ChernSimons theory and the higher genus B-model Andrea Brini - PowerPoint PPT Presentation

Two stringy viewpoints on ω M 3 , K g , h Take A : counts of open holomorphic curves “Arnold’s principle”: smooth invariants for ( M , S ) via symplectic invariants of ( T ∗ M , N ∗ S / M S ) CS on ( S 3 , K ) ← → open A-model on ( T ∗ S 3 , N ∗ K / S 3 K ) [Witten ’92, Ooguri–Vafa ’99] At large ( N , k ) : CS on ( S 3 , K ) ← → open/closed A-model on ( X = Tot ( O ⊕ 2 ( − 1 ) P 1 ) , L K ) [Gopakumar–Vafa ’98, Ooguri–Vafa ’99] (UoB & CNRS) CS theory and the higher genus B-model Warsaw 2018 8 / 39

Two stringy viewpoints on ω M 3 , K g , h Take A : counts of open holomorphic curves K = � : L K = L � = X σ , σ 2 = 1, σ ( ω ) = − ω . � N X , L � ( β, � d ) := d )] vir 1 g , h X , L � ( β,� [ M g , h [Katz–Liu, Li–Song ’01] � � GW X , L � N X , L � ( β, � d ) Q β � d := x g , h g , h � d ,β (UoB & CNRS) CS theory and the higher genus B-model Warsaw 2018 8 / 39

Two stringy viewpoints on ω M 3 , K g , h Take A : counts of open holomorphic curves K = � : L K = L � = X σ , σ 2 = 1, σ ( ω ) = − ω . � N X , L � ( β, � d ) := d )] vir 1 g , h X , L � ( β,� [ M g , h [Katz–Liu, Li–Song ’01] � d = ω S 3 , � � GW X , L � N X , L � ( β, � d ) Q β � := x g , h g , h g , h � d ,β (UoB & CNRS) CS theory and the higher genus B-model Warsaw 2018 8 / 39

Two stringy viewpoints on ω M 3 , K g , h Take B : spectral curves and the topological recursion (0, 1) (1, 1) �� v �� 2 v 1 v v 3 4 �� (0, 0) (1, 0) Fan ( X ) = C (Π X ) Π X M 3 = S 3 , K = � : S � = ( V ( N (Π X )) , d log Y , d log X ) (UoB & CNRS) CS theory and the higher genus B-model Warsaw 2018 9 / 39

Two stringy viewpoints on ω M 3 , K g , h Take B : spectral curves and the topological recursion CEO 0 , 1 [ S � ] = log Y ( x ) X CEO 0 , 2 [ S � ] = B ( X 1 , X 2 ) CEO g , h [ S � ]   � �  CEO g − 1 , h + 1 +  = Res p K CEO [ S � ] CEO g − g ′ , h − h ′ CEO g ′ , h ′ g ′ , h ′ d X ( p )= 0 (UoB & CNRS) CS theory and the higher genus B-model Warsaw 2018 9 / 39

Two stringy viewpoints on ω M 3 , K g , h Take B : spectral curves and the topological recursion CEO 0 , 1 [ S � ] = log Y ( x ) = ω S 3 , � = GW X , L � 0 , 1 0 , 1 X = ω S 3 , � CEO 0 , 2 [ S � ] = B ( X 1 , X 2 ) = GW X , L � 0 , 2 0 , 2 CEO g , h [ S � ] = GW X , L � = ω S 3 , � g , h g , h   � �  CEO g − 1 , h + 1 +  = Res p K CEO [ S � ] CEO g − g ′ , h − h ′ CEO g ′ , h ′ g ′ , h ′ d X ( p )= 0 (UoB & CNRS) CS theory and the higher genus B-model Warsaw 2018 9 / 39

