F-theory and the classification of elliptic Calabi-Yau manifolds

F-theory and the classification of elliptic Calabi-Yau manifolds FRG Workshop: Recent progress in string theory and mirror symmetry March 6-7, 2015 Washington (Wati) Taylor, MIT Based in part on arxiv: 1201.1943, 1204.0283, 1412.6112 D. Morrison, WT arxiv: 1205.0952 WT arxiv: 1404.6300 G. Martini, WT arxiv: 1406.0514 S. Johnson, WT arxiv: 1409.8295 A. Grassi, J. Halverson, J. Shaneson, WT papers to appear w/ J. Halverson and w/ Y. Wang W. Taylor F-theory and elliptic Calabi-Yaus 1 / 33

Basic structure of F-theory: Requires an elliptically fibered CY3 or CY4, π : X 3 B 2 or π : X 4 B 3, π 1 (p) = T 2 Geometry 6D, 4D supergravity. To construct an F-theory model: 1. Choose base B 2. Choose point in CS moduli space (Weierstrass model y 2 = x 3 + fx + g encodes IIB τ = χ + ie φ as fun. on B. A theme of this talk & much research over last five years: Focusing on geometry of B simplifications and insight Primary goals/results: A) Classifying and enumerating elliptic Calabi-Yau threefolds and fourfolds B) Generic features of N = 1 F-theory vacua in 6D and in 4D W. Taylor F-theory and elliptic Calabi-Yaus 2 / 33

What does generic mean? Two distinct aspects: I. Generic features of vacua with fixed B i.e., features present for arbitrary CS moduli ( non-higgsable structure ) II. Features present for most B s 6D: Finite number of B s, fairly clear global picture almost all B s have non-higgsable clusters, certain typical G s 4D: Picture emerging, surprisingly parallel to 6D picture but more complicated technically, many open questions First: I and II in 6D, where story is clear. Next: emerging related 4D story W. Taylor F-theory and elliptic Calabi-Yaus 3 / 33

and elliptic Calabi-Yau threefolds I. Non-Higgsable clusters in 6D Kodaira singularities in EF over divisor S i gauge factor G i from 7-branes determined by ord Si f, g For certain B 2, G nontrivial CS moduli Example: [Morrison/Vafa] F-theory on F 3 heterotic E 8 E 8 on K3 w/ (15, 9) instantons SU(3) with no matter (non-higgsable) In other cases, non-higgsable matter e.g. F-theory on F 7 heterotic w/ (19, 5) instantons G is generically E 7 w/ 1 2 56. Can t Higgs: can t satisfy D-term constraints When no smooth heterotic dual, non-higgsable product groups w/ matter e.g. G 2 SU(2) w/ (7 + 1, 1 2 2) matter W. Taylor F-theory and elliptic Calabi-Yaus 4 / 33

Geometry of non-higgsable groups The base B 2 is a complex surface. Contains homology classes of complex curves C i For C = P 1 = S 2, local geometry encoded by normal bundle O(m) e.g., N C = O(2) = TC : deformation has 2 zeros, C C = +2 If N C = O( n), n > 0, C is rigid (no deformations) For O( n), n > 2, base space is so curved that 7-branes must pile up to preserve Calabi-Yau structure on total space non-higgsable gauge group W. Taylor F-theory and elliptic Calabi-Yaus 5 / 33

NHC s useful in classification of compact (SUGRA) F-theory models (next) and in classification 6D SCFTs [cf. Morrison talk] W. Taylor F-theory and elliptic Calabi-Yaus 6 / 33 Classification of 6D Non-Higgsable Clusters (NHC s) [Morrison/WT] Clusters of curves imposing generic nontrivial codimension one singularities: m (m = 3, 4, 5, 6, 7, 8, 12) -3-2 -3-2 -2-2 -3-2 su(3), so(8), f 4 e 6, e 7, e 8 g 2 su(2) g 2 su(2) su(2) so(7) su(2) Identified using Zariski decomposition of nk e.g., -12 curve: K C = 10 K = (5/6)C + X over Q 4K = 4C + Y eff, 6K = 5C + Z eff e 8 Any other combination including -3 or below (4, 6) at point/curve

