Superconformal Algebras for Twisted Connected Sums and G2 Mirror Symmetry

Superconformal algebras for twisted connected sums and G2 mirror symmetry

JHEP 1812, 011 (2018) [1809.06376]

Marc-Antoine Fiset

University of Oxford

KITP, Santa Barbara, 12 April 2019 (Md , g()M| Ricci(d , g) g) ≈ 0

2-dim Σ2 (complex curve; coords z, z¯)

σ :Σ2 ! Md In coords: x i (z, z¯), i = 1,... d (Field)

Principle to select trajectory σ (Theory) (Polyakov action / Non-linear sigma model) RNS Superstrings RNS Superstrings (IIA, IIB, heterotic): (IIA, IIB, heterotic): Quantization Add fermions Add spinors (Canonical / WZW / α0-expansion)

2-dim2-dimN = QFT 1 SQFTSCFT

Introduction

d d 1,9−d (M , g) M × R Not M-theory! (Md , g()M| Ricci(d , g) g) ≈ 0

2-dim Σ2 (complex curve; coords z, z¯)

σ :Σ2 ! Md In coords: x i (z, z¯), i = 1,... d (Field)

2-dim2-dimN = QFT 1 SQFTSCFT

Introduction

d d 1,9−d (M , g) M × R Not M-theory! (Md , g()M| Ricci(d , g) g) ≈ 0

2-dim Σ2 (complex curve; coords z, z¯)

σ :Σ2 ! Md In coords: x i (z, z¯), i = 1,... d (Field)

2-dim2-dimN = QFT 1 SQFTSCFT

Introduction

d d 1,9−d (M , g) M × R Not M-theory! (Md , g()M| Ricci(d , g) g) ≈ 0

σ :Σ2 ! Md In coords: x i (z, z¯), i = 1,... d (Field)

2-dim2-dimN = QFT 1 SQFTSCFT

Introduction

d d 1,9−d (M , g) M × R Not M-theory! 2-dim Σ2 (complex curve; coords z, z¯) (Md , g()M| Ricci(d , g) g) ≈ 0

2-dim2-dimN = QFT 1 SQFTSCFT

Introduction

d d 1,9−d (M , g) M × R Not M-theory! 2-dim Σ2 (complex curve; coords z, z¯)

σ :Σ2 ! Md In coords: x i (z, z¯), i = 1,... d (Field) (Md , g()M| Ricci(d , g) g) ≈ 0

RNS Superstrings RNS Superstrings (IIA, IIB, heterotic): (IIA, IIB, heterotic): Quantization Add fermions Add spinors (Canonical / WZW / α0-expansion)

2-dim2-dimN = QFT 1 SQFTSCFT

Introduction

d d 1,9−d (M , g) M × R Not M-theory! 2-dim Σ2 (complex curve; coords z, z¯)

σ :Σ2 ! Md In coords: x i (z, z¯), i = 1,... d (Field)

Principle to select trajectory σ (Theory) (Polyakov action / Non-linear sigma model) (Md , g()M| Ricci(d , g) g) ≈ 0

RNS Superstrings RNS Superstrings (IIA, IIB, heterotic): (IIA, IIB, heterotic): Add fermions Add spinors

2-dim2-dimN = QFT 1 SQFTSCFT

Introduction

d d 1,9−d (M , g) M × R Not M-theory! 2-dim Σ2 (complex curve; coords z, z¯)

σ :Σ2 ! Md In coords: x i (z, z¯), i = 1,... d (Field)

Principle to select trajectory σ (Theory) (Polyakov action / Non-linear sigma model)

Quantization (Canonical / WZW / α0-expansion) (Md , g) | Ricci(g) ≈ 0

RNS Superstrings RNS Superstrings (IIA, IIB, heterotic): (IIA, IIB, heterotic): Add fermions Add spinors

2-dim N = 1 SQFTSCFT

Introduction

d d 1,9−d (M , g) M × R Not M-theory! 2-dim Σ2 (complex curve; coords z, z¯)

σ :Σ2 ! Md In coords: x i (z, z¯), i = 1,... d (Field)

