TexasA&Mworkshop,2015 AspectsofAbelianandDiscreteSymmetries inf<theorycompac+fica+on MirjamCve+č
F<TheoryCompac+fica+onswithAbelianSymmetries [Ellip+cFibra+onswithn<ra+onalsec+ons<U(1) n "rnknmordell<weil(mw)group] Based on: arxiv:1303.6970 [hep-th]: M. C., Denis Klevers, Hernan Piragua arxiv:1307.6425 [hep-th]: M. C., D. Klevers, H. Piragua (rnk 2 MW on Calabi-Yau threefolds U(1) 2 ) arxiv:1306.0236 [hep-th]: M. C., Antonella Grassi, D. Klevers, H. Piragua (rnk 2 MW on Calabi-Yau fourfolds; inclusion of flux) arxiv:1310.0463 [hep-th]: M.C., D. Klevers, H. Piragua, Peng Song (rnk 3 MW) arxiv:1503.02068 [hep-th]: M.C., D.Klevers, Damian. M.Peña, (particle physics models) P. Konstantin Oehlmann, Jonas Reuter arxiv:1505.[hep-th]: M.C., D. Klevers, H. Piragua, Wati Taylor (rnk2mwand``un<higgsing ) F-Theory Compactifications with Discrete Symmetries [Elliptic Fibrations with n-section Z n " rnk n Tate-Shafarevich (TS) Group] Based on: arxiv:1502.6970[hep-th]: M. C., Ron Donagi, Denis Klevers, Hernan Piragua, Maximilian Poretschkin (rnk 3 TS on Calabi-Yau threefolds Z 3 ) related
TypeIIBperspec+ve F<THEORYBASICINGREDIENTS
F<theory M-theory On T 2 Limit vol(t 2 ) 0 on S 1 F-theory = Type IIB elliptically fibered Calabi-Yau manifold back-reacted (p,q)7-branes regions with large g s on non-cy space
F<theory M-theory on S 1 Type II A On T 2 Limit vol(t 2 ) 0 E 8 xe 8 Het. Certain setups F-theory = Type IIB elliptically fibered Calabi-Yau manifold Certain setups back-reacted (p,q)7-branes regions with large g s on non-cy space SO(32) Het. S-duality Type I
F<theory<geometricSL(2,Z)invariantformula+onofTypeIIBstringtheory: invariantgeometricobjectistwo<torust 2 (τ) 1 Thestringcoupling(axio<dilaton): C 0 + ig 1 modularparameteroftwo<torust 2 (τ)<sl(2,z) Compac+fca+onisatwo<torusT 2 (τ)<fibra+onoveracompactbasespaceb: τ(x,y,z) Weierstrassform: F<theoryCompac+fica+on:BasicIngredients f,g<func+onfieldsonb y 2 = x 3 + fxz 4 + gz 6 [z:x:y]homog.coords.onp 2 (1,2,3) s B D7 (p,q)7 O7 [Vafa]
F<theory<geometricSL(2,Z)invariantformula+onofTypeIIBstringtheory: invariantgeometricobjectistwo<torust 2 (τ) 1 Thestringcoupling(axio<dilaton): C 0 + ig 1 modularparameteroftwo<torust 2 (τ)<sl(2,z) Compac+fca+onisatwo<torusT 2 (τ)<fibra+onoveracompactbasespaceb: τ(x,y,z) Weierstrassform: F<theoryCompac+fica+on:BasicIngredients y 2 = x 3 + fxz 4 + gz 6 singular T 2 (τ)-fibr. g s # loca+onof7<branes s B D7 (p,q)7 O7 [Vafa]
F<theoryCompac+fica+on:BasicIngredients TotalspaceofT 2 (τ)<fibra+on:singularellip+ccalabi<yaumanifoldx D=4,SUSYvacua:fourfoldX 4 [D=6,SUSYvacua:threefoldX 3 ] X 4 <singulari+esencodecomplicatedset<upofintersec+ng7<branes: Gaugetheory(ADE ) inco<dim.oneinb 3 (7<branes) S Yukawacouplings inco<dim.threeinb 3 pt = S \ S 0 \ S 00 S 0 S pt B S 00 S \ S 0 Majerinco<dim. twoinb 3 [Katz,Vafa] (intersec+ng7<branes) G 4 <fluxatintersec+on induceschiral4dmajer
F<theory&Par+clePhysics MOTIVATION
