U-dual branes and non-geometric string backgrounds

Transcription

1 U-dual branes and non-geometric string backgrounds Athanasios Chatzistavrakidis Institut für heoretische Physik, Leibniz Universität Hannover arxiv: With Fridrik F Gautason, George Moutsopoulos and Marco Zagermann Corfu2013

2 Motivation Windows to non-perturbative aspects of string theory: Branes Dualities Interplay families of non-perturbative extended objects with unusual properties.

3 Motivation Windows to non-perturbative aspects of string theory: Branes Dualities Interplay families of non-perturbative extended objects with unusual properties. Flux compactifications: dualities reveal backgrounds with globally ill-defined geometry, non-geometry. H a abc f a b bc Q ab c c R abc

4 Motivation Windows to non-perturbative aspects of string theory: Branes Dualities Interplay families of non-perturbative extended objects with unusual properties. Flux compactifications: dualities reveal backgrounds with globally ill-defined geometry, non-geometry. H a abc f a b bc Q ab c c R abc he two pictures are related. Can lead to a better understanding of string theory structure and of unconventional flux vacua.

5 Zoo of extended objects tring theory contains: strings (F1) couple to Kalb-Ramond 2-form B 2 ; perturbative. Dp-branes couple to RR forms C p+1 ; tension g 1 s. N5-branes couple to magnetic dual of Kalb-Ramond B 6 ; tension g 2 s. KK Monopoles couple to KK gauge field; also g 2 s.

6 Zoo of extended objects tring theory contains: strings (F1) couple to Kalb-Ramond 2-form B 2 ; perturbative. Dp-branes couple to RR forms C p+1 ; tension g 1 s. N5-branes couple to magnetic dual of Kalb-Ramond B 6 ; tension g 2 s. KK Monopoles couple to KK gauge field; also g 2 s. Dualities map branes to branes: -duality (does not mix NN and RR sectors): Dp D(p± 1) N5 N5 or KKM IIB -duality (mixes NN and RR sectors): D5 N5 KKM KKM

7 Zoo of extended objects tring theory contains: strings (F1) couple to Kalb-Ramond 2-form B 2 ; perturbative. Dp-branes couple to RR forms C p+1 ; tension g 1 s. N5-branes couple to magnetic dual of Kalb-Ramond B 6 ; tension g 2 s. KK Monopoles couple to KK gauge field; also g 2 s. Dualities map branes to branes: -duality (does not mix NN and RR sectors): Dp D(p± 1) N5 N5 or KKM IIB -duality (mixes NN and RR sectors): D5 N5 KKM KKM Q: Is that all?

8 More extended objects Utilizing the full U-duality reveals new families of branes. [Elitzur, Giveon, Kutasov, Rabinovici 97] [Blau, O Laughlin 97] [Hull 97] Co-dimension 2: Defect (or exotic ) branes. [Bergshoeff, Ortin, Riccioni 11] [de Boer, higemori 12] Many objects (especially in lower dimensions). Diversity in: non-perturbativity; tension g α s with α = 1,2,3,4. special transverse directions; 0,1,2,...,7. monodromy properies; defect branes are generically U-folds.

9 More extended objects Utilizing the full U-duality reveals new families of branes. [Elitzur, Giveon, Kutasov, Rabinovici 97] [Blau, O Laughlin 97] [Hull 97] Co-dimension 2: Defect (or exotic ) branes. [Bergshoeff, Ortin, Riccioni 11] [de Boer, higemori 12] Many objects (especially in lower dimensions). Diversity in: non-perturbativity; tension g α s with α = 1,2,3,4. special transverse directions; 0,1,2,...,7. monodromy properies; defect branes are generically U-folds. Focus on IIB fivebranes: KK D5 N

10 Explore: Analogs of DBI action for exotic fivebranes. Couplings to background fields. Exotic branes as sources. Relations to non-geometry. General picture: F abc D5 H abc N5 a f a bc KKM b Q ab c P ab c see also Hassler, Lüst 13

11 tandard world-volume actions D5 brane DBI action: DBI,D5 = D5 d 6 σ e φ det(g ij +B ij +F ij ). WZ action: WZ,D5 = µ D5 e F C 6 (gauge invariant completion of the magnetic coupling to C 2). Polyform: C = C p. As source, modified field eqs. and Bianchi ids.; e.g. df 3 +F 1 H 3 = j D5.

