Gauge theories on circles and the decoding of Ftheory geometries Thomas W. Grimm Max Planck Institute for Physics & Utrecht University Mainly based on works with Andreas Kapfer and Denis Klevers arxiv: 1502.05398 (AK) 1510.04281 (AK & DK) Using results with Federico Bonetti, Stefan Hohenegger, Andreas Kapfer, Jan Keitel arxiv: 1112.1082, 1302.2918, 1305.1929 Caltech, February 2016 1
Introduction / Motivation 2
Effective actions of Ftheory Ftheory compactifications arguably provide the richest class of string theory effective actions original excitement about GUTs in Ftheory not realized in pert. strings numerous examples with exotic matter spectra dualities to heterotic and Type I compactifications new exotic theories, for example, 4D N=3 theories of e.g. talk of Weigand e.g. talks of Anderson, Klevers e.g. talks of Mayrhofer, Lüst [GarcíaEtxebarria,Regalado] e.g. talk Iñaki Two questions arise: How can we infer reliably information about the 2D/4D/6D effective actions of Ftheory? Is there a classification in sight? What do Ftheory geometries classify? 3
Some background material Wellknown slogan: Ftheory compactifications on ellipticallyfibered CalabiYau threefolds yield 6D theories with minimal N=(1,0) supersymmetry pinching of twotorus indicates location of sevenbranes brane and bulk physics encoded by singular complex geometry Sixdimensional theories are perfect to answer the above questions have a rich structure there are many topologically distinct CY threefolds are strongly constraint by anomalies fermions in all N=1 multiplets can contribute to anomalies 4
Goals of this talk (1) Argue that the complete information in the Ftheory geometries actually describe gauge/sugra theories on a circle. This is not unexpected since the Mtheory to Ftheory approach has been suggested already in [Vafa 96]. However, its importance and power might have been underappreciated. Currently this limit is the only reliable way to infer information about Ftheory effective actions. (2) Show that Ftheory geometries can teach us valuable lessons about circlereduced theories. Example: How are anomalies of the higherdimensional theories visible in the lowerdimensional effective theory? (3) Comment on possibility of classifying CalabiYau threefolds (with elliptic fibration) by using the insights from gauge theory. 5
Goals of this talk (1) Argue that the complete information in the Ftheory geometries actually describe gauge/sugra theories on a circle. (2) Show that Ftheory geometries can teach us valuable lessons about circlereduced theories. (3) Comment on possibility of classifying CalabiYau threefolds (with elliptic fibration) by using the insights from gauge theory. 6D gauge theories / Sugra theories on circle Geometry of resolved elliptically fibered CalabiYau threefolds Our systematics is somewhat complementary to the approach and classifications of: [Morrison,Taylor etal.] [Heckman,Morrison,Rudelius,Vafa] 6
Some comments on the Mtheory to Ftheory duality 7
Approaching the problem via Mtheory Viewing Ftheory as an actually 12 dim. theory is problematic (e.g. torus volume unphysical, meaning of nonperturbative states ) Ftheory effective actions via Mtheory (as of now: definition of Ftheory) Consider Mtheory on space T 2 M 9 Ftheory limit: is the complex structure modulus of the T 2, v volume of T 2 (1) Acycle: if small than Mtheory becomes Type IIA (2) Bcycle: Tduality Type IIA becomes Type IIB, is indeed dilatonaxion (3) grow extra dimension: send v! 0 than Tdual Bcycle becomes large [Vafa] 8
Consequences of Mtheory to Ftheory limit First step: approach Mtheory via 11D supergravity on a smooth geometry resolution of singular CalabiYau geometry classification of resolutions at each codimensions in base [almost everyone who has worked on Ftheory, ] codimension 1 in base nonabelian gauge group simple roots codimension 2 in base matter in representation R weights w of R Abelian gauge group factors: n U(1) +1 rational sections of the fibration zero section (assumed to exist throughout this talk) Second step: Mtheory to Ftheory limit shrinks fiber torus and resolutions and grows extra dimension (5D 6D) keep track of M2brane states e.g. M2branes on twotorus fiber correspond to circle KaluzaKlein states 9
