New Calabi-Yau 3-folds and their mirrors via conifold transitions - PowerPoint PPT Presentation

New Calabi-Yau 3-folds and their mirrors via conifold transitions Maximilian Kreuzer / Vienna University of Technology Work with • V. Batyrev (T¨ ubingen): ... conifold transitions / /arXiv:0802.3376 [math.AG] − ′′ − : Integral cohomology & mirror symmetry / • /arXiv:math.AG/0505432 • V. Braun , B.A. Ovrut (U.Penn) and E. Scheidegger (Augsburg) Worldsheet instantons, torsion curves, non-perturbative superpotentials arXiv:hep-th/0703134, hep-th/0703182, 0704.0449 [hep-th] • A. Klemm (Bonn), E. Riegler (Vienna) and E. Scheidegger (Augsburg) Topological strings, CICYs & threshold corrections / /arXiv:hep-th/0410018 Supported by the Austrian Research Fund FWF 1

Content • Motivation to study the web of Calabi-Yau manifolds • Geometry and Combinatorics: Reflexive Polytopes • Torsion in Cohomology: Fundamental MS ← → Brauer group � more torsion in H ∗ • Beyond hypersurfaces: Complete Intersections more general singularities ⊆ Reflexive Cones: ↔ classification rigid CYs • Beyond toric CYs: Mirror Pairs via Conifold Transitions – Results: surprisingly many new CYs with small h 11 1-parameter case: topologies & PF opertors – Construction: combinatorial → 30241 cases with h 11 ≈ 4 • ToDo, ToFindOut & ToApply 2

Why should we classify Calabi-Yau 3-folds? • Math: important part of the classificaiton of 3-folds • Strings: Model building – Heterotic: fundamental groups, torsion / / symmetric CYs (?) – Orientifolds: exceptional divisors / / more generic singularities (?) – F-theory: elliptic 4-folds / / ∃ too many: bottom up classification (?) • What do we know? – only examples (finiteness ?); except: elliptic case [M. Gross] – but Reid’s phantasie: connected by singular transitions • Toric hypersurfaces: + very numerous (largest known list) + very conveneint (combinatorics, MS) − very special !!! + connected ⇒ use as backbone of the web 3

Conifold transitions • Candelas, Green, Hubsch ’90: Other worlds are just around the corner • Greene, Morrison, Strominger ’95: Black hole condensation /N=2 SUSY • Danielsson ’07: ... landscape topography / N=1 • w/Batyrev [0802.3376]: – mirror to Candelas et al., blow down P 1 → flat deformation toric hypersurface → reduce h 11 – generalizing:Batyrev, Ciocan-Fontanine, Kim, van Straten: Mirror symmetry for complete intersections in Grassmannians & Flag manifolds via toric degenerations • Combinatorial conditions: • 2-faces are minimal triangles or squares • smoothing condition: rank of matrix of linear relations 473 800 776 reflexive polyhedra: 4.5GB ... 1 day on desktop / 2-3 days on laptop 30241 new examples of (presumed) mirror pairs 4

Geometry and combinatorics: hep-th/0612307 affine coord. t i = z i /z 0 ∈ R n ⊆ P n • Quintic in P 4 : z 5 0 + . . . + z 5 4 = 0 i on patch U i ∼ homogeneous polynomial p ( z ) ⇒ f ( t j ) = p ( z j ) /z d = R n • C ∗ scaling: line bundle O ( d ): g ij = ( z i z j ) d ... monomial transition function – quasi-homogeneous: weighted projective W P n – multi-quasi-homogeneous: toric variety P Σ (for simplicial fan) c m � t � m � • f ( t i ) = Newton polytope ∆ ∈ M R m ∈ M ∼ = Z n of exponent vectors � m ∈ ∆ ∩ M � z v ij • affine coordinates t i = � v ij = � e i , v j � . . . v j ∈ N = M ∗ j , j � m,v j � t m i • Monomials: ξ m = � = � z polytope ∇ = � v j � ⊆ N R i j i j fan Σ of cones over ∇ ∇ = ∆ ∗ reflexive • { f = 0 } is Calabi-Yau ⇔ � m, v j � ≥ − 1 . . . 5

