Nonabelian Gerbes, Differential Geometry and Stringy Applications

Transcription

1 Nonabelian Gerbes, Differential Geometry and Stringy Applications Branislav Jurčo, LMU München Vietri April 13, 2006

2 Outline I. Anomalies in String Theory II. String group and string structures III. Nonabelian gerbes and their differential geometry (nonabelian B-field) IV. Twisting nonabelian gerbes and M5-brane anomaly III + IV P. Aschieri, L. Cantini, B. Jurčo, CMP, 254 (2005), 367; hep-th/ P. Aschieri, B. Jurčo, JHEP 0410, 068 (2004) (2005), 367; hep-th/ B. Jurčo, math.dg/

3 Heterotic String Theory Perturbative space-time anomalies Green-Schwarz mechanism or dh = TrF 2 TrR 2 p 1 (T ) p 1 (V ) = 0 in H 4 dr (M) = H4 (M, R) V - SO(32) or E 8 E 8 bundle (gauge theory on M) F - curvature T = T M, R - Riemannian curvature 2-form Global world-sheet anomalies in H 4 (M, Z) p 1 (T ) p 1 (V ) = 0

4 Point particle with world-line SUSY I = dt 1 2 g dx i dx j ij dt dt + i d 2 ψi (g ij dt + dxk dt ω ijk)ψ j Canonical quantization M - spin ψ i Γ i Q = ψ i dx i dt dx i dt x i Q Dirac operator on M

5 Path integral D D = i( d dt δi j + dxk dt ωi kj ) D - world-line Dirac operator well defined iff M is spin w 2 = 0 H 2 (M, Z 2 ) nonequivalent spin structures H 1 (M, Z 2 )

6 Spin structure 1 Z 2 Spin(n) SO(n) 1 M - spin if the principal SO(n)-frame bundle lifts to a principal Spin(n)-bundle exp(ii) is well defined M - it not spin w 2 0 exp(ii) exp(ii)hol(a) A - U(1)-gauge field s.t. ambiguity in hol(a) ±1 cancels the ambiguity in D ± 1 M - must be spin C

7 Spin C structure 1 Γ Spin(n) U(1) Spin C (n) 1 Γ Spin(n) U(1) generated by (1, 1) and ( 1, 1) M - spin C if the principal SO(n)-frame bundle lifts to a principal Spin C (n)-bundle exp(ii)hol(a) is well defined there exist a line bundle L such that in H 2 (M, Z 2 ) or in H 3 (M, Z) w 2 (M) = c 1 (L)mod2 W 3 (M) = 0

8 D-branes We can have a D-brane which is not spin C, i.e W 3 0 but a(d) = W 3 (D) [H] D = 0 in H 3 (D, Z) Freed - Witten anomaly recall H 3 (M, Z) [abelian 1-gerbes on M] D-brane - trivialization of an abelian 1-gerbe defined by a(d)

9 if a(d) 0 but na(d) = 0 (torsion) this corresponds to an stack on n D-branes we need at least U(n)- gauge field (bundle) to cancel the anomaly stack of D-banes module of the abelian 1-gerbe in general U(H)

10 M-theory, M5-branes M5-brane with M2-branes ending on it a(m5) = W 4 (M5) [G 4 ] M5 = 0 in H 4 (M5, Z) Diaconescu - Freed - Witten anomaly recall H 4 (M, Z) [abelian 2-gerbes on M] M 5-brane - trivialization of an abelian 2-gerbe defined by a(m5) coinciding M 5-branes a(d) 0 and we need a nonabelian gerbe to cancel the anomaly stack of M5 branes module of the abelian 2-gerbe

11 String stuctures - return to heterotic anomaly p 1 (T ) p 1 (V ) in H 4 (M, Z) Witten, Killingback Dirac-Ramond operator Dirac operator on the loop space LM LM must be spin frame bundle (principal SO(n)-bundle) P M principal LSO(n)-bundle LP LM 1 S 1 LSO(n) LSO(n) 1 obstruction to lift LP into a principal LSO(n)-bundle is the transgression of p 1 (T ) in H 3 (LM, Z) M (TM) has a string structure if this obstruction vanishes (more generally T V makes sense in K-theory, it has a string structure if p 1 (T ) p 1 (V ) vanishes in H 4 (M, Z))

