(Non-) Abelian Discrete Symmetries in String (F-) Theory Mirjam - PowerPoint PPT Presentation

Geometry and Physics of F-theory 2017 ICTP, Trieste, February 27-March 2, 2017 (Non-) Abelian Discrete Symmetries in String (F-) Theory Mirjam Cveti č

Outline (Summary) Progress report since F-theory’16, Caltech I. Abelian discrete gauge symmetries in F-theory multi-sections &Tate-Shafarevich group – highlight Z 3 highlight Heterotic duality and Mirror symmetry II. Non-Abelian discrete gauge symmetries in F-theory relatively unexplored stoop down to weakly coupled regime Non-Abelian discrete symmetries in Type IIB String Explicit construction of CY-threefold, resulting in a four-dimensional Heisenberg-type discrete symmetry

Abelian discrete symmetries in Heterotic/F-theory M.C., A. Grassi and M. Poretschkin, ``Discrete Symmetries in Heterotic/F-theory Duality and Mirror Symmetry,’’arXiv:1607.03176 [hep-th] Non-Abelian discrete symmetries in Type IIB string V. Braun, M.C., R. Donagi and M.Poretschkin, ``Type II String Theory on Calabi-YauManifolds with Torsion and Non-Abelian Discrete Gauge Symmetries,’’ arXiv:1702.08071 [hep-th]

Abelian Discrete Symmetries in F-theory Calabi-Yau geometries with genus-one fibrations These geometries do not admit a section, but a multi-section Earlier work: [Witten; deBoer, Dijkgraaf, Hori, Keurentjes, Morgan, Morrison, Sethi;…] Recent extensive efforts ’14-’16: [Braun, Morrison; Morrison, Taylor; Klevers, Mayorga-Pena, Oehlmann, Piragua, Reuter; Anderson,Garcia-Etxebarria, Grimm; Braun, Grimm, Keitel; Mayrhofer, Palti, Till, Weigand; M.C., Donagi, Klevers, Piragua, Poretschkin; Grimm, Pugh, Regalado; M.C., Grassi, Poretschkin;…] Key features: Higgsing models w/U(1), charge-n < F > ≠ 0 − conifold transition Geometries with n-section Tate-Shafarevich Group Z n Z 2 [Anderson,Garcia-Etxebarria, Grimm; Braun, Grimm, Keitel; Mayrhofer, Palti, Till, Weigand’14] Z 3 [M.C.,Donagi,Klevers,Piragua,Poretschkin 1502.06953]

Tate-Shafarevich group and Z 3 [M.C., Donagi, Klevers, Piragua, Poretschkin 1502.06953 ] Only two geometries: X 1 w/ trisection and Jacobian J(X 1 ) X 1 with tri-section (cubic in P 2 ) Jacobian x P J(X) Jacobian X 1 with tri-section There are three different elements of TS group! (cubic in P 2 ) Shown to be in one-to-one correspondence with three M-theory vacua.

Discrete Symmetries & Heterotic/F-theory Duality [ Morrison,Vafa ‘96; Friedman,Morgan,Witten ’97] Basic Duality (8D): Manifest in stable degeneration limit: K3 surface X splits into Heterotic E 8 x E 8 String on T 2 two half-K3 surfaces X + and X - dual to X − X F-Theory on elliptically fibered X + K3 surface X x Dictionary: X + and X - à background bundles V 1 and V 2 • • Heterotic gauge group G = G 1 x G 2 G i = [E 8 ,V i ] P 1 K3-fibration over • The Heterotic geometry T 2 : at intersection of X + and X - (moduli)

Heterotic/F-theory Duality [Morrison, Vafa ’96], [Berglund, Mayr ’98] Employ toric geometry techniques in 8D/6D to study stable degeneration limit of F-theory Toric polytope: Dual polytope: specifies the ambient specifies the elements of O( -K X ) - space X monomials in ambient space 6D: fiber this construction over another P 1 Study: U(1)’s [M.C., Grassi, Klevers, Poretschkin, Song 1511.08208] at F-theory’16, Caltech Discrete symmetries [M.C., Grassi, Poretschkin 1607.03176] highlights here

