F-theory effective physics via M-theory Thomas W. Grimm Max Planck Institute for Physics (Werner-Heisenberg-Institut) Munich Ahrenshoop conference, July 2014 1
Introduction In recent years there has been vast progress in the study of F-theory effective actions in four and six dimensions using an approach via M- theory. The understanding of geometric properties and the physics in the effective actions are going hand in hand. classical moduli action fluxes and charged spectrum massless and massive U(1) s, gauge theory branches and resolutions 0 corrections... Main motivations: phenomenology: Grand Unified Theories,... general effective theories within string theory (`minimal models ) formulating string theory away from weak coupling (M-theory) 2
F-theory compactifications Type IIB has non-perturbative symmetry rotating interpret Sl(2, Z) = C 0 + ie as complex structure of a two-torus (2 auxiliary dimensions) [Vafa] [Morrison,Vafa] singularities of the fibration are crucial to encode 7-brane physics pinching torus indicates presence of 7-branes magn. charged under minimally supersymmetric F-theory compactifications: Y F-theory on torus fibered Calabi-Yau manifold n with base gauge groups from 8D charged matter from 6D Yukawa-type couplings from 4D B n 1 3
Goals of this talk Answer natural questions arising in this program: How can one derive the effective theory for F-theory? no known fundamental formulation in 12D microscopic formulation poorly understood How can one extract information using techniques for smooth manifolds? interesting 7-brane physics (gauge theory) arises from singularities While not being able to give justice to the recent developments, I will comment on interesting results on the M-theory to F-theory limit that emerged in recent years: Chirality and Chern-Simons terms M 3 S 1, M 5 S 1 universal gravitational instantons of N=1 sugra on M 3 S 1 fluxed circles and fibrations without section 4
F-theory via M-theory 5
F-theory via M-theory F-theory viewed as auxiliary `12 dim. theory (torus volume unphysical) F-theory effective actions has to be studied via M-theory Consider M-theory on space T 2 M 9 F-theory limit: is the complex structure modulus of the T 2, v volume of T 2 (1) A-cycle: if small than M-theory becomes Type IIA (2) B-cycle: T-duality Type IIA becomes Type IIB, is indeed dilaton-axion (3) grow extra dimension: send v 0 than T-dual B-cycle becomes large T 2 can be generalized for singular fibrations: e.g. Taub-NUT D6 D7 6
F-theory / M-theory geometries F-theory geometries can be constructed and analyzed singularities of elliptic fibration induce non-abelian gauge symmetry singularity resolution: (resolution at each codimension) Examples: compact, fully resolved Calabi-Yau three-/fourfolds toric geometry: numerous examples + various types of gauge groups Unification of brane and bulk physics on resolved Calabi-Yau manifolds powerful techniques from algebraic geometry/topology to study gravity coupled gauge theories (and tensor theories) 7
M-theory on resolved CY manifolds physical interpretation of resolution only possible in M-theory/lower-dim. theory moving branes apart on the B-circle: Coulomb branch of the lower-dimensional gauge theory: G U(1) rankg Massive states from M2 branes on geometric 2-cycles: M2-branes on resolution P 1 s over generic points of S massive `W-bosons of G-breaking M2-branes on resolution P 1 s over intersection massive matter multiplets M2-branes on the elliptic fiber + P 1 s massive Kaluza-Klein modes All massive states have to be integrated out to determine Wilsonian effective action in circle compactification also KK-modes are crucial 8
F-theory effective actions via M-theory effective actions can be computed via M-theory / 11-dimensional supergravity on the resolved Calabi-Yau manifolds F-theory on singular M 8/6 M-theory on resolved M 4d/6d effective theory with non-abelian gauge symmetry G and non-abelian tensors 8/6 3d/5d effective theory with only abelian gauge symmetries 1-dim. compactification 3d/5d effective theory pushed to Coulomb branch compare explicitly compute characteristic data determining the action 9