Singularities: the Z n quotient C [ X, Y ] / Z n : X → e 2 πi/n X Y → e 2 πi/n Y ✻ ✍ ✂ ✂ ✍ ✂ ✂ r r r r r r r r ❅ ✂ ✂ ❅ ✂ ✂ Z 3 ✁ ✕ resolution r r r r r r r r ❅ ✲ ✂ ✲ ✂ ✁ ❅ ✂ ✂ ✁ ✒ � r r r r r r r r (subdivision) ❅ ✂ ✂ ✁ � ❅ ✲ ✂ ✲ � ✁ ✂ ✲ r r r r r r r r ˜ X = X 1 , Y 1 = X 2 , . . . Y 3 = ˜ ˜ Invariant X 3 , X 2 Y, XY 2 , Y 3 X = X 3 , Y X ˜ ˜ X 6 , X 5 Y, X 4 Y 2 , . . . Y = X 2 Y , transition: Y 2 = Y 1 /X 1 , . . . new coordinates z j cheating: subdivision of cone σ ⊂ N R with σ ∈ Σ dual to the cone σ ∨ ⊂ M R of monomials X, Y, . . . conifold: xy = zw . . . σ ∨ ∈ M R vs. triangulation of σ ∈ N R , cf. hep-th/0612307 Theorem: • P Σ is smooth iff all cones are simplicial and unimodular • V ol ( θ k ) > 1 for k -face θ k ∈ Σ ⇔ singularity of dimension n − k − 1 because: # of vanishing homogeneous coordinates ⇒ e.g. facet ↔ point 6

Toric (hypersurface) dictionary: • hypersurface f ∆ = 0 ⇔ ∆ is reflexive • (toric) fibration ⇔ ∃ reflexive section of ∇ ⊂ N R v j ∈ Σ (1) • divisors ⇔ • Hodge numbers ⇔ lattice points on θ k • fundamental group ⇔ index of sublattices � pt’s � Z • . . . ⇔ . . . 7

Torsion in (co)homology w/V. Batyrev [math.AG/0505432] • Universal coefficient theorem tor( H i ( X, Z )) ∼ = tor( H i +1 ( X, Z )) ∗ tor( H i ( X, Z )) ∼ = tor( H 2 d − i ( X, Z )) • Poincar´ e duality: • 3-folds ⇒ two independent torsion groups: = tor H 2 ( X, Z ) ∗ (related to fundamental group) tor H 1 ( X, Z ) ∼ = tor H 3 ( X, Z ) ∗ (topological Brauer group) tor H 2 ( X, Z ) ∼ Conjecture: The torsion subgroups of H 2 and H 3 are exchanged under the mirror involution • verified for all 473 800 776 toric Calabi–Yau hypersurfaces: 16 + 16 cases: elliptically fibered: P 2 × P 2 [3 , 3] / Z 3 and P 11133 [9] / Z 3 , P 4 [5] / Z 5 , 13 × π 1 ( X ) = Z 2 (elliptic-K3 fibered) • fundamental group ↔ index of sublattice � codim-2 points � Z (Brauer group) 2 ↔ index of sublattice � codim-3 points � Z i.e. points on edges of ∇ 8