Discrete Symmetry in Heterotic/F-theory Duality [M.C., Grassi, Poretschkin 1607 .03176] Goal: Trace the origin of discrete symmetry D for P 2 (1,2,3) fibration • Conjecture [Berglund, Mayr ’98] X 2 elliptically fibered, toric K3 with singularities (gauge groups) of type G 1 in X + and G 2 in X - its mirror dual Y 2 with singularities (gauge groups) of type H 1 in X + and H 2 in X - with H i =[E 8 , G i ] • Employ the conjecture to construct background bundles with structure group G where D=[E 8 , G] beyond P 2 (1,2,3) • Explore ``symmetric’’ stable degeneration with G 1 =G 2 à symmetric appearance of discrete symmetry D

Example with Z 2 symmetry Dual polytope: Polytope: (monominals of the ambient space) . . . . . . . . . . . 2 - gauge symmetry ) 2 - gauge symmetry 8D: ( 2) ) / Z 2 ) y (( E 7 × SU(2) ) / Z 2 ) . . . . . . . . . . . 2 - vector bundle ( ) 2 - vector bundle bient space P (1 , 1 , 2) . . . . . . . . . y (( E 7 × SU(2) ) / Z 2 ) 2) ) / Z 2 ) bient space P (1 , 1 , 2) . . . . . . . . . . . . . . . . . . . . 6D : - gauge symmetry - gauge symmetry 2) ) / Z 2 ) y (( E 7 × SU(2) ) / Z 2 ) bient space P (1 , 1 , 2) . . . . . . . . . Field theory: Higgsing symmetric U(1) model: only one (symm. comb.) U(1)-massless à only one Z 2 -``massless’’

Example with Z 3 symmetry Dual polytope: Polytope: ularities - gauge symmetry 6D : ( E 6 × E 6 × SU (3)) / Z 3 - gauge symmetry als Z 3 . These examples demonstrate: toric CY’s with MW torsion of order-n, via Heterotic duality related to mirror dual toric CY’s with n-section. Related: [Klevers, Peña, Piragua, Oehlmann, Reuter ‘14]

Non-Abelian Discrete Symmetries – less understood F-theory - limited exploration [Grimm, Pugh,Regalado ’15], c.f., T. Grimm’s talk [M.C., Lawrie, Lin, work in progress] [M.C., Donagi, Lin, work progress] stoop down to weak coupling Type II string compactification Important progress in these directions builds on the work [Camara, Ibanez, Marchesano ’11] Abelian discrete gauge symmetries realized on Calabi-Yau threefolds with torsion. Non-Abelian Heisenberg-type discrete symmetries realized on Calabi-Yau threefolds with torsion classes that have specific non-trivial cup-products. [Berasaluce-Gonzales, Camara, Marchesano, Regalado, Uranga ’12]

