The Power of F-theory:

Transcription

1 The Power o F-ory: Anomalies & Gauge Coupling Functions Thomas W. Grimm Utrecht University & Max Planck Institute or Physics Based on: to appear with Pierre Corvilain, Diego Regalado with Sebastian Greiner String Pheno Conerence, July

2 Outline o Talk Introduction and general comments Result 1: Anomaly cancellation in F-ory Result 2: Computing gauge coupling unctions Conclusions 2

3 Introduction and general comments 3

4 String phenomenology using F-ory F-ory is an ideal arena or string phenomenology: Key is geometrization o bulk and brane physics, including gauge ories with all types o gauge groups SU(N), Sp(N), SO(N), E 6, E 7, E 8 etc. and allows explicit construction o models (GUTs, SM) Using geometry a classiication o appearing models can be approached: e.g. progress on matter spectra, U(1) symmetries, discrete symmetries, The set o constructed models is so large that general claims can be attempted what eatures are allowed/generic in string compactiications A promising uture? I believe: re is currently no or approach to obtain 4D models in string ory that gives such a complete and rich picture (mostly discrete data so ar) computations require beautiul complicated (deep mamatics and powerul) mamatics derivation o continuous coupling unctions will be decisive in judging relevance o F-ory ramework to phenomenology 4

5 What is F-ory? Most practicable deinition: F-ory is M-ory on a two-torus with vanishing volume v Type IIB string ory comes into game via duality (1) reduction on circle 1: M-ory Type IIA (2) T-duality along circle 2: Type IIA Type IIB on circle Important remarks: torus is allowed to pinch (seven-brane in IIB), can take quotient (Klein bottle) shape o torus can vary rom point to point in nine dimensions existence o a higher-dimensional ory is not as trivial as it sounds 5

6 What is F-ory? Well-known slogan: F-ory compactiications on elliptically-ibered Calabi-Yau ourolds yield 4D ories with minimal N=1 supersymmetry pinching o two-torus indicates location o seven-branes in base B brane and bulk physics encoded by singular complex geometry Important remarks: as o now M-ory deinition is only deinition to study general acts about eective actions o F-ory backgrounds eective ories rom M-ory are not in singular limit 3D ories rar push 4D ories into 3D limit complete KK reduction on S 1 6

7 How to study questions in F-ory? First step: approach M-ory via 11D supergravity on a smooth geometry resolution o singular Calabi-Yau geometry [almost everyone who has worked on classiication o resolutions at each co-dimensions in base F-ory, ] resolution trees co-dimension 1 in base non-abelian gauge group co-dimension 2 in base matter in representations Abelian gauge group actors: n U(1) +1 rational sections o ibration Second step: Perorm a complete Kaluza-Klein reduction (4D 3D) - push ory into 3D Coulomb branch, keep track o all massive states need to be integrated out to make match possible 7

8 How to study questions in F-ory? First step: approach M-ory via 11D supergravity on a smooth geometry resolution o singular Calabi-Yau geometry [almost everyone who has worked on classiication o resolutions at each co-dimensions in base F-ory, ] resolution trees co-dimension 1 in base non-abelian gauge group co-dimension 2 in base matter in representations Abelian gauge group actors: n U(1) +1 rational sections o ibration Second step: Perorm a complete Kaluza-Klein reduction (4D 3D) - push ory into 3D Coulomb branch, keep track o all massive states need to be integrated out to make match possible Third step: Compare 3D ories and draw conclusions 7

9 Result 1: Anomaly cancellation in F-ory 8

10 Anomalies in QFTs Anomalies arise i classical symmetry does not persist at quantum level our-dimensional ories have pure gauge and gauge-gravity anomalies A µ chiral ermions A A + A µ axion A A = 0 Green-Schwarz mechanism local anomalies only exist or even-dimensional ories very robust test on quantum consistency o QFT, only at one-loop level In Type II String ory models anomalies are cancelled i Dp-brane tadpoles are cancelled Search or tadpole cancellation conigurations tedious and algorithmically hard to implement 9

