FROM F-THEORY TO DYNAMIC GLSM Physics and Geometry of F-theory - PowerPoint PPT Presentation

FABIO APRUZZI FROM F-THEORY TO DYNAMIC GLSM Physics and Geometry of F-theory (2017), ICTP, Trieste

Based on 1602.04221 & 1610.00718 in collaboration with Falk Hassler, Jonathan Heckman and Ilarion Melnikov and see also 1601.02015 by Sakura Schäfer-Nameki and Timo Weigand

A RECENT HISTORY REVIEW

A RECENT HISTORY REVIEW F on CY4: 4D Theories

A RECENT HISTORY REVIEW F on CY4: 4D Theories … almost everybody in the audience

A RECENT HISTORY REVIEW F on CY4: 4D Theories … almost everybody in the audience F on CY3: 6D (SCFTs) Theories

A RECENT HISTORY REVIEW F on CY4: 4D Theories … almost everybody in the audience F on CY3: 6D (SCFTs) Theories … a large subset of you

A RECENT HISTORY REVIEW F on CY4: 4D Theories … almost everybody in the audience F on CY3: 6D (SCFTs) Theories … a large subset of you F on CY5: 2D (0,2) Theories

A RECENT HISTORY REVIEW F on CY4: 4D Theories … almost everybody in the audience F on CY3: 6D (SCFTs) Theories … a large subset of you F on CY5: 2D (0,2) Theories … a smaller subset of you

MAIN IDEA F-THEORY CY3 6D (SCFTs) Theories CY5 Kähler 4-Manifold + Twist 2D (0,2) Theories Construction of 2D (0,2) CFTs

WHY? MOTIVATIONS ▸ Generating novel 2D SCFTs ▸ 2D Theories are Closely Related to String Theories ▸ UV Completion of Non-Critical Strings

WHY? MOTIVATIONS ▸ Generating novel 2D SCFTs ▸ 2D Theories are Closely Related to String Theories ▸ UV Completion of Non-Critical Strings IR UV 2D GLSM M string D eff >> 10 M KK Energy Extra + Gravity +

WHY? MOTIVATIONS ▸ Generating novel 2D SCFTs ▸ 2D Theories are Closely Related to String Theories ▸ UV Completion of Non-Critical Strings IR UV 2D GLSM M string D eff >> 10 M KK Energy Extra + Gravity + ▸ Relevant for Time-Dependent and de Sitter Backgrounds Hellerman; Maloney, Silverstein; + Strominger

WHAT’S THE PLAN? OUTLINE 1. GLSMs from F-theory on CY5 and Heterotic on CY4 2. Extra Sectors in F-theory 3. Dynamic GLSMs from 6D (1,0) SCFTs on 4-Manifolds 4. Anomaly Polynomials and Central Charges

1. THE ROAD FROM F-THEORY TO 2D F-THEORY ON CY5 CY5

1. THE ROAD FROM F-THEORY TO 2D F-THEORY ON CY5 Elliptically Fibered CY5 B 4 y 2 = x 3 + f ( B 4 ) x + g ( B 4 ) B 4

1. THE ROAD FROM F-THEORY TO 2D F-THEORY ON CY5 Elliptically Fibered CY5 B 4 y 2 = x 3 + f ( B 4 ) x + g ( B 4 ) B 4 O ( − 4 K B 4 ) O ( − 6 K B 4 ) Section: Homogeneous polynomial of a certain degree

1. THE ROAD FROM F-THEORY TO 2D F-THEORY ON CY5 Elliptically Fibered CY5 B 4 y 2 = x 3 + f ( B 4 ) x + g ( B 4 ) B 4 Non-Compact, Gravity decoupled O ( − 4 K B 4 ) O ( − 6 K B 4 ) Section: Homogeneous polynomial of a certain degree

1. THE ROAD FROM F-THEORY TO 2D F-THEORY ON CY5 Elliptically Fibered CY5 B 4 y 2 = x 3 + f ( B 4 ) x + g ( B 4 ) B 4 Non-Compact, Gravity decoupled X 3 O ( − 4 K B 4 ) O ( − 6 K B 4 ) Section: Homogeneous polynomial of a certain degree ∆ = 4 f 3 + 27 g 2 = 0 • 7 Branes wrap 2D Spacetime + Kähler Threefold X 3

1. THE ROAD FROM F-THEORY TO 2D F-THEORY ON CY5 Elliptically Fibered CY5 B 4 y 2 = x 3 + f ( B 4 ) x + g ( B 4 ) B 4 Non-Compact, Gravity decoupled X 3 O ( − 4 K B 4 ) O ( − 6 K B 4 ) Section: Homogeneous polynomial of a certain degree ∆ = 4 f 3 + 27 g 2 = 0 • 7 Branes wrap 2D Spacetime + Kähler Threefold X 3 Gauge theory on 7 Branes ( f, g, ∆ ) Vanishing Degrees of : Kodaira Classification (Compact)