6D II. Classification/enumeration of elliptic Calabi-Yau threefolds Gross: finite number of topologically distinct elliptic CY threefolds (up to birational equivalence) Mathematical minimal model program to classify surfaces: 1. Given surface S, find -1 curve C (C C = 1), blow down S 2. Repeat until done minimal surface Grassi: minimal surfaces for B 2 bases: P 2, F m (m 12), Enriques ( K trivial) Program: start with P 2, F m, blow up to get all bases B 2, constrained by NHC s Given all bases, then consider tuned Weierstrass models all elliptic CY3 s W. Taylor F-theory and elliptic Calabi-Yaus 7 / 33

Found all toric B 2 Generic EF Hodge # s [Morrison/WT, WT] h 21 500 400 300 200 100 0 0 100 200 300 400 500 h 11 61,539 toric bases (some not strictly toric: -9, -10, -11 curves) Reproduces large subset of Kreuzer-Skarke database of CY3 Hodge # s Boundary of shield from generic elliptic fibrations over blowups of F 12. In principle: start with F m, P 2, blow up/tune all EF CY3 s W. Taylor F-theory and elliptic Calabi-Yaus 8 / 33

Computed all bases w/ C -structure [Martini/WT] Preserve only one C ( semi-toric ) 162, 404 distinct bases (including toric) Includes 6 CY s with new Hodge numbers (including above) Includes 13 with nonzero Mordell-Weil rank (r = 1-8, r = 3 above). Lesson: including branching, loops does not significantly increase complexity In principle: can get all B 2 s [Wang/WT to appear: all B s, h 2,1 > 200] W. Taylor F-theory and elliptic Calabi-Yaus 9 / 33

Systematic classification of EFS CY3s: Pick bases B 2, tune G, matter Start at large h 2,1 h 1,1 (X) = r + T + 2 h 2,1 (X) = H neutral 1 = 272 + V 29T H charged. Codimension 1, 2 sing. s G, matter r, V, H ch h 1,1, h 1,2 Blowups, tuning from minimal bases P 2, F m decrease h 2,1 Largest h 2,1 : F 12 (h 1,1 = 11, h 2,1 = 491) [ h 2,1 491 EFS CY3 s] Tune f, g enhanced G? If tune on D = as + bf : if D S 0, (4, 6) pt. only can tune on S = S + 12F S S = 12, S F = 1 S S = +12, F F = 0 SU(2) on S? Anomaly cancellation H ch = 82 2, (h 1,1, h 2,1 ) = (12, 318)... W. Taylor F-theory and elliptic Calabi-Yaus 10 / 33

To find further CY s with 350 h 1,1 491, must blow up (h 1,1, h 2,1 ) = (12, 462) Again, tuning G h 2,1 < 300 (no G on -1 s); blow up again Tune SU(2) on -1 curve C C C = 1 = 1 6 (4 x 2) K C = 1 = 1 3 (8 x 2/2) H ch = 10 2 (same CY: (h 1,1, h 2,1 ) = (13, 433)) (h 1,1, h 2,1 ) = (14, 416 = 433 20 + 3) Continue in this way... W. Taylor F-theory and elliptic Calabi-Yaus 11 / 33

EF CY3 s with h 2,1 350, F m + tuning complete classification [Johnson/WT] 11, 491 untuned Weierstrass SU 2 12, 462 SU 2 x SU 2 SU 3 x SU 2 or [cf. Berglund/Huang/Smith/WT to appear] G_2 x SU 2 SU 2 x SU 2 x SU 2 10, 376 19, 355 20, 350 Matches KS; non-toric + toric at (19, 355); new non-toric below 350 Empirical data on Calabi-Yau s suggests: most (known) CY s are elliptic, particularly at large Hodge numbers W. Taylor F-theory and elliptic Calabi-Yaus 12 / 33

