The Importance of Being Disconnected: Principal Extension Gauge - - PowerPoint PPT Presentation

The Importance of Being Disconnected: Principal Extension Gauge Theories

Antoine Bourget June 5, 2018 Work with Alessandro Pini and Diego Rodriguez-Gomez

Introduction

Gauge theories initially formulated using simple Lie algebras. Possible extensions include:

◮ Products of many Lie algebras / groups

Introduction

[Gaiotto, 0904.2715]

Introduction

Gauge theories initially formulated using simple Lie algebras. Possible extensions include:

◮ Products of many Lie algebras / groups (quivers, ...) ◮ Global structure (π1) of the group

Introduction

[Aharony, Seiberg, Tachikawa, 1305.0318]

Introduction

Gauge theories initially formulated using simple Lie algebras. Possible extensions include:

◮ Products of many Lie algebras / groups (quivers, ...) ◮ Global structure (π1) of the group ◮ Gauging discrete symmetries

Introduction

[Argyres, Martone, 1611.08602]

Introduction

Gauge theories initially formulated using simple Lie algebras. Possible extensions include:

◮ Products of many Lie algebras / groups (quivers, ...) ◮ Global structure (π1) of the group ◮ Gauging discrete symmetries ◮ Start with disconnected continuous gauge group

Last item somewhat less studied.

Introduction

In this work

◮ We consider a special class of non-connected groups ◮ We focus on 4d N = 2 SCFTs ◮ We look at local physics and use algebraic counting tools.

Outline

Counting Operators Principal Extension Groups The Coulomb Index The Higgs Branch of SQCD and the Global Symmetry

Outline

Counting Operators Principal Extension Groups The Coulomb Index The Higgs Branch of SQCD and the Global Symmetry

Space of Vacua

The space of vacua is parametrized by the vev of scalar operators that

◮ Solve the F- and D- terms equations ◮ Are gauge invariant

Mathematically, the coordinate ring is C[Scalar fields]/(Ideal of Vacuum Conditions) Gauge group . We will characterize these rings using their Hilbert series.

What is a Hilbert Series

Formalization of ”operator counting”. HS (C[x], t) = 1 + t + t2 + t3 + ... = 1 1 − t

What is a Hilbert Series

Formalization of ”operator counting”. HS (C[x], t) = 1 + t + t2 + t3 + ... = 1 1 − t HS (C[x1, . . . , xn], t) = 1 (1 − t)n

What is a Hilbert Series

Formalization of ”operator counting”. HS (C[x], t) = 1 + t + t2 + t3 + ... = 1 1 − t HS (C[x1, . . . , xn], t) = 1 (1 − t)n HS

• C[x1, x2, x3]/(x1x2 − x2

3), t

• = 1 + 3t + 5t2 + ... = 1 − t2

(1 − t)3

What is a Hilbert Series

Formalization of ”operator counting”. HS (C[x], t) = 1 + t + t2 + t3 + ... = 1 1 − t HS (C[x1, . . . , xn], t) = 1 (1 − t)n HS

• C[x1, x2, x3]/(x1x2 − x2

3), t

• = 1 + 3t + 5t2 + ... = 1 − t2

(1 − t)3 Under technical assumption, HS (C[Gens]/(Rels), t) =

• Rels
• 1 − tdeg(Rels)
• Gens
• 1 − tdeg(Gens)

It is possible to refine more.

Higgs branch Hilbert Series

Consider the N = 2 SU(N) gauge theory with 2N fundamental hypers. W ∼ Tr ˜ QφQ = ⇒ Q ˜ Q − 1 N (Tr Q ˜ Q)1N = 0

Higgs branch Hilbert Series

Consider the N = 2 SU(N) gauge theory with 2N fundamental hypers. W ∼ Tr ˜ QφQ = ⇒ Q ˜ Q − 1 N (Tr Q ˜ Q)1N = 0 Higgs branch Hilbert series: HS(C[Q, ˜ Q]/(F-terms)) = det

• 1 − t2ΦAdj(X)
• det (1 − tΦF(X))2N det (1 − tΦ¯

F(X))2N .

Higgs branch Hilbert Series

Consider the N = 2 SU(N) gauge theory with 2N fundamental hypers. W ∼ Tr ˜ QφQ = ⇒ Q ˜ Q − 1 N (Tr Q ˜ Q)1N = 0 Higgs branch Hilbert series: HS(C[Q, ˜ Q]/(F-terms)) = det

• 1 − t2ΦAdj(X)
• det (1 − tΦF(X))2N det (1 − tΦ¯

F(X))2N .