11 What about anomalies in F-ory? Problem seems very complicated: M-ory eective actions are odd-dimensional, but connected with F-ory eective actions via circle reductions chirality needs introduction o G 4 -lux in M-ory Our result: Local 4D anomalies are automatically cancelled or a wide class o F-ory backgrounds. [Corvilain,TG,Regalado] to appear ollowing assumptions or argument: (1) re exists a completely smooth geometry describing background somewhere in 3D moduli space (e.g. existence o resolution to use Sugra) (2) chirality is induced by standard G 4 -lux (element o H 4 (Y 4, 1 2 Z) ) 10

12 Sketch o argument: Part I Start with our-dimensional (possibly anomalous) ory dimensionally reduce on a circle + go to 3D Coulomb branch integrate out all massive modes to compute Wilsonian eective action: compute 3D one-loop A µ A actually ininite Chern-Simons terms due to KK-tower massive ermions Tricky part: deal with ininite tower and regularization scheme proper regularization scheme is dictated by KK ansatz In practice: ininitely many KK Pauli-Villars particles in 3D Remark: dimensional reduction includes Green-Schwarz and generalized Chern-Simons terms [Anastasopoulos,Bianchi,Dudas,Kiritsis] 11

13 Sketch o argument: Part II Key Observation: all 3D Chern-Simons terms not invariant under small/large gauge transormation cancel i and only i all 4D anomalies are vanishing Compare with 11D supergravity on elliptically ibered Calabi-Yau ourold: 3D supergravity with ield-independent Chern-Simons terms due to -lux M2-brane states are consistently integrated out comparison shows general anomaly cancellation see Pierre Corvilain s talk Important remarks: more subtle than translating anomaly conditions into geometric relations as done e.g. in [Park],[Cvetic,TG,Klevers],[Bies,Mayrhoer,Weigand] G 4 it is key to assert that F-ory exists: v! 0 limit indeed gives a 4D ory, but uniqueness o 4D ory not needed (all 4D ories are non-anomalous) uniqueness result: [TG,Kaper 15] 12

14 Key message can show general results about F-ory compactiications: (generic gauge groups + matter representations) use: 1) M-ory approach (here 11D sugra)+ 2) mamatical insights about geometries and luxes + 3) results on 3D circle-compactiied ories with ininite modes What about continuous unctions? Maybe this just works because anomalies are so robust and protected! Look at gauge coupling unction next. Holomorphic Next in diiculty level. 13

