New Evidence for (0 , 2) Target Space Duality He Feng Department of - PowerPoint PPT Presentation

New Evidence for (0 , 2) Target Space Duality He Feng Department of Physics, Virginia Tech based on: arXiv:1607.04628[hep-th] Lara B. Anderson and He Feng AMS Special Session - Nov. 13, 2016 He Feng (Virginia Tech) New Evidence for (0 , 2) Target Space Duality 1 / 35

Introduction Goal: further explore target space duality Generate examples with non-trivial D/F term potential ⇒ Count matter spectrum as previous work did Compare effective potential and explore vacuum spaces Study structure group and enhanced symmetries More work Provide complete list of target space dual chains Develop new tools (repeated entry, etc.) He Feng (Virginia Tech) New Evidence for (0 , 2) Target Space Duality 2 / 35

Review of Target Space Duality Abelian, massive 2D theory → (0 , 2) GLSM Multiple U (1) gauge fields A ( α ) with α = 1 , ..., r Chiral superfields: { X i | i = 1 , ..., d } with U (1) charges Q ( α ) , and i { P l | l = 1 , ..., γ } with U (1) charges − M ( α ) . l Fermi superfields: { Λ a | a = 1 , ..., δ } with charges N ( α ) , and a { Γ j | j = 1 , ..., c } with charges − S ( α ) . j Gauge and gravitational anomaly cancellation δ γ d c M ( α ) Q ( α ) S ( α ) � N ( α ) � � � = = a l i j a =1 i =1 j =1 l =1 γ δ c d M ( α ) M ( β ) S ( α ) S ( β ) Q ( α ) Q ( β ) � � N ( α ) N ( β ) � � − = − (1) a a l l j j i i a =1 j =1 i =1 l =1 for all α, β = 1 , ..., r . He Feng (Virginia Tech) New Evidence for (0 , 2) Target Space Duality 3 / 35

Put the above data in a table Γ j x i Q (1) Q (1) Q (1) − S (1) − S (1) S (1) . . . . . . c 1 2 d 1 2 Q (2) Q (2) Q (2) − S (2) − S (2) S (2) . . . . . . c 1 2 d 1 2 . . . . . . ... ... . . . . . . . . . . . . Q ( r ) Q ( r ) Q ( r ) − S ( r ) − S ( r ) S ( r ) . . . . . . c 1 2 1 2 d (2) Λ a p l N (1) N (1) N (1) − M (1) − M (1) − M (1) . . . . . . γ 1 2 δ 1 2 N (2) N (2) N (2) − M (2) − M (2) − M (2) . . . . . . γ 1 2 δ 1 2 . . . . . . ... ... . . . . . . . . . . . . N ( r ) N ( r ) N ( r ) − M ( r ) − M ( r ) − M ( r ) . . . . . . γ 1 2 1 2 δ He Feng (Virginia Tech) New Evidence for (0 , 2) Target Space Duality 4 / 35

GLSM is defined via a superpotential: � � � � � d 2 zdθ Γ j G j ( x i ) + P l Λ a F l S = a ( x i ) (3) j l,a G j and F l a are quasi-homogeneous polynomials with multi-degrees: G j S 1 S 2 . . . S c l F a M 1 − N 1 M 1 − N 2 . . . M 1 − N δ (4) M 2 − N 1 M 2 − N 2 . . . M 2 − N δ . . . ... . . . . . . M γ − N 1 M γ − N 2 . . . M γ − N δ F l a satisfies transversality condition: all F l a ( x ) = 0 only when all x i = 0 F-term potential: � � � � � � � � � 2 + � 2 � G j ( x i ) p l F l V F = a ( x i ) (5) j a l D-term potential: � � 2 r d γ � � � | x i | 2 − | p l | 2 − ξ ( α ) Q ( α ) M ( α ) V D = (6) i l α =1 i =1 l =1 He Feng (Virginia Tech) New Evidence for (0 , 2) Target Space Duality 5 / 35