Two stringy viewpoints on ω M 3 , K g , h Take B : spectral curves and the topological recursion CEO 0 , 1 [ S � ] = log Y ( x ) = ω S 3 , � = GW X , L � 0 , 1 0 , 1 X = ω S 3 , � CEO 0 , 2 [ S � ] = B ( X 1 , X 2 ) = GW X , L � 0 , 2 0 , 2 CEO g , h [ S � ] = GW X , L � = ω S 3 , � g , h g , h   � �  CEO g − 1 , h + 1 +  = Res p K CEO [ S � ] CEO g − g ′ , h − h ′ CEO g ′ , h ′ g ′ , h ′ d X ( p )= 0 ( CEO g , 0 [ S � ] = GW g ( X ) ) (UoB & CNRS) CS theory and the higher genus B-model Warsaw 2018 9 / 39

Yeah but... ...this is just one (trivial) example! Want to: find viewpoint B for S 3 − → M 3 ? � − → K ? (UoB & CNRS) CS theory and the higher genus B-model Warsaw 2018 10 / 39

Yeah but... ...this is just one (trivial) example! Want to: find viewpoint B for S 3 − → M 3 ? (in an arbitrary flat background) � − → K ? (UoB & CNRS) CS theory and the higher genus B-model Warsaw 2018 10 / 39

Part I: S 3 − → M 3 (UoB & CNRS) CS theory and the higher genus B-model Warsaw 2018 11 / 39

S 3 − → M 3 Take M 3 a Clifford–Klein 3-manifold: ∃ g | Ric ( g ) > 0 ⇔ M 3 ≃ S 3 / Γ , Γ ⊂ SO ( 4 ) . Examples: M 3 R P 3 Z / 2 Type A 1 : Γ = ⇒ ≃ M 3 I 120 ⇒ ≃ Type E 8 : Γ = Σ( 2 , 3 , 5 ) Find GW dual X Γ , if any. 1 Find spectral curve dual S Γ , if any. 2 Prove that large N duality, mirror symmetry, and the topological 3 recursion all hold true. (UoB & CNRS) CS theory and the higher genus B-model Warsaw 2018 12 / 39

The quest large N ? ✲ U ( N ) CS on S Γ A -model on...? ❃ ✚ ✻ ✚ ✚ ✚ ? ✚ y r t ✚ e m ✚ m y ✚ Vir-type s r o ✚ constraints? r r i m ✚ ✚ ✚ ✚ ✚ ✚ ❂ ✚ B -model on...? (UoB & CNRS) CS theory and the higher genus B-model Warsaw 2018 13 / 39

S 3 − → M 3 : A-side Geometric transition: V ( x 1 x 4 − x 2 x 3 ) ⊂ A 4 � ❅ � ❅ deform resolve � ❅ � ❅ � ❅ � ❅ � ❅ ✠ � ❅ ❘ � X ≃ T ∗ S 3 X = O ⊕ 2 P 1 ( − 1 ) (UoB & CNRS) CS theory and the higher genus B-model Warsaw 2018 14 / 39

S 3 − → M 3 : A-side Geometric transition: V ( x 1 x 4 − x 2 x 3 ) / Γ � ❅ � ❅ deform resolve � ❅ � ❅ � ❅ � ❅ � ❅ ✠ � ❅ ❘ � X ≃ T ∗ S Γ X orb = [ O ⊕ 2 P 1 ( − 1 ) / Γ] (UoB & CNRS) CS theory and the higher genus B-model Warsaw 2018 15 / 39

S 3 − → M 3 : A-side Remarks [ many things are well defined ]: Γ action is fiberwise 1 X Γ supports a C ⋆ -Calabi–Yau action with compact fixed loci 2 ∃ canonical Lagrangians L Γ ⊂ X Γ ( ⇒ open (orbifold) GW 3 invariants) [Katz–Liu ’01, AB–Cavalieri ’10] orientifold constructions carry through ( SO ( N ) / Sp ( N ) ) 4 [Sinha–Vafa ’00] By (2) above, for Γ � = Z / p Z , X Γ is non toric ⇒ no Hori-Iqbal-Vafa, no obvious mirror spectral curve (UoB & CNRS) CS theory and the higher genus B-model Warsaw 2018 16 / 39