F-theory effective actions via M-theory effective actions can be computed via M-theory / 11-dimensional supergravity on the resolved Calabi-Yau manifolds F-theory on singular M 8/6 M-theory on resolved M 4d/6d effective theory with non-abelian gauge symmetry G and non-abelian tensors 8/6 3d/5d effective theory with only abelian gauge symmetries 1-dim. compactification 3d/5d effective theory pushed to Coulomb branch a) can be a circle (standard approach) b) can be an interval for Spin(7) [Bonetti,TG,Pugh] [Bonetti,TG,Palti,Pugh] compare c) can be an fluxed circle 10
Example 1 Chiral Spectra in F-theory 11
4D F-theory chiral spectrum via M-theory determination of charged chiral spectrum: chirality induced by fluxes on 7-branes, but in M-theory on resolved fourfold there are no chiral fields Chirality formulas for M/F-theory setups: S R (R) = is called matter surface Z G 4 = hdc 3 i S R G 4 flux on resolved fourfold [Braun,Collinucci,Valandro] [Marsano,Schäfer-Nameki] [Krause,Mayrhofer,Weigand] [TG,Hayashi] [Intriligator,Jockers,Mayr, Morrison,Plesser] [Küntzler,Schäfer-Nameki] 3D one-loop Chern-Simons terms linked to 4D chiral index 3D M-theory Chern-Simons terms computes anomaly free chiral spectrum (R) 12
3D Chern-Simons vs. 4D chiral matter Z 3d Chern-Simons terms: S (3) CS = No such terms from classical circle reduction of 4d, N=1 theory generated at one loop by massive fermions 1 loop = 1 X X n(r) q q sign(q ) 2 R q2w (R) A ^ F [TG,Hayashi 11] U(1) rkg U(1) n U(1) =(i, m) Ỹ 4 M-theory on resolved with : classical Chern-Simons term G 4 flux G 4 Ỹ 4 fluxes on count 4D chiral matter spectrum massive in the 3D Coulomb branch determine chiral index 1 loop = flux 4D anomaly induced by chiral matter is captured by 3D Chern-Simons term at one loop in the Coulomb branch anomaly ladder 13
Kaluza-Klein modes at one-loop [Cvetič,TG,Klevers 12] 3D Chern-Simons terms involving the Kaluza-Klein vector Z S (3) CS = 1-loop 0m A 0 ^ F m A m from 4D U(1) symmetry KK-vector in metric reduction term is induced by massive KK-modes charged under U(1), 1 loop 0m = 1 2 X f 1X n= 1 A 0 nqm f sign q f + n r = 1 12 X n(q)q m q A m see also [Landsteiner etal.] [Loganayagam etal.] [Golkar,Son] [Di Pietro,Komargodski] geometric evaluation in M-theory : C 3 = A 0 ^ 0 + A ^ +... A 0 only identified with Kaluza-Klein vector iff: 0 =[B]+ 1 2 c 1(B 3 ) [TG,Savelli] [Park] Z Z 0m = 0 ^ m ^ G 4 = 1 2 c 1 (B 3 ) ^ m ^ G 4 = 1 2 a m Ỹ 4 Ỹ 4 M-theory gives exactly Green-Schwarz term for Abelian/gravitational anomaly 14
Analog analysis in 6D F-theory Five-dimensional N=2 effective theory from M/threefold defining data for the vector sector are: 1 48 2 k Z M 5 A ^ F ^ F 1 384 2 apple Z M 5 A ^ Tr(R ^ R) [Ferrara,...] [Minasian,...] Both and can be determined by reducing 11D Chern-Simons terms: Z Z triple intersections 1 12 k apple M 11 C 3 ^ G 4 ^ G 4 1 96 M 11 C 3 ^ X 8 (R 4 ) k apple second Chern class Comparing to 6D theory on a circle: some Chern-Simons terms classical and some arise at one-loop level by integrating out massive modes early works: [Morrison,Seiberg] [Witten] [Intriligator,Morrison,Seiberg] 15
Self-dual tensors and gravitinos deriving complete 6D N=(1,0) action via M-theory with above logic All Chern-Simons terms that are not purely arising from 6D gauge groups need to be modified. [Bonetti,TG 11] compute one-loop diagram spin-1/2 spin-3/2 tensor k = apple = X mass. states X mass. states k r q q q sign(m) apple r q sign(m) k r apple r 1 5-4 1-19 8 [Bonetti,TG,Hohenegger 13] 5D self-dual tensors are crucial KK modes of 6D self-dual tensors: S tensor = Z M 5 i B ^ (db iqa) r 1 B ^ B [Townsend,Pilch,van Nieuwenhuizen] Comparing M-theory to F-theory also massive Kaluza-Klein modes of 6D tensors and the gravitino have to be included at one loop. 16
Example 2 Gravitational instantons 17
