String Junctions, Abelian Fibrations and Flux/Geometry Duality Michael Schulz Bryn Mawr College String Phenomenology 2008 University of Pennsylvania 31 May 2008 Work performed in collaboration with R. Donagi and P . Gao. String Phenomenology 2008 String Junctions, Abelian Fibrations and Flux/Geometry Duality Michael Schulz – p.1/21
Goal The IIB T 6 /Z 2 orientifold w. N = 2 flux has purely geometric IIA CY duals with no flux. We wish to construct the dual manifolds explicitly. Many properties were deduced by classical sugra dualities (T-duality + M-theory circle swap). We now provide two explicit constructions: Monodromy/junction based description: analogous to F-theory description of K3, but with T 4 rather than T 2 fibers. Explicit algebro-geometric construction (see also Saito’s talk). Relation of CYs to one another? to connected web? String Phenomenology 2008 String Junctions, Abelian Fibrations and Flux/Geometry Duality Michael Schulz – p.2/21
Motivations The T 6 /Z 2 orientifold is the simplest IIB flux compactification. Any insight into this background is likely to shed light on flux compactifications in general (e.g., early analyses of moduli stabilization). The CY duals X m,n have π 1 = Z n × Z n for n = 1 , 2 , 3 , 4 . ⇒ useful for Heterotic phenomenology. Few CYs with nontrivial π 1 are known (talks by Bouchard, Gross). D3 instantons in T 6 /Z 2 (with N = 2 flux) map to WS instantons wrapping P 1 sections of dual CY. ⇒ Nice check of our understanding of D3 instantons & zero mode counting, both at and away from O-planes. String Phenomenology 2008 String Junctions, Abelian Fibrations and Flux/Geometry Duality Michael Schulz – p.3/21
Motivations (continued) ∃ approx CY metric (valid for small fiber), precisely dual to classical sugra description of T 6 /Z 2 . Harmonic forms and low lying massive modes can be given explicitly in this approx CY metric. ⇒ Can in principle deduce warped KK reduction of the dual flux compactification using the duality (cf. talks by Shiu, Torroba). Flux breaks N = 4 → 2 . ⇒ Precise parametrization of extended SUSY breaking by CY topology. Connection to D(imensional) duality (via relative Jacobian of second CY construction). String Phenomenology 2008 String Junctions, Abelian Fibrations and Flux/Geometry Duality Michael Schulz – p.4/21
Known properties of IIA CY duals X m,n Abelian surface ( T 4 ) fibration over P 1 , with 8 + N singular fibers. ( N = number of D3-branes in T 6 /Z 2 ). Hodge #’s: h 11 = h 21 = N + 2 , where N + 4 mn = 16 . H ∧ F = 1 � ( F 3 ∼ 2 m, H 3 ∼ 2 n, N D3 + 4 N O3 in dual.) Generic D N lattice of sections (mod torsion) (since N D-branes + O-plane can coalesce to SO (2 N ) in T 6 /Z 2 dual). π 1 = Z n × Z n , isometry = Z m × Z m . (For nonminimal flux m, n , higgsing only partially breaks sugra U (1) s in T 6 /Z 2 ). The case m = n = 0 gives K3 × T 2 instead of CY. String Phenomenology 2008 String Junctions, Abelian Fibrations and Flux/Geometry Duality Michael Schulz – p.5/21
Known properties (continued) Approximate metric, harmonic forms (small parameter = fiber/base). J fiber ∝ mdx 1 ∧ dx 2 + ndx 3 ∧ dx 4 . Polarization: H 2 · A = 2 mn , H · E I · E J = − mδ IJ Intersections: (from sugra EFT or explicit harmonic forms). H · c 2 = 8 + N (from F 1 topological amplitude and Green-Schwarz). String Phenomenology 2008 String Junctions, Abelian Fibrations and Flux/Geometry Duality Michael Schulz – p.6/21
Construction 1: Monodromy/junction based description String Phenomenology 2008 String Junctions, Abelian Fibrations and Flux/Geometry Duality Michael Schulz – p.7/21
Warm-up: F-theory on K3 Recall IIB encoding of the geometry of a T 2 fibration over P 1 (e.g., K3): τ = cpx modulus of T 2 H γ F 1 = 1 unit RR charge τ → τ + 1 about γ τ → τ + 1 about γ . ` a ` a aα + bβ cycle in T 2 : ´ ´ → K , ( τ = C 0 + i/g s dilaton-axion) b b “ ” Here D7 = object on which 0 − 1 K = monodromy matrix. 1 1 fundamental strings can end. pα + qβ (instead of α ) cycle shrinks, Similarly ( p, q ) 7-brane = object on 1+ pq − p 2 “ ” which ( p, q ) string can end. K [ p,q ] = . q 2 1 − pq String Phenomenology 2008 String Junctions, Abelian Fibrations and Flux/Geometry Duality Michael Schulz – p.8/21
