Introduction and Motivation Brane constructions and quivers Coulomb branch of 3d N = 4 Summary and Conclusions Coulomb Branch and the Moduli Space of Instantons Giulia Ferlito Imperial College London March 24, 2015 Based on work done in collaboration with Amihay Hanany, Stefano Cremonesi, Noppadol Mekareeya Giulia Ferlito Coulomb Branch and the Moduli Space of Instantons
Introduction and Motivation Brane constructions and quivers Coulomb branch of 3d N = 4 Summary and Conclusions Introduction and Motivation Supersymmetric Gauge Theories in 3d with N = 4 are subject to a peculiar duality: Mirror Symmetry 3d mirror symmetry exchanges Coulomb branch and Higgs branch of two dual theories. C H H C 3 pieces of jargon that become confusing under mirror symmetry: ◮ Coulomb branch: moduli space parametrised by scalars in the V-plet ◮ Higgs branch: moduli space parametrised by scalars in the H-plet ◮ Moduli Space of Instantons Giulia Ferlito Coulomb Branch and the Moduli Space of Instantons
Introduction and Motivation Brane constructions and quivers Coulomb branch of 3d N = 4 Summary and Conclusions Introduction and Motivation Supersymmetric Gauge Theories in 3d with N = 4 are subject to a peculiar duality: Mirror Symmetry 3d mirror symmetry exchanges Coulomb branch and Higgs branch of two dual theories. C H H C 3 pieces of jargon that become confusing under mirror symmetry: ◮ Coulomb branch: moduli space parametrised by scalars in the V-plet ◮ Higgs branch: moduli space parametrised by scalars in the H-plet ◮ Moduli Space of Instantons Goal: Moduli Space of G-instanton ∼ = Higgs branch of specific theory Giulia Ferlito Coulomb Branch and the Moduli Space of Instantons
Introduction and Motivation Brane constructions and quivers Coulomb branch of 3d N = 4 Summary and Conclusions Introduction and Motivation Supersymmetric Gauge Theories in 3d with N = 4 are subject to a peculiar duality: Mirror Symmetry 3d mirror symmetry exchanges Coulomb branch and Higgs branch of two dual theories. C H H C 3 pieces of jargon that become confusing under mirror symmetry: ◮ Coulomb branch: moduli space parametrised by scalars in the V-plet ◮ Higgs branch: moduli space parametrised by scalars in the H-plet ◮ Moduli Space of Instantons Goal: Moduli Space of G-instanton ∼ = Higgs branch Coulomb branch of specific theory of dual theory dual Giulia Ferlito Coulomb Branch and the Moduli Space of Instantons
Introduction and Motivation Brane constructions and quivers Coulomb branch of 3d N = 4 Summary and Conclusions Introduction and Motivation Supersymmetric Gauge Theories in 3d with N = 4 are subject to a peculiar duality: Mirror Symmetry 3d mirror symmetry exchanges Coulomb branch and Higgs branch of two dual theories. C H H C 3 pieces of jargon that become confusing under mirror symmetry: ◮ Coulomb branch: moduli space parametrised by scalars in the V-plet ◮ Higgs branch: moduli space parametrised by scalars in the H-plet ◮ Moduli Space of Instantons Goal: Moduli Space of G-instanton ∼ ∼ = = Higgs branch Coulomb branch of specific theory of dual theory dual Giulia Ferlito Coulomb Branch and the Moduli Space of Instantons
Introduction and Motivation Brane constructions and quivers Coulomb branch of 3d N = 4 Summary and Conclusions Outline Introduction and Motivation 1 Brane constructions and quivers 2 ADHM quivers Hilbert series for Higgs branch Dualities on brane construction Overextended Quivers Coulomb branch of 3d N = 4 3 Fields Monopole operators The Coulomb branch formula for HS Non simply laced quivers Summary and Conclusions 4 Giulia Ferlito Coulomb Branch and the Moduli Space of Instantons
Introduction and Motivation ADHM quivers Brane constructions and quivers Hilbert series for Higgs branch Coulomb branch of 3d N = 4 Dualities on brane construction Summary and Conclusions Overextended Quivers ADHM quivers Instantons have a well-known realization in terms of branes: Dp-branes inside D(p+4)-branes with or without O(p+4)-planes (orientifolds) To realize a 3d theory choose p = 2 ⇒ D2 branes in the background of D6-branes k D 2 instanton of charge k on C 2 SU ( N ) flavour group N D 6 This brane construction can be associated to a quiver gauge theory U ( k ) SU ( N ) Adj The Higgs branch of this quiver gauge theory is isomorphic to the moduli space of k SU ( N ) instantons on C 2 To engineer other groups need a background with orientifolds Giulia Ferlito Coulomb Branch and the Moduli Space of Instantons
