F-theory, M5-branes and N=4 SYM with Varying Coupling Sakura Sch - PowerPoint PPT Presentation

F-theory, M5-branes and N=4 SYM with Varying Coupling Sakura Sch¨ afer-Nameki Geometry and Physics of F-theory, ICTP, February, 2017 1610.03663 with Benjamin Assel 1612.05640 with Craig Lawrie, Timo Weigand

Plan Goal: Understanding 4d N = 4 SYM with varying coupling, i.e. D3-branes in F-theory, via M5-branes on elliptic three-folds. I. D3s in F/M5s in M II. 4d N=4 SYM with varying coupling and Duality Defects III. New chiral 2d (0 , 2) Theories

I. D3s in F/M5s in M

4d N = 4 SYM with varying τ F-theory is IIB with varying τ , where there is also a self-duality group SL 2 Z , which descends upon D3-branes to the Montonen-Olive duality group of N = 4 SYM. 4d N = 4 SYM has an SL 2 Z duality group acting on the complexified coupling τ = θ 2 π + i 4 π τ → aτ + b g 2 , cτ + d , ad − bc = 1 and integral. Incidentally: the gauge group G maps to the Langlands dual group G ∨ . Usually, we consider τ constant in the 4d spacetime. Coming from F-theory, it’s very natural to ask whether we can define a version of N = 4 SYM with varying τ , compatible with the SL 2 Z action. ⇒ Network of 3d walls, 2d and 0d duality defects in N = 4 .

Duality Defects Variation of τ without singular loci are trivial. So the interesting physics will happen along the 4d space-time where τ is singular. ⇒ around such singular loci, τ will undergo an SL 2 Z monodromy. Usual lore: τ as the complex structure of an elliptic curve E τ ⇒ Lift to M5-branes ⇒ Setup: elliptic fibration over the 4d spacetime with N = 4 SYM in the bulk and duality defects (2d), which can intersect in 0d.

M5-brane point of view { 6d (2 , 0) theory on E τ × R 4 } = { N = 4 SYM on R 4 with coupling τ } So the setup that we will study is: { 6d (2 , 0) theory on a singular elliptic fibration } = { 4d N = 4 SYM with varying τ and duality defects } X 4 Y 3 B 2 M 3

Setups: # Setup 1: τ varies over 4d space (with B. Assel) ⇒ Y 3 elliptic three-fold ⊂ elliptic CY4 # Setup 2: τ varies onver a 2d space: 2d (0 ,p ) scfts (with C. Lawrie, T. Weigand) ⇒ D3s on curves in the base of CYn. In both setups: M5-brane point of view will be instrumental.

Advantages of the M5-brane point of view Various advantages in considering the M5-branes on elliptic surface � C instead of D3 on C : # 3-7 modes: Automatically included as chiral modes from B 2 reduced along (1 , 1) forms from singular fibers. # Topological Twist: 4d N = 4 with varying τ on C requires topological duality twists (TDT) [Martucci] Will see: corresponds to M5-brane on � C with standard ‘geometric’ topological twist. [Assel, SSN] # Non-abelianization: Bonus symmetry, and so TDT, exists for U (1) N = 4 SYM From M5-brane: 6d → 5d + non-abelianization approch exists see e.g. [Kugo], [Cordova, Jafferis], [Assel, SSN, Wong], [Luo, Tan, Vasko, Zhao] Similar considerations apply to the M2-brane duals, which give rise to a 1d N = 2 , 4 SQM. Non-abelianization possible there using BLG theory. For K3: [Okazaki]

The 6d (2 , 0) Theory # Lorentz and R-symmetry: SO (1 , 5) L × Sp (4) R ⊂ OSp (6 | 4) # Tensor multiplet: B MN : ( 15 , 1 ) with selfduality H = d B = ∗ 6 H n : m � Φ � ( 1 , 5 ) m : (¯ ρ � 4 , 4 ) # Abelian EOMs: H − = d H = 0 , n = 0 , m = 0 . ∂ 2 Φ � m � ∂ρ � /

II. 4d N = 4 SYM with varying coupling and Duality Defects [Assel, SSN]

M5-branes on Elliptic 3-folds An elliptic fibration E τ → Y 3 → B (Y not CY) has metric � 2 dy 2 � Y = 1 ( dx + τ 1 dy ) 2 + τ 2 µν db µ db ν . ds 2 + g B τ 2 Pick a frame e a for the base B and 1 e 5 = √ τ 2 dy . e 4 = ( dx + τ 1 dy ) , √ τ 2 Let Y 3 be a K¨ ahler three-fold, so the holonomy is reduced to U (3) L : SO (6) L → U (3) L 4 → 3 1 ⊕ 1 − 3 . On a curved space: Killing spinor equation with ∇ M connection ( ∇ M − A R M ) η = 0 R-symmetry background ⇒ constant spinor wrt twisted connection.