Principle to select trajectory σ (Theory) (Polyakov action / Non-linear sigma model)

Quantization (Canonical / WZW / α0-expansion)

2-dim QFT (Md , g) | Ricci(g) ≈ 0

2-dim2-dimN = QFT 1 SCFT

Introduction

d d 1,9−d (M , g) M × R Not M-theory! 2-dim Σ2 (complex curve; coords z, z¯)

σ :Σ2 ! Md In coords: x i (z, z¯), i = 1,... d (Field)

2-dim N = 1 SQFT (Md , g)

2-dim2-dimN = QFT 1 SQFT

Introduction

d d 1,9−d (M , g) | Ricci(g) ≈ 0 M × R Not M-theory! 2-dim Σ2 (complex curve; coords z, z¯)

σ :Σ2 ! Md In coords: x i (z, z¯), i = 1,... d (Field)

2-dim N = 1 SCFT • Not surjective: • Not injective: ∃ other constructions of SCFTs Mirror symmetry, T-duality

Mirror symmetry for TCS G2? [Braun, Del Zotto ’17; ’18]

Landau-Ginzburg, Gepner, etc.

Introduction

(Md , g) | Ricci(g) ≈ 0

2-dim N = 1 SCFT • Not injective: Mirror symmetry, T-duality

Mirror symmetry for TCS G2? [Braun, Del Zotto ’17; ’18]

Introduction

(Md , g) | Ricci(g) ≈ 0

• Not surjective: ∃ other constructions of SCFTs

Landau-Ginzburg, Gepner, etc. 2-dim N = 1 SCFT Introduction

(Md , g) | Ricci(g) ≈ 0

• Not surjective: • Not injective: ∃ other constructions of SCFTs Mirror symmetry, T-duality

Mirror symmetry for TCS G2? [Braun, Del Zotto ’17; ’18]

Landau-Ginzburg, Gepner, etc. 2-dim N = 1 SCFT Operator algebras TCS Contents

1 Operator algebras Formalism Examples (various Md )

2 TCS Algebras for TCS G2 mirror symmetry

M.-A. Fiset 1809.06376 ∞-dimensional Z2-graded Lie algebra “Operator algebra” N =1 {Gn , Tn}n∈Z , c Lie bracket Vir c := G 3 (z) , T(2)(z) OPEs 2 c : central charge 0 0 0 0 c/2 2T (z ) ∂z T (z ) 1 3 T (z)T (z ) ∼ + + [Tm, Tn] = (m − n)Tm+n + (m − m)c (z − z0)4 (z − z0)2 z − z0 12 T (z)G(z0) ∼ ... [Tm, Gn] = ... 0 [Gm, Gn] = ... G(z)G(z ) ∼ ...

Generators(weights)

Operator algebras Formalism TCS Examples Operator algebras: formalism

(Md , g) | Ricci(g) ≈ 0 2-dim N = 1 SCFT

Virasoro N = 1 acts on the Hilbert space.

M.-A. Fiset 1809.06376 “Operator algebra” N =1 Vir c := G 3 (z) , T(2)(z) OPEs 2

c/2 2T (z0) ∂0 T (z0) T (z)T (z0) ∼ + + z (z − z0)4 (z − z0)2 z − z0 T (z)G(z0) ∼ ... G(z)G(z0) ∼ ...

Generators(weights)

Operator algebras Formalism TCS Examples Operator algebras: formalism

(Md , g) | Ricci(g) ≈ 0 2-dim N = 1 SCFT

Virasoro N = 1 acts on the Hilbert space.

∞-dimensional Z2-graded Lie algebra

{Gn , Tn}n∈ , c Lie bracket Z c : central charge

1 3 [Tm, Tn] = (m − n)Tm+n + (m − m)c 12 [Tm, Gn] = ...

[Gm, Gn] = ...

M.-A. Fiset 1809.06376 Operator algebras Formalism TCS Examples Operator algebras: formalism

(Md , g) | Ricci(g) ≈ 0 2-dim N = 1 SCFT

Virasoro N = 1 acts on the Hilbert space.