F-theory on K3 = IIB on T 2 /Z 2 Let ( p, q ) charges A = (1 , 0) , B = (1 , − 1) and C = (1 , 1) . Perturbative description of T 2 /Z 2 orientifold: 16 D7s + 4 O7s. Nonperturbative description: each O7 resolves to BC pair. (B,C 7-branes are determined [up to equivalences] by factorization of K O7 into K [ p,q ] ’s.) So, F-theory manifold: base P 1 ∼ = T 2 /Z 2 and 24 singular fibers A 16 ( BC ) 4 , with monodromies � � � � � � 1 − 1 − 1 2 − 1 K A = , K B = , K C = . 1 1 2 1 These nonperturbative IIB data define the topology of K3. String Phenomenology 2008 String Junctions, Abelian Fibrations and Flux/Geometry Duality Michael Schulz – p.9/21
CY duals of T 6 /Z 2 are abelian fibrations CY duals X m,n are T 4 fibration over P 1 . Why? No flux: T 6 /Z 2 orientifold IIA on K3 × T 2 ↔ ( K3 = T 2 fibration over P 1 ) (both dual to type I or het-SO on T 6 ). With N = 2 flux F 3 ∼ 2 m, H 3 ∼ 2 n : T 6 /Z 2 orientifold ↔ IIA on CY X m,n ( X m,n = T 4 fibration over P 1 ) Roughly, twists mix previous T 2 factor with T 2 fiber of K3. String Phenomenology 2008 String Junctions, Abelian Fibrations and Flux/Geometry Duality Michael Schulz – p.10/21
Monodromy matrices for CY duals N D3s + O3s of T 6 /Z 2 A N B 1 C 1 B 2 C 2 B 3 C 3 B 4 C 4 ↔ singular T 4 fibers of X m,n . Can again determine the monodromy matrices explicitly. We find 0 1 − 1 | 1 1 | A = ( old K A ) ⊗ ( identity ) on T 2 × T 2 , K A = B −− −− −− −− C @ | 1 | 1 but B i , C i differ for i = 1 , 2 , 3 , 4 . For example, 0 − 1 | − m 1 1 2 | m A = ( old K B ) ⊗ ( identity ) on T 2 × T 2 + m, n twists. K B 1 = B −− −− −− −− C @ − n − n | 1 − m | 1 The monodromies uniquely determine the topology of X m,n . String Phenomenology 2008 String Junctions, Abelian Fibrations and Flux/Geometry Duality Michael Schulz – p.11/21
CY dual interpretation of RR tadpole Since the base of X m,n is P 1 , a loop that encloses all singular fibers is contractible (to the point at infinity). ⇒ Total monodromy must be unity: 1 = K total N = K C 4 K B 4 . . . K C 1 K B 1 K A � 1 0 0 0 � 1 − Q 0 = , 1 0 1 where Q = N − 16 + 4 mn . Topological constraint reproduces T 6 /Z 2 D3 charge constraint Q = 0 . String Phenomenology 2008 String Junctions, Abelian Fibrations and Flux/Geometry Duality Michael Schulz – p.12/21
String junctions & Mordell-Weil lattice String junctions: are W-bosons of 7-brane gauge theory, encode homology of F-theory elliptic fibration, (DeWolfe et al.) equivalence classes (charges) form lattice. H 2 is generated by: generic fiber, components of singular fibers, string junctions, H 1 ( P 1 , R 1 π ∗ Z ) ← sections. Mordell-Weil lattice of sections = junction lattice/null loops (Fukae et al.). String Phenomenology 2008 String Junctions, Abelian Fibrations and Flux/Geometry Duality Michael Schulz – p.13/21
MW and junction lattice for X m,n a ( p, q, r, s ) 1-cycle in T 4 fiber shrinks at each A , B i , C i . In CY X m,n : Obtain 2-cycles in X m,n from S 1 [ p,q,r,s ] fibration over ( p, q, r, s ) junction graphs in base P 1 . Again, MW lattice of sections = junction lattice/null loops. A N � 4 Obtain D N from A N B i C i i =1 B i C i ⇒ ( A + A = B i + C i ) but not E N +1 from A N B i C i C j ( C i � = C j ). D N = free part of MW lattice. Z m × Z m = torsion part of MW lattice = isometry group. String Phenomenology 2008 String Junctions, Abelian Fibrations and Flux/Geometry Duality Michael Schulz – p.14/21
Relations between CYs N + 4 mn = 16 . Complete set of 8 X m,n is { X 1 , 1 , X m, 1 , X 1 ,n , X 2 , 2 } . � � X m, 1 / Z m × Z m = X 1 ,m , Relations: � � X 4 , 1 / Z 2 × Z 2 = X 2 , 2 Z 2 × Z 2 ⊂ Z 4 × Z 4 . with When singular fibers coalesce, additional isometries can develop. Have new MW torsion from “weakly integer” junctions. For example, a (1 , 0) string can end on a coalesced A 2 pair: “ (1 / 2 , 0) on each.” Junction description suggests that all X m,n are related to X 1 , 1 by quotienting (though not necessarily by free action). Do not appear to have extremal transitions to other interesting CYs. (Cpx def away from singular loci → new section; Kähler resolution → new Kodaira component; same Hodge numbers either way.) String Phenomenology 2008 String Junctions, Abelian Fibrations and Flux/Geometry Duality Michael Schulz – p.15/21
Construction 2: Explicit algebro-geometric construction String Phenomenology 2008 String Junctions, Abelian Fibrations and Flux/Geometry Duality Michael Schulz – p.16/21
Recommend
More recommend
