Bosonization in 3d and 2d David Tong ICTP, April 2018 Work with - - PowerPoint PPT Presentation

Bosonization in 3d and 2d

David Tong ICTP, April 2018

Work with Andreas Karch and Carl Turner

Progress in Understanding 3d Gauge Theories

2016-2018: Aitken, Aharony, Bashmakov, Benini, Benvenuti, Cordova, Gaiotto, Gomis, Hsin, Kachru, Komargodski, Jensen, Metlitski, Mulligan, Seiberg, Senthil, Sharon, Son, Radicevic, Robinson, Tong, Torroba, Turner, Vishwanath, Wang, Wang, Witten, Wu, Xu, You

Bosonization in 3d

Early (slightly wrong) conjecture by Polyakov, 1988. Real evidence has come only more recently…

SA = Z d3x |Dµ|2 ||4 + 1 4⇡✏µνρaµ@νaρ

Theory A: Theory B:

SB = Z d3x i ¯ ψ / ∂ψ

U(1)1 + WF boson = free fermion

Bosonization in 3d

Boson + 2p flux Fermion An old story: non-relativistic flux attachment:

=

Wilczek 1982, Jain 1989

ρscalar + f 2π = 0

SA = Z d3x |Dµ|2 m2||2 ||4 + 1 4⇡✏µνρaµ@νaρ

Bosonization in gapless, relativistic theories is much more subtle.

Evidence from Holography

Higher spin theories in the bulk = Simple theories on the boundary

Vasiliev, 1980s and 1990s Klebanov and Polyakov 2002 Sezgin and Sundell 2002 Giombi and Yin 2009

Evidence from Holography

Theory A: U(N) Yang-Mills + WF boson + CS = k Tested beyond all reasonable doubt at large N and k Theory B: SU(k) Yang-Mills + fermion + CS = -N+1/2

Minwalla et al. 2011-2015 Aharony et al. 2011-2015

Evidence From Things Working Out Nicely

U(1)1 + WF boson = free fermion

1 2⇡✏µνρfνρ ! jµ

Step 1: Identify currents on both sides of duality Step 2: Couple to background fields Step 3: Make background fields dynamical Step 4: Repeat

Karch and Tong 2016 Seiberg, Senthil, Wang and Witten 2016

Evidence From Things Working Out Nicely

U(1)1 + WF boson = free fermion

Z Da Zscalar[a] eiSCS[a]+iSBF [a;A] = Zfermion[A] e− i

2SCS[A]

dynamical gauge field background gauge field Promote A to a dynamical gauge field

Z DADa Zscalar[a] eiSCS[a]+iSBF [a;A]−iSBF [A;C] = Zscalar[C]eiSCS[C]

Z DA Zfermion[A] e− i

2SCS[A]−iSBF [A;C]

Equation of motion for A says that da = dC

] eiSCS[a]+iSBF [a;A]−iSBF [A;C] = Zscalar[C]eiSCS[C]

Z DA Zfermion[A] e− i

2SCS[A]−iSBF [A;C]

=

U(1)1/2 + fermion = WF boson

Barkeshli, McGreevy, 2012

Z Da Zscalar[a] eiSBF [A;a] = Zscalar[A]

Evidence From Things Working Out Nicely

Promote A to a dynamical gauge field. Right-hand side becomes On the left-hand-side something pretty happens. After integrating out A, we find But this is just the time-reversal of the original duality

Z Da Zfermion[a] e− i

2 SCS[a]−iSBF [a;A]−iSCS[A] = Zscalar[A]

Now start from:

Z DA Zscalar[A] eiSBF [A,C]

Z Da Zfermion[a] e+ i

2SCS[a]+iSBF [a;A]+iSCS[A]

Peskin 1978, Dasgupta Halperin 1981

This is particle-vortex duality!

Evidence From Things Working Out Nicely

Karch and Tong 2016 Seiberg, Senthil, Wang and Witten 2016

U(1)1 + WF boson = free fermion

• U(1)-1/2 + fermion = WF boson
• Bosonic particle vortex duality
• Fermionic particle vortex duality
• An infinite number of new dualities...
• e.g. U(1) + 2 fermions is self-dual with emergent

SU(2) x SU(2) global symmetry.

Barkeshli, McGreevy, 2012 Peskin 1978, Dasgupta Halperin 1981 Son; Senthil, Wang; Metlitski, Vishwanath 2015 You, Xu 2016

Evidence from the Lattice

Chen,Son, Wang, Raghu, 2017 Karthik and Narayanan, 2016-18

Analytic Numeric

Evidence from Supersymmetry

3d mirror symmetry is “supersymmetric particle-vortex duality”

Intriligator and Seiberg 1996 Aharony, Hanany, Intriligator, Seiberg, Strassler 1997

Mirror symmetry for 3d N=2 CS theories:

U(1)1/2 + charged chiral = free chiral

Dorey and Tong, 1999 Tong 2000

Evidence from Supersymmetry

3d mirror symmetry is “supersymmetric particle-vortex duality”

Intriligator and Seiberg 1996 Aharony, Hanany, Intriligator, Seiberg, Strassler 1997

Mirror symmetry for 3d N=2 CS theories:

Dorey and Tong, 1999 Tong 2000

U(1)1/2 + charged chiral = free chiral U(1)1 + WF boson = free fermion

break supersymmetry

Kachru, Mulligan, Torroba, Wang 2016

Question What happens if we reduce to 2d?

Reduction to 2d

3d mirror symmetry U(1)1/2 + chiral = free chiral 3d bosonization

break susy break susy reduce to 2d reduce to 2d

? ?

