Super Topological Recursion Kento Osuga University of Sheffield - PowerPoint PPT Presentation

Intro/Motivation Airy Structures Super Topological Recursion Super Topological Recursion Kento Osuga University of Sheffield Joint with Vincent Bouchard (U. Alberta) November 18, 2020 Strings and Fields 2020, YITP

Intro/Motivation Airy Structures Super Topological Recursion Origin of Topological Recursion (TR) 1-Hermitian matrix model of rank N with couplings t k � d +1 M k �� � � � Z ( N ; t k ) = dM exp − N t k Tr . H N k =0 N 2 − 2 g � Tr( M k 1 )Tr( M k 2 ) · · · Tr( M k n ) � ( g ) � c g ≥ 0 Any efficient technique to compute correlation functions? Ward identity (loop equations) & large N limit → all � Tr( M k 1 ) � (0) are encoded in a hyperelliptic curve C where C = { ( x, y ) ∈ C 2 | y 2 = P 2 d ( x ; g k ) } Topological recursion = utilize the geometry of C to compute � Tr( M k 1 )Tr( M k 2 ) · · · Tr( M k n ) � ( g ) c

Intro/Motivation Airy Structures Super Topological Recursion Topological Recursion has its own life! Spectral curve Invariants → TR → (Σ , , , ... ) (correlation functions) Kontsevich-Witten • Airy curve → TR → Intersection numbers ( y 2 = x ) (minimal Liouville gravity) Volumes of moduli spaces • Mirzakhani curve ( y = sin √ x ) → TR → of hyperbolic surfaces (JT gravity) Gromov-Witten invariants • Mirror curve (to a toric CY X ) → TR → on the mirror X (topological string amplitudes)

Intro/Motivation Airy Structures Super Topological Recursion Is it just a happy coincidence...? Why is such a simple formalism so ubiquitous in computations of a variety of invariants...? Any fundamental structure underlying the topological recursion? → Abstract loop equations (significant abstraction of the matrix-model Ward identity) Topological recursion is “solving” abstract loop equations dual Topological recursion ← − − → Airy structures

Intro/Motivation Airy Structures Super Topological Recursion Story of Topological Recursion Abstract loop equations (system of infinitely many equations) ւ ց solve geometrically: solve algebraically: ◦ pants decomposition ◦ vertex operator algebras ◦ residue analysis ◦ Virasoro constraints ↓ ↓ Topological recursion Airy structures Let’s decorate this story with supersymmetry!

Intro/Motivation Airy Structures Super Topological Recursion Airy Structures

Intro/Motivation Airy Structures Super Topological Recursion Question Let x 1 , ..., x n be a set of n variables and � be another parameter. We then define a particular set D of differential operators in the form D := C [ x i , � ∂ i , � ]. � ∂ x + � xx + � 2 x∂ x ∈ D , 1 , ∂ x , x 3 ∂ x / e.g., ∈ D . We consider a power series F = � log Z ∈ C [[ x i , � ]] and a set of differential operators L 1 , ..., L m ∈ D . Then, one may ask: What is the condition on F and L i such that a solution of differential equations L i Z = 0 exists and it is unique ? Airy structures provide a sufficient (but not necessary) condition

Intro/Motivation Airy Structures Super Topological Recursion Airy structures The formalism of Airy structures states that if: • n = m (# x i = # L i ), [ L i , L k ] = � D k for some D k • ∀ i, j ∈ Z , ij · L k , ij ∈ D , • F 0 , 1 ( i 1 ) = F 0 , 2 ( i 1 , i 2 ) = 0 ∀ i 1 , i 2 ∈ Z where d � g F g,n ( i 1 , ..., i n ) x i 1 · · · x i n , � � F = n ! g,n ≥ 0 i 1 , ··· ,i n =1 • one more technical condition on { L n } (omitted), then there exists a unique solution of L i Z = 0. Concretely, one can recursively compute all F g,n ( i 1 , ..., i n ) to recover Z from the data of { L n } .

Intro/Motivation Airy Structures Super Topological Recursion Vertex Operator Algebra Any example of Airy structures such that F g,n are meaningful? Twisted module of the free boson VOA (chiral CFT) α r � φ ( z ) = z r +1 r ∈ Z + 1 2 α − r = x r + 1 2 , α r = � ∂ r + 1 2 , [ α r , α s ] = � rδ r + s L n = 1 : α i α n − i : + � � ∀ n, m ≥ − 1 , 16 δ 0 ,n , 2 r ∈ Z + 1 2 [ L n , L m ] = � ( n − m ) L n + m L Φ n = Φ L n Φ − 1 omitted technical condition ⇒ dual Z annihilated by { L Φ n } ← − − → Topological recursion 1 : 1 Φ ← − − → a spectral curve

Intro/Motivation Airy Structures Super Topological Recursion Super Topological Recursion

Intro/Motivation Airy Structures Super Topological Recursion Story of Super Topological Recursion Abstract super loop equations (system of infinitely many equations) ւ ց solve geometrically: solve algebraically: ◦ pants decomposition ◦ vertex operator superalgebras ◦ residue analysis ◦ super Virasoro constraints ↓ ↓ Super Topological Recursion Super Airy Structures

Intro/Motivation Airy Structures Super Topological Recursion Super Airy Structures It is now clear how to incorporate supersymmetry into Airy structures • introduce additional fermionic variables θ 0 , ..., θ n and corresponding differential operators G 0 , ..., G m . • find appropriate conditions such that a solution of L i Z = G r Z = 0 exists and it is unique . • consider twisted modules of the free VOSA (chiral SCFT) and the super Virasoro subalgebra generated by { L n , G r } where n ≥ − 1 , r ≥ − 1 2 . Recall: Airy structures provide an algebraic way of solving abstract loop equations. Can we find abstract super loop equations that can be solved by super Airy structures ??

Intro/Motivation Airy Structures Super Topological Recursion Super Topological Recursion Vincent Bouchard and I have proposed the definitions of • a local super spectral curve, • abstract super loop equations, • super topological recursion, and showed a relation to dual super Airy structures. How did we find them? 1. recursive structure of supereigenvalue models (supersymmetric analogues of Hermitian matrix models) 2. correspondence with super Airy structures All the details are discussed in arXiv:2007.13186 ;-)

Intro/Motivation Airy Structures Super Topological Recursion Open Questions Summary Starting with appropriate initial data, it computes correlation functions of • supereigenvalue models both in the NS and R sector • (2 , 4 l )-minimal superconformal models coupled to Liouville supergravity Reduction to the standard topological recursion is proven for a certain family of local super spectral curves. (Super JT gravity) Open Questions Full classification? Any other applications in physics? Enumerative invariants with odd cohomology classes? Moduli spaces of super Riemann surfaces? Super algebraic curves and super quantum curves?

Intro/Motivation Airy Structures Super Topological Recursion References Topological Recursion: Chekhov–Eynard–Orantin (math-ph/0603003) Eynard–Orantin (math-ph/0702045, 0811.3531) Applications: Bouchard–Klemm–Mari˜ no–Pasquetti(0709.1453) Dunin-Barkowski–Orantin–Shadrin–Spitz (1211.4021) and many more... Airy Structures: Kontsevich–Soibelman (1701.09137) Andersen–Borot–Chekhov–Orantin (1703.03307) Super Topological Recursion: Bouchard–Ciosmak–Hadasz–O–Ruba–Su� lkowski (1907.08913) Bouchard–O (2007.13186)

Intro/Motivation Airy Structures Super Topological Recursion Thank you