1. THE STRATEGY 1 Z 10 SYM: Tr( F IJ F IJ ) + 2 i χ Γ I D I χ d 10 x � � L 10 D = 4 g 2 Y M (4D) Beasley, Heckman, Vafa; Donagi-Wijnholt 7 Branes • Isometry SO (1 , 9) → SO (1 , 7) × U (1) R → SO (1 , 1) × U (3) × U (1) R X 3

1. THE STRATEGY 1 Z 10 SYM: Tr( F IJ F IJ ) + 2 i χ Γ I D I χ d 10 x � � L 10 D = 4 g 2 Y M (4D) Beasley, Heckman, Vafa; Donagi-Wijnholt 7 Branes • Isometry SO (1 , 9) → SO (1 , 7) × U (1) R → SO (1 , 1) × U (3) × U (1) R X 3 • Topological Twist SO (1 , 1) × U (3) × U (1) R → SO (1 , 1) × SU (3) × U (1) X × U (1) R

1. THE STRATEGY 1 Z 10 SYM: Tr( F IJ F IJ ) + 2 i χ Γ I D I χ d 10 x � � L 10 D = 4 g 2 Y M (4D) Beasley, Heckman, Vafa; Donagi-Wijnholt 7 Branes • Isometry SO (1 , 9) → SO (1 , 7) × U (1) R → SO (1 , 1) × U (3) × U (1) R X 3 • Topological Twist SO (1 , 1) × U (3) × U (1) R → SO (1 , 1) × SU (3) × U (1) X × U (1) R J top = J X + 3 2 J R

1. THE STRATEGY 1 Z 10 SYM: Tr( F IJ F IJ ) + 2 i χ Γ I D I χ d 10 x � � L 10 D = 4 g 2 Y M (4D) Beasley, Heckman, Vafa; Donagi-Wijnholt 7 Branes • Isometry SO (1 , 9) → SO (1 , 7) × U (1) R → SO (1 , 1) × U (3) × U (1) R X 3 • Topological Twist SO (1 , 1) × U (3) × U (1) R → SO (1 , 1) × SU (3) × U (1) X × U (1) R χ (10) J top = J X + 3 A (10) 2 J R I V (0 , 0) Vector Multiplet: µ − v + { D (0 , 1) ¯ ψ + , (0 , 1) ∂ A Chiral Superfield (CS) Multiplets: φ (3 , 0) Φ (3 , 0) χ + , (3 , 0) Λ − , (0 , 2) λ − , (0 , 2) Fermi Multiplet:

1. THE STRATEGY 1 Z 10 SYM: Tr( F IJ F IJ ) + 2 i χ Γ I D I χ d 10 x � � L 10 D = 4 g 2 Y M (4D) Beasley, Heckman, Vafa; Donagi-Wijnholt 7 Branes • Isometry SO (1 , 9) → SO (1 , 7) × U (1) R → SO (1 , 1) × U (3) × U (1) R X 3 • Topological Twist SO (1 , 1) × U (3) × U (1) R → SO (1 , 1) × SU (3) × U (1) X × U (1) R χ (10) J top = J X + 3 A (10) 2 J R I V (0 , 0) Vector Multiplet: µ − v + { D (0 , 1) ¯ ψ + , (0 , 1) ∂ A Chiral Superfield (CS) Multiplets: φ (3 , 0) Φ (3 , 0) χ + , (3 , 0) Form degree on Λ − , (0 , 2) λ − , (0 , 2) the 3fold Fermi Multiplet:

1. THE EFFECTIVE THEORY: BULK ACTION ON 7 BRANES Z (Kinetic Terms) + (DTerms) − | E | 2 − | J | 2 � � L 2 D = X 3 J (3 , 1) = ∂ W top E (0 , 2) = ∂ W top = D (0 , 1) Φ (3 , 0) F (0 , 2) = [ D (0 , 1) , D (0 , 1) ] = F (0 , 2) ∂ Λ 0 , 2 ∂ Λ 3 , 1 Z Z � � � � Λ (0 , 2) ∧ D (0 , 1) Φ (3 , 0) Λ (3 , 1) ∧ F (0 , 2) W top = Tr + Tr X 3 X 3 ¯ ∂ A φ (3 , 0) = 0 F (0 , 2) = 0 ω X 3 ∧ ω X 3 ∧ F (1 , 1) + [ φ (3 , 0) , φ (0 , 3) ] = 0 2D GLSM Action & Equations of Motion

1. THE EFFECTIVE THEORY: BULK ACTION ON 7 BRANES Z (Kinetic Terms) + (DTerms) − | E | 2 − | J | 2 � � L 2 D = X 3 J (3 , 1) = ∂ W top E (0 , 2) = ∂ W top = D (0 , 1) Φ (3 , 0) F (0 , 2) = [ D (0 , 1) , D (0 , 1) ] = F (0 , 2) ∂ Λ 0 , 2 ∂ Λ 3 , 1 Z Z � � � � Λ (0 , 2) ∧ D (0 , 1) Φ (3 , 0) Λ (3 , 1) ∧ F (0 , 2) W top = Tr + Tr X 3 X 3 ¯ E.o.M: ∂ A φ (3 , 0) = 0 F (0 , 2) = 0 ω X 3 ∧ ω X 3 ∧ F (1 , 1) + [ φ (3 , 0) , φ (0 , 3) ] = 0 2D GLSM Action & Equations of Motion

