[PPT] - A new look at integrable -models and their deformations Dmitri PowerPoint Presentation

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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

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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 – Fn1,...,nS = SU(N) S(U(n1) × · · · × U(nS)),

S

i=1

ni = N Complex structure ↔ ordering of n1, . . . , nS. Complex definition: F = GL(N, C)/P P = parabolic subgroup stabilizing the flag 0 ∈ L1 ⊂ . . . ⊂ LS = CN, dimC Lk := dk =

k

i=1

ni . (Can be generalized to other groups)

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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]

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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−1dg = Jh + Jm

Metric G: the ‘Killing metric’ ds2 = Tr(JmJm) Complex structure J B-field: fundamental Hermitian form of the metric B = G ◦ J Action: S[G, J] :=

Σ

d2z Gmn ∂U m∂U n

[DB, 2014]

Note: metric G is only K¨ ahler iff G

H is symmetric (= Grassmannian)

Geodesics of G are homogeneous [Alekseevsky, Arvanitoyeorgos ’07].

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The gauged linear sigma-model (GLSM).

K¨ ahler case: GLSM ↔ K¨ ahler quotients. Grassmannian: Gr(m, N) = Hom(Cm, CN)/ /U(m) . Lagrangian L = Tr((DU)†(DU)), DU := ∂U − i U A , U †U = 1m .

[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].

U1 US−2 V1 VS−2 US−1 VS−1

CN

L1 L2 LS−2 LS−1 · · ·

Flag manifold with ‘Killing metric’ (not K¨ ahler for S > 2): a ‘gauge field’ [DB ’17] A = ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗                

d1 d2 dS−1

, A = (A)† , A ‘reduced’ gauge field! I will explain the meaning of this later on in the talk.

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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] L = Ψa / DΨa + (rs)cd

ab

Ψa 1 + γ5

2 Ψc Ψd 1 − γ5 2 Ψb

,

where Ψa =

Ua

V a

,

a = 1, . . . , N ‘Dirac boson’ rs is the classical r-matrix of [Belavin, Drinfeld ’80]: rs =

s 1−s π+ + 1 1−s π− + 1 2 1+s 1−s π0 (solution of classical Yang-Baxter equation)

σ-model = chiral gauged Gross-Neveu model (in bosonic incarnation)! (Fermionic version: [Gross, Neveu ’74, Witten ’78, Andrei, Lowenstein ’80])

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Elementary example

Chiral symmetry (λ ∈ C×) [Zumino, 1977; Mehta 1990] U → λU, V → λ−1V ensures that the interaction is quadratic in U and in V , so that we can integrate

ut V .

Massless Thirring model [Thirring ’58], cf. [Swieca ’77] for a review N = 1, ungauged: A = 0, undeformed: (rs)cd

ab = δc a δd b

L = Ψ/ ∂Ψ + 1

2 (ΨγµΨ)2 = V ∂U − V ∂U + |U|2|V |2

Integrate out V : L = ∂U ∂U

UU

→ cylinder C× with multiplicative coordinate U Vanishing β-function: bosonic Thirring vs. free boson (with a linear dilaton)

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The β-function [DB, ’20]

Feynman rules:

z1 z2

1 z2−z1

i z1 z2

1 z2−z1

j i j k l

(rs)kl

ij

Diagrams contributing to the β-function at one loop:

k i j l l i j k

β-function: βkl

ij = N

p,q=1
(rs)kq

ip (rs)ql pj − (rs)ql ip(rs)kq pj

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Ricci flow [DB ’20]

The Ricci flow equation ˙ rkl

ij = βkl ij has a remarkably simple solution s = eNτ

(was conjectured in [Costello, Yamazaki 2019]). Alternatively, return to the σ-model and solve the geometric Ricci flow equations − ˙ gij = Rij + 1 4HimnHjm′n′gmm′gnn′ + 2 ∇i∇jΦ , − ˙ Bij = −1 2 ∇k Hkij + ∇kΦ Hkij , − ˙ Φ = const. − 1 2 ∇k∇k Φ + ∇kΦ∇kΦ + 1 24 HkmnHkmn CP1: the ‘sausage’ solution s = e2τ (N = 2) [Fateev, Onofri, Zamolodchikov, ’1994] ds2 = (s−1−s)|dW |2

(s+|W |2)(s−1+|W |2)

0 < s < 1 Length ∼ | log s| CPN−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.