F<theoryandPar+clePhysics Abroaddomainofnon<perturba+vestringtheorylandscape withnewpromisingpar+clephysics&cosmology Recent past, primary focus on [SU(5)] GUT s ( 08.): Local model building: Global model building: [Donagi,Wijnholt;Beasley,Heckman,Vafa; Font,Ibanez; Hayashi,Kawano,Tsuchiya,Watari,Yamazaki; Dudas,Palti; Cecotti,Cheng,Heckman,Vafa; ] [Blumenhagen,Grimm,Jurke,Weigand; Marsano,Saulina,SchäferNameki;Grimm,Krause,Weigand; M.C.,Halverson,Garcia-Etxebarria;...] Standard Model [Lin,Weigand] 3-family Standard, Pati-Salam, Trinification Models [M.C., Klevers, Peña, Oehlmann, Reuter] [Vafa; Vafa,Morrison, ] Employinggeometrictechniquesforellip+callyfibered Calabi<Yaumanifoldsand/orduali+estodetermine Gauge symmetries, matter repres.& multiplicities, Yukawa couplings,
F<theory&AbelianGaugeSymmetries MOTIVATION
AbelianSymmetriesinF<theory Physics:importantingredientoftheStandardModelandbeyond Mul+pleU(1) sdesirable Formaldevelopments:newCYellip+cfibra+onswithra+onalsec+ons While non-abelian symmetries extensively studied ( 96 ) [Kodaira; Tate;Morrison,Vafa; Bershadsky,Intriligator,Kachru,Morrison,Sadov,Vafa; Candelas,Font, ] : Abelian sector rather unexplored A lot of recent progress ( 11-15): [ Morrison,Park;.Lawrie,Schäfer-Nameki; Borchmann,Mayrhofer,Palti,Weigand; U(1) M.C.,Klevers,Piragua; Grimm,Kapfer,Keitel;Braun,Grimm,Keitel; MC,Grassi,Klevers,Piragua; U(1) 2 Borchman,Mayrhofer,Palti,Weigand; M.C.,Klevers,Piragua,Song; U(1) 3.Morrison,Taylor; ]
Non<AbelianGaugeSymmetry 1.Weierstrassformforellip+cfibra+onofX y 2 = x 3 + fxz 4 + gz 6 2.SeverityofsingularityalongdivisorSinB: [Kodaira;Tate;Vafa;Morrison,Vafa; Bershadsky,Intriligator,Kachru,Morrison,Sadov,Vafa; ] [ord S (f),ord S (g),ord S (Δ)] Singularity type of fibration of X 3.Resolu+on:singularitytypestructureofatreeof"""" s"overs" """"""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""( "Dynkin"diagram)" I n -singularity (" SU(n)): P 1 1 P 1 2 P1 3 P 1 4 Recent refinements: [Esole,Yau; ;Hayashi,Lawrie,Schäfer-Nameki,Morrison; ] - Cartan generators for A i gauge bosons: in M-theory via Kaluza-Klein (KK) reduction of C 3 forms along (1,1)-forms i $ P 1 on X C 3 A i i i - Non-Abelian generators: light M2-brane excitations on s P 1 P 1 B [Witten]
U(1) s<abeliansymmetry U(1)gaugebosonsA m shouldalsoariseviakk<reduc+on C 3 A m m. (1,1)<formsonX Forbidnon<Abelianenhancement(byM2 swrapping s):onlyi 1 <fibers P 1 m
U(1) s<abeliansymmetry&ra+onalsec+ons U(1)gaugebosonsA m shouldalsoariseviakk<reduc+on C 3 A m m. (1,1)<formsonX m [Morrison,Vafa ] (1,1) - form rational section Ra+onalpointQonellip+ccurveEwithzeropointP m issolu+oninfieldkofweierstrassform (x Q,y Q,z Q ) y 2 = x 3 + fxz 4 + gz 6 Ra+onalpointsformagroup(underaddi+on)onE Mordell<Weilgroupofra+onalpoints P Q
U(1) s<abeliansymmetry&mordell<weilgroup 2. Q on E induces a rational section ŝ Q : B X of elliptic fibration B Q ŝ Q ŝ Q B ŝ Q gives rise to a second copy of B in X: new divisor B Q in X
U(1) s<abeliansymmetry&mordell<weilgroup 2. Q on E induces a rational section ŝ Q : B X of elliptic fibration B Q ŝ Q ŝ Q B ŝ Q gives rise to a second copy of B in X: new divisor B Q in X (1,1)-form m constructed from divisor B Q (Shioda map) indeed (1,1) - form m rational section