Susy Reduction to 2d

U(1) N=2 Chern-Simons theories

reduce to 2d 3d mirrors 2d mirrors

GLSM = Landau-Ginzburg

Dorey and Tong, 1999 Tong 2000 Hori, Vafa 2000 Aganagic, Hori, Karch, Tong 2001 Aharony, Razamat, Willet 2016

Susy Reduction to 2d

compactify

Aganagic, Hori, Karch, Tong 2001

3d mirrors 2d mirrors

U(1)1/2 + chiral = free chiral

+ 1 4e2fµνf µν

add irrelevant

• perators

+ 2⇡2 e2 jµjµ

S1 of radius R

N=(2,2) Cigar = N=(2,2) Liouville

Hori, Kapustin 2001

W ∼ e−Y

= e2R

ds2 = 1 4⇡

• dy2 + d✓2

ds2 = 4⇡

• d⇢2 + tanh2⇢ d2

Kapustin, Strassler 1999

Reduction to 2d

3d mirror symmetry U(1)1/2 + chiral = free chiral 3d bosonization

break susy break susy reduce to 2d reduce to 2d

?

2d mirror symmetry Cigar = Liouville

Breaking Susy in 2d

N=(2,2) Cigar

γ 1

• Identify global U(1) symmetry on both sides
• Couple to background vector multiplet V
• Turn on background parameters m and D-term.
• D-term will break supersymmetry s

D-term gives masses to all scalars. We get

LThirring = i ¯ / @ + m ¯ − ⇡ ( ¯ µ )( ¯ µ )

Breaking Susy in 2d

• Identify global U(1) symmetry on both sides
• Couple to background vector multiplet V
• Turn on background parameters m and D-term.
• D-term will break supersymmetry s

γ ⌧ 1

N=(2,2) Liouville

V (Y ) =

• µ

2e−y−iθ + m

• 2

+ Dy e−y ⇡ p 2D µ

D-term gives Only the periodic scalar remains light

LSG = 1 4⇡ (@i✓)2 + √ 2D|m| cos ✓

But there is a shift of the kinetic term at one-loop

LSG = ✓ 1 4πγ + 1 8π ◆ (∂θ)2 + √ 2Dm cos θ

LThirring = i ¯ / @ + m ¯ − g( ¯ µ )( ¯ µ )

2d Bosonization

Massive Thirring = Sine-Gordon

Coleman 1975 Mandelstam 1976

Massive Thirring: Sine-Gordon:

LSG = 2 2 (@i✓)2 + √ 2D|m| cos ✓

This is the shift that arises at one-loop? Open question: Higher loops?!

2⇡22 = ⇡ 2 + g

Reduction to 2d

3d mirror symmetry U(1)1/2 + chiral = free chiral 3d bosonization

break susy break susy reduce to 2d reduce to 2d

2d mirror symmetry Cigar = Liouville 2d bosonization

?

Non-Susy Reduction to 2d

3d bosonization

U(1)1 + WF boson free fermion

Work with weakly coupled UV theory

S3d = Z d3x 1 4e2fµνf µν + |Dµ|2 m2

B||2 ||4 + 1

4⇡✏µνρaµ@νaρ

maps to Thirring coupling for fermion not clear what weak coupling here corresponds to for fermion!

=

✓ = I a 2 [0, 2⇡)

Non-Susy Reduction to 2d

compactify 3d bosonization 2d theories

U(1)1 + WF boson free fermion

Work with weakly coupled UV theory S1 of radius R

U(1) Abelian Higgs at Thirring

S3d = Z d3x 1 4e2fµνf µν + |Dµ|2 m2

B||2 ||4 + 1

4⇡✏µνρaµ@νaρ

maps to Thirring coupling for fermion not clear what weak coupling here corresponds to for fermion! 1-loop potential for Wilson line fixes

θ = π θ = π

6=

=

ˆ λ = λR ⌧ 1

= e2R ⌧ 1

Non-Susy Reduction to 2d

U(1)1 + WF boson free fermion

We must work with the strongly coupled theory

=

1 2⇡✏µνρ@νaρ ! jµ

V = ¯

µ

Upon compactification, the low-energy fermionic theory has two emergent currents

ji

V = ¯

i ji

A = ✏ijjV j

and The bosonic theory must have the same currents. These involve the Wilson line

ji

V = ✏ij@j✓

ji

A = @i✓

Unlike at weak coupling, the Wilson line must remain massless!

✓ = I a

Use matching of currents:

Non-Susy Reduction to 2d

Matching of symmetries in 3d bosonization are enough to ensure that, upon reduction on S1

L = i ¯ / @ L = 2 2 (@i✓)2

2 = 1 4⇡

symmetries also fix Question: is there a way to fix this value directly from 3d dynamimcs?

Summary

3d mirror symmetry U(1)1/2 + chiral = free chiral 3d bosonization

break susy break susy reduce to 2d reduce to 2d

2d mirror symmetry Cigar = Liouville 2d bosonization

Non-Susy Reduction to 2d

From this, everything else follows:

U(1)1 + WF boson free fermion

=

+ 1 2e2f 2

i3

+ 2π2 e2 jiji

In 3d: S1 of radius R Mass terms also fixed by symmetries

= e2R

L = i ¯ ψ / ∂ψ − π γ ( ¯ ψγiψ)( ¯ ψγiψ)

L = 1 2 ✓ 1 4π + 1 2πγ ◆ (∂iθ)2

In 2d: