PDF Gauge Theory III - ICTS

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A Mathematical Introduction to 3d N = 4 Gauge Theory III

Mathew Bullimore

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Recap I

We consider a 3dN = 4 supersymmetric gauge theory:

1. A compact Lie groupG

2. A unitary representationρ:G→U(n)acting onR=Cⁿ. Global symmetries:

I R-symmetrySU(2)H×SU(2)C I Flavour symmetryGH×GC

GH =NU(n)(ρ(G))/ρ(G) GC⊃TC =Hom(π1(G), U(1))

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Recap 2

Introduce mass and FI parameters:

I Mass parametersm∈tH I FI parametersζ∈tC

Study supersymmetric vacua at points on the parameter spacetH×tC.

I Assume isolated massive vacuavαat generic points.

I Moduli spaces open up on loci Wαβ where domain walls become tensionless.

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v↵

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v

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W_↵

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tH⇥tC

v

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Today

What is the general structure of supersymmetric vacua?

Two extreme cases:

I m,ζboth generic: isolated massive vacua.

I m= 0,ζ= 0: intricate moduli space with intersecting branches or strata of various types - see Hanany’s lectures.

Intermediate cases:

1. m= 0,ζ generic→Higgs branch X 2. ζ= 0, mgeneric→Coulomb branchX^!

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Broad Picture

From the moduli spacesX,X^! we can study the two extreme limits.

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⇣!0 <latexit sha1_base64="G6COlEM1ACouu2SRnk+gLzCmM4c=">AAAB73icbVBNS8NAEJ34WetX1aOXxSJ4KomKepKCF48V7Ae0oWy2k3bpZpPuboQS+ie8eFDEq3/Hm//GbZuDtj4YeLw3w8y8IBFcG9f9dlZW19Y3Ngtbxe2d3b390sFhQ8epYlhnsYhVK6AaBZdYN9wIbCUKaRQIbAbDu6nffEKleSwfzThBP6J9yUPOqLFSKyIdiSPidktlt+LOQJaJl5My5Kh1S1+dXszSCKVhgmrd9tzE+BlVhjOBk2In1ZhQNqR9bFsqaYTaz2b3TsipVXokjJUtachM/T2R0UjrcRTYzoiagV70puJ/Xjs14Y2fcZmkBiWbLwpTQUxMps+THlfIjBhbQpni9lbCBlRRZmxERRuCt/jyMmmcV7yrysXDZbl6m8dRgGM4gTPw4BqqcA81qAMDAc/wCm/OyHlx3p2PeeuKk88cwR84nz8DVo9L</latexit>

m6= 0

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X

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m!0

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⇣6= 0

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X^!

I Turning on m,ζ leads to isolated massive vacua again.

I Turning off ζ, mleads to conical spaces forming strata of a larger intricate moduli space atm= 0,ζ= 0.

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Some Terminology

Higgs and Coulomb refer to the infrared behaviour of the gauge theory.

Higgs branch:

I Gauge symmetry Gbroken.

I Hypermultiplet fields X, Y have non-zero expectations values.

Coulomb branch

I Gauge symmetry broken to maximal torus T ⊂G.

I Hypermultiplet fields X, Y vanish.

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Higgs Branch X

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Higgs Branch

To access the Higgs branchX,

I m= 0

I ζ generic

Generic means in the complement of the hyperplanesWαβ intC.

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X

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tC

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Classical Vacua

Settingm= 0, the equations for classical vacua become:

µ_R−ζ= 0 µ_C= 0 (σ·J)·X = 0 (ϕ·J)·X = 0 (σ·J)·Y = 0 (ϕ·J)·Y = 0

[ϕ, ϕ^†] = 0 [σ, ϕ] = 0

Under our assumptions, for genericζ:

I Only solutions haveσ, ϕ= 0.

I Gcompletely broken.

We can therefore focus on the top line.

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Quotient Construction

The Higgs branchX parametrises solutions to µ_R−ζ= 0 µ_C= 0

moduloGgauge transformations.

I This is a hyper-K¨ahler or holomorphic symplectic quotient X =µ⁻_R¹(ζ)∩µ⁻_C¹(0)/G

=T^∗R ///ζG

I A non-renormalisation theorem ensures there are no quantum corrections to this description!

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Some Geometry

The hyper-K¨ahler quotient is defined for any quaternionic representation and produces a hyper-K¨ahler manifoldX.