∞-dimensional Z2-graded Lie algebra “Operator algebra” N =1 {Gn , Tn}n∈Z , c Lie bracket Vir c := G 3 (z) , T(2)(z) OPEs 2 c : central charge 0 0 0 0 c/2 2T (z ) ∂z T (z ) 1 3 T (z)T (z ) ∼ + + [Tm, Tn] = (m − n)Tm+n + (m − m)c (z − z0)4 (z − z0)2 z − z0 12 T (z)G(z0) ∼ ... [Tm, Gn] = ... 0 [Gm, Gn] = ... G(z)G(z ) ∼ ...

Generators(weights)

M.-A. Fiset 1809.06376 Operator algebras Formalism TCS Examples Operator algebras: formalism

Assemble your favorite name! (closely related concepts):

Chiral (Super-)Conformal Vertex Operator W Field Algebra Lie conformal / Current Λ-bracket

• Vertex Operator Algebra [Borcherds ’86; Frenkel, Lepowsky, Meurman ’88; Kac ’98; Frenkel, Ben-Zvi ’01]

M.-A. Fiset 1809.06376 Operator algebras Formalism TCS Examples

Operator Algebra [Thielemans ’95 hep-th/9506159] Mathematica package

1 Vector space V of operators denoted {1, A, B,...}. We think of elements of V as operators acting on the Hilbert space of the QFT. b f b 2 Z2-grading V = V ⊕ V and 1 ∈ V .

3 Even linear map ∂ : V ! V (e.g. ∂z ). 4 Sequence of bilinear pairings ·· : V ⊗ V V, n ∈ , compatible with n ! Z the grading and s.t. (terms in OPEs) (0) ∀ A, B ∈ V, ∃ n(A, B) ∈ such that AB = 0 . Z n≥n(A,B) (1)( unity): 1A = δ A ∀A ∈ V n 0,n (2)( commutativity): m |A||B| X (−1) (m−n) BA = (−1) ∂ AB ∀n ∈ Z n (m − n)! m m≥n (3)( associativity):

X n − 1 A BC = (−1)|A||B| B AC + AB C m n l − 1 l n m l≥1 m+n−l

Normal ordering: :AB: := AB 0

M.-A. Fiset 1809.06376 Operator algebras Formalism TCS Examples Calabi-Yau: Virasoro N = 2

(Md , g) | Ricci(g) ≈ 0 2-dim N = 1 SCFT

N =1 Vir c := G 3 , T(2) | OPEs 2 ∩ N =2 ˜ Vir c := G 3 , T(2) , J(1) , G 3 ) | OPEs 2 2 Properties / applications: • Spacetime supersymmetry • Spectral ﬂow: Isomorphism NS / R • Chiral ring of special NS ﬁelds (chiral primaries) related to Ramond ground states and Dolbeault cohomology • Topological twists • etc. [Lerche, Vafa, Warner ’89; Banks, Dixon, Friedan, Martinec ’88, etc.]

M.-A. Fiset 1809.06376 N =4 4 • e.g. n = 2 (K3): Odakec=6 = Vir c=6 (little) [Eguchi, Ooguri, Taormina, Yang ’89; Aspinwall, Morrison ’94, etc.]

6 • e.g. n = 3: Odakec=9

Operator algebras Formalism TCS Examples Calabi-Yau: Odake2n

Calabi-Yau manifolds have enhanced operator algebra. [Odake ’89] Complex dimension n ∈ N≥0 ⇒ discrete sequence of such algebras.

N =2 Vir c ∩ 2n ˜ Odakec=3n := G 3 , T(2) , J(1) , G 3 , A n + iB n , C n+1 + iD n+1 2 2 2 2 2 2

M.-A. Fiset 1809.06376 6 • e.g. n = 3: Odakec=9

Operator algebras Formalism TCS Examples Calabi-Yau: Odake2n

Calabi-Yau manifolds have enhanced operator algebra. [Odake ’89] Complex dimension n ∈ N≥0 ⇒ discrete sequence of such algebras.