Massive states in five dimensions 11D Sugra on smooth geometry 5D effective theory Both types of theories need to be computed and then compared 6D supergravity theory (gauge theory) on a circle push to Coulomb branch integrate out all massive modes 5D gauge group in Coulomb branch: U(1) rankg U(1) n U(1) mass of 5D state at KaluzaKlein level descending from 6D state in representation R of G with weights and U(1)charges w n q m m = m CB + m KK = w I I + q m m + n r Coulomb branch vevs (blowup vevs) circle radius (torus fiber volume) 10
Anomaly cancellation and circle compactifications 11
Mtheory on CalabiYau threefolds effective action has been studied long ago: 5D N=2 theory important to us are the ChernSimons terms: 1 48 2 K ABC Z M 5 A A ^ F B ^ F C 1 384 2 c B Z M 5 A B ^ Tr(R ^ R) [Ferrara, ] [Minasian, ] triple intersection Z numbers: second Z Chern class: A =1,...,h 1,1 (Y 3 ) K ABC =! A ^! B ^! C c A =! A ^ c 2 (Y 3 ) Y 3 Y 3 arise from 11D sugra action including terms up to 8 derivatives comparison to ChernSimons terms obtained after circle reduction classical and oneloop corrections required early works: [Morrison,Seiberg][Witten][Intriligator, Morrison,Seiberg] 12
ChernSimons terms on circle side Classical and oneloop ChernSimons terms classical terms depend on, b, a (6D tensor coupling, anomaly coefficients), b, a fixed by geometry (intersection numbers base, location of branes, canonical class of base) there are many more ChernSimons terms in Mtheory [Bonetti,TG 11] oneloop CSterms in the effective theory induced by integrating out: K ABC = X mass. states k r q A q B q C sign(m) c A = X mass. states apple r q A sign(m) massive spin 1/2 fermions massive spin 3/2 fermions massive selfdual tensors [Bonetti,TG,Hohenegger 13] massive KaluzaKlein states of all 6D fields that carry chirality and contribute to the 6D anomaly 13
Jumping ChernSimons terms match was still not possible for certain geometries mass n=2 n=1 n=0 n=1 m CB mass m CB [TG,Kapfer,Keitel 13] n=0 regularization of infinite sum over KK modes in oneloop CS terms gets modified While oneloop CS terms are independent of the precise numerical value of the mass of a state, they do depend on (1) sign of CB mass sign(m CB ) (2) hierarchy of m n KK = n/r and associate an integer label `w,q to each massive state: m CB is between mass of and +1 KaluzaKlein state `w,q `w,q m CB 14
Extending box graphs Box graphs have been introduced to systematically classify the signinformation and realized Coulomb branch phases [Hayashi, Lawrie, Morrison, SchäferNameki][Braun,SchäferNameki] Example: antisymmetric representation 10 of SU(5) + + + + boxes are for the weights w of a representation colors indicate signinformation clear rules and allowed connections studied 15
Extending box graphs Box graphs have been introduced to systematically classify the signinformation and realized Coulomb branch phases [Hayashi, Lawrie, Morrison, SchäferNameki][Braun,SchäferNameki] Example: antisymmetric representation 10 of SU(5) + + + + add information about jump levels `w 16
Extending box graphs Box graphs have been introduced to systematically classify the signinformation and realized Coulomb branch phases [Hayashi, Lawrie, Morrison, SchäferNameki][Braun,SchäferNameki] Example: antisymmetric representation 10 of SU(5) + + + + `w add information about jump levels What are the rules for such diagrams? What are the inherent symmetries? 17
Large gauge transformations large gauge transformations precisely induce certain types of jumps 6D gauge transformation: ˆ I (x, y) = k I y 0 ˆ m (x, y) = circle coordinate k m y I = I + ki r, m = m + km r, shift of Coulomb branch vevs: mix of 5D vectors: à I = A I k I A 0 mixingin the 1 KaluzaKlein à m = A m k m A 0 vector A 0 rearrangement of whole KaluzaKlein tower large gauge transformations act on nontrivially on Symmetry? `w,q Yes! But only if one takes into account 6D anomalies! Oneloop corrected effective 5D theories differing by the above trans formation are identified if and only if 6D anomalies are cancelled. 18
Discovering the 6D anomaly conditions Recall the form of the 6D anomaly cancellation conditions: e.g. pure Abelian: pure nonabelian: (b mnb pq + b mpb nq + b mqb np ) = 3 b (G) b (G) = X R X F 1/2 (q) q m q n q p q q q F 1/2 (R) C R X X tr R ˆF 4 = B R tr f ˆF 4 + C R (tr f ˆF 2 ) 2 differ from oneloop ChernSimons terms: e.g. pure Abelian: pure nonabelian: K mnp = 1 2 K IJK = 1 2 X F 1/2 (q) q m q n q p (2`q + 1)sign(m q CB ) q X F 1/2 (R) X w I w J w K (2`w + 1)sign(m w CB) w2r R 19
