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 ( π 1 ) 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 ( π 1 ) 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 ( π 1 ) 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”. 1 HS ( C [ x ] , t ) = 1 + t + t 2 + t 3 + ... = 1 − t

What is a Hilbert Series Formalization of ”operator counting”. 1 HS ( C [ x ] , t ) = 1 + t + t 2 + t 3 + ... = 1 − t 1 HS ( C [ x 1 , . . . , x n ] , t ) = (1 − t ) n

What is a Hilbert Series Formalization of ”operator counting”. 1 HS ( C [ x ] , t ) = 1 + t + t 2 + t 3 + ... = 1 − t 1 HS ( C [ x 1 , . . . , x n ] , t ) = (1 − t ) n � � = 1 + 3 t + 5 t 2 + ... = 1 − t 2 C [ x 1 , x 2 , x 3 ] / ( x 1 x 2 − x 2 3 ) , t HS (1 − t ) 3

What is a Hilbert Series Formalization of ”operator counting”. 1 HS ( C [ x ] , t ) = 1 + t + t 2 + t 3 + ... = 1 − t 1 HS ( C [ x 1 , . . . , x n ] , t ) = (1 − t ) n � � = 1 + 3 t + 5 t 2 + ... = 1 − t 2 C [ x 1 , x 2 , x 3 ] / ( x 1 x 2 − x 2 3 ) , t HS (1 − t ) 3 Under technical assumption, � 1 − t deg( Rels ) � � Rels � 1 − t deg( Gens ) � HS ( C [ Gens ] / ( Rels ) , t ) = � Gens It is possible to refine more.

Higgs branch Hilbert Series Consider the N = 2 SU ( N ) gauge theory with 2 N fundamental hypers. Q − 1 W ∼ Tr ˜ Q ˜ N ( Tr Q ˜ ⇒ Q φ Q = Q ) 1 N = 0

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

Higgs branch Hilbert Series Consider the N = 2 SU ( N ) gauge theory with 2 N fundamental hypers. Q − 1 W ∼ Tr ˜ Q ˜ N ( Tr Q ˜ ⇒ Q φ Q = Q ) 1 N = 0 Higgs branch Hilbert series: � � 1 − t 2 Φ Adj ( X ) det HS ( C [ Q , ˜ Q ] / ( F-terms )) = F ( X )) 2 N . det (1 − t Φ F ( X )) 2 N det (1 − t Φ ¯ � d η SU ( N ) ( X ) HS ( C [ Q , ˜ H = Q ] / ( F-terms )) .

Example: SU (3) Use the Weyl integration formula: � � − t 2 ( z 2 2 + z 2 � z 2 + z 2 z 1 + z 1 z 2 z 1 + z 2 1 z 1 + 1 + 2) PE 1 2 z 2 z 2 � � d η SU (3) ( z 1 , z 2 ) . − t ( z 1 z 2 + z 1 + z 2 + 1 z 2 + z 2 z 1 + 1 z 1 ) SU (3) PE = 1 + 36 t 2 + 40 t 3 + 630 t 4 + ...

Example: SU (3) Use the Weyl integration formula: � � − t 2 ( z 2 2 + z 2 � z 2 + z 2 z 1 + z 1 z 2 z 1 + z 2 1 z 1 + 1 + 2) PE 1 2 z 2 z 2 � � d η SU (3) ( z 1 , z 2 ) . − t ( z 1 z 2 + z 1 + z 2 + 1 z 2 + z 2 z 1 + 1 z 1 ) SU (3) PE = 1 + 36 t 2 + 40 t 3 + 630 t 4 + ... Refining with respect to the SU (6) global symmetry, one finds 1+ t 2 ( χ 10001 + 1)+2 t 3 χ 00100 + t 4 ( χ 01010 + χ 10001 + χ 20002 + 1)+ ...