H =

• dηSU(N)(X)HS(C[Q, ˜

Q]/(F-terms)) .

Example: SU(3)

Use the Weyl integration formula:

• SU(3)

dηSU(3)(z1, z2) PE

• −t2( z2

1

z2 + z2z1 + z1 z2

2 + z2 2

z1 + 1 z2z1 + z2 z2

1 + 2)

• PE
• −t( z1

z2 + z1 + z2 + 1 z2 + z2 z1 + 1 z1 )

• .

= 1 + 36t2 + 40t3 + 630t4 + ...

Example: SU(3)

Use the Weyl integration formula:

• SU(3)

dηSU(3)(z1, z2) PE

• −t2( z2

1

z2 + z2z1 + z1 z2

2 + z2 2

z1 + 1 z2z1 + z2 z2

1 + 2)

• PE
• −t( z1

z2 + z1 + z2 + 1 z2 + z2 z1 + 1 z1 )

• .

= 1 + 36t2 + 40t3 + 630t4 + ... Refining with respect to the SU(6) global symmetry, one finds 1+t2 (χ10001 + 1)+2t3χ00100+t4 (χ01010 + χ10001 + χ20002 + 1)+...

Outline

Counting Operators Principal Extension Groups The Coulomb Index The Higgs Branch of SQCD and the Global Symmetry

Outer Automorphisms

AN−1 · · · P DN · · · P E6 P

Definition

Definition :

• SU(N) = SU(N) ⋊ϕ {1, P}

(X, 1) (X, P)

Representations ( SU(3) example)

Representations (the bifundamental)

If x, y ∈ CN this representation is given by ΦF¯

F(X, 1)

x y

• =

X ¯ X x y

• =

Xx ¯ Xy

• and

ΦF¯

F(1, P)

x y

• =
• A

A−1 x y

• =
• Ay

A−1x

• ,

X = A−1X PA , AT = (−1)N−1A and det A = 1 .

Example: orthogonal groups

• SO(2N) = O(2N) .

Example of O(2): cos θ − sin θ sin θ cos θ

• ∪

cos θ sin θ sin θ − cos θ

• Diagonalize (using z = eiθ):

z z−1

• ∪

1 −1

• Rotations

Reflexions

Weyl Integration Formula

Weyl Integration Formula

For a class function f ,

• SU(N)

dη

SU(N)(X)f (X) = 1

2

• dµ+

N(z)f (z) +

• dµ−

N(z)f (zP)

• With

dµ+

N(z) = N−1

• j=1

dzj 2πizj

• α∈R+(AN−1)

(1 − z(α)) , and N even: dµ−

N(z) = N/2

• j=1

dzj 2πizj

• α∈R+(BN/2)

(1 − z(α)) . N odd: dµ−

N(z) = (N−1)/2

• j=1

dzj 2πizj

• α∈R+(C(N−1)/2)

(1 − z(α)) .

Outline

Counting Operators Principal Extension Groups The Coulomb Index The Higgs Branch of SQCD and the Global Symmetry

Coulomb branch Hilbert Series

The Coulomb branch Hilbert series appears as a limit of the SCI I =

• dηG(X) PE
• i∈multiplets

f Ri χRi(X)

• ;

f V = − σ τ 1 − σ τ − ρ τ 1 − ρ τ + σ ρ − τ 2 (1 − ρ τ) (1 − σ τ) , f

1 2 H =

τ (1 − ρ σ) (1 − ρ τ) (1 − σ τ) .

Coulomb branch Hilbert Series

Take τ → 0, ρσ =: t . f V = t, f

1 2 H = 0 .

ICoulomb

G

(t) =

• G

dηG(X) 1 det (1 − tΦAdj(X)) , Molien’s formula for the invariants of the adjoint representation.

The Coulomb Index

Computation for SU(N): ICoulomb

SU(N)

(t) = 1

N

• i=2

(1 − ti) , corresponds to C[φij]SU(N) ∼ = C[Tr(φk)k=2,...,N] , polynomial ring without any relation.

Basic Invariant Theory

Particular case: freely-generated ring C[x]G ∼ = C[I1, . . . , Im] , HS(C[x]G, t) = 1

m

• i=1

(1 − tdeg Ii) .

Basic Invariant Theory

Particular case: freely-generated ring C[x]G ∼ = C[I1, . . . , Im] , HS(C[x]G, t) = 1

m

• i=1

(1 − tdeg Ii) . In general, Hironaka decomposition for invariant rings: C[x]G ∼ =

p

• j=1

JjC[I1, . . . , Im] HS(C[x]G, t) =

p

• j=1

tdeg Jj

m

• i=1

(1 − tdeg Ii) .