Duality web large N ? ✲ U ( N ) CS on S Γ A -model on X Γ ✚ ❃ ✻ ✚ ✚ ✚ ? ✚ y r t ✚ e m ✚ m y ✚ Vir-type s geometric r o ✚ constraints? r engineering r i m ✚ ✚ ✚ ✚ ✚ ✚ ✚ ❂ ❄ N = 1 ( G Γ , ∅ ) B -model on...? SYM in d = 5 (UoB & CNRS) CS theory and the higher genus B-model Warsaw 2018 17 / 39

Duality web large N ? ✲ U ( N ) CS on S Γ A -model on X Γ ✚ ❃ ✻ ✚ ✚ ✚ ? ✚ y r t ✚ e m ✚ m y ✚ loop s geometric r o ✚ equations r engineering r i m ✚ ✚ ✚ ✚ ✚ ✚ ❂ ✚ ❄ ✛ N = 1 ( G Γ , ∅ ) B -model on relativistic � SW/IS correspondence SYM in d = 5 G Γ -Toda (UoB & CNRS) CS theory and the higher genus B-model Warsaw 2018 18 / 39

G Γ / � G Γ -relativistic Toda Phase space: P Γ ≃ ( C ⋆ ) r Γ x × ( C ⋆ ) r Γ y , { x i , y j } = C Γ ij x i y j Dynamics: L : ( P , { , } ) → ( G Γ / T Γ , { , } DJOV ) {L ∗ H i , L ∗ H j } = 0 , H i ∈ O ( G Γ ) G Γ . Type A: non-periodic/periodic Ruijsenaars system [Fock–Marshakov ’97-’14, Williams ’12, Kruglinskaya–Marshakov ’14] (UoB & CNRS) CS theory and the higher genus B-model Warsaw 2018 19 / 39

Relativistic Toda Γ spectral curves Toda Γ = ( C Γ , L , Ω 1 , Ω 2 ) , with: ρ � = 1 ∈ Rep ( G Γ ) : 1 P Γ ,ρ � det ρ ( µ id − L ( λ )) ∈ Z [ µ ][ Tr ω 1 L ( λ ) , . . . , Tr ω r Γ L ( λ )] . C Γ = {P Γ ,ρ = 0 } ; 2 ∃ a canonical correspondence on C Γ , lifting to projectors 3 π 1 : Pic ( 0 ) ( C Γ ) → T Γ and π 2 : H 1 ( C Γ , Z ) → L ; motion linearises on T Γ . [Kanev ’87, Donagi ’88, McDaniel–Smolinsky ’92-97, Martinec–Warner ’95] Spectral differentials: (Ω 1 , Ω 2 ) = ( d log λ, d log µ ) . 4 [Krichever–Phong ’96-’97, D’Hoker–Krichever–Phong ’02] � � λ + C Also: Tr ω i L ( λ ) = u i + δ i , ¯ , C = Casimir . Problem completely 5 i λ solved by determining ∧ n ρ = p n , Γ ( ρ ω 1 , . . . , ρ ω r Γ ) ∈ Rep ( G Γ ) . (UoB & CNRS) CS theory and the higher genus B-model Warsaw 2018 20 / 39

ADE 6 , 7 Problem is trivial in type A n ( ρ = � ) , a back-of-the-envelope calculation for type D n ( ρ = ( 2n v ) ), and computable in reasonably short time on Mathematica for ( E 6 , 27 ) (runtime: 30mins) and ( E 7 , 56 ) (1/2 day). (UoB & CNRS) CS theory and the higher genus B-model Warsaw 2018 21 / 39

Toda Γ spectral curves: type A 15 10 5 1 −1 (UoB & CNRS) CS theory and the higher genus B-model Warsaw 2018 22 / 39

