Anomalous discrete symmetries in D-brane models

Transcription

1 Anomalous discrete symmetries in D-brane models Shohei Uemura Maskawa Institute for Science and Culture, Kyoto Sangyo University based on a work in progress with Tatsuo Kobayashi (Hokkaido University), and Junji Yamamoto (Kyoto University), Aug 26, 2017, Fuji Calm

2 Introduction Discrete symmetries are phenomenologically important. ex. Flavor symmetry, R parity, etc. In string EFT, discrete symmetries can appear in some models. String vacuum can have 4D chiral fermions and a symmetry can be anomalous. Anomalous symmetry broken by non-perturbative effects In fact, anomalous U(1), which is canceled by axion shift, can be broken by D-brane instanton. It is controlled by axion shift symmetry. How about discrete one?? Green-Schwartz mechanism

3 Plan to Talk 1. D6-brane models on a toroidal orbifold. 2. Discrete symmetries in D6-brane models. 3. Anomaly of the Discrete Symmetries, GS-like-mechanism and nonperturbative effects.

4 1. D6-brane models on a toroidal orbifold

5 How to get realistic models from type II string theory Real world = the Standard Model 4 dimensional, chiral non-abelian gauge symmetry, non-susy, generation... Type IIA String theory : 10 dimensional N=2 SUGRA, gauge symmetry = U(1) Orientifold + Compactified on CY 3-fold X 3 4 dim N=1 SUGRA Gauge group : given by D6-branes wrapping 3-cycles on the compact space. Chiral matters appear at the intersection points of D-branes. Intersecting D6-brane models

6 Intersecting D6-brane model Compact space

7 D6-branes on torus Compact space : T 6 = T 2 T 2 T 2 T 2 = {z i / : z i z i + 1, z i ~z i + τ i }

8 D6-branes on torus Compact space : T 6 = T 2 T 2 T 2 T 2 = {z i / : z i z i + 1, z i ~z i + τ i } We set 1-cycles [e x i ], [e y i ] as, [e y i ] [e x i ]

9 D6-branes on torus A D6-brane wraps 3 cycles Π a denoted by 6 integer winding numbers, Π i i i a = (n a e x + m i a [e i y ]) i On each T 2 i, D6-brane a intersect D6-brane b I ab = (n i a m i b m i a n i b ) times. Π i a = ( ei x + 2[e i y ]) i Π b = ( ei x [e i y ]) i I ab = 1 2 = 3 T 2 3 a and b have I ab = ς i=1 i I ab intersection points I ab chiral fermions

10 Toroidal orbifolds For more concrete model, we need toroidal orbifold = torus modded by a space group. Here, we concentrate on Z 2 Z 2 toroidal orbifold with discrete torsion. Θ Z 2 : T 2 T 2 T 2 T 2 T 2 T 2 Θ Z 2 : T 2 T 2 T 2 T 2 T 2 T 2 This orbifold has collapsed 3-cycles on the fixed points of Z 2 Z 2. The cycles are distinguished not only by the winding numbers, but also the fixed points wrapped by the cycles.

11 2. Discrete symmetries in D6-brane models

12 Discrete Symmetries on Torus D6-brane models on torus can have discrete flavor symmetry. First, a 6 dimensional tori has U 1 6 isometries, = U(1) U(1)

13 Discrete Symmetries on Torus D6-brane models on torus can have discrete flavor symmetry. First, a 6 dimensional tori has U 1 6 isometries, One D6-brane discretizes them to some abelian group. = Z 2

14 Discrete Symmetries on Torus D6-brane models on torus can have discrete flavor symmetry. First, a 6 dimensional tori has U 1 6 isometries, One D6-brane discretizes them to some abelian group. = As the result, the U 1 are discretized to Z N, (N = g. c. d. I ab ).