Can we complete classification at all h 2,1? Limitations on existing technology Constructing all non-toric bases: General algorithm w/wang, issues remain at large h 1,1, small h 2,1. U(1) s [Mordell-Weil]: From global information, hard to compute. Much recent work. [Aluffi/Esole, Morrison/Park, Cvetic/Klevers/Piragua, Mayrhofer/Palti/Weigand... ] Morrison/Taylor, Cvetic/Klevers/Piragua/WT: U(1) s from Higgsing Tuning high rank I N 1 singularities: Algebraically difficult at largest values of N but seems tractable Classifying codimension two singularities: No systematic theory (most challenging math problem?) W. Taylor F-theory and elliptic Calabi-Yaus 13 / 33

Codimension 2 singularities: tricky, not completely classified Simple case: rank one enhancement [Katz-Vafa] e.g. SU(N) enhanced SU(N + 1) SU(N + 1) adjoint SU(N) Similar for e.g. A N 1 D N : SU(N) Realized by embedding of Dynkin diagrams in singularity Other cases more complex [Sadov, Morrison/WT, Esole/Yau] A N 1 A 2N 1 gives SU(N) + or adj + 1 Analyze through explicit resolution in specific cases [MT, Esole/Yau] [Grassi/Halverson/Shaneson: alternative: local deformation?] W. Taylor F-theory and elliptic Calabi-Yaus 14 / 33

Topology of matter representations Some reps are not understood in F-theory but seem okay from supergravity Example: SU(N) Could be associated with A 3 ˆD 6 [MT] But explicit singularity not known. Genus classification from anomaly structure [Kumar/Park/WT] 2g 2 = (K + C) C = R x R g R 2 g R = 1 12 (2C R + B R A R ) e.g. for box, g box = (N 1)(N 2)/2; = 3 for N = 4. (group theory coefficients: tr R F 2 = A R trf 2, tr R F 4 = B R trf 4 + C R (trf 2 ) 2 ) Suggests arithmetic genus 3 singularity in curve C on B 2 Confirm? Structure suggests Kodaira: Rep. theory cod. 2 sing. s [ HLMS boxes?] W. Taylor F-theory and elliptic Calabi-Yaus 15 / 33

Most bases B 2 have NHC s The only bases B 2 that lack NHC s are weak Fano = (generalized) del Pezzo Ten corresponding Hodge pairs (2 + T, 272 29T), T = 0,..., 9 16 toric bases out of 61,539 lack NHC s, 27 semi-toric from 162,404 h 21 500 400 toric bases, those lacking NHC s in orange 300 200 100 0 0 100 200 300 400 500 h 11 W. Taylor F-theory and elliptic Calabi-Yaus 16 / 33

Upshot for a 6D phenomenologist Physics of a typical 6D SUSY F-theory compactification T 25, rank G 35, G (G 2 SU(2)) 3 E 2 8 F 4 SU(3) SO(8) Non-Higgsable gauge groups are generic in two senses: G + matter are non-higgsable persist throughout CS moduli space Almost all B s have non-higgsable G, matter. No tuning necessary Geometric moduli space = physical moduli space, all branches connected Now how about 4D? Remarkably, story is closely parallel at level of geometry W. Taylor F-theory and elliptic Calabi-Yaus 17 / 33

4D F-theory compactifications Story parallel in many ways: Compactify on Calabi-Yau fourfold, base B 3 = complex threefold Underlying moduli space of EF CY fourfolds on rational B 3 {EFCY3 s} But: G-flux superpotential, lifts flat directions Brane worldvolume DOF nontrivial can e.g. break gauge group Today: focus on underlying fourfold geometry, geometric non-higgsable structure W. Taylor F-theory and elliptic Calabi-Yaus 18 / 33

4D non-higgsable clusters [Morrison/WT] (see also: Anderson/WT, Grassi/Halverson/Shaneson/WT) At level of geometry, similar to 6D but more complicated Toric bases: straightforward to compute using dual monomials to rays v i f, g span {m : m, v i 4, 6} Generally: 7-branes wrap divisors = surfaces S. Orders of vanishing of f, g depend on normal bundle N S. Expanding in coordinate z, S = {z = 0}, Up to leading non-vanishing term, f = f 0 + f 1 z + f 2 z 2 + f k O( 4K S + (4 k)n S ) g k O( 6K S + (6 k)n S ) Can do computations using geometry of surfaces W. Taylor F-theory and elliptic Calabi-Yaus 19 / 33