Toda Γ spectral curves: type D 15 10 5 1 −1 (UoB & CNRS) CS theory and the higher genus B-model Warsaw 2018 23 / 39

Toda Γ spectral curves: type E 6 25 20 15 10 5 −2 −1 0 1 2 (UoB & CNRS) CS theory and the higher genus B-model Warsaw 2018 24 / 39

Toda Γ spectral curves: type E 7 50 40 30 20 10 −2 −1 0 1 2 3 −3 (UoB & CNRS) CS theory and the higher genus B-model Warsaw 2018 25 / 39

E 8 ( E 8 , ρ = e 8 ) is completely unwieldy at face value, but it’s related to the Poincaré sphere, it’s immoral to leave anyone behind, and particularly frustrating when the most exceptional case is kept untreated. ∃ there’s a semi-numerical way to break up the computation into a big number of smaller pieces of at most the size of the E 7 problem. Runtime grand total: 110 months, however code is easily “parallelisable”. With N ≃ 75 cores at once, this was reduced to about 1.5 months on a couple of small departmental clusters; see tiny.cc/E8Char (UoB & CNRS) CS theory and the higher genus B-model Warsaw 2018 26 / 39

Toda Γ spectral curves: type E 8 250 200 150 100 50 0 −5 5 (UoB & CNRS) CS theory and the higher genus B-model Warsaw 2018 27 / 39

At the end of the day... Theorem (Borot–AB ( ADE 6 , 7 ), AB ( E 8 )) The B-model Gopakumar–Vafa duality holds in all genera for Clifford–Klein 3-manifolds in a reducible flat background – and for them alone. (UoB & CNRS) CS theory and the higher genus B-model Warsaw 2018 28 / 39

At the end of the day... SOTP: CS on Seifert manifolds reduces to a trigonometric (multi-)eigenvalue model ∃ P CS ∈ C [ x , y ] | P CS ( z , ω S 3 / Γ 0 , 1 ( z )) = 0 [AB-Eynard–Mariño ’11, Borot–Eynard ’14, Borot-AB ’15, AB ’17] P CS = det ( µ − L ( λ )) | u ( t ) ⇒ ω S 3 / Γ = CEO 0 , 1 [ Toda Γ ] 0 , 1 [Borot-AB ’15, AB ’17] ω S 3 / Γ = CEO 0 , 2 [ Toda Γ ] 0 , 2 ⇒ all genus/colorings Extra sphaericos nulla salus (UoB & CNRS) CS theory and the higher genus B-model Warsaw 2018 28 / 39

Application: mirrors of DZ Frobenius manifolds Two meaningful operations: replace Ω 1 = d log λ → d λ (Dubrovin’s almost duality) 1 restrict to degenerate leaf C → 0 2 Combining both: bona-fide (conformal, with flat-unit) FM structure F Γ Toda on a suitable Hurwitz space. Dubrovin–Zhang constructed Frobenius structures on orbit spaces of affine Weyl groups. The resulting Frobenius manifold depends furthermore on a choice of a marked simple root of g Γ . [Dubrovin–Zhang ’96-’97] (UoB & CNRS) CS theory and the higher genus B-model Warsaw 2018 29 / 39

Theorem For all simple Lie groups, we have F Γ Toda ≃ DZ ( Weyl Γ ) 1 choices of marked root in DZ correspond to choices of dual 2 marked fundamental weights in F Γ Toda for simply-laced Γ and canonical choice of roots, 3 F Γ Toda ≃ QH orb ( P 1 Γ ) [Rossi ’08, Zaslow ’92] the higher genus GW potential of P 1 Γ coincides with the CEO 4 higher genus free energies on F Γ Toda . Upon reversing Dubrovin’s almost duality on the same leaf: Toda ≃ QH orb ([ C 2 / Γ]) ≃ QH ( � � F Γ C 2 / Γ) (UoB & CNRS) CS theory and the higher genus B-model Warsaw 2018 30 / 39

