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 tk Z(N;tk) =

• HN

dM exp

• −N

d+1

• k=0

tkTr

• Mk

.

• g≥0

N2−2g Tr(Mk1)Tr(Mk2) · · · Tr(Mkn)

(g) c

Any efficient technique to compute correlation functions? Ward identity (loop equations) & large N limit → all Tr(Mk1)(0) are encoded in a hyperelliptic curve C where C = {(x, y) ∈ C2|y2 = P 2d(x; gk)} Topological recursion = utilize the geometry of C to compute Tr(Mk1)Tr(Mk2) · · · Tr(Mkn)(g)

c

Intro/Motivation Airy Structures Super Topological Recursion

Topological Recursion has its own life!

Spectral curve (Σ, , , ...) → TR → Invariants (correlation functions)

• Airy curve

(y2 = x) → TR → Kontsevich-Witten Intersection numbers (minimal Liouville gravity)

• Mirzakhani curve

(y = sin √x) → TR → Volumes of moduli spaces

• f hyperbolic surfaces

(JT gravity)

• Mirror curve

(to a toric CY X) → TR → Gromov-Witten invariants

• n 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

• f 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 Topological recursion

dual

← − − → 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:

• pants decomposition
• residue analysis

solve algebraically:

• vertex operator algebras
• 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 x1, ..., xn be a set of n variables and be another

• parameter. We then define a particular set D of differential
• perators in the form D := C[xi, ∂i, ].

e.g., ∂x + xx + 2x∂x ∈ D, 1, ∂x, x3∂x / ∈ D. We consider a power series F = log Z ∈ C[[xi, ]] and a set of differential operators L1, ..., Lm ∈ D. Then, one may ask: What is the condition on F and Li such that a solution of differential equations LiZ = 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 (#xi = #Li),
• ∀ i, j ∈ Z,

[Li, Lk] = Dk

ij · Lk,

for some Dk

ij ∈ D,

• F0,1(i1) = F0,2(i1, i2) = 0

∀ i1, i2 ∈ Z where F =

• g,n≥0

g n!

d

• i1,··· ,in=1

Fg,n(i1, ..., in)xi1 · · · xin,

• one more technical condition on {Ln} (omitted),

then there exists a unique solution of LiZ = 0. Concretely, one can recursively compute all Fg,n(i1, ..., in) to recover Z from the data of {Ln}.

Intro/Motivation Airy Structures Super Topological Recursion

Vertex Operator Algebra

Any example of Airy structures such that Fg,n are meaningful? Twisted module of the free boson VOA (chiral CFT) φ(z) =

• r∈Z+ 1

2

αr zr+1 α−r = xr+ 1

2 ,

αr = ∂r+ 1

2 ,

[αr, αs] = rδr+s ∀ n, m ≥ −1, Ln = 1 2

• r∈Z+ 1

2

: αiαn−i : + 16δ0,n, [Ln, Lm] = (n − m)Ln+m

• mitted technical condition

⇒ LΦ

n = ΦLnΦ−1

Z annihilated by {LΦ

n} dual

← − − → Topological recursion Φ

1 : 1

← − − → 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:

• pants decomposition
• residue analysis

solve algebraically:

• vertex operator superalgebras
• 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 θ0, ..., θn and

corresponding differential operators G0, ..., Gm.

• find appropriate conditions such that a solution of

LiZ = GrZ = 0 exists and it is unique.

• consider twisted modules of the free VOSA (chiral SCFT)

and the super Virasoro subalgebra generated by {Ln, Gr} 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, 4l)-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