N =2 Vir c ∩ 2n ˜ Odakec=3n := G 3 , T(2) , J(1) , G 3 , A n + iB n , C n+1 + iD n+1 2 2 2 2 2 2

N =4 4 • e.g. n = 2 (K3): Odakec=6 = Vir c=6 (little) [Eguchi, Ooguri, Taormina, Yang ’89; Aspinwall, Morrison ’94, etc.]

M.-A. Fiset 1809.06376 Operator algebras Formalism TCS Examples Calabi-Yau: Odake2n

Calabi-Yau manifolds have enhanced operator algebra. [Odake ’89] Complex dimension n ∈ N≥0 ⇒ discrete sequence of such algebras.

N =2 Vir c ∩ 2n ˜ Odakec=3n := G 3 , T(2) , J(1) , G 3 , A n + iB n , C n+1 + iD n+1 2 2 2 2 2 2

N =4 4 • e.g. n = 2 (K3): Odakec=6 = Vir c=6 (little) [Eguchi, Ooguri, Taormina, Yang ’89; Aspinwall, Morrison ’94, etc.]

6 • e.g. n = 3: Odakec=9

M.-A. Fiset 1809.06376 Operator algebras Formalism TCS Examples Parallel tensors and algebra generators

2n CY ; g , ω(2) , Ω(n)

z }| { z }| { z }| { 2n ˜ Odakec=3n := G 3 , T(2) , J(1) , G 3 , A n + iB n , C n+1 + iD n+1 2 2 2 2 2 2

N =1 Metric g, ∇i g = 0 Generators (G 3 , T(2)) of Vir 2

0 p-form φ, ∇i φ = 0 Generators (Φ p , Φ p+1 ) 2 2

(Symmetries leading to these currents understood for general (1, 0) non-linear sigma-models. [Howe, Papadopoulos ’93; de la Ossa, Fiset ’18])

M.-A. Fiset 1809.06376 Operator algebras Formalism TCS Examples Free

1 1 R or S ; g = ﬂat , dx

z }| { 1 D 1 1 E Free := (ψ(1/2) , j(1)) | ψ(z)ψ(w) ∼ z−w , j(z)j(w) ∼ (z−w)2 S

N =1 D 1 1 E Vir c=3/2 = :jψ: , :ψψ: + :jj: |{z} 2 2 G Free | {z } T Free

M.-A. Fiset 1809.06376 S N =1 D iΦ X E Vir c=7/10 = √ , − Tricritical Ising sector 15 5 |{z} | {z } 3-Ising G 3-Ising T

Properties / applications (some conjectural): • Spacetime supersymmetry • Spectral ﬂow: Isomorphism NS / R • Chiral ring of special NS ﬁelds (chiral primaries) related to Ramond ground states and DolbeaultCheck cohomology [Fernandez, Ugarte ’98] • Topological twists, etc??? [Shatashvili, Vafa ’95; de Boer, Naqvi, Shomer ’05]

Operator algebras Formalism TCS Examples

Shatashvili-Vafa algebra (for G2)

Operator algebras associated to G2 & Spin(7) holonomy [Shatashvili, Vafa ’95] (c.f. talk by S. Shatashvili 12 Sep 2017 @ Simons Center)

7 G2-manifold M ; g , ϕ(3) , ∗ ϕ(4)

z }| { z }| { z }| { G2 SVc=21/2 := (G 3 , T(2)), (Φ 3 , K(2)), (X(2) , M 5 ) | OPEs 2 2 2

M.-A. Fiset 1809.06376 Operator algebras Formalism TCS Examples

Shatashvili-Vafa algebra (for G2)

Operator algebras associated to G2 & Spin(7) holonomy [Shatashvili, Vafa ’95] (c.f. talk by S. Shatashvili 12 Sep 2017 @ Simons Center)

7 G2-manifold M ; g , ϕ(3) , ∗ ϕ(4)

z }| { z }| { z }| { G2 SVc=21/2 := (G 3 , T(2)), (Φ 3 , K(2)), (X(2) , M 5 ) | OPEs S 2 2 2 N =1 D iΦ X E Vir c=7/10 = √ , − Tricritical Ising sector 15 5 |{z} | {z } 3-Ising G 3-Ising T

M.-A. Fiset 1809.06376 Operator algebras Algebras for TCS TCS G2 mirror symmetry Algebra for TCS [Kovalev ’03; Corti, Haskins, Nordstr¨om,Pacini ’13, ’15]

1 M7 = R

1 S 6 6 CY 4 CY− CY+

1 S [Goette] | {z }

4 3 4 3 Odakec=6 × Free Odakec=6 × Free S S 6 6 Odakec=9 × Free Odakec=9 × Free S S SVG2

M.-A. Fiset 1809.06376 t!∞ 2 2 g −! gS + dt + dθ t!∞ Wedge ω −! dt ∧ dθ + ωS t!∞ Ω −! (dθ − idt) ∧ Ω S Normal ordering 6 CY+

| {z }

6 Odakec=9 × Free ⊂ = E D G , −! GS + :jt ψt : + :jθψθ: =: G+ J , −! :ψt ψθ: + JS =: J+ A + iB , −! :(ψθ − iψt )(AS + iBS ): =: A+ + iB+ ψξ

Operator algebras Algebras for TCS TCS G2 mirror symmetry Algebra for TCS

1 R

1 Sθ

CY 4 = S

1 Sξ | {z }

4 3 D E Odakec=6 × Free = GS , JS , AS + iBS , ψt , ψθ , ψξ & partners

M.-A. Fiset 1809.06376 t!∞ 2 2 g −! gS + dt + dθ t!∞ Wedge ω −! dt ∧ dθ + ωS t!∞ Ω −! (dθ − idt) ∧ Ω S Normal ordering

⊂

−! GS + :jt ψt : + :jθψθ: =: G+ −! :ψt ψθ: + JS =: J+ −! :(ψθ − iψt )(AS + iBS ): =: A+ + iB+

Operator algebras Algebras for TCS TCS G2 mirror symmetry Algebra for TCS

1 R

1 Sθ

CY 4 = S CY 6 + 1 Sξ | {z } | {z }

6 4 3 D E Odakec=9 × Free Odakec=6 × Free = GS , JS , AS + iBS , ψt , ψθ , ψξ = & partners E D G , J , A + iB , ψξ

M.-A. Fiset 1809.06376 Operator algebras Algebras for TCS TCS G2 mirror symmetry Algebra for TCS

1 R

t!∞ 2 2 1 g −! gS + dt + dθ Sθ t!∞ Wedge ω −! dt ∧ dθ + ωS t!∞ Ω −! (dθ − idt) ∧ Ω 4 S CY = S Normal ordering CY 6 + 1 Sξ | {z } | {z }

6 4 3 D E Odakec=9 × Free ⊂ Odakec=6 × Free = GS , JS , AS + iBS , ψt , ψθ , ψξ = & partners E D G , −! GS + :jt ψt : + :jθψθ: =: G+ J , −! :ψt ψθ: + JS =: J+ A + iB , −! :(ψθ − iψt )(AS + iBS ): =: A+ + iB+ ψξ

M.-A. Fiset 1809.06376 | {z }

6 GM := G + :jξψξ: Odakec=9 × Free [Figueroa O’Farrill ’96] Φ := :ψξJ: + A S X := 1 :JJ: − :ψ B: 2 ξ SVG2

Operator algebras Algebras for TCS TCS G2 mirror symmetry Algebra for TCS

1 M7 = R

1 Sθ

6 CY 4 CY+

1 Sξ

2 gM := g + dξ ϕ := dξ ∧ ω + Re(Ω) 1 ψ := 2 ω ∧ ω − dξ ∧ Im(Ω)

M.-A. Fiset 1809.06376 Operator algebras Algebras for TCS TCS G2 mirror symmetry Algebra for TCS