The Coulomb Index

Computation for SU(N): ICoulomb

• SU(N)

(t) =

• k1<···<kr odd

tk1+···+kr

• i even

(1 − ti)

i odd

(1 − t2i) , Why? Tr((φP)k) = (−1)kTr(φk) . There are ”holes” in the structure of invariants.

The Coulomb Index

Invariant theory interpretation: 1. The primary invariants Ik for 2 ≤ k ≤ N defined by Ik =

• Tr(φk)

for k even Tr(φk)2 for k odd . 2. The secondary invariants Jk1,...,kr =

r

• i=1

Tr(φki) , for k1, . . . , kr odd and 3 ≤ k1 < · · · < kr ≤ N, with r even (r = 0 corresponds to the trivial invariant 1). Relations (among others): J2

k1,...,kr − Ik1 . . . Ikr = 0 ,

Outline

Counting Operators Principal Extension Groups The Coulomb Index The Higgs Branch of SQCD and the Global Symmetry

The Higgs Branch of SQCD

Come back to the Higgs branch of SCQD: H

SU(N) =

• dη

SU(N)(X)

det

• 1 − t2ΦAdj(X)
• det
• 1 − tχFlav

10...0 ⊗ ΦF¯ F(X)

, What is the flavor symmetry group?

The Higgs Branch of SQCD

Come back to the Higgs branch of SCQD: H

SU(N) =

• dη

SU(N)(X)

det

• 1 − t2ΦAdj(X)
• det
• 1 − tχFlav

10...0 ⊗ ΦF¯ F(X)

, What is the flavor symmetry group? Mesons satisfy symmetry / antisymmetry relations depending on the parity of N.

The Higgs Branch of SQCD

SU(N) U(2N)

• SU(N)

SO(2N)

• SU(N)

USp(2N) Even N Odd N

Examples

HSU(3) = 1 + 36t2 + 40t3 + 630t4 + 1120t5 + 7525t6 + ... H

SU(3) = 1 + 21t2 + 20t3 + 336t4 + 560t5 + 3850t6 + ...

The mesons satisfy a (anti-)symmetry property depending on the parity of N. H

SU(3)

= 1 + [2, 0, 0]C3 t2 +

• [0, 0, 1]C3 + [1, 0, 0]C3
• t3

+

• 2 [0, 1, 0]C3 + 2 [0, 2, 0]C3 + [4, 0, 0]C3 + 2
• t4 + ... .

Examples

HSU(4) = 1 + 64t2 + 2156t4 + 49035t6 + ... H

SU(4) = 1 + 28t2 + 1106t4 + 24381t6 + ...

The mesons satisfy a (anti-)symmetry property depending on the parity of N. H(4, 8) = 1 + [0, 1, 0, 0]D4 t2 +

• 2 [0, 0, 0, 2]D4 + 2 [0, 0, 2, 0]D4

+2 [0, 2, 0, 0]D4 + 2 [2, 0, 0, 0]D4 + [4, 0, 0, 0]D4 + 2

• t4

+...

Conclusion

◮ Representation theory of principal extension allows to

construct interesting Lagrangian N = 2 SCFTs

◮ They have non freely generated CBs in general ◮ They are type A theories with orthogonal / symplectic global

symmetries

Conclusion

Further explorations:

◮ Spectrum of line operators, other extended operators

Conclusion

Further explorations:

◮ Spectrum of line operators, other extended operators ◮ Quivers and brane realization

Conclusion

Further explorations:

◮ Spectrum of line operators, other extended operators ◮ Quivers and brane realization ◮ 3d mirror symmetry, monopole formula

Conclusion

Further explorations:

◮ Spectrum of line operators, other extended operators ◮ Quivers and brane realization ◮ 3d mirror symmetry, monopole formula ◮ Compactification on a circle and relation with affine gauge

symmetry

Conclusion

Further explorations:

◮ Spectrum of line operators, other extended operators ◮ Quivers and brane realization ◮ 3d mirror symmetry, monopole formula ◮ Compactification on a circle and relation with affine gauge

symmetry

◮ Global anomalies

Conclusion

Further explorations:

◮ Spectrum of line operators, other extended operators ◮ Quivers and brane realization ◮ 3d mirror symmetry, monopole formula ◮ Compactification on a circle and relation with affine gauge

symmetry

◮ Global anomalies ◮ Non conformal theories