M5-branes on Elliptic 3-folds: Twist # Standard geometric twist: U (1) L with U (1) R Sp (4) R → SU (2) R × U (1) R 4 → 2 1 ⊕ 2 − 1 . # Topological Twist T U (1) twist = ( T U (1) L − 3 T U (1) R ) implies that the supercharge decomposes as SO (6) L × Sp (4) R → SU (3) L × SU (2) R × U (1) twist × U (1) R ( 4 , 4 ) → ( 3 , 2 ) − 2 , 1 ⊕ ( 3 , 2 ) 4 , − 1 ⊕ ( 1 , 2 ) − 6 , 1 ⊕ ( 1 , 2 ) 0 , − 1 ⇒ ( 1 , 2 ) 0 , − 1 give two scalar supercharges

Now specialize the 6d spacetime to be E τ → Y 3 → B 2 with coordinates x 0 , ··· ,x 5 , and ( x 4 ,x 5 ) the directions of the elliptic fiber. The spin connection along U (1) L is Ω U (1) L = − 1 6(Ω 01 + Ω 23 + Ω 45 ) , and the twist corresponds to turning on the background gauge field A U (1) R = − 3Ω U (1) L . The base B 2 is K¨ ahler as well, so the holonomy lies in U (1) ℓ × SU (2) ℓ ⊂ U (3) L with the U (1) generators given by T L = T ℓ + 2 T 45 Key: SO (2) 45 rotation is along the fiber, and the non-trivial fibration is characterized through a connection in this SO (2) 45 direction and the spin connection is A D = ω D = − ∂ a τ 1 e a 4 τ 2

Duality Twist This means: from the 4d point of view the topological twisting requires A D = ω D = − ∂ a τ 1 e a 4 τ 2 The associated U (1) is in fact what is known as the ”bonus symmetry” of abelian N = 4 SYM [Intrilligator][Kapustin, Witten] and we recovered the duality twist of N=4 SYM [Martucci] from the M5-brane theory. The bonus symmetry exists for the abelian N = 4 SYM and acts as follows on the supercharges for ab − cd = 1 m → e − i Q ˙ 2 α ( γ ) Q ˙ m e iα ( γ ) = cτ + d where Q m → e 2 α ( γ ) ˜ | cτ + d | ˜ i Q m i → φ i , � � + → e − i i λ ˙ m 2 α ( γ ) λ ˙ m λ m 2 α ( γ ) λ m φ + , − → e − � F ± ⋆F � F ( ± ) ≡ √ τ 2 F ( ± ) → e ∓ iα ( γ ) F ( ± ) µν µν 2

Duality Twisted N = 4 SYM from 6d 6d topological twist + dim reduction to B gives an N = 4 SYM with varying τ over a K¨ ahler base B � total = 1 S U (1) τ 2 F 2 ∧ ⋆F 2 − iτ 1 F 2 ∧ F 2 4 π B � + 8 ¯ (1 , 0) ∧ ρ (0 , 2) α − ∂ A ⋆ ˜ α + ¯ ∂ A ˜ ∂ ⋆ ψ α (1 , 0) χ (0 , 0) α − ∂ψ α ψ ˙ α ψ ˙ α (0 , 1) ˜ χ (0 , 0) ˙ (0 , 1) ∧ ˜ ρ (2 , 0) ˙ α π B � − 1 α ∧ ⋆∂ϕ α ˙ ¯ α + 2¯ ∂ϕ α ˙ ∂ A σ (2 , 0) ∧ ⋆∂ A ˜ σ (0 , 2) 4 π B and non-abelian extension (see paper with Ben Assel). The twisted fields are form fields and sections of the A D bundle specified by the charges: F ( ± ) ˜ ϕ α ˙ α χ α χ ˙ α ψ α ψ ˙ α ρ α ρ ˙ α σ (2 , 0) σ (0 , 2) ˜ ˜ ˜ 2 (0 , 0) (0 , 0) (1 , 0) (0 , 1) (0 , 2) (2 , 0) L q/ 2 ∓ 2 0 − 2 2 0 − 2 0 2 0 − 2 D