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Anomalies [DB ’20]

Remember U ∈ CN, so to pass to CPN−1 we need to gauge the chiral symmetry. However the symmetry is typically anomalous: recall Schwinger’s effective action S

eff. = ξ

2

dz dz Fzz 1

△Fzz , Fzz = i (∂A − ∂A)

[Schwinger ’1962]

Not invariant under the complexified gauge transformations A → A + ∂α, A → A + ∂α. To cancel the anomaly one can add fermions minimally: L = Ψa / DΨa + (rs)cd

ab

Ψa 1 + γ5

2 Ψc Ψd 1 − γ5 2 Ψb

+ Θa /

DΘa , Ψ, Θ ∈ Hom(C, C2 ⊗ CN) . 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

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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 Ggauge (Global) action of a complex group Gglobal Complex moment map µ for Gglobal Φ L =

V · DU + U · DV
+ κ Tr(µ µ)

Complex symplectic quotient instead of K¨ ahler quotient! Anomaly cancellation conditions: StrW(TaTb) = 0 (Ta ∈ ggauge) StrW(Ta) = 0

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Examples [’20, to appear]

CPN−1 model with minimally coupled fermions. Phase space Φ(s) = T∗CPN−1|N = T∗V, where V = Π(O(1) ⊕ · · · ⊕ O(1)

N times

) is a vector bundle over CPN−1 (the ‘fermionic conifold’). Gglobal = SL(N, C) × 1 ⊂ PSL(N|N) acting on CN|0 but trivially on C0|N. Choosing a different Gglobal, one gets different couplings of the fermions: Gglobal = 1 × SL(N, C): fermionic Gross-Neveu-model Gglobal = PSL(N|N): CPN−1|N-model [Read, Saleur ’01, Witten’03, Schomerus et.al.’10] Worldsheet supersymmetric CPN−1 model. Phase space Φ(s) = T∗V, where V = Π(TCPN−1), Gglobal = SL(N, C)

U V

CN C1|1 Ggauge =

g =
λ

χ λ

∈ SL(1|1), λ ∈ C×
U ∈ Hom(C1|1, CN) and V ∈ Hom(CN, C1|1).

– Super-quivers –

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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)

SO(2)

[Brodbeck & Zagermann ‘00])

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]

· · ·

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Inhomogeneous gauge [DB ’20]

Instead of working in a standard gauge like UU = 1, we can choose UN = 1 (inhomogeneous gauge) Varying the action w.r.t. A, we get UV = 0, i.e. VN = −

N−1

j=1

UjVj.

Substituting in the Lagrangian, we get

L =

N−1

k=1
Vk∂Uk − V k∂U k + β|Vk|2

+ z1 z2 i −δj

i β log |z1 − z2|2 ,

+

N−1

l,m=1

alm |Ul|2|Vm|2 + γ

N−1
p=1

UpVp

2

quartic vertices

+ α N−1

k=1

|Uk|2

N−1
p=1

UpVp

2

sextic vertices

p p′ q r q′ r′ −α δpp′δqq′δrr′

Instead of a σ-model we obtained a theory with polynomial interactions! Parallel with Ashtekar variables:

Direct derivation for SL(2,R)

SO(2)

[Brodbeck & Zagermann ‘00]

Interactions are polynomial
Degenerations are allowed (compare with nilpotent orbits)

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The Costello-Yamazaki approach [Costello-Yamazaki ’19]

A semi-holomorphic 4D Chern-Simons theory on Σ × C , where Σ = ‘topological plane’ (z, z) → worldsheet to-be C = complex curve (w, w) with a holomorphic differential ω = dw = 0. KC = 0 implies C ≃ C, C∗, Eτ. The action: SCS = 1

Σ×C

ω ∧ Tr

A ∧ (dA + 2

3A ∧ A)

,

where A = Azdz + Azdz + Awdw. One couples this theory to two βγ systems, with target space T ∗M, where M is a complex homogeneous space: Sdef =

Σ

d2z (piD(w1)

z

qi + piD(w2)

z

qi) , where D(w1)

z

qi = ∂zqi −

a

(A(w1)

z

)avi

a . Dmitri Bykov | Flag manifold sigma-models

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The Costello-Yamazaki approach [Costello-Yamazaki ’19]

‘Light-cone’ gauge Aw = 0, solve for Az, Az. In this gauge the equations are ∂zAz − ∂zAz + [Az, Az] = 0, ∂wAz = δ(2)(w − w1)

a

pivi

a τa

∂wAz = δ(2)(w − w2)

a

pivi

a τa .

Family Az(w), Az(w) of flat connections, depending meromorphically on w! Key observation: Green’s function ∂

−1 w

= classical r-matrix [Belavin, Drinfeld ’80] Rational case: r(w) =

τa⊗τa w

∈ g ⊗ g, i.e. r(w) = 1

w ∈ End(g).

Trigonometric case: let g = g+⊕g− (complex structure on G, Manin triple, etc.), then rcomp.(u) =

τ+

a ⊗τ− a

1−u

−

τ−

a ⊗τ+ a

1−u−1

∈ g ⊗ g, i.e. rcomp.(z) =

Π+ 1−u − Π− 1−u−1 ∈ End(g). Dmitri Bykov | Flag manifold sigma-models

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The Costello-Yamazaki approach [Costello-Yamazaki ’19]

Upon integrating out Az, Az, we get (rational case) S =

d2z
pi∂zqi + pi∂zqi + rw1−w2
pivi

a τa, pivi a τa)

=

=

d2z
pi∂zqi + pi∂zqi +

1 w1 − w2

|pivi

a|2

=

= integrate out p, p (the fiber of T ∗M) ∼ ∼

d2z
Gij∂zqi∂zqj

with Gij =

a

vi

avj a

−1 Invertibility ↔ Homogeneous space Rational case: the flag manifold σ-model described earlier. [DB, 2019] But this also provides deformations of those models, trigonometric and elliptic. We pass over to this topic, starting with deformations of the CPn−1 model.

