A new look at integrable σ -models and their deformations Dmitri Bykov Max-Planck-Institut f¨ ur Physik (M¨ unchen) Steklov Mathematical Institute (Moscow) RAQIS, Annecy, 1 September 2020 Based on several papers of the speaker, including arXiv:2006.14124 and an upcoming one
Plan – Flag manifold σ -models from spin chains – – Gauged linear σ -models → gauged chiral bosonic Gross-Neveu systems – – Ricci flow: the explicit universal solution – – Chiral anomalies, inclusion of fermions, SUSY – S � SU ( N ) F n 1 ,...,n S = S ( U ( n 1 ) × · · · × U ( n S )) , n i = N i =1 Complex structure ↔ ordering of n 1 , . . . , n S . Complex definition: F = GL ( N, C ) /P P = parabolic subgroup stabilizing the flag � k 0 ∈ L 1 ⊂ . . . ⊂ L S = C N , dim C L k := d k = n i . i =1 (Can be generalized to other groups) Dmitri Bykov | Flag manifold sigma-models 2 /13
Flag manifolds. They arise as effective continuum theories of spin chains with SU ( N ) -symmetry. ( SU (2) -case: [Haldane ’83] , [Affleck ’85] ) The idea is that the flag manifold is the space of N´ eel vacua of the classical chain: – Geometric theory: [DB ’11-’12] – Analysis of spin chains: [Affleck et.al. ’17 ( SU (3) ), ’19, ’20 ( SU ( N ) )] – Discrete ‘t Hooft anomalies [Tanizaki & Sulejmanpasic ’18, Seiberg et.al. ’18] predict gapless spectrum ↔ match with rigorous spin chain results [Lieb, Schultz, Mattis ’61, Affleck, Lieb ’86] Dmitri Bykov | Flag manifold sigma-models 3 /13
The σ -model in conventional terms. In this talk I will discuss mostly the flag σ -models that are (conjecturally) integrable. Symmetric space examples date back to [Pohlmeyer ’76, Zakharov, Mikhaylov ’78, Eichenherr, Forger ’79] In general, take the flag manifold G H , define g = h ⊕ m , J = − g − 1 dg = J h + J m Metric G : the ‘Killing metric’ ds 2 = Tr( J m J m ) Complex structure J B -field: fundamental Hermitian form of the metric B = G ◦ J � d 2 z G mn ∂U m ∂U n Action: S [ G , J ] := [DB, 2014] Σ ahler iff G Note: metric G is only K¨ H is symmetric (= Grassmannian) Geodesics of G are homogeneous [Alekseevsky, Arvanitoyeorgos ’07] . Dmitri Bykov | Flag manifold sigma-models 4 /13
The gauged linear sigma-model (GLSM). K¨ ahler case: GLSM ↔ K¨ ahler quotients. Grassmannian: Gr ( m, N ) = Hom( C m , C N ) / /U ( m ) . Lagrangian L = Tr(( DU ) † ( DU )) , U † U = 1 m . DU := ∂U − i U A , [Cremmer, Scherk ’78, D’Adda, L¨ uscher, di Vecchia ’78] . Flag manifold with K¨ ahler metric: GLSM ↔ Nakajima (quiver) varieties [Nakajima ’94, Nitta ’03, Donagi, Sharpe ’08] . V S − 2 V S − 1 V 1 C N · · · L S − 2 L S − 1 L 1 U 1 L 2 U S − 2 U S − 1 Flag manifold with ‘Killing metric’ (not K¨ ahler for S > 2 ): a ‘gauge field’ [DB ’17] d S − 1 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ A ‘reduced’ gauge field! ∗ ∗ ∗ ∗ A = ( A ) † , A = , I will explain the meaning of this ∗ ∗ ∗ ∗ later on in the talk. ∗ ∗ ∗ ∗ d 1 d 2 Dmitri Bykov | Flag manifold sigma-models 5 /13
A new look at σ -models [DB ’20] Same models may be obtained by the approach of [Costello, Yamazaki ’2019] from a coupling of two βγ -systems. Together with the GLSM approach this leads to the following alternative definition [DB ’20] � � � � Ψ a 1 + γ 5 Ψ d 1 − γ 5 D Ψ a + ( r s ) cd L = Ψ a / Ψ c Ψ b , ab 2 2 where � � U a Ψ a = , a = 1 , . . . , N ‘Dirac boson’ V a r s is the classical r -matrix of [Belavin, Drinfeld ’80] : 1 − s π − + 1 1 1+ s s r s = 1 − s π + + 1 − s π 0 (solution of classical Yang-Baxter equation) 2 σ -model = chiral gauged Gross-Neveu model (in bosonic incarnation)! (Fermionic version: [Gross, Neveu ’74, Witten ’78, Andrei, Lowenstein ’80] ) Dmitri Bykov | Flag manifold sigma-models 6 /13
Elementary example Chiral symmetry ( λ ∈ C × ) [Zumino, 1977; Mehta 1990] U → λ U, V → λ − 1 V ensures that the interaction is quadratic in U and in V , so that we can integrate out V . Massless Thirring model [Thirring ’58] , cf. [Swieca ’77] for a review N = 1 , ungauged: A = 0 , undeformed: ( r s ) cd ab = δ c a δ d b 2 (Ψ γ µ Ψ) 2 = V ∂U − V ∂U + | U | 2 | V | 2 ∂ Ψ + 1 L = Ψ / cylinder C × with multiplicative coordinate U L = ∂U ∂U Integrate out V : → UU Vanishing β -function: bosonic Thirring vs. free boson (with a linear dilaton) Dmitri Bykov | Flag manifold sigma-models 7 /13