Introduction and Motivation ADHM quivers Brane constructions and quivers Hilbert series for Higgs branch Coulomb branch of 3d N = 4 Dualities on brane construction Summary and Conclusions Overextended Quivers ADHM quivers G Brane configurations ADHM quiver k D 2 U ( k ) SU ( N ) Adj A N − 1 N D 6 O 6 − � k D 2 images k D 2 USp ′ (2 k ) SO (2 N + 1) B N A N D 6 images N D 6 O 6 + k D 2 images k D 2 O (2 k ) USp (2 N ) C N S N D 6 images N D 6 O 6 − k D 2 images k D 2 USp (2 k ) SO (2 N ) D N A N D 6 images N D 6 Giulia Ferlito Coulomb Branch and the Moduli Space of Instantons
Introduction and Motivation ADHM quivers Brane constructions and quivers Hilbert series for Higgs branch Coulomb branch of 3d N = 4 Dualities on brane construction Summary and Conclusions Overextended Quivers Hilbert series for Higgs branch Higgs branch of ADHM quiver theories ∼ = the moduli space of k G-instantons on C 2 ∼ = � on C 2 M ADHM M k,G H To study moduli space of G-instantons ⇒ calculate Hilbert Series for the Higgs branch. What is the Hilbert Series (HS)? ◮ It is a partition function that counts chiral gauge invariant operators Why do we care? ◮ The chiral gauge invariant operators parametrise the moduli space ◮ Hilbert Series encodes: dimension of moduli space, generators, relations Giulia Ferlito Coulomb Branch and the Moduli Space of Instantons
Introduction and Motivation ADHM quivers Brane constructions and quivers Hilbert series for Higgs branch Coulomb branch of 3d N = 4 Dualities on brane construction Summary and Conclusions Overextended Quivers Hilbert series for Higgs branch Higgs branch of ADHM quiver theories ∼ = the moduli space of k G-instantons on C 2 ∼ = � on C 2 M ADHM M k,G H To study moduli space of G-instantons ⇒ calculate Hilbert Series for the Higgs branch. What is the Hilbert Series (HS)? ◮ It is a partition function that counts chiral gauge invariant operators Why do we care? ◮ The chiral gauge invariant operators parametrise the moduli space ◮ Hilbert Series encodes: dimension of moduli space, generators, relations How do we calculate it? HS for the space C 2 / Z 2 C 2 with action of Z 2 : ( z 1 , z 2 ) ← → ( − z 1 , − z 2 ) Holomorphic functions invariant under this action: z 2 1 , z 2 2 , z 1 z 2 , z 4 1 , ... All monomials constructed from 3 generators subject to 1 relation ◮ X = z 2 1 , Y = z 2 2 , Z = z 1 z 2 ◮ XY = Z 2 Giulia Ferlito Coulomb Branch and the Moduli Space of Instantons
Introduction and Motivation ADHM quivers Brane constructions and quivers Hilbert series for Higgs branch Coulomb branch of 3d N = 4 Dualities on brane construction Summary and Conclusions Overextended Quivers Hilbert series for Higgs branch HS for the space C 2 / Z 2 Collect the infinitely many invariants in 1 function Isometry group of C 2 : U (2) Cartan subalgebra: U (1) 2 Choose counters or fugacities t 1 , t 2 ◮ t 1 is the U (1) charge of z 1 ◮ t 2 ” ” z 2 � ∞ HS( t 1 , t 2 ) = 1 + t 2 1 + t 2 t i 1 t j 2 + t 1 t 2 + ... = with j = i mod 2 2 i,j =0 Giulia Ferlito Coulomb Branch and the Moduli Space of Instantons
Introduction and Motivation ADHM quivers Brane constructions and quivers Hilbert series for Higgs branch Coulomb branch of 3d N = 4 Dualities on brane construction Summary and Conclusions Overextended Quivers Hilbert series for Higgs branch HS for the space C 2 / Z 2 Collect the infinitely many invariants in 1 function Isometry group of C 2 : U (2) Cartan subalgebra: U (1) 2 Choose counters or fugacities t 1 , t 2 ◮ t 1 is the U (1) charge of z 1 ◮ t 2 ” ” z 2 � ∞ HS( t 1 , t 2 ) = 1 + t 2 1 + t 2 t i 1 t j 2 + t 1 t 2 + ... = with j = i mod 2 2 i,j =0 Can unrefine: t 1 = t 2 = t − → count all monomials at given degree � � t i + j = 1 + 3 t + 5 t 2 + ... = (2 k + 1) t 2 k HS( t ) = i,j... k =0 1 − t 4 = (1 − t 2 ) 3 Dimension of moduli space = pole of unrefined HS Giulia Ferlito Coulomb Branch and the Moduli Space of Instantons
Introduction and Motivation ADHM quivers Brane constructions and quivers Hilbert series for Higgs branch Coulomb branch of 3d N = 4 Dualities on brane construction Summary and Conclusions Overextended Quivers Hilbert series for Higgs branch HS for the space C 2 / Z 2 Can refine: t 1 = yt and t 2 = t/y HS( t ; y ) = 1 + ( y 2 + 1 + y − 2 ) t 2 + ( y 4 + y 2 + 1 + y − 2 + y − 4 ) t 4 + ... � t 2 k SU(2) = χ [2 k ] y � k =0 1 − t 4 = (1 − t 2 y 2 )(1 − t 2 )(1 − t 2 y − 2 ) ◮ where y 2 + 1 + y − 2 = χ [2] SU(2) � y y 4 + y 2 + 1 + y − 2 + y − 4 = χ [4] SU(2) � y generators: triplet of SU (2) − → X, Y, Z at degree 2 relation: numerator → quadratic in generators Giulia Ferlito Coulomb Branch and the Moduli Space of Instantons
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