StructureofEllip+cFibra+onswithRa+onalSec+ons Weierstrass form (WSF) w/ rat. point Q= Conifold singularity at codimension two in B: -Factorization: (y y Q )(y + y Q )=(x x Q )(x 2 + x Q x + fz 4 Q + x 2 Q) y Q = fz 4 Q +3x 2 Q =0 - Singularity: WSF singular at Q [x Q : y Q : z Q ] I 2 fiber Singular fiber fz 4 Q +3x 2 Q =0 Q Q resolved Q y Q =0 B c mat w/isolated (matter) curve Section ŝ Q implies U(1)-charged matter
Ellip+ccurveswithrank<2Mordell<Weilgroup EXAMPLEWITHU(1) 2 [TWORATIONALSECTIONS]
Rnk2:Concreteexample Ellip+ccurveEwithtwora+onalpointsQ,R Consider line bundle M=O(P+Q+R) of degree 3 on E (non-generic cubic in P 2 ) naturalrepresenta+onashypersurfacep=0indelpezzodp 2 [u:v:w:e 1 :e 2 ] homogeneouscoordinatesofdp 2 (blow<upofp 2 w/[u :v :w ]at2points:u =ue 1 e 2,v =ve 2,w =we 1 ) [M.C., Klevers,Piragua] uvwe 1 e 2 P : E 2 \ p =[ s 9 : s 8 : 1 : 1 : 0], Q : E 1 \ p =[ s 7 :1:s 3 : 0 : 1], R : D u \ p =[0:1:1: s 7 : s 9 ]. Pointsrepresentedbyintersec+onsof differentdivisorsindp 2 withp [generalizations: M.C., Klevers, Piragua, Taylor]
Construc+onofCYEllip+cFibra+ons [M.C.,Klevers,Piragua;M.C.,Grassi,Klevers,Piragua] CY-fibrations X with rank 2 curve in dp 2 Related work: [Borchmann,Mayrhofer,Palti,Weigand] threesec+ons ŝ P, ŝ Q, ŝ R p=0 in dp 2 dp 2 coordinates [u:v:w:e 1 :e 2 ] and coefficients s i lifted to sections on B CYmanifoldX"topologicallydeterminedbydivisors, S 7 =(s 7 ) S 9 =(s 9 ).
ClassifyallvacuawithfixedEindP 2 &chosenbasebind=6andd=4 Example:D=4, : Construc+onofCYEllip+cFibra+ons B = P 3 1.Xgeneric[alls i exist,generic]:u(1)xu(1) S 7 = n 7 H P 3 S 9 = n 9 H P 3 2.X non<generic[s i, realizesu(5)att=0]:su(5)xu(1)xu(1) s 1 = t 3 s 0 1 s 2 = t 2 s 0 2 Can construct and study entire family of CY s explicitly s 3 = t 2 s 0 3 s 5 = ts 0 5
ChargedMajer U(1) 2 Chargedmajeratcodim<twosingulari+esinB Majerlocusat: y Q = fzq 4 +3x 2 Q =0 TypeA:majeratlociinBwheresec+onsareill<defined TypeB:majeratlocicharacterizedbyaddi+onalconstraints TypeA:lociinBwherethesec+onsareill<defined sec+onsnolongerpointsine, wrapen+rep 1 insmoothx Representa+on Ill<definedsec+on iv. iv.(q 1,q 2 )=(<1,1) Q fz 4 Q +3x 2 Q =0 y Q =0 c mat v. B c mat v.(q 1,q 2 )=(0,<2) vi.(q 1,q 2 )=(<1,<2) R zerosec+onp vi. c mat
F<theory&DiscereteGaugeSymmetries MOTIVATION
F<theoryandDiscreteSymmetries Physics:importantingredientofthebeyondtheStandardModelphysics forbid terms for fast proton decay and other R-parity violating terms, e.g., R-parity (Z 2 ), baryon triality (Z 3 ) and proton hexality (Z 6 ) Discrete symmetries must be realized as discrete gauge symmetries, typically arising as a remnant of a broken U(1) gauge symmetry Geometry:newCalabi<Yaugeometrieswithgenus<onefibra+ons These geometries do not admit a section, but only a multi-section Earlier work:[witten; deboer,dijkgraaf, Hori, Keurentjes, Morgan, Morrison, Sethi;.] Recent extensive efforts ( 14-15): [Braun, Morrison; Morrison, Taylor; Klevers, Pena, Oehlmann, Piragua, Reuter; Anderson,Garcia-Etxebarria, Grimm; Braun, Grimm, Keitel; Mayrhofer, Palti, Till, Weigand; ] & This talk [M.C., Donagi,Klevers,Piragua,Poretschkin]