The β -function [DB, ’20] Feynman rules: ( r s ) kl k 1 1 ij z 2 − z 1 z 2 − z 1 z 1 z 2 z 1 z 2 j i j i l Diagrams contributing to the β -function at one loop: i k i l j j l k β -function: N � � � β kl ( r s ) kq ip ( r s ) ql pj − ( r s ) ql ip ( r s ) kq ij = pj p,q =1 Dmitri Bykov | Flag manifold sigma-models 8 /13
Ricci flow [DB ’20] r kl ij = β kl ij has a remarkably simple solution s = e Nτ The Ricci flow equation ˙ (was conjectured in [Costello, Yamazaki 2019] ). Alternatively, return to the σ -model and solve the geometric Ricci flow equations 4 H imn H jm ′ n ′ g mm ′ g nn ′ + 2 ∇ i ∇ j Φ , g ij = R ij + 1 − ˙ B ij = − 1 2 ∇ k H kij + ∇ k Φ H kij , − ˙ Φ = const . − 1 2 ∇ k ∇ k Φ + ∇ k Φ ∇ k Φ + 1 − ˙ 24 H kmn H kmn CP 1 : the ‘sausage’ solution s = e 2 τ ( N = 2 ) [Fateev, Onofri, Zamolodchikov, ’1994] ( s − 1 − s ) | dW | 2 ds 2 = Length ∼ | log s | 0 < s < 1 ( s + | W | 2 )( s − 1 + | W | 2 ) CP N − 1 : Ricci flow interpolates between a cylinder ( C × ) N − 1 in the UV (asymptotic freedom) and a ‘round’ projective space of vanishing radius in the IR. Dmitri Bykov | Flag manifold sigma-models 9 /13
Anomalies [DB ’20] Remember U ∈ C N , so to pass to CP N − 1 we need to gauge the chiral symmetry. However the symmetry is typically anomalous: recall Schwinger’s effective action � eff . = ξ dz dz F zz 1 △ F zz , F zz = i ( ∂ A − ∂ A ) S [Schwinger ’1962] 2 Not invariant under the complexified gauge transformations A → A + ∂α , A → A + ∂α . To cancel the anomaly one can add fermions minimally: � � � � Ψ a 1 + γ 5 Ψ d 1 − γ 5 D Ψ a + ( r s ) cd L = Ψ a / + Θ a / Ψ c Ψ b D Θ a , ab 2 2 Ψ , Θ ∈ Hom( C , C 2 ⊗ C N ) . Incidentally these are the same fermions that cancel the anomaly in L¨ uscher’s nonlocal charge [L¨ uscher, Pohlmeyer ’78, Abdalla et.al., 1981-84] (i.e. an anomaly in the Yangian [Bernard ’91] )! Conjecture: all such flag manifold models with fermions are quantum integrable Dmitri Bykov | Flag manifold sigma-models 10 /13
General setup [’20, to appear] Super phase space Φ = complex symplectic (quiver) super-variety Matter fields U, V in reps. W ⊕ W ∨ of a complex gauge group G gauge (Global) action of a complex group G global Complex moment map µ for G global � Φ � � L = V · D U + U · D V + κ Tr( µ µ ) Complex symplectic quotient instead of K¨ ahler quotient! Anomaly cancellation conditions: Str W ( T a T b ) = 0 ( T a ∈ g gauge ) Str W ( T a ) = 0 Dmitri Bykov | Flag manifold sigma-models 11 /13
Examples [’20, to appear] CP N − 1 model with minimally coupled fermions. Phase space Φ ( s ) = T ∗ CP N − 1 | N = T ∗ V , ) is a vector bundle over CP N − 1 where V = Π ( O (1) ⊕ · · · ⊕ O (1) N times (the ‘fermionic conifold’). G global = SL ( N, C ) × 1 ⊂ PSL ( N | N ) acting on C N | 0 but trivially on C 0 | N . Choosing a different G global , one gets different couplings of the fermions: G global = 1 × SL ( N, C ) : fermionic Gross-Neveu-model G global = PSL ( N | N ) : CP N − 1 | N -model [Read, Saleur ’01, Witten’03, Schomerus et.al.’10] Worldsheet supersymmetric CP N − 1 model. Phase space Φ ( s ) = T ∗ V , where V = Π ( T CP N − 1 ) , G global = SL ( N, C ) C N � � � � V 0 λ ∈ SL (1 | 1) , λ ∈ C × G gauge = g = χ λ U C 1 | 1 U ∈ Hom( C 1 | 1 , C N ) and V ∈ Hom( C N , C 1 | 1 ) . – Super-quivers – Dmitri Bykov | Flag manifold sigma-models 12 /13
Results/Outlook. • Integrable sigma-models beyond symmetric target spaces [DB ’14 + , Costello-Yamazaki 2019] • GLSM formulation beyond K¨ ahler target spaces [DB ’17] • σ -models = gauged chiral Gross-Neveu models [DB ’20] • The one-loop β -function is universal for all of these models (one-loop exact?) • (Complicated) Ricci flow eqs. have a simple (ancient) solution, interpolating between the homogeneous metric and a cylinder • Super phase spaces, anomaly cancellation conditions [’20, to appear] • New way of obtaining SUSY σ -models [’20, to appear] • Interactions are polynomial → relation to Ashtekar variables of GR (direct derivation for SL (2 , R ) [Brodbeck & Zagermann ‘00] ) SO (2) • Poisson brackets of Lax operators are ultralocal [Delduc et.al. ’19] • Should be possible to construct the full quantum theory: S/R-matrix, spectrum (thermodynamic Bethe ansatz), analyze resurgence, etc. Possibly using the ODE/IQFT approach [Bazhanov, Lukyanov, Zamolodchikov 98 + , Bazhanov, Kotousov, Lukyanov ’17] • · · · Dmitri Bykov | Flag manifold sigma-models 13 /13
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