ELEMENTARY E-STRINGS Choi, Kang Sin Ewha Womans University and IBS In collaboration with Soo-Jong Rey (IBS, SNU) IBS-KIAS Joint Workshop at High1 JeongSeon, Feb. 7, 2017 The Standard Model from string theory Large symmetry predicted by string theory e.g. heterotic string: 10D, N=4 SUSY, E8E8 SYM Symmetry breaking associated with geometric symmetry of internal space Small observed symmetry of SM
4D, N=1 SUSY, SU(3)SU(2)U(1) with observed field contents. Unification via E-chain E8 [Ramond] Gauge bosons unified in a single 248 extra dimension E6 Matter and Higgs in a single 27 SO(10) Quarks and leptons in a single 16
SU(5 ) Simple gauge group SUSY gauge coupling unif SU(3) x SU(2) x U(1) The SM E8 x E8 Heterotic string 10D closed string (6 extra dim.) 2D worldsheet: different SUSY (0,1) between left- and right-movers. E8 x E8 unification
Embraces all the above features and SUSY breaking in the hidden sector E8 Questions How two E8s communicate? [Sonnenschein, Theisen, Yankielowitz] [Horava, Witten] Origin? Non-perturvative effect Possible vavcua? 6D SCFT The largest SCFT with Poincare symm.: 6D N = (2,0) SUSY. [Nahm] Non-Abelian structure of ADE is responsible by tensor multiplets. Bosonic: BMNSD, 5 scalars of SO(5)R, no gravity. No Lagrangian description: Lorentz covariance vs. self-duality, non-Abelian structure
is a worldvolume theory of M5-branes d.f. on M5-brane. M-theory dual: a stack of M5-branes d.f. M2 ending between M5-branes M2 cf. F1 ending between D-branes For #M5 > 1 very little is understood in M-theory Properties can be studied in the IIB side Local models are classified as IIB on C2 / ADE Analyzed by resolutions C2 / ADE = separated M5s M5 M5
Swampland A local bottom-up vacuum may not be UV completed. Ex. 1 electron in the closed universe e- M5 M5 (2,0) AN SCFT = theory of N linear M5-branes D, E type SCFT in the simply connected space Once we incorporate gauge group, we should put charged objects.
To avoid swampland First principle (string theory) Global consistency condition: Divergence free at one-loop cf. anomaly Constrained vacua (1,0) SCFT 6D SCFT with (1,0) SUSY is less constrained and hence richer and more interesting. Still M5 is described by (1,0) tensor multiplet with BMNSD and one scalar. Recently classified by F-theory. [Heckman, Morrison, Vafa] [Del Zotto, Heckm an, Tomasiello, Vafa] 1.
Disconnected ex. E-string and TN. 2. Missing theories (irreducibility} Bottom-up: globally not allowed theories 3. Also, (2,0) SCFT can be a part of the theory and destined to break. (1,0) SCFT are elementary generators. Strongly coupled heterotic string M-theory on S1 / Z2 = heterotic string theory [Horava, Witten] At the ends of interval we have separated two E8s M2-brane stretched between them. String tension & gauge coupling is proportional to R
R E8 E8 248 248 M2 H M9 M9 Small instantons
Solution to self-dual equation F2 = *F2 : codimR = 4 Global consistency condition (12-n) + (12+n) = 24 E8 G + commutant unbroken group 1. Shrinking to zero size = E8 unbroken 2. Phase transition: NS5 detached into bulk. [Ganor, Hanany] E8 E8 M2 E-string
M5 M9 M9 dH = F1 F1 + F2 F2 R R = 0 ch2(FE8,1) + ch2(FE8,2) = ch2(R) Various configurations Various strings from M2 E8 E8 M
E E H M9 M5 Independent d.f.? Origin? Probing the structure of SCFTs. Global consistency condition? M9 Type IIA M theory on S1 = type IIA string theory
S2 can be viewed as S1 fibration over an interval I = S1 / Z2 radius ~ gauge coupling = VEV of dilaton E8s located at two poles E8s located at two ends Endpoints: (7+1)D hypersurface with a special value of coupling = 7-brane Type IIB Type IIB string theory on S2 + varying axio-dilaton = heterotic on T2 E8 E8
D3 D3 D3 D3 7 AN Dynkin diagram Heterotic string NS5 ? 7 F-theory Axio-dilaton of IIB has SL(2,Z) symmetry = the symm of T2
K3 7-branes are singularities classified [Kodaira] Everything is geometry: algebraic geometry M5? Understanding on small instantons Singularity vs gauge group Mathematician had classified [Du Val] [Neron] [Kodaira] codimensionC two singularities. Intersection numbers are identical to Cartan matrix of Lie algebras. [McKay] A, D, E algebra. Connection to physics: M/F-theory M2-branes wrapping on spheres gives charged W bosons.
Further compactification and duality F-theory on Heterotic string on T2 K3 Calabi-Yau 3 T2 S2 Hirzeburch Fn K3 S2
cf. Global consistency condition (12-n) + (12+n) = 24 Small instantons (discriminant locus) In the global construction, the complete 7-branes are E8+E8+Dinst [Aspinwall, Morrison] II (1,1,2) 12-n 12+n E8 II* (4,5,10) Blow-up Blowing up and extracting M5
(5,6,12) (4,6,12) = (1,0,0) E8 II* (4,5,10) Dinst II (1,1,2) @intersection (5,6,12) Small instantons (discriminant locus) In the global construction, the complete 7-branes are E8+E8+Dinst [Aspinwall, Morrison] II (1,1,2) 12-n 12+n
E8 II* (4,5,10) E-string E + E H Transition from E + E to H. [Haghighat, Lockhart, Vafa] E8 E8 M E E
H M9 Origin? Structure? Global consistency? M5 M9 Ep Eq M M2 between M5-branes = E-string + conjugate E-string E-string worldsheet theory: O(n) with symmetric repr E8
[J.-h. Kim, S. Kim, K.M. Lee, J.Park, Vafa] Ep Conjugate E-string has different theory USp(n) with antisymmetric repr. M -Eq Anomaly check [S. Kim, H.C. Kim, J. Park] [Shimizu, Tachikawa] (2,0) M-string limit M9 M5
E E = G E-string cycles Ep span the integral lattice (on which ADE roots are de fined) ADE quantum numbers are identified by D3 wrapped on the correspo nding root. E + E, etc are necessary for D, E type (2,0) SCFT. In the global construction, E + E does not have genus zero, calculated by adjunction formula; modified by f. We need E + E f ~ E E a new string we call G-string. Anomaly cancellation E8 E8 M
E E H M9 M5 M9 Consistency checks: 6D: E-string cycles span the integral lattice satisfying Dirac quantization and s elf-duality. [W. Taylor, Seiberg] 2. 2D: Chain of strings [S. Kim, H.C. Kim, J. Park] [Shimizu, Tachikawa]: this is the only possibility. We know how to cancel 10D anomaly of heterotic string
1. Conclusion Origin of M5-brane Induced from intersection of 7-branes Structure of SCFT is not flat (1D chain) but round (2D structure), especially for D, E types. Cf. topological vertex calc. of elliptic genera. E-strings are elementary Four kinds of E-strings H = E + E, M = E E, G = E E , Construction and classification of vacua of non-perturvative heterotic string theory Many SCFTs and
non-perturbative heterotic string vacua cannot be made.
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