� � 8 − i S na = σ (0 , 2) ] − [˜ ψ ˙ (0 , 1) ∧ ⋆ψ α α Tr 16 f (0 , 0) [ σ (2 , 0) ∧ ˜ (1 , 0) ] ϕ α ˙ π √ τ 2 α B + 1 α ] ∧ σ (2 , 0) − 1 4 [˜ (0 , 1) ∧ ˜ ψ ˙ α 4 [ ψ α (1 , 0) ∧ ψ (1 , 0) α ] ∧ ˜ ψ (0 , 1) ˙ σ (0 , 2) χ ˙ α (0 , 0) ∧ ⋆χ α ρ ˙ α (2 , 0) ∧ ρ α + [˜ (0 , 0) ] ϕ α ˙ α + [˜ (0 , 2) ] ϕ α ˙ α � χ ˙ α σ (0 , 2) + [ χ α − [˜ (0 , 0) , ˜ α ] ∧ ˜ (0 , 0) ,ρ (0 , 2) α ] ∧ σ (2 , 0) ρ (2 , 0) ˙ � 1 α ][ ϕ β ˙ 2[ ϕ α ˙ α ,σ (2 , 0) ] ∧ [ ϕ α ˙ σ (0 , 2) ] + [ ϕ α ˙ α ,ϕ β ˙ β ,⋆ϕ α ˙ + Tr α , ˜ β ] 16 πτ 2 � α ,ϕ β ˙ + [ ϕ α ˙ β ][ ϕ β ˙ α ,⋆ϕ α ˙ β ] + [ σ (2 , 0) ∧ ˜ σ (0 , 2) ] ⋆ ([ σ (2 , 0) ∧ ˜ σ (0 , 2) ]) , Here A = a (1 , 0) + a (0 , 1) and dA = F 2 implies f (2 , 0) = √ τ 2 ∂a (1 , 0) , f (0 , 2) = √ τ 2 ¯ f (1 , 1) + f (0 , 0) ∧ j = √ τ 2 (¯ ∂a (0 , 1) , ∂a (1 , 0) + ∂a (0 , 1) ) So far: this describes the ‘4d bulk’ theory on B 2 with varying τ . Loci of interest: singularities in the fiber, which give duality defects.

Singular Elliptic Curves and Defects We can describe the elliptic fibration by E τ in terms of a Weierstrass model y 2 = x 3 + fx + g f and g sections K − 2 / − 3 and the singular loci are B ∆ = 4 f 3 + 27 g 2 = 0 . Close to a singular locus z 1 = 0 , τ ∼ i log z 1 + ··· with a branch-cut in the complex plane z 1 . For the M5 this is relevant along ∆ ∩ B : z 1 W γ τ γτ 1 =0 B 2 z 2 C

Gauge theoretic description of walls and defects Locally we can cut up B = ∪ B i and W ij 3d walls between these regions, where τ has a branch-cut. Define F D = τ 1 F + iτ 2 ⋆ F then the action of γ ∈ SL 2 Z monodromy on the gauge field is � � � � ( F ( j ) W ij = γ ( F ( i ) D ,F ( j ) ) D ,F ( i ) ) � � W ij � This maps the gauge part S F = − i B F ∧ F D to itself, except for an 4 π offset on the 3d wall (see also [Ganor]) � � � W ij = − i A ( i ) ∧ F ( i ) D − A ( j ) ∧ F ( j ) S γ D 4 π W ij E.g. γ = T k this is a level k CS term.

Chiral Duality Defects The wall action S γ is neither supersymmetric nor gauge invariant. At the boundary of the wall ∂W = C this induces chiral dofs: e.g. for the T k wall this is simply a chiral WZW model with β i , i = 1 , ··· ,k , with ⋆ 2 dβ i = idβ i [Witten] � � k � − 1 ⋆ 2 ( dβ i − A ) ∧ ( dβ i − A ) − i S C = β i F 8 π 4 π C C i =1 Under gauge transformations A → A + d Λ , β i → β i + Λ this generates � F Λ which cancels the anomaly from the 3d wall.