Defects and Dualities

Home , AGT correspondence, G2 (mathematics)

Hee-Cheol Kim Perimeter Institute

Based on arXiv:1412.2781 with Davide Gaiotto (Perimeter) arXiv:1412.6081 with Mathew Bullimore (IAS), Peter Koroteev (Perimeter) Introduction

What is a defect ? A singularity supporting a submanifold L k D ( D>k ) in ⇢ M spacetime manifold D . M

D M

(Codimension D k defect) Introduction

What types of defects appear in physics ?

Wilson (or ’t Hooft) line heavy vortex, cosmic string Domain wall

z 1,3 2 1,3 R R R

1,2 = R R C

… (many other defects) Introduction

How are defects deﬁned ? ( )=v 1 + 1. Topological defect

v v+ ( )=v 1

2. Singular behavior of ﬁelds m xi F dxµ dx⌫ ✏ dxj dxk µ⌫ ^ ⇠ 4 ijk x 3 ^ | | 3. Orbifold on spacetime D D /ZN M ! M

4. Coupling to defect theory

SD[ D]+Sk[ k]+SD,k[ D, k]

5. RG ﬂow in Higgs branch B(x, y) (x + iy)r h i⇠ Introduction

Why are defects important ? • They serves as order parameters in phase transition. Symmetry breaking, conﬁning phase, Higgs phase, … • They can observe symmetries or dualities. Global structure of symmetries, electric-magnetic duality, mirror duality, … • They can provide a zoo of theories in lower dimensions after compactiﬁcation.

Class S theories, Class R theories, AGT correspondence, … • so on …. Contents

• Introduction • Line defects and surface defects in gauge theory • Defects and duality in 5d gauge theory • Conclusion Defects in gauge theory Line defects

1,3 R Wilson loop C

W =Tr exp i A dxµ R : Rep. of gauge group R RP µ G ✓ IC ◆

- Heavy electrically charged particle

- In conﬁning phase, it exhibits area law : being an order parameter of conﬁning phase

⌧V (L) W e ,V(L) L ⌧ h i⇠ ⇠ C

L Line defects

’t Hooft loop (in 4d) .

- Disorder operator : deﬁned by singularity m xi F dxµ dx⌫ ✏ dxj dxk µ⌫ ^ ⇠ 4 ijk x 3 ^ m : magetic charge | | - Heavy magnetic monopole

- In Higgs phase, it exhibits area law.

⌧V (L) T e ,V(L) L ⌧ h i⇠ ⇠ C

L Duality on Line defects

Electric-magnetic duality

(E,~ B~ ) (B,~ E~ ) ,e g = 4⇡/e ! !

Wilson loop ’t Hooft loop

- Duality in 4d gauge theories - Strong-Weak duality, so hard to check - Some checks have been done with supersymmetric partition functions Codimension 2 defects

z r ( Surface defects in 4 dimensions ) 2 ✓ R

1. Singularity of ﬁelds as r 0 (Gukov-Witten defect) ! µ Aµdx diag(m1, ,m1,m2, ,m2, ,ms ,ms)d✓ mi : monodromy parameters ⇠ ··· ··· ··· ··· s n1 n2 ns ni = N i=1 | {z } | {z } | {z } X - Gauge symmetry (suppose SU ( N ) gauge group) is broken, near the

defect, to Levi subgroup L = S ( U ( n 1 ) U ( n s )) . ⇥ ···⇥ - Classiﬁed by or ⇢ =[n1,n2, ,ns] L ··· Codimension 2 defects

z r 2 2. Geometric singularity on R ✓ 2⇡ 2 - Orbifolding : ✓ ✓ + R /Zs ⇠ s

- We can turn on discrete Z s holonomies. 2 2 s s H = diag(!, , !, ! , , ! , , ! , ! ) s ··· ··· ··· ··· (! = 1) n1 n2 ns

- Gauge symmetry| {z }is| broken{z }to L |= S{z(U(n}1) U(ns)) ⇥ ···⇥ 3. Coupling to defect theory - Consider a D 2 dimensional theory (maybe gauge theory) with global symmetry SU ( N ) and couple it to the bulk SU ( N ) gauge ﬁeld.

D D 2 D D 2 SU(N) SU(N) … SU(N) … Codimension 2 defects

a Q↵ 4. Higgsing SU(N) SU(N) UV SU(N) SU(N)

Q ⇠ detQ (x + iy)r h i⇠ !1 h i⇠ p

SU(N) w/o defect IR SU(N) w/ defect

- Consider a UV SU ( N ) SU ( N ) theory with a bifundamental matter Qa ⇥ ↵ and move far along Higgs branch. Q ⇠ - We give a large vacuum expectation value, h i⇠ !1 . - Renormalization group ﬂow leads to SU ( N ) theoryp in low energy (IR). - If we instead give a large coordinate dependent vev, det Q ( x + iy ) r , h i⇠ the IR SU ( N ) theory accommodates codim. 2 defect at x = y =0 . Duality on Codimemsion 2 defects

• Mirror symmetry in 3d : Wilson line Codimension 2 defect

• S-duality in 4d : Codimension 2 defect Codimension 2 defect

• What about 5d ? Defect and Duality in 5d gauge theory Duality in 5d gauge theory

• 5d gauge theory is non-renormalizable.