1. THE EFFECTIVE THEORY: BULK ACTION ON 7 BRANES Z (Kinetic Terms) + (DTerms) − | E | 2 − | J | 2 � � L 2 D = X 3 J (3 , 1) = ∂ W top E (0 , 2) = ∂ W top = D (0 , 1) Φ (3 , 0) F (0 , 2) = [ D (0 , 1) , D (0 , 1) ] = F (0 , 2) ∂ Λ 0 , 2 ∂ Λ 3 , 1 Z Z � � � � Λ (0 , 2) ∧ D (0 , 1) Φ (3 , 0) Λ (3 , 1) ∧ F (0 , 2) W top = Tr + Tr X 3 X 3 ¯ E.o.M: ∂ A φ (3 , 0) = 0 F (0 , 2) = 0 ω X 3 ∧ ω X 3 ∧ F (1 , 1) + [ φ (3 , 0) , φ (0 , 3) ] = 0 Susy variations match the Action & E.o.M. 2D GLSM Action & Equations of Motion

1. THE EFFECTIVE THEORY: BULK ACTION ON 7 BRANES Note: we presented here an action with all the KK modes, but at low energy only 0-modes appear, counted by bundle cohomologies on X 3 Z (Kinetic Terms) + (DTerms) − | E | 2 − | J | 2 � � L 2 D = X 3 J (3 , 1) = ∂ W top E (0 , 2) = ∂ W top = D (0 , 1) Φ (3 , 0) F (0 , 2) = [ D (0 , 1) , D (0 , 1) ] = F (0 , 2) ∂ Λ 0 , 2 ∂ Λ 3 , 1 Z Z � � � � Λ (0 , 2) ∧ D (0 , 1) Φ (3 , 0) Λ (3 , 1) ∧ F (0 , 2) W top = Tr + Tr X 3 X 3 ¯ E.o.M: ∂ A φ (3 , 0) = 0 F (0 , 2) = 0 ω X 3 ∧ ω X 3 ∧ F (1 , 1) + [ φ (3 , 0) , φ (0 , 3) ] = 0 Susy variations match the Action & E.o.M. 2D GLSM Action & Equations of Motion

1. THE EFFECTIVE THEORY FROM HETEROTIC ON CY4 Y 4 ∼ = O ( K X 3 ) → X 3 Relation to F-theory Z (Kinetic Terms) + (DTerms) − | E | 2 − | J | 2 � � L 2 D = Y 4 E (0 , 2) = ∂ W top ∂ W top = F ( even ) = F ( odd ) J (0 , 2) = Z 2 : Ω (4 , 0) 7! � Ω (4 , 0) (0 , 2) (0 , 2) ∂ Λ ( odd ) ∂ Λ ( even ) (0 , 2) (0 , 2) Z � � F (0 , 2) → F ( even ) + F ( odd ) Ω (4 , 0) ∧ Tr Λ (0 , 2) ∧ F (0 , 2) W top = (0 , 2) (0 , 2) Y 4 F (0 , 2) = ω Y 4 x F (1 , 1) = 0 E.o.M. 2D GLSM Action & Equations of Motion

1. BACK TO F-THEORY: LOCALIZED MATTER some bundles on X 1 3 X 2 3 X 2 Q ∈ K 1 / 2 3 ⊗ R 1 ⊗ R 2 S Q c ∈ K 1 / 2 ⊗ R ∨ 1 ⊗ R ∨ 2 S B 4 Ψ ∈ Ω (0 , 1) ( K 1 / 2 ⊗ R 1 ⊗ R 2 ) S S X 1 Ψ c ∈ Ω (0 , 1) ( K 1 / 2 ⊗ R ∨ 1 ⊗ R ∨ 2 ) 3 S S Z h Q c , Λ (0 , 2) Q i + Ψ c , Q c , ⌦ � � ↵ ⌦ � � ↵ W top,S = ∂ + A 1 + A 2 Q ∂ + A 1 + A 2 Ψ + S S Surface • Triple and Quartic Intersections localizing on curves and points Z f αβγ δ Ψ ( α ) δ Q ( β ) δ Q ( γ ) W top, Σ = Σ ⇣ δ Ψ ( α ) δ Q ( β ) δ Q ( γ ) δ Q ( δ ) ⌘ | p W top,p = h αβγδ

2. AN EXAMPLE y 2 = x 3 + f ( v, X 3 ) x + g ( v, x 3 ) B 4 = N v → X 3 g ( v, X 3 ) ∼ v 5 g ( X 3 ) f ( v, X 3 ) ∼ v 4 f ( X 3 ) X 3 = P 2 × P 1