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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
2/13
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).
3/13
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.
4/13
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
5/13
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
6/13
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.
7/13
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.
8/13
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.
9/13
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
10/13
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
11/13
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
12/13
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?
13/13
Thank you!
13/13