F<theoryandDiscreteSymmetries AnaturalobjectajachedtosuchF<theorycompac+fica+onsisthe Tate<Shafarevichgroup,adiscretegroupthatorganizesinequivalent genus<onegeometrieswhichsharethesamejacobian. ExpectedthattheTate<Shafarevichgroupcoincideswiththeresul+ng discretesymmetry,though,f<theoryeffec+veac+onisthesameforall thesegeometries. In contrast, M-theory can distinguish between these geometries, due to a discrete flux that can be turned on the compactification circle from, say, 6D F-theory compactification to 5D. [default 6D5D] Expected that the different M-theory vacua are in one to one correspondence with the elements of the TS group.
Abelian&DiscreteGaugeSymmetryinF<theory F<theorycompac+fica+onwithnsec+ons(AbelianU(1) (n<1) ) [Morrison, Taylor; Anderson, García-Etxebarria, Grimm, Keitel; Braun, Grimm, Keitel] connectedviaconifoldtransi+on F<theorycompac+fica+onwithann<sec+on(discreteZ n symmetry)
Abelian&DiscreteGaugeSymmetryinF<theory F<theorycompac+fica+onwithnsec+ons(AbelianU(1) (n<1) ) Independent Sections connectedviaconifoldtransi+onto F<theorycompac+fica+onwithann<sec+on(discreteZ n symmetry) I 2 fibers occur at certain co-dimension two loci in the base of the elliptic fibration
F<theorycompac+fica+onwithnsec+ons(AbelianU(1) (n<1) ) Abelian&DiscreteGaugeSymmetryinF<theory Independent Sections blow-down (P 1 in the geometry with multiple sections collapses) F<theorycompac+fica+onwithann<sec+on(discreteZ n symmetry) Deformation (S 3 glues several sections to a multi-section) Conifold transition - Geometry
F<theorycompac+fica+onwithnsec+ons(AbelianU(1) (n<1) ) Abelian&DiscreteGaugeSymmetryinF<theory I Independent Sections I 2 -fiber blow-down (P 1 in the geometry with multiple sections collapses) appearanceofmasslessfield F<theorycompac+fica+onwithann<sec+on(discreteZ n symmetry) Deformation (S 3 glues several sections to a multi-section) massless field acquires VEV Conifold transition - Effective theory
Abelian&DiscreteGaugeSymmetryinF<theory F<theorycompac+fica+onwithnsec+ons(AbelianU(1) (n<1) ) blow-down connectedviaconifoldtransi+onto deformation Geometry Effective field theory F<theorycompac+fica+onwithann<sec+on(discreteZ n symmetry) Since Abelian symmetries better understood (c.f., recent works) most efforts focus on the geometry and spectrum of U(1) (n-1), to deduce, primarily via effective field theory, implications for Z n. Approach also taken here
WhyZ n symmetrywithn>2? RecentworkprimarilyfocusedonthesimplestcaseofZ 2. [TheTate<Shafarevichgroupconsistsonlyoftwoelementsassociatedwith onegenus<onefibra+onanditsjacobian.] Goal:toextendtheanalysisbeyondZ 2.FocusprimarilyonZ 3. ExpectthesenewfindingswillplayakeyroleforZ n (n>3),too. FocusonobtainingdifferentM<theoryvacuabygeometrically iden+fying(onthemul+plesec+onside)thescalarfieldswhosevev s giverisetothethreeinequivalentfieldtheoriesin5d.
