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 U(1) 1 + WF boson = free fermion Z d 3 x |D µ � | 2 � | � | 4 + 1 Theory A: 4 ⇡✏ µ νρ a µ @ ν a ρ S A = Z d 3 x i ¯ ψ / S B = ∂ψ Theory B: Early (slightly wrong) conjecture by Polyakov, 1988. Real evidence has come only more recently…

Bosonization in 3d An old story: non-relativistic flux attachment: d 3 x |D µ � | 2 � m 2 | � | 2 � | � | 4 + 1 Z 4 ⇡✏ µ νρ a µ @ ν a ρ S A = = ρ scalar + f 2 π = 0 Boson + 2 p flux Fermion Wilczek 1982, Jain 1989 Bosonization in gapless, relativistic theories is much more subtle.

Evidence from Holography Higher spin theories in the bulk = Simple theories on the boundary Klebanov and Polyakov 2002 Vasiliev, 1980s and 1990s Sezgin and Sundell 2002 Giombi and Yin 2009

Evidence from Holography Theory A: U(N) Yang-Mills + WF boson + CS = k Theory B: SU(k) Yang-Mills + fermion + CS = -N+1/2 Tested beyond all reasonable doubt at large N and k Minwalla et al. 2011-2015 Aharony et al. 2011-2015

Evidence From Things Working Out Nicely U(1) 1 + WF boson = free fermion Step 1: Identify currents on both sides of duality 1 2 ⇡✏ µ νρ f νρ j µ ! 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 D a Z scalar [ a ] e iS CS [ a ]+ iS BF [ a ; A ] = Z fermion [ A ] e − i 2 S CS [ A ] dynamical gauge field background gauge field Promote A to a dynamical gauge field Z Z D A D a Z scalar [ a ] e iS CS [ a ]+ iS BF [ a ; A ] − iS BF [ A ; C ] = Z scalar [ C ] e iS CS [ C ] D A Z fermion [ A ] e − i 2 S CS [ A ] − iS BF [ A ; C ] Equation of motion for A says that da = dC ] e iS CS [ a ]+ iS BF [ a ; A ] − iS BF [ A ; C ] = Z scalar [ C ] e iS CS [ C ] Z = D A Z fermion [ A ] e − i 2 S CS [ A ] − iS BF [ A ; C ] U(1) 1/2 + fermion = WF boson Barkeshli, McGreevy, 2012

Evidence From Things Working Out Nicely Z 2 S CS [ a ] − iS BF [ a ; A ] − iS CS [ A ] = Z scalar [ A ] D a Z fermion [ a ] e − i Now start from: Z D A Z scalar [ A ] e iS BF [ A,C ] 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 Z D a Z fermion [ a ] e + i 2 S CS [ a ]+ iS BF [ a ; A ]+ iS CS [ A ] But this is just the time-reversal of the original duality Z D a Z scalar [ a ] e iS BF [ A ; a ] = Z scalar [ A ] Peskin 1978, This is particle-vortex duality! Dasgupta Halperin 1981

Evidence From Things Working Out Nicely U(1) 1 + WF boson = free fermion • U(1) -1/2 + fermion = WF boson Barkeshli, McGreevy, 2012 Peskin 1978, • Bosonic particle vortex duality Dasgupta Halperin 1981 Son; Senthil, Wang; • Fermionic particle vortex duality Metlitski, Vishwanath 2015 • An infinite number of new dualities... • e.g. U(1) + 2 fermions is self-dual with emergent SU(2) x SU(2) global symmetry . You, Xu 2016 Karch and Tong 2016 Seiberg, Senthil, Wang and Witten 2016

Evidence from the Lattice Analytic Chen,Son, Wang, Raghu, 2017 Numeric Karthik and Narayanan, 2016-18

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: U(1) 1/2 + charged chiral = free chiral Dorey and Tong, 1999 Tong 2000 break supersymmetry U(1) 1 + WF boson = free fermion Kachru, Mulligan, Torroba, Wang 2016

Question What happens if we reduce to 2d?

Reduction to 2d break susy 3d mirror symmetry 3d bosonization U(1) 1/2 + chiral = free chiral reduce reduce to 2d to 2d ? ? break susy

Susy Reduction to 2d Aganagic, Hori, Karch, Tong 2001 Aharony, Razamat, Willet 2016 3d mirrors U(1) N=2 Chern-Simons theories Dorey and Tong, 1999 Tong 2000 reduce to 2d 2d mirrors GLSM = Landau-Ginzburg Hori, Vafa 2000