For a representation of cotangent typeT^∗R with complex coordinates (X, Y)it is natural to regardX as a complex manifold together with:

I A K¨ahler formω. This is a closed (1,1)-form onX inherited from ω=dX∧dX¯ +dY ∧dY .¯

I A holomorphic symplectic formΩ. This is a closed(2,0)-form onX inherited from

Ω =dX∧dY .

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ζ Dependence

Under our assumptions, for genericζ,X is a smooth holomorphic symplectic manifold.

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⇣!0

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X

I When ζ→0,X develops a conical singularity where the gauge symmetry Gis unbroken.

I This forms part of a larger moduli space of vacua at the superconformal pointm= 0,ζ= 0.

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Example: Supersymmetric QED

Consider supersymmetric QED withG=U(1)andwhypermultiplets.

w

X

j=1

(|Xj|²− |Yj|²) =ζ

w

X

j=1

XjYj= 0

I Supposeζ >0and look for solutions withYj= 0,

w

X

j=1

|Xj|²=ζ .

I This describes complex projective space CP^w⁻¹=S^2w⁻¹/U(1)as a quotient along fibers of the Hopf fibration.

I Including Yj gives the cotangent bundle,X=T^∗CP^w⁻¹.

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Example: Supersymmetric QED

The FI parameter space istC=Rwith hyperplane at{ζ= 0}

I When ζ= 0,X is the closure of a minimal nilpotent orbit X =N^min⊂sl(w,C).

I This is parametrised by the meson matrix Mij =XiYj.

I Turning on ζgives a holomorphic symplectic resolution.

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T^⇤CP^w ¹

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T^⇤CP^w ¹ <latexit sha1_base64="0dsIJZ8NwRmBtpMHUPAlFB64igo=">AAACDHicbVDLSsNAFJ34rPVVdekmWARXJVHxsSu4cSUV7AOaUCbTSTt0MgkzN2IZ8gFu/BU3LhRx6we482+ctFlo64GBwznnMveeIOFMgeN8WwuLS8srq6W18vrG5tZ2ZWe3peJUEtokMY9lJ8CKciZoExhw2kkkxVHAaTsYXeV++55KxWJxB+OE+hEeCBYygsFIvUrVi42dT2svwjAkmOubrKc9oA+gIyayLDMpp+ZMYM8TtyBVVKDRq3x5/ZikERVAOFaq6zoJ+BpLYITTrOyliiaYjPCAdg0VOKLK15NjMvvQKH07jKV5AuyJ+ntC40ipcRSYZL6wmvVy8T+vm0J44WsmkhSoINOPwpTbENt5M3afSUqAjw3BRDKzq02GWGICpr+yKcGdPXmetI5r7lnt5Pa0Wr8s6iihfXSAjpCLzlEdXaMGaiKCHtEzekVv1pP1Yr1bH9PoglXM7KE/sD5/ACabnPA=</latexit>

Nmin

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⇣

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⇣= 0

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Global Symmetries

Turning on a generic FI parameterζ, the supersymmetric gauge theory flows to a sigma model with target spaceX.

We would like to understand global symmetries from this perspective:

I R-symmetries

I Flavour symmetries

We also want to re-introduce the mass parametersmand recover the hyperplane arrangement from this perspective.

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R-symmetry

If we regardX is a hyper-K¨ahler manifold:

I SU(2)H acts by isometries of the hyper-K¨ahler metric onX, rotating theCP¹ family of complex structures.

If we regardX as a holomorphic symplectic manifold:

I U(1)H acts by isometries ofX, induced by (X, Y)→(e^iθ/2X, e^iθ/2Y)

I It is a Hamiltonian isometry of ω.

I It transformsΩwith weight +1: Ω→e^iθΩ.

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Flavour Symmetries

The flavour symmetriesTH,TC are also encoded in geometry and topology ofX.

Under the assumptions in lecture 2:

1. The topological symmetryTC is recovered from topology ofX, tC=H2(X,R).

2. The flavour symmetry GH is the group of Hamiltonian isometries of X that leave invariant the holomorphic symplectic formΩ.

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Example: Supersymmetric QED

Recall that for supersymmetric QED,X =T^∗CP^w⁻¹

I Topological symmetry: H2(X,R) =Rgenerated by the class [CP^w⁻¹], in agreement withTH =U(1).

I Hamiltonian isometries preserving the holomorphic symplectic form on X=T^∗CP^w⁻¹ are preciselyGH=P SU(w).

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Mass Parameters

Let us now re-introduce the mass parametersm∈tH.

From the perspective of a sigma model toX, this is a deformation by a real superpotential

hm=m·µH,R

I hm:X →Ris the moment map for the U(1)m⊂TH Hamiltonian isometry generated by m.