Calabi-Yau threefold X 6 withTorsion Torsion ( H 5 ( X 6 , Z )) Torsion ( H 2 ( X 6 , Z )) Z )) = Z k , ' example Torsion ( H 4 ( X 6 , Z )) Torsion ( H 3 ( X 6 , Z )) Z )) = Z k 0 ' [Camara, Ibanez, Marchesano ’11] w/ Let ρ 2 , β 3 , ˜ ω 4 , and ζ 5 represent the generators of the torsion cohomologies orsion Z , Torsion Z , Torsion and Torsion Z = d ˜ ρ 4 = k ζ 5 , d γ 1 k ρ 2 , k − 1 and k ′ − 1 torsion linking numbers k 0 ˜ d ω 2 = k 0 β 3 , = d α 3 ω 4 , (consequence of re γ 1 , ω 2 , α 3 and ˜ ρ 4 are non-closed forms satisfying: expressions for torsion ectively, and they satisfy: linking numbers) Z Z Z Z γ 1 ^ ζ 5 = ρ 2 ^ ˜ ρ 4 = α 3 ^ β 3 = ω 2 ^ ˜ ω 4 = 1 X 6 X 6 X 6 X 6 Upon Type IIB KK reduction of C 2 , B 2 , C 4 gauge potentials on X 6 � G ij η i µ η µ j à Z k x Z k’ discrete symmetry, realized in the Stückelberg mass [Berasaluce-Gonzales, Camara, Marchesano, Regalado, Uranga ’12] When: , M non-vanishing ρ 2 ^ ρ 2 = M ˜ ω 4 , M 2 Z Upon KK reduction, Heisenberg discrete symmetry specified by k, k’, M: ∂ µ b i � k A i η i = i = 1 , 2 , µ , � G ij η i µ η µ j w/ µ ∂ µ b 3 � k 0 A 3 µ � Mb 2 ( ∂ µ b 1 � k A 1 η 3 = µ ) µ

Calabi-Yau threefold X 6 withTorsion Torsion ( H 5 ( X 6 , Z )) Torsion ( H 2 ( X 6 , Z )) Z )) = Z k 1 ⇥ Z k 2 , ' another example Torsion ( H 4 ( X 6 , Z )) Torsion ( H 3 ( X 6 , Z )) Z )) = Z k 3 , ' When: , w/ M non-vanishing ρ 2 ^ ρ 2 = M ˜ ω 4 , M 2 Z à Heisenberg discrete symmetry specified by k 1 , k 2 , k 3 and M: ∂ µ b 2 ( i ) � τ∂ µ b 1 A 2 µ ( i ) � τ A 1 � � = ( i ) + k i , i = 1 , 2 η µ ( i ) � G IJ ∗ η I µ η µ J ∗ µ ( i ) w/ ∂ µ b 3 + k 3 A 3 η 3 b 2 (1) � τ b 1 k 2 A 1 � � = µ � M µ (2) . (1) µ , which take th and τ = C 0 +ie − φ deno [Grimm, Pugh, Regalado ’15] This structure resul Dilaton-axion coupling

Non-Abelian Discrete Symmetry in Type IIB Requires the study of Calabi-Yau threefolds with torsion by determining torsion cohomology groups and their cup-products à technically challenging Choose a specific Calabi-Yau threefold X 6 : free quotient of T 6 by a fixed point free action of Z 2 × Z 2 : [part of a general classification [ Donagi, Wendland’09 ]] � z 0 + 1 � g 1 : ( z 0 , z 1 , z 2 ) 7! 2 , � z 1 , � z 2 , � � z 0 , z 1 + 1 2 , � z 2 + 1 � g 2 : ( z 0 , z 1 , z 2 ) 7! st known example of 2 =x i + 𝜐 i y i C / ( Z + τ i Z ) 3 z i , action of the group i=0,1, 2

Computation of cup-products: Strategy: relate X 6 to a submanifold Y 0 : Y 0 , ! X, ( x 0 , x 1 , x 2 , y 0 ) 7! ( x 0 + ⌧ 0 y 0 , x 1 , x 2 ) , z i =x i + 𝜐 i y i four-dimensional sub-torus quotient, invariant under Z 2 × Z 2 . Cup-products in Y 0 could be done explicitly by hand, but obtained as part of a computational scheme (cellular model à co-chains of cubical cells) that gives, among others, full integer cohomology of X 6 . Restriction i ∗ : H ∗ (X 6 , Z) → H ∗ (Y 0 , Z) - surjective, and exhibits H ∗ (Y 0 , Z) as a direct summand of H ∗ (X 6 , Z), along with the multiplicative structure of the cohomlogy ring of Y 0 Determine cup-products of H 2 (X 6 , Z) torsion classes that are non-vanishing in H 4 (X 6 , Z).