Topological symmetry and (de)confinement in gauge theories and spin - PowerPoint PPT Presentation

Topological symmetry and (de)confinement in gauge theories and spin systems Mithat ¨ Unsal, SLAC, Stanford University based on arXiv:0804.4664 QCD* parts with M. Shifman Thanks to Eun-ah Kim, B. Marston, M. Shifman, J. Harvey, M. Headrick, E.Poppitz, M. Mulligan, L.G. Yaffe, O. Aharony. for useful communications. 1

• In zero temperature asymptotically free or super- renormalizable (non-)abelian gauge theories, is there a symmetry associated with confinement/ deconfinement? • IR gapped, IR gapless, IR CFT ? Is there a distinguishing (continuum) notion beyond perturbation theory? • Obvious answer: No. There is no symmetry in microscopic theory related to confinement. 2

The theories and the goal Polyakov Representation : nothing, fundamental or adjoint P( R ) on R 3 Small non-thermal circle with stabilized QCD( R ) ∗ on R 3 × S 1 center (if center is not already stable quantum mechanically) Frustrated spin systems in d=2 space dimensions Lattice or continuum compact QED d=2+1dimensions AMBIGUOUS! (will be discussed, can be resolved.) 3

The goal All these theories have long distance regimes where they are described in terms of a compact abelian gauge theory. All has monopole-instantons. What is their IR physics? ‘t Hooft, Mandelstam, Nambu, Polyakov 70’s Gapped, ungapped, interacting CFT Surprisingly, all of these are possible! A sharp and useful notion of emergent (IR) topological symmetry is at work. It is the goal of this talk to make it precise. 4

Reminder: Abelian duality and Polyakov model Free Maxwell theory is dual to the free scalar theory. F = ∗ d σ The masslessness of the dual scalar is protected by a continuous shift symmetry U (1) flux : σ → σ − β Topological current vanishes by Bianchi Noether current of dual theory: identity. J µ = ∂ µ σ = 1 2 ǫ µ νρ F νρ = F µ Its conservation implies the absence of magnetic monopoles in original theory ∂ µ J µ = ∂ µ F µ = 0 5

The presence of the monopoles in the original theory implies reduction of the continuous shift symmetry into a discrete one. ∂ µ J µ = ∂ µ F µ = ρ m ( x ) The dual theory 2 ( ∂σ ) 2 − e − S 0 ( e i σ + e − i σ ) L = 1 Discrete shift symmetry: σ → σ + 2 π U (1) flux if present, forbids (magnetic) flux carrying operators. 6

Reminder: Add a massless adjoint Dirac fermion P(adj) Affleck, Harvey, Witten 82 Microscopic theory has U(1) fermion number symmetry. ψ → e − i β ¯ ψ → e i β ψ , ¯ ψ Monopole operator: e − S 0 e i σ ψψ I α i = (dim ker / D α i − dim ker / D α i ) Jackiw-Rebbi 76, Callias 78 The invariance under U(1) fermion number symmetry demands ψ → e i β ψ , σ − → σ − N f I α 1 β = σ − 2 β 7

Symmetry of the long distance theory U (1) ∗ : U (1) − N f I α i U (1) flux Forbids any pure flux operators e iq σ such as Current: K µ = ¯ ψσ µ ψ − n f I α 1 ∂ µ σ = ¯ ψσ µ ψ − n f I α 1 J µ AHW concludes: Fermion number breaks spontaneously and photon is NG boson. Current conservation = Local version of Callias index theorem 8

Massless Fundamental fermions and IR CFT N f 4-component Dirac spinors or 2 n f 2-component Dirac spinors: � � � � ψ a ψ a Ψ a = 1 2 Ψ a = , , ¯ ¯ ψ a ψ a 2 1 Microscopic symmetries: G M , P(F) = SO (3) L × C × P × T × Z 2 × U (1) V × U (1) A × SU ( n f ) 1 × SU ( n f ) 2 non-anomalous on R 3 This theory and certain frustrated spin systems in two spatial (and one time) dimensions share universal long distance physics. (to be discussed). Microscopic symmetries identical with QCD and QCD* except the underlined symmetry. HOW DOES THIS THEORY FLOWS INTO A CFT? 9

Perturbation theory: 3d QED with massless fermions � 1 � � S P(F) µ ν + i ¯ F 2 pert . = Ψ a γ µ ( ∂ µ + iA µ ) Ψ a 4 g 2 R 3 3 Is the masslessness destabilized non-perturbatively? Monopole operators: e − S 0 e i σ det 2 + e − S 0 e − i σ det ¯ 1 ¯ a,b ψ a 1 ψ b ψ a ψ b 2 a,b U (1) ∗ : ψ 1 → e i β ψ 1 , ψ 2 → e i β ψ 2 , → σ − 2 n f β . σ − Topological symmetry: U (1) ∗ : U (1) A − N f I α i U (1) flux 10

N f ≥ 2 No relevant flux (monopole) operators in the original electric theory!! IR theory quantum critical due to the absence of relevant or marginal destabilizers. Integrate out a thin momentum slice of massless fermions: Applequist et.al 88 µ ν + g 2 � � 1 µ ν → 1 3 n f 1 F 2 F 2 F µ ν F µ ν √ g 2 g 2 8 � 3 3 Move into deep IR: 1 L ∼ F µ ν � − 1 / 2 F µ ν + i ¯ Ψ a γ µ ( ∂ µ + i A µ ) Ψ a √ n f Dimensionless coupling of CFT 11