I Induces a potential proportional to |dhm|².

I The supersymmetric vacua are now critical points of h.

I They are equivalently fixed points of the U(1)maction.

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Domain Walls and Hyperplanes

Ifmis generic the critical locus consists of isolated points vα∈X.

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x³

v↵

v

I We can again study 2dN = (2,2) domain walls.

I They are now solutions to gradient flow forhmonX.

I The tension is|hm(α)−hm(β)|.

I The critical locus jumps when{hm(α)−hm(β)}= 0.

I This reproduces gauge theory answer: h(α) =hm(α).

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Coulomb Branch X

^!

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Coulomb Branch

To access the Coulomb branch,

I ζ= 0

I m generic

Generic now means in the complement of hyperplanesWαβin tH.

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X

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tC

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Classical Description

The equations for classical supersymmetric vacua become:

µ_R= 0 µ_C= 0

(σ·J+m·JH)·X= 0 (ϕ·J)·X = 0 (σ·J+m·JH)·Y = 0 (ϕ·J)·Y = 0

[ϕ, ϕ^†] = 0 [σ, ϕ] = 0

Under our assumptions, for genericm:

I Only solutions haveX, Y = 0.

I σ, ϕin a common Cartan subalgebrat⊂g.

I For genericσ, ϕ, gauge symmetryGis broken to maximal torusT.

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Classical Description

What about the gauge connectionA?

I If the gauge symmetry is broken to a maximal torusT ⊂G, the connection can be dualised to a periodicscalar fieldγcalled the dual photon,

∗dAa=dγa a= 1, . . . ,rk(G).

I Near generic values of the vectormultiplet scalars, the Coulomb branch therefore looks like (R³×S¹)^rk(G), parametrised by (σa, ϕa, γa).

I This classical picture receives dramatic quantum corrections!

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Quantum Corrections

The Coulomb branchX^! receives quantum corrections.

I When Gis abelian, there are only 1-loop quantum corrections, which were completely understood 25 years ago¹.

I When Gis non-abelian, there are also non-perturbative corrections.

I Many predictions from mirror symmetry and brane constructions.

I There has been much recent progress using tools from algebraic geometry ²- see lectures of Hanany and Nakajima.

1Seiberg, Seiberg-Witten, Seiberg-Intriligator

2Cremonesi-Hanany-Zaffaroni, Nakajima, Nakajima-Braverman-Finkelberg, MB-Dimofte-Gaiotto, and many others

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Example: Supersymmetric QED

In supersymmetric QED,G=U(1).

I The classical result would be R³×S¹.

I In fact, it is the resolution of A-type singularity,X =C^²/Zw.

I The classical and quantum cross-sectionϕ= 0 are illustrated below forw= 2.

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m1

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m2

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General Structure

The general structure ofX^! is the same as X with the R-symmetries and flavour symmetries interchanged.

I For genericm, it is a smooth holomorphic symplectic manifold of complex dimension

dim_CX^!= 2rk(G).

I Becomes holomorphic symplectic cone asm→0.

I Flavour symmetries:

I tH =H2(X,R).

I tC=Hamiltonian isometries ofX^! preserving holomorphic symplectic form.

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Towards Symplectic Duality

(29)

Comparisons

To a supersymmetric gauge theory, we have assigned two spacesX,X^!. Features of the theory can be understood from the perspective of both.

For example, flavour symmetries

I tH ={Hamiltonian isometries ofX}=H2(X^!,R)

I tH ={Hamiltonian isometries ofX^!}=H2(X^,R) and supersymmetric vacua

I Vacua vα={TH-fixed points ofX}={TC-fixed points of X^!}.

I Supersymmetric CS levels / vector central charge:

h(α) =hm(α) =h^!_t(α)

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Symplectic Duality

Pairs of holomorphic symplectic spacesX,X^! with these properties are known as symplectic dual³.

I There are non-trivial equivalences between mathematical structures associated toX, X^!.

I The idea is to find observables in a supersymmetric gauge theory that can be equivalently described in sigma models toX andX^!.

I Important implications in representation theory.

3Braverman-Licata-Proudfoot-Webster

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Mirror Symmetry

(32)

Mirror Symmetry

A related but distinct phenomenon is 3d mirror symmetry.

This is an infrared duality:

I Two theories T,T^∨that flow to the same superconformal fixed point with flavour and R-symmetries exchanged,

SU(2)H ↔SU(2)C GH↔GC.

I This exchanges the Higgs and Coulomb branch X(T)↔X^!(T^∨) X(T^∨)↔X^!(T) Brane constructions provide a plethora of mirror pairs!