Part II: � − → K (UoB & CNRS) CS theory and the higher genus B-model Warsaw 2018 31 / 39

B-model for K The aim: Recall that for K = � , the full set of quantum invariants of the unknot were computed from the rational version of the topological recursion: ω S 3 , � = CEO g , h [ S � ] g , h Ideally, we’d like to have exactly the same for all knots. (UoB & CNRS) CS theory and the higher genus B-model Warsaw 2018 32 / 39

B-model for K Four things we know/expect: spectral curves : 1-dimensional mirrors S K via the knot DGA and 1 the augmentation polynomial of K : → Aug K ∈ Z [ X , Y , Q ] N X (Π) = 1 − X − Y + QXY − [Aganagic–Vafa ’12, AENV ’13, Ng ’04] topological recursion : open/closed B-model on conic bundles 2 zw = P ( X , Y ) ∈ C 2 × ( C ∗ ) 2 solved by Eynard–Orantin recursion; [BKMP ’07, Dijkgraaf–Vafa ’08, Gu–Jockers–Klemm–Soroush ’14] the closed sector : CEO g , 0 [ S K ] = GW g ( X ) regardless of K ; 3 [GJKS ’14] symplectic invariance : if φ : (( C ⋆ ) 2 , ω ) → (( C ⋆ ) 2 , ω ) is 4 symplectic, then CEO g , 0 [ S ] = CEO g , 0 [ φ ∗ S ] (UoB & CNRS) CS theory and the higher genus B-model Warsaw 2018 33 / 39

The Main Conjecture (crude form) Piecing everything together... Conjecture There exists a set-theoretic correspondence K → φ K ∈ SCr ( 2 ) (the symplectic Cremona group of the plane ) such that ω K g , h = CEO g , h [ φ ∗ K N (Π X )] ( ⋆ ) (cfr: mutations of LG potentials) [Galkin–Usnich ’10, Akhtar–Coates–Galkin–Kasprzyk ’13] (UoB & CNRS) CS theory and the higher genus B-model Warsaw 2018 34 / 39

The symplectic Cremona group of the plane Let ω = d log X ∧ log Y be the standard holomorphic symplectic form on the 2-torus. CP 2 Cr ( 2 ) = Aut C ( C ( X , Y )) : → ∨ ֒ (( C ∗ ) 2 , ω ) SCr ( 2 ) = { φ ∈ Cr ( 2 ) | φ ∗ ω = ω } : � �� ∨ � � ( C ⋆ ) 2 , SL 2 ( Z ) , Z / 5 = � P � � � SL 2 ( Z ) = � C , I | C 3 = I 4 = 1 , I 2 C = CI 2 � , P : ( X , Y ) → Y , Y X + 1 X SCr ( 2 ) / ( C ⋆ ) 2 ≃ � C , I , P | C 3 = I 4 = P 5 = 1 , I 2 C = CI 2 , I = PCP � [Usnich ’08, Blanc ’10] (UoB & CNRS) CS theory and the higher genus B-model Warsaw 2018 35 / 39

The Main Conjecture (somewhat improved form) Conjecture There exists a set-theoretic correspondence K → φ K ∈ SCr ( 2 ) (the symplectic Cremona group of the plane ) such that ω K g , h = CEO g , h [ φ ∗ K N (Π X )] ( ⋆ ) Two (necessary) tweaks: conformal φ K : φ ∗ K ω = α K ω , α k ∈ Z 1 symmetry action on the open string sector: ∆ : S K → S K (cfr: 2 Γ -correspondences for 3-manifolds) (UoB & CNRS) CS theory and the higher genus B-model Warsaw 2018 36 / 39