12 String Group π k O(n) Z 2 Z 2 0 Z Z SO(n) 0 Z 2 0 Z Z Spin(n) Z Z String(n) Z not Lie; defined up to homotopy 1 K(Z, 2) String(n) Spin(n) 1 such that the boundary map is given by π 3 Spin(n) π 2 K(Z, 2) = Z p 1 /2 H 4 (BSpin(n)) Hom(π 3 Spin(n), Z) more generally Spin(n) G-compact simply connected String(n) String G String structure lifting of a principal G-bundle P to a principal String G -bundle ( 1 2 p 1(P ) = 0) models of String - Brylinski; Teichner, Stolz; Baez, Crans, Schreiber, Stevenson; Henriquez

13 Crossed Module (H D) H,D Lie groups. H is a crossed D-module if there is a group homomorphism α : H D and an action of D on H, (d, h) d h, such that and α(h) h = hh h 1 for h, h H α( d h) = dα(h)d 1 for h H, d D. holds true. BCSS constuction of String H = ΩG and D = P 0 G

14 Nonabelian (bundle) gerbes {O α } open covering of M Local description - Nonabelian Čech 2-cocycle {d αβ, h αβγ } d αβ : O α O β D h αβγ : O α O β O γ H fulfilling the following cocycle condition d αβ d βγ = α(h αβγ )d αγ on O α O β O γ and h αβγ h αγδ = d αβh βγδ h αβδ on O α O β O γ O δ.

15 Stable equivalence {d αβ, h αβγ } and {d αβ, h αβγ } are related as and d αβ = d αα(h αβ )d αβ d 1 β h αβγ = dα h αβ d α d αβ h βγ d α h αβγ d α h 1 αγ with d α : O α D and h αβ : O α O β H.

16 [Principal String G -bundles] [( ΩG P 0 G)-gerbes] [Principal G-bundles] [(ΩG P 0 G)-gerbes] String structure - lift of a (ΩG P 0 G)-gerbe to a ( ΩG P 0 G)- gerbe

17 Gerbe connection {A α, a αβ } A α Ω 1 (O α ) Lie(D) a αβ Ω 1 (O α O β ) Lie(H) fulfilling A α = d αβ A β d 1 αβ + d αβdd 1 αβ + α(a αβ) on O α O β and a αβ + d αβa βγ = h αβγ a αγ h 1 αβγ +h αβγdh 1 αβγ +T A α (h 1 αβγ ) on O α O β O γ. T A (h) - Lie(H)-valued one-form. For X Lie(D) we put T X (h) = [h exp(tx) (h 1 )], [ ] - tangent vector to the curve at the group identity 1 H. For Lie(D)-valued one form A = A ρ X ρ, with {X ρ } a basis of Lie(D), put T A A ρ T X ρ. curvature F α Ω 2 (O α ) Lie(D) F α = da α + A α A α

18 Nonabelian B-field {B α, δ αβ } B α Ω 2 (O α ) Lie(H) δ αβ Ω 2 (O αβ ) Lie(H) such that B α = d αβb β + δ αβ on O α O β and δ αβ + d αβδ βγ = h αβγ δ αγ h 1 αβγ + B α h αβγ B α h 1 αβγ on O α O β O γ. H α = db α + T Aα (B α )

19 Twisting by an abelian 2-gerbe H = ΩG, D = P 0 G h αβγ h αγδ λ αβγδ = d αβh βγδ h αβδ a αβ + d αβ a βγ α αβγ = h αβγ a αγ h 1 αβγ + h αβγdh 1 αβγ + T A α (h 1 αβγ ) B α + β αβ = d αβb β + δ αβ H α γ α = db α + T Aα (B α ) λ αβγδ (U(1)-valued) is the obstruction to the String structure - abelian 3-cocycle - abelian 2-gerbe {λ αβγδ, α αβγ, β αβ, γ α } is a Deligne class - abelian 2-gebre with curvings (α, β, γ are u(1)-valued).

20 Diaconescu-Freed-Witten anomaly of a stack of M5- branes can be canceled by a twisted nonabelian gerbe with G = E 8 field content of a higher gauge theory gauge theory with a nonabelian 2-form gauge potential B on the M5 world-volume? Action? nonabelian B couples to the ends of M2-branes ending in M5-branes similarly as in one degree lower nonabelian gauge potential A couples to the ensds of srings ending on the D-branes.