1 M7 = R

1 Sθ

6 CY 4 CY+

1 Sξ

2 gM := g + dξ ϕ := dξ ∧ ω + Re(Ω) 1 ψ := 2 ω ∧ ω − dξ ∧ Im(Ω) | {z }

6 GM := G + :jξψξ: Odakec=9 × Free [Figueroa O’Farrill ’96] Φ := :ψξJ: + A S X := 1 :JJ: − :ψ B: 2 ξ SVG2

M.-A. Fiset 1809.06376 Operator algebras Algebras for TCS TCS G2 mirror symmetry Algebra for TCS

1 M7 = R

1 S 6 6 CY 4 CY− CY+

1 S

| {z }

4 3 Odakec=6 × Free S 6 Odakec=9 × Free S SVG2

M.-A. Fiset 1809.06376 Operator algebras Algebras for TCS TCS G2 mirror symmetry Algebra for TCS

1 1 R R

1 1 Sθ Sξ

4 r 4 6 6 CY = S+ S− = CY CY− CY+

1 1 Sξ Sθ

∗ gS+ = r (gS−) ∗ ∗ ∗ ωS+ = r Re(ΩS−) Re(ΩS+) = r ωS− Im(ΩS+) = −r Im(ΩS−) (θ, ξ, t) 7−! (ξ, θ, −t) | {z } Preserve OPEs (“Automorphism”)

GS 7−! GS JS 7−! AS AS 7−! JS BS 7−! −BS (ψθ, ψξ, ψt ) 7−! (ψξ, ψθ, −ψt )

M.-A. Fiset 1809.06376 G Φ+ = :ψξJS : + :ψξψt ψθ: + :ψθAS : + :ψt BS : All SV 2 generators map to themselves.

Operator algebras Algebras for TCS TCS G2 mirror symmetry Algebra for TCS

1 M7 = R

1 S 6 6 CY 4 CY− CY+

1 S

| {z }

4 3 4 3 Odakec=6 × Free Odakec=6 × Free S S 6 6 Odakec=9 × Free Odakec=9 × Free S S SVG2 SVG2

M.-A. Fiset 1809.06376 S SVG2

Operator algebras Algebras for TCS TCS G2 mirror symmetry Algebra for TCS

1 M7 = R

1 S 6 6 CY 4 CY− CY+

1 S

| {z } G Φ+ = :ψξJS : + :ψξψt ψθ: + :ψθAS : + :ψt BS : All SV 2 generators map to themselves.

4 3 4 3 Odakec=6 × Free Odakec=6 × Free S S 6 6 Odakec=9 × Free Odakec=9 × Free S S SVG2

M.-A. Fiset 1809.06376 Operator algebras Algebras for TCS TCS G2 mirror symmetry Algebra for TCS: a subtlety

• The OPEs of SVG2 fail to deﬁne an associative operator algebra (axiom 3). However associativity is achieved by setting [Figueroa-O’Farrill ’97]

2 0 = 4:GX : − 2:ΦK: − 4∂z M − ∂z G (and the ideal it generates).

• Similarly, Odake6 is only associative up to the ideal generated by [Odake ’89] 0 = ∂z A − :JB: 0 = ∂z B + :JA: .

• Inclusions above hold up to these ideals.

M.-A. Fiset 1809.06376 Operator algebras Algebras for TCS TCS G2 mirror symmetry Automorphisms

j ψ Free: ξ ξ T-dualityT ξ − −

T G G˜ J A B C D Odake2n: Mirror symmetryM + + − − + − + − Phase Phφ=π + + + + − − − −

G T G Φ X K M SV 2 : GK mirror symmetry M + + − + − +

First observed in [Becker, Becker, Morrison, Ooguri, Oz, Yin ’96] . Interpreted as mirror symmetry for Joyce orbifolds in [Gaberdiel, Kaste ’04].

M.-A. Fiset 1809.06376 3 1 T4 [Braun, Del Zotto ’17] : T-dualities along T ﬁbres and external Sξ. Here purely quantum mechanical statement.

Operator algebras Algebras for TCS TCS G2 mirror symmetry Automorphisms

4 4 Odake × Freej × Freex × Freet Odake × Freej × Freex × Freet S S 6 6 Odake × Freex Odake × Freex Legend: S S T-duality SVG2 Mirror symmetry Phase GK mirror symmetry