Elliptic loci of SU(3) vacua

Based on 2010.06598, with Elias Furrer & Jan Manschot

Johannes Aspman

May 11, 2021

Trinity College Dublin

Coee talk

Quantum Gravity and Modularity, HMI workshop

Overview

• Introduction and motivation

• SU(3) Seiberg-Witten theory

• Restricting to certain subloci

• Summary and outlook

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Introduction and motivation

Introduction

• In 1994 Seiberg & Witten found the exact non-perturbative eective

action for pure N = 2 super-Yang-Mills in four dimensions with

gauge group SU(2).

• One of their main results was that the N = 2 theory shows a version

of the famous Montonen-Olive duality of N = 4, which is captured

by the fact that the quantum moduli space is parametrised by a

modular function of a discrete subgroup of SL(2,Z).

• Their results were quickly generalised to other gauge groups and it

has been conjectured that the modularity under a congruence

subgroup of SL(2,Z) should be generalised to some subgroup of

Sp(2N − 2,Z) for gauge group SU(N).

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Our motivation

• A topological version of N = 2 SYM can be formulated through

topological twisting. This theory has been shown to connect to

Donaldson invariants of four manifolds.

• Recently, Korpas et al. ('17, '19) showed that the path integral can

be written as an integral over the fundamental domain of a certain

modular object,

ΦJµ(p,x) =

∫H/Γ0(4)

dτ ∧ dτ∂τH,

where the integrand is a total derivative of a mock modular form of

the corresponding duality group.

• We are interested in generalising these results to the case of higher

ranked gauge groups where much less is known.

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Quick review of SU(2) SW theory

y2 =(x2 − u)2 − 1, ∆ = u2 − 1, τ =θ

π+

8πi

g2,

u(τ) =1

2

ϑ42 + ϑ4

3

ϑ22ϑ

23

(τ), Γu = Γ0(4) ⊂ SL(2,Z).

−1 0 1 2 3 4

F TF T 2F T 3F

SF T 2SF

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SU(3) Seiberg-Witten theory

Coulomb branch of the SU(3) SYM

y2 =(x3 − ux− v)2 − 1, Ω =

(τ11 τ12

τ12 τ22

)∈ H2,

u =u2 =1

2〈Tr(φ2)〉R4 , v = u3 =

1

3〈Tr(φ3)〉R4 ,

∆ =(4u3 − 27(v + 1)2

) (4u3 − 27(v − 1)2

).

Re(v)

Im(u)

Re(u)Eu

Ev

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Special points

• We will be particularly interested in the points

(u, v) =(e2πin/3, 0), n = 1, 2, 3,

(u, v) =(0,±1).

These are the points where two or more singular lines intersect.

• The rst set has two mutually local dyons becoming massless and

are typically referred to as the multi-monopole points.

• The second set of points are superconformal xed points of

Argyres-Douglas type. Here, three mutually non-local dyons become

massless.

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Restricting to certain subloci

The subloci L2

• The moduli space of genus two curves,M2, contains

two-dimensional loci, L2 ⊂M2, for which the genus two curve can

be mapped to genus one curves by a degree 2 map.

• The locus L2 can be characterised as the zero locus of a weight 30

polynomial in the genus two Igusa invariants, J2, J4, J6 and J10.

These are the analogues of the gi of the elliptic curves.

• The curves described by L2 can be written on the form

Y 2 = X6 − s1X4 + s2X

2 − 1,

where s1 and s2 are complex coordinates on L2.

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Elliptic loci of SU(3) vacua

• The SU(3) SW moduli space intersects with L2 in three

one-dimensional loci,

Eu : v = 0,

Ev : u = 0,

E3 : 784u9 − 24u6(297v2 + 553)− 15u3(729v4 + 5454v2 − 4775)

+ 8(27v2 − 25)3 = 0.

• The special points of interest all lie in either Eu or Ev, and not in E3.We therefore, focus on these.

• One can show that on the loci Eu and Ev we have τ11 = τ22. We

therefore dene τ± = τ11 ± τ12. This will turn out to correspond to

the good modular parameters of the elliptic subcovers.

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Eu : v = 0, τ := τ−

u−(τ) =η(τ9

)3η(τ)3

+ 3, Γu− = Γ0(9) ⊂ SL(2,Z).

−5 −4 −3 −2 −1 0 1 2 3 4 5

F TF T 2F T 3F T 4FFT−1FT−2FT−3FT−4F

SF T 3SFT−3SF

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Eu : v = 0, τ := τ+

u+(τ) =

√E4(τ)

3√E4(τ)3/2 − E6(τ)

, Γu+ * SL(2,Z).

−2 −1 0 1 2

F

SF

TF

TSF

T−1F

T−1SF

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Ev : u = 0, τ := τ±

v(τ) =

(η( τ3 )

η(τ)

)6

− 27

(η(τ)

η( τ3 )

)6

,

Fricke invol.: v(−3/τ) = −v(τ), Γv ⊂ SL(2,R).

−3 −2 −1 0 1 2 3

τAD,1 τAD,2

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Summary and outlook

• We studied special loci of the moduli space of N = 2 SU(3) SYM

described by families of elliptic curves.

• On these loci, the moduli parameters u and v can be expressed as

modular functions of certain discrete subgroups of SL(2,R).

• Some open questions include:

• Finding expressions for u and v away from these special loci in terms

of Siegel modular forms.

• Applying it to the topological theory as indicated in the introduction.

• Generalisation to other gauge groups, or to addition of matter.

• Can similar methods be applied in other settings, like string

compactications or amplitudes for example?

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Thank you!

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