Example: non-higgsable SU(2) Take S = P 2, K S = 3H. Choose normal bundle N = 4H (analogue of -4 curve ) f 0 O( 4K S + 4N S ) = O( 4H), f 1 O(0) g 0 O( 6H), g 1 O( 2H), g 2 O(2H) O(nH) only has non-vanishing elements when n 0 (nh effective) f 0 = g 0 = g 1 = 0 (1, 2) vanishing SU(2) For global model, consider P 1 bundle over S = P 2, twist = 4H Threefold F 4, 3D analogue of Hirzebruch surfaces Single group clusters: SU(2), SU(3), G 2, SO(7), SO(8), F 4, E 6, E 7, E 8 In particular, cannot have: non-higgsable SU(5), SO(10) Note: SO(7) nontrivial, f 2 O(2X) 0, g 3 O(3X) = 0, X = 2K + N. W. Taylor F-theory and elliptic Calabi-Yaus 20 / 33

2-factor non-higgsable group products Either as an isolated 2-group cluster, or within larger clusters, the only 2-factor products that can appear are: G 2 SU(2), SO(7) SU(2), SU(2) SU(2), SU(3) SU(2), SU(3) SU(3) Example of constraints: why no G 2 SU(3)? ((2, 3) + (2, 2) = (4, 5)) Follows from monodromy conditions SU(3) : g 2 must be a perfect square. Non-Higgsable g 2 = cu 2, unique u G 2 : must have f 2 = cv 2, g 3 = dv 3, unique v Assume SU(3) on divisor A = {z = 0}, G 2 on divisor B = {w = 0} Must have leading term in g = z 2 w 3, on A, g 2 = w 3 u 2. W. Taylor F-theory and elliptic Calabi-Yaus 21 / 33

4D: clusters can have complicated structure Basic issue: On curve, all points homologous On surface, many distinct curves Can support independent matter Branching Unlike in 6D, cluster structure can have branching in quiver diagram e.g. (higher branchings also possible) W. Taylor F-theory and elliptic Calabi-Yaus 22 / 33

Clusters can also have long chains Example: multiple blowup of P 1 F 4 Similar for F 8 : SU(2) SU(3) 11 SU(2) W. Taylor F-theory and elliptic Calabi-Yaus 23 / 33

And closed loops Multiple blowup of P 1 P 1 P 1 Upshot: local structure restricted (9 groups, 5 products), but global structure can be very complex W. Taylor F-theory and elliptic Calabi-Yaus 24 / 33

Classification of elliptic Calabi-Yau fourfolds Mathematical minimal models Mori theory. No proofs, some exploration: w/halverson: P 1 bundles over toric bases B 2 w/wang: Monte Carlo exploration by blowing up/down toric bases w/huang, Wang: bases with large h 3,1 Structure seems closely analogous to EF CY threefolds minimal models F m but more complex Blow up curves, points: h 3,1, h 1,1 [Lynker, Schimmrigk, Wisskirchen] Hypersurfaces in weighted P 5 (1998) Generic EF CY 4: Lots of G, matter at level of geometry W. Taylor F-theory and elliptic Calabi-Yaus 25 / 33

Exploring threefold bases B 3 : P 1 bundles [Halverson/WT, to appear] Consider B 3 = P 1 bundle over a surface S. Characterized by twist T = c 1 (L), B 3 projectivization of L T = normal bundle of section = S. Smooth heterotic dual when S = generalized del Pezzo. Give non-higgsable G 1 G 2 on sections, G i E 8 (no common matter); Classification of dual smooth heterotic/f-theory models: Anderson/WT More generally, take S = B 2, which supports an elliptic CY3 Enumerated 177,703 distinct B 3 for toric B 2 Each represents distinct divisor S + N geometry Exploration of generic fourfolds and of local geometry W. Taylor F-theory and elliptic Calabi-Yaus 26 / 33