12 tandard world-volume actions D5 brane DBI action: DBI,D5 = D5 d 6 σ e φ det(g ij +B ij +F ij ). WZ action: WZ,D5 = µ D5 e F C 6 (gauge invariant completion of the magnetic coupling to C 2). Polyform: C = C p. As source, modified field eqs. and Bianchi ids.; e.g. df 3 +F 1 H 3 = j D5. N5 brane use -duality see also Eyras, Janssen, Lozano 98 DBI: DBI,N5 = N5 d 6 σe φ τ det(g ij τ 1 F ij ), F = C 2 +dã1. WZ: WZ,N5 = µ N5 e F C 6, (gauge invariant completion of the magnetic coupling to B 2). new polyform C = C0 τ 2 B 2 (C 4 C 2 B 2 )+(B C 2 B 2 B 2 ). As source, dh 3 = j N5 (NN source).

13 imilarly for RR sector. Exotic DBI actions brane use yz trategy: KK decomposition -duality rules application of duality rules. ds 2 = Ĝµνdx µ dx ν +G mn η m η n, B = 1 2 ˆB µν dx µ dx ν B mnη m η n +(η m 1 2 Am ) θ m, where η m = dx m +A m and θ mµ = B mµ B mn A n µ. Rules: yz G mn G mn det(g mn) = det(g mn +B mn) Gmn, A m µ Ĝ µν B mn θ mµ ˆB µν yz θ mµ, yz Ĝµν, yz B mn det(b mn) = det(g mn +B mn) (B 1 ) mn, yz A m µ, yz ˆB µν.

14 Applying the rules: DBI,5 2 2 = M 5 22 d 6 σ e φ τ det(e mn ) det(ĝµν i X µ j X ν + G mn η im η jn τ 1 F ij ). Indices: i: parallel, M: 10D, m: isometries. As usual, E = G +B. Also, modulus τ = (C yz B yz C 0 )+i det(e mn )e φ, η im = i X m +B mµ i X µ B mn A n µ ix µ, Gauge invariant 2-form: F ij = 2 [i à j] +(ζ µνyz ζ µν B yz ) i X µ j X ν +2ǫ nm ζ µm [i X µ η j]n ( ) + ǫ mn ζ 0 + B mn (ζ yz B yz ζ 0 ) η im η jn. Using -duality, also obtain the brane action. imilar actions for cases with 1 special direction were studied already in the 90s. [Bergshoeff, Eyras, Janssen, Lozano, Ortin...]

15 Couplings of exotic branes Not so straightforward task... Where does the couple to? N5 couples to B 6. -duality does not mix NN and RR sectors. it should couple to some magnetic dual of B 2. Our strategy: Use definition of B 6 as magnetic dual of B 2 and the KK decomposition in the η m basis to find the components of its (double) -dual that couple to the brane. H 7 = db 6 +(RR terms) = e 2φ db 2. he components of B 6 that couple to the N5 brane are We find the rule: (1 P z )(1 P y )B 6, P y = dy ι y. (1 P z )(1 P y )B 6 yz ι y ι z B yz couples to the double contraction of an 8-form magnetic dual of B 2. he same result was predicted before using different methods. Bergshoeff, Ortin, Riccioni 11

16 Modified Bianchi identity As for D5 and N5 sources, the should modify the field equations. Consider = NN +µ ι y ι z B yz 8. hould express everything in terms of the same variables. Generalized geometry comes into play. Generalized metric (general parametrization [cf. Aldazabal, Baron, Marques, Nunez 11]): ( ) g Bg H = 1 B Bg 1 +gβ g 1 B βg g 1, β = β βgβ ij i j 2-vector. Rewritten NN action (for B = 0) [Andriot et al. 11] [cf. Blumenhagen et al.] NN = d 10 x ( ) g e 2 φ R+4 d φ QNR M QM NR + d( ), with Q 2 1 = dβ, and magnetic dual e 2φ dβ = dβ 2 8. hen: d(q1 MN g My dy g Nz dz) = j is source for Q.