Tate<ShafarevichGroupandF<Theory The Tate-Shafarevich group is defined as = {Genus-one fibered Calabi-Yau manifolds X i ; so that the Jacobian of X i is J(X) } An element X i of consists of a triple: X i = (X i ; f i ; a i ): X i is a genus-one fibration over a base B f i : X i B is the fibration map a i : J(X) x X i X i is the Jacobian action Two geometries with different actions of the Jacobian differ as elements in
Tate<ShafarevichgroupandZ 3 inf<theory Jacobian X 1 with tri-section x P J(X) Jacobian Three different elements of TS group X 1 with tri-section But only two geometries J(X) and X 1
[Anderson,Garcia-Etxebarria, Grimm; Braun,Grimm,Keitel; Mayrhofer, Palti, Till, Weigand] F<Theory/M<TheoryFluxedCircle Compac+fica+onDiagramZ 3 [M.C.,Donagi,Klevers,Piragua,Poretschkin]
F/M-theory Fluxed Circle Compactification (Z 3 ) 6D F-theory [flux-wilson line] [massive flux] M-theory 5d KK [J(X)] [X 1 ] [X 1 ] 6D matter w/ charge 3 KK expanded tower w/ KK mass
F<THEORYwithU(1)&CHARGE3MATTER
F<THEORYwithU(1)&Charge3majer [Klevers, Mayorga-Pena, Oehlmann, Piragua, Reuter] Toricblow<upofsingulardP 1 [cubicinp 2 ] U(1) KK U(1) 5D c 1 c 2 U(1) 6D In M-theory two correct matter charges for Higgsing (with flux ξ= -1/3; -2/3)
Inves+ga+ngallM<theoryphases Problem: 3curvesneeded,butI2<fiberonlyhastwora+onalcomponents Onlytwocurves,c 1 andc 2, withcharges(2,<1)and(<1,2)manifest curvewithcharge(3,0)isnotvisibleinthetoricresolu+on [curveintheclass[c 1 +T 2 ]wouldhavetherightcharges] Computa+onoftheGromov<Wijeninvariantreveals N c 1+T 2 =1 Where is the geometry with the charge (3,0) curve manifest?
Inves+ga+ngallM<theoryphases Strategy: Lookatdifferentphaseswherezerosec+onis holomorphic,i.e.lookatalldifferentresolu+onphases (relatedtotheoriginalonebyfloptransi+on).[witten] ExemplifyhereforZ 3. Thisstrategyisexpectedtomakeallcurvesmanifestand shouldapplytoz n (w/n>3).
FindingtheThirdCurve Resolu+onsofthesingulardP 1 viacompleteintersec+on U(1) KK U(1) 5D U(1) 6D In M-theory, correct matter charges for Higgsing (with flux ξ=0) to U(1) x Z 3
Summary SpecificTechnicalResults: Explicitlyconstructthethreedifferentcurvesthatyieldthefields withdesiredchargesandconsidertheblow<down;crucialto considertwodifferentresolu+onsofthesingulardp 1 geometry, relatedbyfloptransi+on. Whereasonecanexpecttoseeatmosttwovacuainone resolu+onexplicitly,wecandetectthehiddenonebycompu+ng thegromov<wijeninvariantofthecorrespondingcurveclass (whichwasdone). WealsodemonstratethatthethreedifferentM<theoryvacuado notleadtothreedifferentgeometriesbutmustbedis+nguished bytheac+onofthejacobianontheremainingtwonon<trivial elementsofthetate<shafarevichgroup.
OpenQues+ons Physicalinterpreta+onoftheJacobianac+ononthegenus<one geometries?[crucialtodifferen+atethetwoelementsofthetate< Shafarevichgroupwhicharegeometricallyiden+cal.] Temp+ngtoiden+fythesumopera+onofTSgrouptotheaddi+onofthe discretefluxesonthecompac+fica+oncircle. AsthethirdM<theoryvacuumcorrespondstodeforma+onviaVEV ofthemodewithcharges(3,0),itexhibitsaz 3 in5d expectedthatthecorrespondinggeometryhasnontrivial torsionalhomology;geometricalinsightss+lllacking [work in progress: M.C.,Donagi,Poretschkin] Generaliza+ontoZ n (n>3);expectdifferentn<sec+ongeometries; Fieldtheoryofhigherchargefields(AbelianU(1) 3 constr.z 4 ) [M.C.,Klevers,Piragua,Song]