15 Result 2: Computing gauge coupling unctions 14

16 a ú F = i Êa. (2.37) Õ gauge on D7-brane i.e. F = 0, which a S = S fi2fi (S), where lux minus sign stands isorzero, orientation reversal. p,q p,qaccording p,q condition hetransorm symmetries (2.28) since, in addition to constant shits with, one also invariant since it shits to eq. (2.26 ü to split cohomologies H (S ) = H (S ) H according to (2.26), we ind that gauge coupling unction + + (S+ ) under. The + a version oembedding (2.35) actually its with a holomorphic in G and. ield However, (2.35) is isonly valid when image pair can dimensional gauge A and o D7-brane omorphic and invariant under whole set o shit symmetries (modulo a 3 ux on D7-braneas is [6, zero, i.e. F = 0, which as noted above, is not a gauge i 7] a b maginary shit), as it should. ˆ = D7invariant T + ik ab G + K t condition since it shits according to eq. (2.26). Thus, D7 gauge 2 1,0 p p immediately[jockers,louis],owhen including Wilson line moduli or D7-brane, we Gauge A P + a + a, p = 1,..., ha as(s+ ) coupling unction oraa = stack o D7-branes: (2.35) is actually D7 we deined p p worldvolume where luxes 2,0 lem. At irst, one might think that gauge coupling unction K is = given K K 3 4 = s + s, 1,..., h K K (S+ ) i 1 a b a b a ú ˆD7 = T D7 T + ika abquadratic G + term K ab in, Wilson lines (2.36) by (2.36), where F = i Êa. lux:(2.15). contains Õ 2fi 2 where o P is ad7-brane unction that is equal to +1 onsuch S and 1 on (S). The ac he dimensional reduction action does not give a term a Since se transorm2,0according to (2.26), we ind that 1,0 eagain deined worldvolume luxes as have to expanded into H quadratic (S+ ) and in rom (2.34). moduli As argued inbe [6], a contribution inh Wilson linesno luxollows + ), respectively, (S Kähler axionic moduli dilaton-axion is both holomorphic and invariant under whole set o a a a open ollowing 1= c on isorientiold string states. d at one loop in gs o and reore natural that it is not captured by a ú G b imaginary shit), as it should. F = i Êaconstant. (2.37) Õ 8 2fi p -Ineld action, which is Itonly valid at tree level in open string amplitudes. is important to stress thatwhen notion o being 1) moduli impliesor tht Finally, including Wilson(0, line k tionstransorm were computed in [9,to10] in which show that indeed, a Calabi-Yau s hese according (2.26), we indmodels, that gauge coupling unction depend on toroidal complex structure moduli z o ambient ace a problem. At irst, one might think that gau p ˆ p 1,0 erm arises atand one loop level. Itdependence is reore natural to split D7 A in = AD7 Pshit + + a as (modulo Include Wilson line moduli: holomorphic invariant under whole set o 1,.a..,ahquadratic (S+ ) te pwe can pexpand make this more explicit, this case byasymmetries (2.36), where Tp = contains t imaginary shit), as itˆshould. 1 loophowever, dimensional reduction o D7-brane act red ˆ ˆ p pq 1 = +, sqr(2.38) ˆr ), D7 D7 on D7-divisor: D7 = Re (ˆ i (1,0)-orm q 2 and indd7-brane, again (2.34). argued in [6], a contribution ally, when including Wilson line moduli orwe we As immediately ab natural t complex structure dependence captured by holomorphic is generated at one loop in g and is reore sroblem. obtained by direct dimensional reduction o D7-brane action. Compars 1 At irst, one might gauge unction given where (ˆ pthink, ˆp ) isthat a real basis o Hcoupling (S). Here pq is a is holomorphic unction in Dirac-Born-Ineld action, which is only valid at tree lev with (2.36) onewhere is lead to making ansatz k case by (2.36), T contains a quadratic term in Wilson lines (2.15). structure moduli z. For an appropriate basis, its real part Re pq is inver Suchpqcorrections were computed in [9, 10] Tin toroidal mod First term: redeine r, dimensional reduction o inverse D7-brane action does not give such a termin F-ory denote by Re. This ansatz can be justiied 1 1 loop pq ˆ quadratic term arises at one loop(2.39) level. It is reore na = d a (a + a ) + log, p q q D7 D7 ind again (2.34). Asargued argued [6], 15, a contribution in inwilson lines term: discretein [ 21] and wasquadratic recentlysecond used Type restores IIB orientiolds 2 inin [13, ˆD7 ˆred ˆ1 loop, a = + ated at one loop in come gs andback is reore natural that it is not captured by shit-symmetries o p D7 can D7 also ex to that F-ory treatment in shit section 3. Clearly, one a holomorphic unction. Note our analysis o symmetries 15 p tree levelredin open string amplitudes.8 ˆ orn-ineld action, which is only valid at real basis (ˆ, ) such that ˆ Gauge coupling unction in IIB orientiolds

17 Gauge coupling unction in IIB orientiolds Kinetic mixing between D7-brane U(1) and R-R gauge ield : that one obtains rom reducing D7-bra A apple [Jockers,Louis] ˆ ŸD7 =Re pq Re Ÿ (M q Ÿ i qr M rÿ ) a p seemingly not holomorphic act that in (2.29) couplings (and not What about F-ory approach: ˆ appled Can it solve puzzle about holomorphicity o? pq Can it help to compute? Closed string moduli dependence? Even open moduli dependence? Can we understand correction? log Yes! 16