Fayet-Iliopoulos (FI) parameter controls the phase, consider a single U(1): For ξ > 0, not all x i are zero thus not all F a are zero, G j ( x i ) = 0 and < p > = 0 ⇒ “geometric” phase (X, V) where X is a CY and V is a bundle, V = ker ( F l a ) i ) in the monad: im ( E a γ � δ � ⊗ E a ⊗ F l 0 → O ⊕ r V i a − − − → O M ( N a ) − − − → O M ( M l ) → 0 (7) M a =1 l =1 For ξ < 0, < p > � = 0 thus all < x i > = 0 ⇒ “nongeometric” phase Landau-Ginzburg orbifold with a superpotential: � � W ( x i , Λ a , Γ i ) = Γ j G j ( x i ) + Λ a F a ( x i ) (8) j a For multiple U(1)’s, hybrid phase He Feng (Virginia Tech) New Evidence for (0 , 2) Target Space Duality 6 / 35

Target space duality For Landau-Ginzburg orbifold with a superpotential: � � W ( x i , Λ a , Γ i ) = Γ j G j ( x i ) + Λ a F a ( x i ) (9) j a Observation (Distler, Kachru): An exchange/relabeling of the functions G j and F a will not affect the Landau-Ginzburg model, as long as anomaly cancellation conditions are satisfied. Procedure: Geometric to nongeometric phase: find phase with one � p l � � = 0 for some l , say l = 1. Γ j i := � p 1 � Λ a i s.t. � i || G j i || = � Rescale: ˜ Λ a i := Γ ji � p 1 � , ˜ 1 || . i || F a i Move to a region where Λ a i appear only with P 1 , i.e. choose F l a i = 0 ∀ l � = 1, i = 1 , . . . k . Leave non-geometric phase: || ˜ Λ a i || = || Γ j i || − || P 1 || and || ˜ Γ j i || = || Λ a i || + || P 1 || , return to a generic pt. and get new ( ˜ X, ˜ V ). He Feng (Virginia Tech) New Evidence for (0 , 2) Target Space Duality 7 / 35

Example Γ j Λ a x i p l (10) 0 0 0 1 1 1 1 − 2 − 2 1 0 0 2 − 3 1 1 1 2 2 2 0 − 4 − 5 0 1 1 6 − 8 Here || G 1 || = (2 , 4), || G 2 || = (2 , 5), || F 1 1 || = (2 , 8), || F 1 2 || = (3 , 7), || F 1 3 || = (3 , 7), || F 1 4 || = (1 , 2). Sum of third and fourth F equals sum of two G’s. Γ 1 = � p 1 � Λ 3 , ˜ Γ 2 = � p 1 � Λ 4 , ˜ Λ 3 = Γ 1 Λ 4 = Γ 2 Redefine: ˜ � p 1 � , ˜ � p 1 � , ˜ 3 , ˜ 4 , ˜ 3 = G 1 , ˜ G = F 1 G 2 = F 1 F 1 F 1 4 = G 2 then the new geometry is given by: || ˜ G 1 || = (3 , 7), || ˜ G 2 || = (1 , 2), || ˜ F 1 3 || = (2 , 4), || F 1 4 || = (2 , 5) Γ j Λ a x i p l (11) 0 0 0 1 1 1 1 − 3 − 1 1 0 1 1 − 3 1 1 1 2 2 2 0 − 7 − 2 0 1 4 3 − 8 He Feng (Virginia Tech) New Evidence for (0 , 2) Target Space Duality 8 / 35

Compare degree of freedom: Γ j Λ a x i p l (10) 0 0 0 1 1 1 1 − 2 − 2 1 0 0 2 − 3 1 1 1 2 2 2 0 − 4 − 5 0 1 1 6 − 8 dim ( M 0 ) = h 1 , 1 ( X ) + h 2 , 1 ( X ) + h 1 ( End 0 ( V )) = 2 + 68 + 322 = 392, h ∗ ( V ) = (0 , 120 , 0 , 0) Γ j Λ a x i p l 0 0 0 1 1 1 1 − 3 − 1 1 0 1 1 − 3 (11) 1 1 1 2 2 2 0 − 7 − 2 0 1 4 3 − 8 dim ( � M 0 ) = h 1 , 1 ( � X ) + h 2 , 1 ( � X ) + h 1 ( End 0 ( � V )) = 2 + 95 + 295 = 392, h ∗ ( � V ) = (0 , 120 , 0 , 0) Landscape scan by Blumenhagen + Rahn, agreement in nearly all ∼ 80,000 examples. He Feng (Virginia Tech) New Evidence for (0 , 2) Target Space Duality 9 / 35