Torsion curves for the “Heterotic standard model” with V. Braun, B. Ovrut and E. Scheidegger • Schoen CY: (3,3) parameter, fiber product of two elliptic fibers over P 1 • complete intersection: P 2 × P 2 × P 1 � 3 0 1 � ⇒ Batyrev-Borisov mirror 0 3 1 • free Z 3 phase (toric) × Z 3 permutation (non-toric) group action • Self-mirror! ⇒ Z 3 × Z 3 torsion curves (spectral sequence computation) • Application: torsion curves cannot be holomorphic vs. Beasley–Witten no-go single curve in homology class! → SUSY breaking & moduli stabilization Lessons for CYs: • need codimension > 1 for having both fundamental + Brauer group • for large torsion in H ∗ Candelas et al’s ’88 list of CICYs in products of P n ’s is a good starting point (don’t need complicated polytopes) 9

Apropos complete intersections • f 1 = . . . = f r = 0 ⇒ Minkowski sum ∆ = ∆ 1 + . . . + ∆ r • Batyrev-Borisov mirror duality: ∇ = ∇ 1 + . . . + ∇ r where ∆ ∗ = �∇ 1 , . . . , ∇ r � conv ∆ = ∆ 1 + . . . + ∆ r � ∆ i , ∇ j � ≥ − δ ij ∇ ∗ = � ∆ 1 , . . . , ∆ r � conv ∇ = ∇ 1 + . . . + ∇ r hypersurface � t i f i = 0 complement Cayley trick: CICY f i = 0 ↔ Reflexive Gorenstein cones of dimension n + r for CY n − r dual M R ⊇ ˜ ˜ ∇ = Conv( e j , ∇ j ) ⊆ ˜ ˜ N R ∼ = R n + r ∆ = Conv( e i , ∆ i ) ↔ Batyrev, Nill: Combinatorial aspects of mirror symmetry [math/0703456]: • Split Gorenstein cone ⇒ CY n − r fold (codimension r) • Rigid CY: only one cone is split, the mirror is “generalized” CY in the sense of 1990s • ∃ classification algorithm (work in progress) 10

Nef partitions & Batyrev–Borisov duality r r r ∆ ◦ ∈ N i z � m,v i � → coordinates z i � sections ∼ m ∈ ∆ Π r r r i r r r r r r r r ∆ ∈ M = N ∗ → line bundles ↔ equations r r r r r r r r r r ⇒ CICY: decompose ∆ = ∆ 1 + ∆ 2 (Minkowski sum) V.V. Batyrev & L.A. Borisov [alg-geom/9412017]: − NEF partitions: piecewise linear convex “support functions” ϕ j ( e i ) = δ ij numerically effective → ample line bundles − combinatorial duality ↔ mirror symmetry ... 4 reflexive polytopes:  ∆ ∗ = �∇ 1 , ∇ 2 � ∆ = ∆ 1 + ∆ 2 ≥ − 1 if i = j  � ∆ i , ∇ j � = ∇ ∗ = � ∆ 1 , ∆ 2 � ∇ = ∇ 1 + ∇ 2 ≥ 0 if i � = j  ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ 11

Mirror symmetry: duality extends to Hodge data V.V.Batyrev, L.A.Borisov: alg-geom/9509009 ¯ ( − ) ρ x t ρ y t ( − 1) p + q h pq t p ¯ t q = � � y , t ¯ t ) B ( I ; t − 1 , ¯ t ) S ( C ∗ S ( C x , t ) ( t ¯ t ) r I =[ x,y ] − C x , C y ∈ face lattice of Gorenstein cone spanned by ( e i , ∆ i ) − B ( I ) encodes combinatorics of the sublattice I = [ x, y ] with x < y m ∈ C x t deg( m ) − S ( C x , t ) = (1 − t ) ρ x � related to the Erhart polynomial nef.x ( ∈ PALP) by Erwin Riegler [math.AG/0103214, math.CS/0204356]: Batyrev’s formula for codimension r = 1: codim 1 - divisors do not intersect � � h 11 = l (∆ ∗ ) − 1 − d − l ∗ ( θ ∗ ) + l ∗ ( θ ∗ ) l ∗ ( θ ) cd ( θ ∗ )=1 cd ( θ ∗ )=2 for r > 1 a combinatorial characterization of intersecting divisors is missing ! 12