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

Outer Automorphisms · · · A N − 1 P · · · D N P E 6 P

Definition Definition : � SU ( N ) = SU ( N ) ⋊ ϕ { 1 , P} ( X , 1) ( X , P )

Representations ( � SU (3) example)

Representations (the bifundamental) If x , y ∈ C N this representation is given by � x � � X � � x � � Xx � 0 Φ F¯ F ( X , 1) = = ¯ ¯ y 0 X y Xy and � x � � � � x � � � 0 A Ay F (1 , P ) Φ F¯ = = , A − 1 A − 1 x 0 y y A T = ( − 1) N − 1 A X = A − 1 X P A , det A = 1 . and

Example: orthogonal groups � SO (2 N ) = O (2 N ) . Example of O (2): �� cos θ �� �� cos θ �� − sin θ sin θ ∪ − cos θ sin θ cos θ sin θ Diagonalize (using z = e i θ ): �� z �� �� 1 �� 0 0 ∪ z − 1 0 0 − 1 Rotations Reflexions

Weyl Integration Formula

Weyl Integration Formula For a class function f , �� � � � SU ( N ) ( X ) f ( X ) = 1 d µ + d µ − N ( z ) f ( z P ) d η � N ( z ) f ( z ) + 2 � SU ( N ) With N − 1 � � d z j d µ + N ( z ) = (1 − z ( α )) , 2 π iz j j =1 α ∈ R + ( A N − 1 ) and N / 2 � � d z j d µ − N even: N ( z ) = (1 − z ( α )) . 2 π iz j α ∈ R + ( B N / 2 ) j =1 ( N − 1) / 2 � � d z j d µ − N odd: N ( z ) = (1 − z ( α )) . 2 π iz j j =1 α ∈ R + ( C ( N − 1) / 2 )

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 � � � � f R i χ R i ( X ) I = d η G ( X ) PE ; i ∈ multiplets σ ρ − τ 2 σ τ ρ τ f V = − 1 − σ τ − 1 − ρ τ + (1 − ρ τ ) (1 − σ τ ) , τ (1 − ρ σ ) 1 2 H = (1 − ρ τ ) (1 − σ τ ) . f

Coulomb branch Hilbert Series Take τ → 0 , ρσ =: t . f V = t , 1 2 H = 0 . f � 1 I Coulomb ( t ) = d η G ( X ) det (1 − t Φ Adj ( X )) , G G Molien’s formula for the invariants of the adjoint representation.

The Coulomb Index Computation for SU ( N ): 1 I Coulomb ( t ) = , SU ( N ) � N (1 − t i ) i =2 corresponds to C [ φ ij ] SU ( N ) ∼ = C [ Tr ( φ k ) k =2 ,..., N ] , polynomial ring without any relation.

Basic Invariant Theory Particular case: freely-generated ring 1 C [ x ] G ∼ HS ( C [ x ] G , t ) = = C [ I 1 , . . . , I m ] , . � m (1 − t deg I i ) i =1

Basic Invariant Theory Particular case: freely-generated ring 1 C [ x ] G ∼ HS ( C [ x ] G , t ) = = C [ I 1 , . . . , I m ] , . � m (1 − t deg I i ) i =1 In general, Hironaka decomposition for invariant rings: p � C [ x ] G ∼ J j C [ I 1 , . . . , I m ] = j =1 � p t deg J j j =1 HS ( C [ x ] G , t ) = . � m (1 − t deg I i ) i =1

The Coulomb Index Computation for � SU ( N ): � t k 1 + ··· + k r k 1 < ··· < k r odd I Coulomb � (1 − t i ) � ( t ) = (1 − t 2 i ) , � SU ( N ) i even i odd Why? Tr (( φ P ) k ) = ( − 1) k Tr ( φ k ) . There are ”holes” in the structure of invariants.

The Coulomb Index Invariant theory interpretation: 1. The primary invariants I k for 2 ≤ k ≤ N defined by � Tr ( φ k ) for k even I k = for k odd . Tr ( φ k ) 2 2. The secondary invariants � r Tr ( φ k i ) , J k 1 ,..., k r = i =1 for k 1 , . . . , k r odd and 3 ≤ k 1 < · · · < k r ≤ N , with r even ( r = 0 corresponds to the trivial invariant 1). Relations (among others): J 2 k 1 ,..., k r − I k 1 . . . I k r = 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: � � � 1 − t 2 Φ Adj ( X ) det H � SU ( N ) = d η � SU ( N ) ( X ) � � , 1 − t χ Flav 10 ... 0 ⊗ Φ F¯ F ( X ) det What is the flavor symmetry group?

The Higgs Branch of SQCD Come back to the Higgs branch of SCQD: � � � 1 − t 2 Φ Adj ( X ) det H � SU ( N ) = d η � SU ( N ) ( X ) � � , 1 − t χ Flav 10 ... 0 ⊗ Φ F¯ F ( X ) det 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 (2 N ) Even N Odd N � � SO (2 N ) USp (2 N ) SU ( N ) SU ( N )