TS duality with extra U(1) Add a new coord y 1 with multi-degree B and a hypersurface of degree B . Perform previous procedure (e.g. || B || = || F 1 1 || + || F 1 2 || − S 1 ) Resolve singularities (Distler, Greene, Morrison) by formally adding a P 1 (another coord y 2 ) Set constraint G B = y 1 = 0 to eliminate y 1 . Use additional U (1) and D-term to fix y 2 to a real constant. ↔ X × a single pt. Γ 1 Γ c Γ B x 1 . . . x d y 1 y 2 . . . 0 . . . 0 1 1 0 . . . 0 − 1 Q 1 . . . Q d B 0 − S 1 . . . − S c − B Λ 1 Λ 2 Λ δ . . . p 1 p 2 . . . p γ 0 0 . . . 0 − 1 0 . . . 0 N 1 N 2 . . . N δ − M 1 − M 2 . . . − M γ End up with new geometry: ˜ ˜ Γ 1 Γ c Γ B x 1 . . . x d y 1 y 2 . . . 0 . . . 0 1 1 − 1 . . . 0 − 1 Q 1 . . . Q d B 0 − ( M 1 − N 1 ) . . . − S c − ( M 1 − N 2 ) Λ 1 ˜ Λ 2 ˜ Λ δ . . . p 1 p 2 . . . p γ 1 0 . . . 0 − 1 0 . . . 0 0 M 2 − B . . . N δ − M 1 − M 2 . . . − M γ He Feng (Virginia Tech) New Evidence for (0 , 2) Target Space Duality 10 / 35

More TS Duality with Redundant Entry We consider V = ker ( F l a ) defined by a short exact sequence γ � δ � F l a 0 → V → O M ( N a ) − − → O M ( M l ) → 0 (12) a =1 l =1 Adding a redundant entry can lead to non trivial results after TS duality F 0 → V → B − → C → 0 F ′ 0 → V ′ → B ⊕ L − → C ⊕ L → 0 (13) where the new defining map F ′ is given by � F � α F ′ = (14) β C This repeated L is bounded, because of well-defined map F l a ; Too many L ’s won’t enroll in the transformation so keep redundant. He Feng (Virginia Tech) New Evidence for (0 , 2) Target Space Duality 11 / 35

Bundle stability/holomorphy and D/F-term N = 1 Supersymmetry in 4 D ⇒ Hermitian-Yang Mills Eqns F ab = F ab = g ab F ba = 0 (15) g ab F ba = 0 ⇔ Donaldson-Uhlenbeck-Yau Thm: V is stable (poly-stable). F ab = F ab = 0 ⇔ V is holomorphic. Stability ⇔ 4 D D-terms Holomorphy ⇔ 4 D F-terms Our work: Test TS duality with bundles not stable/holomorphic everywhere See if the stability/holomorphy properties (etc.) carry through He Feng (Virginia Tech) New Evidence for (0 , 2) Target Space Duality 12 / 35

D-term and stability Thanks to recent progress (Sharpe, Anderson, Gray, Lukas, Ovrut) The slope, µ ( V ), of a vector bundle is � 1 µ ( V ) ≡ c 1 ( V ) ∧ ω ∧ ω (16) rk( V ) X where ω = t k ω k is the Kahler form on X ( ω k a basis for H 1 , 1 ( X )). V is Stable if for every sub-sheaf F ⊂ V s.t. µ ( F ) < µ ( V ) V is Poly-stable if V = � i V i , where V i stable s.t. µ ( V ) = µ ( V i ) ∀ i . Problem: hard to find all sub-sheaves. V is stable if ∀ sub-line bundles L , µ ( L ) < µ ( ∧ k V ) = 0, where 0 < k < n . If there is a sub-bundle L = O ( a, b ), where ab < 0, then V is stable in the region 1 1 1 ( L ) t j t k = rk ( L ) d ijk c i rk ( L ) s i c i µ ( L ) = 1 ( F ) = s 1 a + s 2 b < 0 (17) He Feng (Virginia Tech) New Evidence for (0 , 2) Target Space Duality 13 / 35