Universal M5-brane instantons [TG,Savelli 11] M-theory on an elliptically fibered Calabi-Yau fourfold: Calabi-Yau condition implies four supercharges, but also: topological conditions: h 1,0 (B) =h 2,0 (B) =h 3,0 (B) =0 M5-brane instanton on B can contribute to three-dimensional superpotential [Witten] W 3d = Ae T 0 M-theory/F-theory duality: M5-brane Taub-NUT instanton in F-theory on : vol(b) = r 2 S 1 Re T 0 = vol(b) universal gravitational instanton in N=1 supergravity on a circle recently the supergravity analysis of Taub-NUT instantons, including oneloop corrections, was carried out [Tong,Turner] in 18
Example 3 Fluxed circle reductions 19
F-theory on manifolds without section most F-theory works have focused on geometries with a section: globally identify one point on the torus fiber embedding of base Kaluza-Klein vector more than one point multiple sections Abelian gauge groups can be brought into Weierstrass form: y 2 = x 3 + fxz 4 + gz 6 well studied Consider: Calabi-Yau threefold with genus-one fibration without section Möbius strip: boundary Z 2 - fiber over S 1 does not admit a section dilaton-axion (u) can be still extracted at each point of B 2 F-theory well-defined on this space, despite having no Weierstrass model? M-theory on Calabi-Yau threefolds without sections is well defined and can be analyzed using M-theory 5d effective description of F-theory setup 20
Proposed effective theory [Anderson,García-Etxebarria,TG,Keitel 14] (1) F-theory setup contains geometrically massive U(1)s Z dualize C : shift gauging of axions 6 C 2 c S St = C 6 ^ F U(1) D 6 c = dc + ma purely geometric D7 U(1) with charged spectrum under these U(1)s (2) Derivation from M-theory requires a fluxed circle reduction from 6d 5d new massive and massless 5d combinations of circle Kaluza-Klein vector and 6d massive U(1) A 0 D 5 c = dc + ma U(1) + na 0 [Jockers,Louis] L mass = f 2 ma U(1) + na 0 A mass agreement of 5d effective theories after integrating out massive fields [V.Braun,Morrison] [Morrison,Taylor] recent analysis of geometries without section 21
Local check via T-duality metric for T 2 -fibration Y 3 without section can locally only brought to form: ds 2 (X) =g i du i dū j + v0 Im X Y 2 non-trivial field strength in simplest situation X = dx + X Y = dy + Ỹ non-trivial T 2 KK-vectors hdxi = n hdy i =0 M-theory on connects to Type IIB theory on extra : S 1 [Witten] Y 3 non-vanishing flux G3 = F3 H3 : F 3 = n ^ dy flux along the circle Fluxed circle reduction induces gauging with Kaluza-Klein vector Dc = dc + n A 0 C 2 gauging of the axion arising from the expansion in 0 A 22
Global checks by computing 1-loop CS terms compare 1-loop Chern-Simons terms with intersection numbers and second Chern class for Calabi-Yau examples without section Y 3 concrete examples with one massive 6d U(1) have a bi-section to check proposal in global models compute hypermultiplet spectrum charged under massive 6d U(1) by using conifold transition to models with two sections massless U(1) in 6d find perfect agreement many works on F-theory with multiple U(1)s: [TG,Weigand], [Morrison,Park] [Borchmann,Mayrhofer,Palti,Weigand], [V.Braun,TG,Keitel], [Cvetic,Klevers,Piragua,Song] [A.Braun,Collinucci,Valandro] [Kuntzler,Schäfer-Nameki]... to apply proposal: compute hypermultiplet spectrum by using Chern-Simons terms, i.e. intersection numbers and Chern-classes of Y 3 extension to Calabi-Yau fourfolds with four-form flux is an important open problem Yukawa couplings and U(1) selection rules? [García-Etxebarria,TG,Keitel, to appear] 23
Conclusions F-theory compactifications are interesting both from a conceptional as well as phenomenological point of view embedding of Grand Unified Theories or Standard Model into string theory interplay of recent developments in geometry with gauge/gravity theories effective F-theory physics has to be studied via M-theory many intriguing insights due to trans-dimensional treatment - Example 1: Chern - Simons terms vs. higher-dimensional chirality (relations to AdS-CFT and S 3 S 1,S 5 S 1 partition functions) strategy also seems to apply in situations where 6d theory is hard to formulate itself: (1,0) tensor theories (non-abelian?) - Example 2: M5-brane instantons vs. Gravitational instantons - Example 3: fluxed circle reductions vs. no-section geometries many challenges, in particular, in deriving continues data / couplings 24