15 Discrete Symmetries on Torus One Z N permute the intersection points = chiral fermions. The other represents the winding number. The transformation rule is represented by matrices, Since C, Z 0, the symmetry group is non-abelian group : Z N Z N Z N. For Z 2 Z 2 toroidal orbifold, C, Z is not commute orbifolding. Only Z 2 sub group remains Z 2 Z 2 Z 2 = D 4

16 2. Anomaly of the Discrete Symmetries, GSlike-mechanism and nonperturbative effects.

17 GS-mechanism in D6-brane model. In string theory, gauge anomalies are controlled by GS-mechanism. For the D6-brane model, U(1) mixed anomaly does not vanish automatically, but it is canceled by axion shifts as, CS term, M4 (Π a +Π a ) Tr F F C 3 + Tr F C 5 Anomaly is canceled by c a = Πa C 3 c a + N a Π b Π b Π a The axion shift is equal to the sum of U(1) charges of the zero modes between virtual D6-brane wrapping the corresponding cycle and the other D-branes.

18 Axion Shift under Z 2 We would like to define axion shifts under Z 2 flavor symmetry by the sum of the charges of the charged zero mode of the virtual D6- brane wrapping the corresponding cycle. axion Π i : a 3-cycle on the compact space, (Π i : orientifold image) c i c i + A i π, where A i = σ m D brane s zero modes q (m) Z T2 2 (R (m) )

19 Anomaly cancellation By definition, this shift can trivially cancel the discrete anomaly. Anomaly is calculated as, A Z2 G 2 = m (m) (m) q Z2 T2 (R G ) Which is the same to the definition of the Z 2 shift, it trivially cancel the anomaly. cancel

20 It may cancel the discrete anomaly, but, there are cycles which do not respect the Z 2. How to define Z 2 charge of the zero modes?

21 b a b a Π i Π j Π i respects the Z 2 symmetry. Zero-modes construct the Z 2 rep. Π j does not respect the Z 2 symmetry. It may cancel the discrete anomaly, but, there are cycles which do not respect the Z 2. How to define Z 2 charge of the zero modes?

22 Can we define Z 2 charge? Since the D-brane system has Z 2 symmetry, it is always possible to find Π j which have the same zero mode structure. b a b a Π j Π j Π j Thus we can deal all the zero modes as representation of the Z 2.

23 Non-perturbative effects (D-brane instanton) The non-perturbative effects induced by D-brane instanton E is written as O E e S DBI,E+S CS,E where O E = ς i dα i e S int(α i,ψ i ) When E respects the Z 2 symmetry, S int is invariant under Z 2, the transformation law under Z 2 of O E is given by the transformation of the zero-modes α i, O E e πa i =π σ (m) m q Z2 T2 (R (m) ) OE and it is absorbed by the axion shift. e S CS,E e S CS,E πa i Non-perturbative effects are controlled by this flavor symmetry!

24 Conclusion The Flavor symmetry in D6-brane models on the toroidal orbifolds can be anomalous. We find the proper Z 2 shifts of axions to cancel the anomaly. This Z 2 shifts properly control the non-perturbative effects induced by D-brane instantons, too.

25 Thank you

26 Intersecting D6-brane models Gauge group : given by D6-brane s degree of freedom. Chiral matter : given at intersection points between D6-branes

27 Anomaly of discrete symmetries We have chiral fermions charged under discrete symmetries and gauge fields in D-brane models Anomaly? string Green-Schwartz mechanism We would like to study the relationship between this symmetry and the RR 3-forms.

28 Discrete Symmetries of toroidal models D6-brane models on torus can have discrete flavor symmetry. A 6 dimensional tori has U 1 6 isometries, One D6-brane discretize them to some abelian group.

29 Axion shift under U(1) Axion zero mode of RR 3-fields C 3 Homology class Π a H 3 (M, Z) Axion shift under U(1) the sum of charges of zero modes between the D-brane wrapping Π a and other D-branes.

30 Anomalous Z 2 model We can build model having an anomalous Z 2 discrete symmetry. Table. One example of anomalous Z 2 model. Since the intersection numbers are even on the first and the second T 2, we have two Z 2, Z 2 SU 2 2 a3 is anomalous.