Big enhancement of spacetime and global symmetries (conformal symmetry) × G IR , P(F) ∼ C × P × T × U (1) V × U (1) flux × SU (2 n f ) Theories interpolate between weakly and strongly coupled interacting CFTs as the number of flavors is reduced. Anticipating ourselves a bit: The same dynamics as the IR of the frustrated spin systems with no broken symmetries. SU ( n f ) N f = 1 Perhaps one relevant flux operator, photon remains massless due to U(1)*. 12

Sub-conclusions The infrared of Polyakov models with complex fermions Take two electric charges at ( x , y ) ∈ R 2  | x − y | pure Polyakov or with heavy fermions      | x − y | − 1 V non − pert . ( | x − y | ) ∼ with massless fundamental fermions ,     log | x − y | with massless adjoint fermions ,  Respectively, P: Gapped, linear confinement Z 1 P(F): Interacting CFT, massless photon U (1) ∗ P(adj): massless photon, NG boson U (1) ∗ Recall: All of these theories have monopole-instantons! 13

YM* and QCD* Yaffe, M.U. 08 Shifman, M.U. 08 QCD( R ) ∗ P( R ) � � � � � � � � 1 2 3 1 2 3 4 1 R R/ (2 Z) � a a a a a a a a a a 1 2 4 3 1 2 3 4 1 2 1) Alter the topology of adjoint Higgs scalar into a compact one. An extra topological excitation moves in from infinity. (associated with affine root) S 1 × R 3 2) Take YM or QCD(R) on small and add center stabilizing double trace deformations. (Different theory from QCD? See below.) 1 and 2 are the same. 14

YM* theory at finite N β =1/ Τ * YM S YM ∗ = S YM + � R 4 R 3 × S 1 P [ U ( x )] confined ⌊ N/ 2 ⌋ 1/ Λ 2 1 n 4 | tr ( U n ) | 2 � P [ U ] = A π 2 β 4 n =1 deconfined 0 Deformation, A IDEA: Connect large and small circle physics in a smooth way! Then solve in regime where we have theoretical control. � 1 � g N � � 2 S dual = � � ( ∂σ ) 2 − ζ cos( α i · σ ) . 2 L 2 π R 3 i =1 15

At large N, the difference of YM and YM* is sub-leading in N. Volume independence (valid EK reduction) via center stabilizing deformations. ordinary Yang − Mills deformed Yang − Mills ∞ ∞ deformation equivalence orbifold equivalence L L c combined deformation − orbifold equivalence 0 0 Large N dynamics on =Large N quantum mechanics R 4 16

The notion of continuum compact 3d QED is ambiguous. Option 1: YM noncompact adjoint Higgs field, Polyakov model QCD* Option 2: YM compact adjoint Higgs field, In the presence of certain number of massless complex fermions, the first class always remains ungapped and the latter develops a gap for gauge fluctuations. Why? AHW and the first part of the talk: [ U (1)] N − 1 SU ( N ) U (1) − → − → ���� ���� � �� � � �� � ���� Higgsing nonperturbative with noncompact scalar compact QED 3 CFT or free photon Shifman, MU and the second part of the talk [ U (1)] N − 1 SU ( N ) nothing − → − → ���� ���� � �� � � �� � � �� � Higgsing nonperturbative gapped gauge bosons with compact scalar compact QED 3 17

QCD(R)* On locally three manifolds, there is no chiral anomaly. On locally four manifolds, due to chiral anomaly, the axial U(1) symmetry reduce to a discrete one. Quantum theory: U (1) A → Z 2 h σ → σ − 2 π Discrete topological shift ψ → e i 2 π 2 h ψ , symmetry: h U (1) ∗ ( Z h ) ∗ − → � �� � � �� � non − compact Higgs or P( R ) compact Higgs or QCD( R ) ∗ 18

QCD(F)* with one flavor Shifman, M.U. 08 M 2 ( x ) = e − S 0 e − i σ M 1 ( x ) = e − S 0 e i σ ψ 1 ψ 2 , ( Z 1 ) ∗ M 1 ( x ) = e − S 0 e − i σ ¯ ψ 1 ¯ M 2 ( x ) = e − S 0 e + i σ ψ 2 , Structure of the zero modes dictated by Callias index theorem, observed beautifully on lattice by Bruckmann, Nogradi, Pierre van Baal 03. BNvB also introduced the notion of zero mode hopping as the boundary conditions are changed for fermions. 19

IR of QCD(F)* with one flavor � 1 � � µ ν + i ¯ F 2 S = Ψ γ µ ( ∂ µ + iA µ ) Ψ 4 g 2 R 3 3 + c 1 e − S 0 e i σ ψ 1 ψ 2 + c 1 e − S 0 e − i σ ¯ ψ 1 ¯ ψ 2 + c 2 e − S 0 ( e − i σ + e i σ ) + . . . Mass gap for gauge fluctuations and fermions. Chiral condensate (which does not break any symmetry). 20

�� � QCD(adj)* R 3 × S 1 F ˜ � S 2 F, F Magnetic Magnetic Monopoles Bions Rebbi-Jackiw fermionic zero modes ( Z 2 ) ∗ BPS KK (1, 1/2) ( � 1, 1/2) e − S 0 e i σ det I,J ψ I ψ J , (2,0) ( � 2, 0) BPS KK e − 2 S 0 ( e 2 i σ + e − 2 i σ ) ( � 1, � 1/2) (1, � 1/2) e − S 0 e i σ det I,J ¯ ψ I ¯ ψ J ψ I → e i 2 π 8 ψ I Discrete shift symmetry : σ → σ + π 21