18 Gauge coupling unction in F-ory What about F-ory approach: ˆ appled Can it solve puzzle about holomorphicity o? pq Can it help to compute? Closed string moduli dependence? Even open moduli dependence? Can we understand correction log? Yes! In act, I would not be able to answer irst two sets o se questions without using F-ory. 17

19 Gauge coupling unction in F-ory Key observation: axionic moduli Wilson line moduli R-R bulk gauge ields G a a p A apple arise in F-ory (via M-ory) by expanding C 3 = N A A 2 H 2,1 (Y 4 ) A + N A A Mamatical object to consider: intermediate Jacobian change o complex structure o is captured by holomorphic unction AB (z) J 3 (Y 4 )= H2,1 (Y 4 ) H 3 (Y 4, Z) = T 2h 2,1 Y 4 dilaton-axion bulk complex structure Y 3 D7-brane moduli complex torus 18

20 Results on three-orm cohomology Computed complete 3D action in F-ory rame: Intersection structure induces constraints such that kinetic mixing is holomorphic. ˆ appled7 = (Mapple A + i apple M A [Corvilain,TG,Regalado 16] )N A AB Function can be computed explicitly or hypersuraces in toric ambient spaces closed/open moduli dependence [Greiner,TG 17] [Greiner,TG 15] see Sabastian Greiner s talk Correction captured by Riemann ta-unction on torus log ( AB,N A ) J 3 (Y 4 ) [Witten], [Kerstan,Weigand], [Corvilain,TG,Regalado 16] 19

21 Computing unction AB Observation or hypersuraces in toric ambient space: [Greiner,TG 17] one shows that all non-trivial three-orms stem rom resolved singularities over Riemann suraces in R Y 4 [Batyrev], [Klemm,Lian,Roan,Yau] h 2,1 (Y 4 ) = Xn 2 =1 ( ) ( genus o Riemann surace number o resolution divisors) toric resolution spaces singularities over Riemann surace (1,0)-orm on Riemann surace singularities are resolved by toric resolution tree (note: ibers are two- dimensional, P 2 ) 20

22 Computing unction AB Observation or hypersuraces in toric ambient space: [Greiner,TG 17] one shows that all non-trivial three-orms stem rom resolved singularities over Riemann suraces in R Y 4 [Batyrev], [Klemm,Lian,Roan,Yau] h 2,1 (Y 4 ) = Xn 2 =1 ( ) ( genus o Riemann surace number o resolution divisors) toric resolution spaces AB (z) unction and metric on space o (2,1)-orms is determined by: 1. periods o Riemann suraces 2. intersection pattern o resolution divisors singularities over Riemann surace both are explicitly computable or a given example ourold 20

23 Computing unction AB Observation or hypersuraces in toric ambient space: [Greiner,TG 17] one shows that all non-trivial three-orms stem rom resolved singularities over Riemann suraces in R Y 4 [Batyrev], [Klemm,Lian,Roan,Yau] h 2,1 (Y 4 ) = Xn 2 =1 ( ) ( genus o Riemann surace number o resolution divisors) toric resolution spaces Can exploit mirror symmetry! mirror ourold AB (z) =M ABK z K + O(e z ) [Greiner,TG 15] singularities over Riemann surace 20

24 Conclusions F-ory provides powerul tools to study a large class o models discrete data has been studied intensely in last years here: example o anomaly cancellation we have provided an argument that shows automatic anomaly cancellation, or a large set o geometries Continuous unctions: gauge coupling unction initiated derivation o gauge couplings in F-ory (via M-ory) ocused here on dependence on axionic moduli, Wilson lines, and discussed kinetic mixing explicit computations o gauge coupling unction on closed and open moduli by geometric methods many open and tractable problems on F-ory eective action 21

25 Thank you or your attention! 21