17 Defect branes are U-folds he argument: [de Boer, higemori 12] 3D viewpoint: point-particle states, moduli undergo monodromies (elements of U-duality group E 8,8 (Z)) when transported around branes. 10D viewpoint: moduli are components of sugra background fields. monodromies multivalued background fields. i.e. non-geometry ; fields cannot be patched locally with diffeos and gauge trafos, need to use dualities as transition functions. In the flux compactifications language such cases are known as U-folds. [Hull 04] the connection to non-geometry is better seen in the supergravity description.

18 upergravity description upergravity solution associated to brane [Lozano-ellechea, Ortin 00] ds 2 = H(dr 2 +r 2 dθ 2 )+HK 1 (dx 89 ) 2 +(dx ) 2, e 2φ = HK 1, B 2 = θk 1 dx 89, K = H 2 +θ 2. θ θ +2π leads to problems globally. E.g. B 89 (r,θ +2π) = θ+2π H 2 +(θ+2π) 2 B 89 (r,θ)+δb 89. his is essentially the same situation as for the -dual of the 3D nilmanifold. b Q ab c -fold. [Hull 04] [cf. Hassler, Lüst 13] f a bc

19 Explicit expressions Explicit form of the generalized metric: θ H 5 2 2,(yz) = θ 0 H 0 θ K 0 ds2 yz = H 1 (dy 2 +dz 2 ), θ 0 0 K β = θ y z. well-behaved. Constant Q flux. Magnetic dual: β yz 8 = Hdx dy dz ι y ι z β yz 8 = Hdx Remarks: Makes no sense to compute B 6 in the non-geometric case. In N5 case, B 2 = θdy dz B 6 = Hdx Note the correspondence of B(N5) β(5 2 2 ) and B 6(N5) ι y ι z β yz 8 (52 2 ).

20 RR non-geometry -duality mediates non-geometry to the RR sector. upergravity solution associated to brane ds 2 = (HK 1 ) 1 2(dr 2 +r 2 dθ 2 )+(HK 1 ) 1 2(dx 89 ) 2 +(HK 1 ) 1 2(dx ) 2, e 2φ = (HK 1 ) 1, C 2 = K 1 θdx 89, K = H 2 +θ 2. As before θ θ +2π leads to C 89 (r,θ +2π) C 89 (r,θ)+δc 89. -dual of the -fold, Q ab c P ab c U-fold. [Aldazabal, Camara, Font, Ibanez 06] couples to an exotic dual of C 2. [cf. Bergshoeff, Ortin, Riccioni] Its treatment goes through similar lines, but......extended Generalized Geometry (structure group extended further to U-duality group) comes into play 2-vector γ = γ ij i j (cousin of C 2). [Aldazabal, Andres, Camara, Grana] Flux: P 89 θ = θ γ 89.

21 General picture Inclusion of -duality enhances the standard flux chain H a abc f a b bc Q ab c to F abc H abc a f a bc b Q ab c P ab c

22 General picture Inclusion of -duality enhances the standard flux chain H a abc f a b bc Q ab c to F abc H abc a f a bc b Q ab c P ab c hese fluxes are associated to brane sources F abc D5 = H a abc f a b bc Q ab c N5 = KKM = P ab c 5 2 3

23 ummary Main messages Plethora of non-perturbative objects in string theory due to U-duality. hey couple to exotic duals of the standard gauge potentials. trongly related to non-geometric backgrounds and to modern techniques to study (unconventional but generic) flux vacua. he study of such situations is very useful in order to gain a more complete understanding of string vacua.