Hodge numbers of P 1 bundle B 3 s Most of these bases have NHC s, those without have small Hodge # s Set has atypically small h 1,1, generically expect more NHC s Contains many minimal threefold bases B 3 W. Taylor F-theory and elliptic Calabi-Yaus 27 / 33

Exploring threefold bases B 3 : Monte Carlo [Wang/WT, to appear] Start with e.g. B 3 = P 3, blow up and down randomly (Random walk on graph: samples vertices proportional to incident edges) Note: does not connect to all bases, need singular intermediaries but expect as in 6D, may give decent sampling of bulk Average rank of NHC 50 160 140 120 100 80 60 60 80 100 120 140 160 W. Taylor F-theory and elliptic Calabi-Yaus 28 / 33

Some statistics P 1 bundles P 1 bdl. S ± Monte Carlo SU(2) 0.39 0.33 0.47 G 2 0.27 0.18 0.31 F 4 0.16 0.17 0.09 SU(3) 0.002 0.04 0.11 SO(8) 0.003 0.04 0.01 E 8 0.07 0.11 0.01 Frequency of single non-higgsable factors Monte Carlo frequency of two-group product factors: G 2 SU(2) : 54% SU(2) SU(2) : 30% SU(3) SU(2) : 8% SU(3) SU(3) : 7% Statistics are very rough. Systematic issues with each, only covering subsets W. Taylor F-theory and elliptic Calabi-Yaus 29 / 33

Issues with non-higgsable clusters for fourfolds At level of geometry, 6D: geometric moduli space physical moduli space 4D: G-flux (Curvature of C 3 in M-theory description) superpotential W(φ) lifts moduli may drive vacua to enhanced symmetry points with G G NHC Additional DOF in F-theory not understood Branes on compact surface: DOF on world-volume Fluxes in w-volume can break non-higgsable G [e.g. in Anderson/WT] Non-commuting DOF [X, X] for adjoint scalars higher-dimensional branes Branes in curved compact spaces not well described No clear picture yet for how to merge these DOF with F-theory geometry One approach: T-branes [Cecotti/Cordova/Heckman/Vafa, Anderson/Heckman/Katz,Collinucci/Savelli,... ] W. Taylor F-theory and elliptic Calabi-Yaus 30 / 33

Realizing the standard model in F-theory Ignoring the outstanding issues of G-flux and seven-brane DOF, what are the options for realizing G = SU(3) SU(2) U(1)? 1. Tune the whole thing but not on divisors with NHC s 2. Tune part of G, get part from NHC; e.g. NH SU(3), tune SU(2) U(1) 3. Get all of G from non-higgsable structure NHC SU(3) SU(2) reasonably common, 8% of G 1 G 2 in MC study NHC U(1) open question (some NH U(1) s in 6D [Morrison/Park/WT]) Matter spectrum of NH realization of SU(3) SU(2) matches well to SM, given U(1), anomaly cancellation Unification SU(5), SO(10) cannot appear as NHC s. Can t enhance NHC SU(5) E 6,... possible for NHC s, could break e.g. from fluxes on branes. W. Taylor F-theory and elliptic Calabi-Yaus 31 / 33

Dark matter candidates Two possibilities: I) hidden sector dark matter, e.g. from a disconnected cluster II) WIMP dark matter (from SU(2) G, G = SU(2), SU(3), SO(7), SO(8)) W. Taylor F-theory and elliptic Calabi-Yaus 32 / 33

Conclusions: Systematic control of elliptic Calabi-Yau threefolds Can systematically construct elliptic threefolds starting from F m, P 2, blowing up and tuning by fixing Weierstrass coefficients, h 3,1, h 1,1 Constructions of toric, semi-toric, non-toric bases match KS data well Clear bounds on set of threefolds, systematic enumeration from large h 2,1 Some technical issues: Mordell-Weil, codimension 2/matter Story for fourfolds appears similar, but additional complexity. Generic appearance of gauge groups and matter leads to new physics questions Can we realize natural phenomenological scenarios with generic gauge/matter? Natural mechanisms to produce U(1)? Signals of non-higgsable dark matter? How does G-flux and the superpotential interact with non-higgsable clusters? W. Taylor F-theory and elliptic Calabi-Yaus 33 / 33