field and B-field from Freed-Witten anomaly Raffaele Savelli SISSA/ISAS - Trieste Based on a work with L. Bonora and F. Ferrari Ruffino (arxiv: ) Seminal paper by Freed & Witten: hep-th/

2 Plan of the talk What is Freed-Witten anomaly and how to get rid of it? Case by case analysis of the physical consequences on the B-field and A-field on a single D-brane Comments on fractional branes, Page charge and on how these ideas open the road to K-theory Summary and outlook 2

3 Freed-Witten anomaly is a global anomaly of the open superstring path integral measure In type II theory, let X be the target space and Y be a single D-brane Given the open string embedding such that φ : Σ X φ Σ : Σ Y After integrating over all the fermionic matter fields, the path integral contains pfaff D φ exp ( 2πi Σ ) φ B exp ( 2πi Σ ) φ A sign ambiguity! it is a section of Pfaff LY holonomy over open surfaces! it is a section of L B LY can provide the right trivialization to get well-definedness 3

4 Want a well-defined function to integrate, thus: Pfaff L B must be geometrically trivial This means: can be trivialized by means of a global nowhere-zero section (of unit norm) s s can be chosen to be parallel (killed by the covariant derivative) pfaff D φ exp ( 2πi Σ φ B ) exp ( 2πi Σ φ A ) s is the function we are looking for, to be integrated over the bosonic part of the parameter space. Notice: s is defined only up to an overall constant (immaterial for the path integral) How can we construct such an s? 4

5 Some mathematical properties of Pfaff and L B c 1 (Pfaff) = moreover, the ones of pfaff(d) are in the class: call c 1 (L B ) = φ( Σ) ρ α φ( Σ) G = (g αβγ, Λ αβ, B α ) W 3 (T Y ) H 2 (LY, Z) c 1 (G) H 3 (Y, Z) such that torsion line bundle the corresponding parallel local section c 1 (G) [pfaff(d) αβ ] = φ( Σ) constant transition functions w 2 (T Y ) H 1 (LY, S 1 ) is the first Chern class of the underlying 1-gerbe: ˇδ 2 g αβγ = 0 B α B β = dλ αβ ˇδ 1 Λ αβ = (2πi) 1 d log g αβγ moreover a trivialization of G induces a trivialization of we call the local parallel one which we will induce σ α L B Anomaly cancellation occurs iff W 3 (T Y ) = c 1 (G) 5

6 Achieve geometric triviality of Pfaff L B simply requiring: For the gerbe choose ρ α σ α = s [σ αβ ] = [ρ αβ ] = [pfaff(d) αβ ] G 2 g αβγ this means: ending up with (η αβγ, 0, B + F ) constant (possible, since the gerbe must be torsion!) as a reparameterization of the B-gerbe choose the following (geometrically trivial) gerbe, thanks to the FW anomaly condition: (g 1 αβγ η αβγ, Λ αβ, da α F α ) with η αβγ σ αβ with inducing and B+F being invariant under gauge tranformations dλ Notice: anomaly could not be detected by tori, such as Σ C the condition on LY c 1 (Pfaff) = c 1 (L B ) could not be enough! We expect necessity of cohomological condition on Y W 3 (T Y ) = c 1 (G) g 1 η = ˇδ 1 h αβ [g 1 η] S 1 = [exp 2πi F ] S 1 Λ αβ = A β A α (2πi) 1 d log h αβ since it is the only 6

7 Simplest case: What can we say about the fields B and F? The restriction of H to the brane must be trivial in de Rham cohomology. Take H=0 on Y also as representative 3-form w 2 = 0 ( W 3 = c 1 (G) = 0) so that ρ α and σ α global Trivialize the B-gerbe with transitions equal to 1: (1, Λ αβ, B α ) (1, Λ αβ, da α F α ) = (1, 0, B + F ) with [G] S 1 = [exp 2πi B] = HolB H 2 (Y, S 1 ) ˇδ 1 h αβ = 1 Fα Z α A β = A α + (2πi) 1 d log h αβ + Λ αβ F is the integral Chern class of a non-canonical gauge bundle, due to (large) gauge transformations Φ (locally dλ) which are integral 2-classes Moreover, B+F H 2 (Y, R) { G torsion Ω 1 R acyclic and that means: is not quantized, but it admits a connection: = Λ αβ = A β A α A α (A A ) α 7