Susy Reduction to 2d Aganagic, Hori, Karch, Tong 2001 3d mirrors U(1) 1/2 + chiral free chiral = + 2 ⇡ 2 add irrelevant 1 e 2 j µ j µ 4 e 2 f µ ν f µ ν + operators Kapustin, Strassler 1999 S 1 of radius R � = e 2 R compactify 2d mirrors = N=(2,2) Cigar N=(2,2) Liouville Hori, Kapustin 2001 ds 2 = � 1 d ⇢ 2 + tanh 2 ⇢ d � 2 � ds 2 = dy 2 + d ✓ 2 � � � 4 ⇡ 4 ⇡� W ∼ e − Y

Reduction to 2d break susy 3d mirror symmetry 3d bosonization U(1) 1/2 + chiral = free chiral reduce reduce to 2d to 2d 2d mirror symmetry ? Cigar = Liouville break susy

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 N=(2,2) Cigar D-term gives masses to all scalars. We get γ � 1 − ⇡ L Thirring = 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 p � � µ 2 2 D � 2 e − y − i θ + m � N=(2,2) Liouville D-term gives e − y ⇡ V ( Y ) = + Dy � � � µ γ ⌧ 1 Only the periodic scalar remains light 1 √ 4 ⇡� ( @ i ✓ ) 2 + L SG = 2 D | m | cos ✓ But there is a shift of the kinetic term at one-loop ✓ 1 4 πγ + 1 ◆ √ ( ∂θ ) 2 + L SG = 2 Dm cos θ 8 π

2d Bosonization Coleman 1975 Mandelstam 1976 Massive Thirring = Sine-Gordon L Thirring = i ¯ @ + m ¯ − g ( ¯ � µ )( ¯ Massive Thirring: / � µ ) L SG = � 2 √ 2 ( @ i ✓ ) 2 + Sine-Gordon: 2 D | m | cos ✓ 2 ⇡ 2 � 2 = ⇡ 2 + g This is the shift that arises at one-loop? Open question: Higher loops?!

Reduction to 2d break susy 3d mirror symmetry 3d bosonization U(1) 1/2 + chiral = free chiral reduce reduce ? to 2d to 2d 2d mirror symmetry 2d bosonization Cigar = Liouville break susy

Non-Susy Reduction to 2d = free fermion U(1) 1 + WF boson 3d bosonization d 3 x � 1 B | � | 2 � � | � | 4 + 1 Z 4 e 2 f µ ν f µ ν + |D µ � | 2 � m 2 Work with weakly 4 ⇡✏ µ νρ a µ @ ν a ρ S 3 d = coupled UV theory not clear what weak coupling maps to Thirring here corresponds to for fermion! coupling for fermion

Non-Susy Reduction to 2d = free fermion U(1) 1 + WF boson 3d bosonization d 3 x � 1 B | � | 2 � � | � | 4 + 1 Z 4 e 2 f µ ν f µ ν + |D µ � | 2 � m 2 Work with weakly 4 ⇡✏ µ νρ a µ @ ν a ρ S 3 d = coupled UV theory not clear what weak coupling maps to Thirring here corresponds to for fermion! coupling for fermion � = e 2 R ⌧ 1 S 1 of radius R compactify ˆ λ = λ R ⌧ 1 I θ = π 1-loop potential for Wilson line fixes ✓ = a 2 [0 , 2 ⇡ ) U(1) Abelian Higgs 2d theories 6 = Thirring at θ = π

Non-Susy Reduction to 2d We must work with the strongly coupled theory = free fermion U(1) 1 + WF boson 1 V = ¯ j µ 2 ⇡✏ µ νρ @ ν a ρ � µ ! Use matching of currents: Upon compactification, the low-energy fermionic theory has two emergent currents V = ¯ j i A = ✏ ij j V j j i � i and I ✓ = The bosonic theory must have the same currents. These involve the Wilson line a j i V = ✏ ij @ j ✓ j i A = @ i ✓ Unlike at weak coupling, the Wilson line must remain massless!

Non-Susy Reduction to 2d Matching of symmetries in 3d bosonization are enough to ensure that, upon reduction on S 1 L = � 2 L = i ¯ 2 ( @ i ✓ ) 2 / @ � 2 = 1 symmetries also fix 4 ⇡ Question: is there a way to fix this value directly from 3d dynamimcs?

Summary break susy 3d mirror symmetry 3d bosonization U(1) 1/2 + chiral = free chiral reduce reduce to 2d to 2d 2d mirror symmetry 2d bosonization Cigar = Liouville break susy

Non-Susy Reduction to 2d From this, everything else follows: = free fermion U(1) 1 + WF boson In 3d: 1 + 2 π 2 2 e 2 f 2 + e 2 j i j i i 3 � = e 2 R S 1 of radius R In 2d: ✓ 1 L = 1 1 ◆ ∂ψ − π L = i ¯ γ ( ¯ ψγ i ψ )( ¯ ( ∂ i θ ) 2 ψ / ψγ i ψ ) 4 π + 2 2 πγ Mass terms also fixed by symmetries