What ( ⋆ ) would be f 1 ։ T � { φ ∈ PL ( S 1 ) | Disc ( φ ) dyadic partition, slopes ∈ 2 Z } SCr ( 2 ) [Usnich ’07] (UoB & CNRS) CS theory and the higher genus B-model Warsaw 2018 37 / 39

What ( ⋆ ) would be f 1 ։ T � { φ ∈ PL ( S 1 ) | Disc ( φ ) dyadic partition, slopes ∈ 2 Z } SCr ( 2 ) [Usnich ’07] 1.0 0.8 0.6 0.4 0.2 0.2 0.4 0.6 0.8 1.0 (UoB & CNRS) CS theory and the higher genus B-model Warsaw 2018 37 / 39

ChernSimons theory and the higher genus B-model Andrea Brini - PowerPoint PPT Presentation

ChernSimons theory and the higher genus B-model Andrea Brini University of Birmingham & CNRS Quantum fields, knots and strings, Sep 2018 (UoB & CNRS) CS theory and the higher genus B-model Warsaw 2018 1 / 39 This talk Main thread

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ChernSimons theory and the higher genus B-model Andrea Brini - PowerPoint PPT Presentation

ChernSimons theory and the higher genus B-model Andrea Brini University of Birmingham & CNRS Quantum fields, knots and strings, Sep 2018 (UoB & CNRS) CS theory and the higher genus B-model Warsaw 2018 1 / 39 This talk Main thread

Multi-Hamiltonian formulations of the extended Chern-Simons theory with higher derivatives

Chern-Simons vector models and duality in 3 dimensions 19. Aug. 2013 @ YITP String theory and

Conformal Chern-Simons Gravity ESI Workshop on Higher Spin Hamid R. Afshar Vienna University of

Complete factorization in minimal N=4 Chern-Simons matter theory 7. Aug. 2017 @ YITP Strings and

Entanglement Entropy in 2+1 Chern-Simons Theory Shiying Dong UIUC With: Eduardo Fradkin, Rob

Higher spin theories, holography and Chern-Simons vector models Simone Giombi Perimeter

A simplicial approach to the non-Abelian Chern-Simons path integral Atle Hahn Group of

Asymptotics of higher genus string integrands Boris Pioline String theory from a worldsheet

On the Universality of the Chern-Simons Diffusion Rate Aldo L. Cotrone Florence University

ESI Workshop on Higher Spin Gravity Based on: hep-th/1006.4788 [Ammon, Gutperle, Kraus, EP]

Anomalies, Chern-Simons Terms, and Black Hole Entropy ! Tatsuo Azeyanagi (ENS) ! ! Based on

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Rational Points on Curves of Higher Genus Michael Stoll Universit at Bayreuth Pacific

Introduction M : an oriented closed 3-manifold : 1 ( M ) PSL(2 , C ) : a rep. of the

Brane Tilings, M2-Branes and Chern-Simons Theories NOPPADOL MEKAREEYA Theoretical Physics Group,

Applications of Elliptic genera A.Libgober July 7, CAT09, Warsaw 1.Elliptic genus in non

Geometric Rank Functions and Rational Points on Curves Eric Katz (University of Waterloo) joint

Computing Cyclic Isogenies in Genus 2 with Applications in Cryptography Alina Dudeanu 1 Dimitar

Hirzebruch genera and functional equations Victor M. Buchstaber Steklov Mathematical Institute,

Fast computation of isomorphisms of hyperelliptic curves and explicit descent Reynald Lercier,

The Strong Symmetric Genus of Almost All D -type Generalized Symmetric Groups Michael A. Jackson

A Non-Compact Elliptic Genus Sujay Ashok Jan Troost 1004.3649 and CNRS Ecole Normale

Logistics The Renderman Shading Language Checkpoint 3 Grading underway Checkpoint 4

KNL KNL KNL KNL KNL KNL KNL Example code: Check available memory [Xajacks@eln4

Symmetry Breaking in Quantum Curves & Super Chern-Simons Matrix Models Sanefumi Moriyama