Discovering the 6D anomaly conditions We were able to show: pure Abelian: Properties of 1loop CSterms: pure nonabelian: (1) additional q r, w I from large gauge transformation acting on 6D anomalies: X F 1/2 (q) q m q n q p q q q X F 1/2 (R) C R R r L (2`q + 1)sign(m q CB )! q r (2`w + 1)sign(m w CB)! w L [TG,Kapfer 15] pure Abelian: (2) identities involving four weights to pure obtain nonabelian: C R Change in classical and 1loop Chern Simons terms cancel if and only if 6D anomalies are cancelled 20
Arithmetic structures and anomaly cancellation A proof anomaly cancellation for Ftheory geometries requires to show that the large gauge transformations are actually a geometric symmetry. [TG,Kapfer 15] for models with only Abelian gauge symmetries this is possible large gauge transformations MordellWeil group of rational sections MW(Y 3 ) = Z n U(1) Z k1... Z kntor key: in the Mtheory to Ftheory limit we are free to pick the zerosection that specifies the KaluzaKlein vector (any choice works fine!) define new arithmetic structures for nonabelian large gauge transformations define arithmetic structure on geometries with multisections mathematical meanings? Higgsing might connect nonabelian models to Abelian models arithmetic structures need to be compatible [TG,Kapfer,Klevers 15] 21
Comments on the classification of threefolds 22
Characteristic topological data Wall s classification theorem: The homotopy types of CalabiYau threefolds are classified by the following numerical characteristics: Hodge numbers: Triple intersection numbers: Second Chern class: h 1,1 (Y 3 ), h 2,1 (Y 3 ) c A K ABC Spaces differing in these numerical data cannot be continuously deformed into each other without encountering singularities. Are these data systematically specified by the Ftheory effective actions on an additional circle? 23
Information from gauge theories on circles (1) Hodge numbers are determined by spectrum (in the following: only nonabelian) h 1,1 (Y 3 )=T + 2 + rank(g) h 2,1 (Y 3 )=H neut 1 (2) Intersection numbers and second Chern class determined by: classical data:, b, a K 0 = K IJ = C IJ b c = 12 a oneloop data: spectrum plus circle info (sign table, jump levels) + + + + extended box graphs as convenient object for classification? 24
( Intersection and Chern class data Intersection numbers and Chern classes (nonabelian groups only) = 1 2(T sd T asd ) F1/2 5F3/2 + 1 X F1/2(R) X K lw 2 (l w +1) 2 000, 120 4 R w2r2 = 1 X K F1/2(R) X 00I l w (l w +1)(2l w +1)w I sign(m w 6 CB), R w2r K 0IJ K IJK c 0 = 1 X F1/2(R) X (1 + 6 l w (l w +1))w I w J, 12 R w2r = 1 X F1/2(R) X (2l w +1)w I w J w K sign(m w 2 CB), R w2r = 1 X 19F3/2 F1/2 4(T sd T asd ) F1/2(R) X 6 R w2r l w (l w +1), c I = X R F1/2(R) X w2r (2l w +1)w I sign(m w CB), 25
Two classification problems (1) given a 6D N=(1,0) anomalyfree theory in tensor Coulomb branch construct classifying topological data of CalabiYau threefolds Systematics: start with gauge theories of nonhiggsable clusters successive unhiggsing of gauge groups in field theory generate tree of topological data [Morrison, Taylor] Challenges: moding out symmetries detecting additional geometric constraints extending analysis to nonresolvable geometries String Universality: Is Ftheory a theory of Frything in 6D? (2) given an set of CalabiYau geometries, as provided e.g. by the KreuzerSkarke list classify 6D N=(1,0) theories associated to elliptic fibrations 26
Conclusions The map between the CalabiYau threefold geometry and a 6D sugra theory on a circle has been established at classical and oneloop level. Anomalies can be inferred from ChernSimons terms, if the latter are probed by large gauge transformations. Large gauge transformations map to arithmetic structures on ell. fibrations. for Abelian gauge groups one encounters the MordellWeil group general proof of anomaly cancellation for viable Ftheory geometries new arithmetic structures for nonabelian groups and geometries with multisections. Extension to 3D/4D has been worked out for chiral spectrum induced by fluxes Ftheory provides a novel way to approach classification problems. topological data of resolved CalabiYau threefolds are robust and corse information directly related to corse information about gauge theories Are there mathematical restrictions on viable Hodge numbers, intersection numbers Chern classes? ruling out 6D gauge theories coupled to gravity? 27
Thank you for your attention! 28