8 If Hol B=0, we can choose the preferred gauge B=0 and the gauge bundle on the brane will be canonical! W 3 = c 1 (G) = 0) (1, Λ αβ, B α ) (η αβγ, Λ αβ, da α F α ) = (η αβγ, 0, B+F ) and The B-gerbe is now trivialized with w2 - transitions ˇδ 1 h αβ = η αβγ Fα x 2 + Z x = 0 if w 2 = 0 α x = 1 if w 2 0 A β = A α + (2πi) 1 d log h αβ + Λ αβ that means B as before (it still exists, that can realize Hol B!), while F is half - quantized, since it is the Chern class of a half - bundle (its square is a true bundle), canonical only if Hol B=0. U(1)-charged spinors exist on the brane! The gauge invariant B+F is not quantized, since it represents Hol B - w2. When Hol B = w2, it will be an integral 2-class 8

9 W 3 = c 1 (G) 0) no global trivializations for the B-gerbe and no real class realizing the holonomy! (g αβγ, Λ αβ, B α ) ((g 1 η) αβγ, Λ αβ, da α F α ) = (η αβγ, 0, B+F ) ˇδ 1 h αβ = (g 1 η) αβγ so we have [exp 2πi F α ] = HolB w 2 α that means A β = A α + (2πi) 1 d log h αβ We only have the gauge invariant field strength B + F (locally da) as before not quantized and it is the real Chern class of a canonical generalized bundle with connection on the brane If we turn on the H-flux, H=d(B+F) on Y, we cannot any more gauge away B and B+F no longer represents a class. So the (generalized) gauge bundle will never be canonical! 9

10 Fractional branes and Page charge Consider the Wess-Zumino action of a Dp brane with flat B: S W Z C p+1 + (B + F ) C p 1 Y lower dimensional D-brane charges naturally appear and from FW anomaly we learned that, in general, mod integers: never quantized, Q RR = but gauge invariant S 2 B + F = (2πi) 1 log (S 2, HolB w 2 ) R in the BPS limit this is also equal to the tension of D(p-2) within Dp In particular, fractional branes from ADE-orbifolds are naturally taken into account: HolB = d I / Γ and w 2 = 0 10

11 When B is curved B+F is not closed any more, but F is! Q P = S 2 F Page charge, never gauge invariant (if db=0, it could be such only if HolB=0 on the brane) but (half-)quantized for every Spin C brane! (otherwise it encodes info of the restricted B-gerbe) Actually, for D2-branes homologically trivial in the target: B = H S 2 = B 3 and QRR should be the number of S 2 B 3 D-particles left after the collapse but F = B + F H is now gauge invariant! S 2 S 2 B 3 [Taylor, 00] correction from the bulk sugra action Only in this trivial situation (but with non-zero HolB on Y, because of curvature), Page charge is gauge invariant already in homology 11

12 In general, only the K-theory class of Page charge is gauge invariant! To qualitatively see that, consider this concrete example: Brane D5 X X X X X X D3 X X X X MMS5 X X X X X X S 2 X X S 3 X X X at the level of representative forms, a l.g.t. for F on the D5 leads to an effective quantized D3 charge: Φ δ 4 (D5) the transformation of B is just the opposite and it is related to the de Rham class of the B-gerbe in H 3 (X) S 2 Φ = S 3 H Z 12

13 the D3 effective charge becomes now: H δ 3 (MMS5) passing to cohomology classes, this term is exact under the d3 differential of Atiyah-Hirzebruch spectral sequence AHSS leads from Cohomology to K-theory through a series of quotients. At the 2 nd step, classes outside Ker(d3) are ruled out because FW-anomalous, while those in Im(d3) lift to 0 charge This is good, since, at a closer look, D-brane charges take values in K-theory rather than (co)homology groups 13

14 Conclusions Freed-Witten anomaly tells us which are the allowed brane configurations in type II superstrings The right mathematical framework to deal with its cancellation is the theory of gerbes with connection The brane provides the suitable trivialization of the B-gerbe to get rid of the ambiguity of the Pfaffian FW anomaly accounts for fractional (or even irrational) RR charges and for quantized but not gauge invariant Page charges It opens the road to the K-theoretical classification of D- brane charges via the AHSS 14

15 Perspectives After the recent work by Distler, Freed and Moore on the type I generalization of FW anomaly [ ], it would be interesting to reproduce such a classification in the presence of orientifolds Use similar techniques for higher rank gerbes to investigate on the quantization conditions of unimproved RR field strengths in type II supergravity theories Inspired by F-theory models, analyse the effects on FW anomaly of both S-duality (anomaly for general p,q branes or even for bound states) and of general gauge group enhancing: g 1 η = ˇδ 1 h αβ non-commutative bundle [ 1 ] = [e In the presence of non-abelian gauge groups, the brane provides a non trivial gerbe, and the bundle transitions close modulo the center of the enhanced group 15