• Class of theories admit UV completion as 5d CFT (or 6d CFT), and we are interested in these theories.

• Such theories (at least some of them) have string theory origin and their dualities inherit string theory duality.

ex) Duality in 5-brane web diagram Duality in 5d gauge theory

5d pure SU (2) gauge theory w/o matter ﬁeld preserving 8 supersymmetries.

6 2⇡2 g2 + a 5-brane realization 5

0123456789 a D5 ------NS5 ------{ 5d theory a : Coulomb branch parameter g2 : classical gauge coupling Duality in 5d gauge theory

5d pure SU (2) gauge theory w/o matter ﬁeld preserving 8 supersymmetries.

6 2⇡2 g2 + a 5-brane realization 5

0123456789 a D5 ------NS5 ------{ 5d theory a : Coulomb branch parameter g2 : classical gauge coupling

2 2⇡2 2 Duality in 5d ﬁeld theory : (˜a, g˜ ) ( 2 + a, g ) ! g = 90 o rotation of 5-brane diagram

(It corresponds to S SL (2 , Z ) action of Type IIB) ⇢ Codimension 2 defect of SU (2) gauge theory

1,2 R Simplest codimension 2 defect ( Type I ) r

µ 1) Gukov-Witten type : Aµdx diag(m, m)d✓ ✓ ⇠ 1,4 R 2 2 2) Z 2 orbifold : H = diag(!, ! ) , ! =1

3) Coupling to 3d U (1) gauge theory with 2 fundamental chiral multiplets

5d 3d SU(2) U(1)

4) Higgsing SU (2) SU (2) gauge theory by giving a vev det Q ( x + iy ) r =1 ⇥ h i⇠ All same defects !! Codimension 2 defect of SU (2) gauge theory

Codimension 2 defect ( Type II )

1) Higgsing SU (3) gauge theory with 2 fundamental matters q, q˜ by giving a position dependent vev to mesonic operator M = q q˜ =0 . h i h i6 q M =0 h i6 SU(3) U(2) SU(2) q˜ UV IR 2) Coupling to a couple of 3d free chiral multiplets.

5d 3d SU(2) U(1)

Same defects !! Brane realization of codimension 2 defects

Type I Type II 6 m 5 m 7 D3-branes (defects)

5d theory { 0123456789 D5 ------NS5 ------D3 --- - { 3d theory Brane realization of codimension 2 defects

Type I Type II 6

Duality in ﬁeld theory = 90 o rotation of brane-diagram Defect partition function and Duality test

1 4 We have computed defect partition functions (on S R ) and tested duality. ⇥

Exact partition functions can be computed using supersymmetric localization.

Supersymmetric localization :

1. Deform path integral in SUSY manner w.r.t supercharge Q .

3d/5d 3d/5d S [ ] tQV 2 Z Z (t)= e E (Q V = 0) ! D Z 2. The result is independent of the deformation

d 3d/5d S [ ] tQV Z (t) Q e E V =0 dt ⇠ D Z ⇣ ⌘ 3. Send t and do gaussian integral around saddle point of QV =0 , !1 which gives rise to Exact Result since Z ( t = 0) = Z ( t = ) . 1 Defect partition function and Duality test

a : Coulomb branch parameter 4⇡2 g2 : gauge coupling Type I (GW) Z(I) = Z(I)(a, g2 ; m) m : 3d monodromy parameter

a˜ : Coulomb branch parameter 2 4⇡ g˜2 : gauge coupling Type II (free chirals) Z(II) = Z(II)(˜a, g˜2 ;˜m) m˜ : 3d mass parameter

2 2⇡2 2 Duality transforms (˜a, g˜ ) ( 2 + a, g ) ! g

We have checked that two partition functions completely agree.

Z(I) = Z(II)

2 up to (˜a = 2⇡ + a, g˜2 = g2, m˜ = m) g2 Duality in 3d/5d system

New duality by 3d S SL (2 , Z ) transformation ⇢ • 3d S SL (2 , Z ) action (not duality in general) ⇢ 1. Gauging a U (1) global symmetry : U(1) U(1)

2. Adding a mixed Chern-Simons term : 3 d xAnewdAold Z 3d 3d In our example U(2) SU(2) U(1) S - transform

• The same S action on 3d/5d system becomes duality !

4⇡2 4⇡2 4⇡2 Z(II)(a, g2 ; m) Z(I)(a, g2 ; m)=Z(II)(˜a, g˜2 ;˜m) S - transform Conclusion Conclusions

• There are various defects in physics and they play important roles.

• We have studied interesting properties of defects under dualities.

- Combination of different types of defects. - Dualities on defect theories and those of bulk theories. - Integrability : ˆ Z =0 becomes Baxter equations in integrable systems. O - Implications to realistic system.