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Integrability.

The conjecture of integrability of the models is based on the following evidence:

The zero-curvature representation

Au = 1 + u 2 Kzdz + 1 + u−1 2 Kzdz, u ∈ C∗ , where K = Noether current (flat).

Involutivity of the integrals of motion [Delduc et. al. ’19]
Explicit classical solutions
U(3)

U(1)3

[DB ’16], generalizing [Din, Zakrzewski ’80]
Analogy with the case of symmetric spaces (review: [Zarembo ’17])

Symmetric spaces of SU(N): Grassmannians G(m, N) :=

SU(N) S(U(m)×U(N−m)) Dmitri Bykov | Flag manifold sigma-models

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Generalized Einstein metrics on flag manifolds [DB ’20]

Consider the Ricci flow for the metric − ˙ gij = Rij + 1 4HimnHjm′n′gmm′gnn′ + ∇iDjΦ + ∇jDiΦ , DΦ = dΦ − E . The structure of the blow-up (strong coupling) is as follows: gij ∼ (1 − s) (ghom.)ij. For the homogeneous metric DiΦ = 0. Since s = eNτ, in the limit − ˙ gij → N (ghom.)ij Homogeneous (Killing) metric ghom. satisfies a generalized Einstein condition Rij + 1 4HimnHjm′n′gmm′

hom.gnn′

hom. = N (ghom.)ij .

For Grassmannians H = 0, i.e. Rij = N (ghom.)ij. N= first Chern number of the tangent bundle c1(G(m, N)) = N [C ] (C =generator) – independent of m! β-function for a symmetric space = dual Coxeter number (independent of H in G

H ).

We observe this for non-symmetric spaces – can be proven directly!

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Flag manifold models from the PCM: nilpotent orbits

Let g ∈ SU(N), g−1dg = i (Φ dz + Φ dz). ∂Φ + i [Φ, Φ] = 0 . Flatness of Noether current, Principal Chiral Model Impose the condition ΦS = 0 (closure of nilpotent orbit in glN) Assume ΦS = 0, and ΦS−1 = 0 . The map Φ(z, z) satisfying the E.O.M. defines a flag 0 ⊂ Ker(Φ) ⊂ Ker(Φ2) ⊂ · · · ⊂ Ker(ΦS) ≃ CN This is a point in F := U(N) U(κ1) × · · · × U(κS), where κj = dim Ker(Φj)/Ker(Φj−1) = number of Jordan blocks of size at least i . Claim: the map Σ → F is a solution of a flag manifold sigma-model This is a map to a single orbit: type does not change due to e.o.m. [DB, 2019] I will now describe the model.

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The sausage model [Fateev, Onofri, Zamolodchikov, 1994]

Simplest deformation: 2D target space with a U(1)-isometry [DB-L¨

ust, 2020]

ds2 =

2

i,j=1

Gij dXidXj = 1 4 g(µ) dµ2 + g(µ) dφ2 . For what g(µ) is the model integrable? Mechanical reduction always integrable (2 integrals of motion). Generalized Pohlmeyer map: set Gij ∂Xi∂Xj = cosh χ. Sinh-Gordon equation replaced by ∂∂χ − 2g′′(µ) sinh χ = 0 . Add the equation for µ: ∂∂µ − 2 g′(µ) cosh χ = 0 . The two eqs. follow from a single Lagrangian if g(µ) = b + a cosh µ : L = 1 2∂µ∂µ + 1 2∂χ∂χ + 2a cosh µ cosh χ = = 1 2∂ µ ∂ µ + a cosh ( √ 2 µ)

+

1 2∂ χ ∂ χ + a cosh ( √ 2 χ)

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Ricci flow in the K¨ ahler case.

The (K¨ ahler) metric of the ‘sausage’ [Fateev, Onofri, Zamolodchikov, 1994] ds2 = 1 s − s

|dW|2

(s + |W|2)( 1

s + |W|2) ,

0 < s < 1 . CPN−1 also has a generalized K¨ ahler deformation [Demulder et.al. 2020], constructed along the lines of [Delduc, Magro, Vicedo 2013]. The B-field has the form B =

i

bi ∧ dφi, so T-dualizing all angles we get rid of it. T-dual geometry is K¨ ahler with potential [DB-L¨

ust, 2020]

K =

N

j=1

(ZjZj−1 − ZjZj−1) + 2

N

j=1

P(tj − tj−1 − 2 τ), tj = Zj + Zj , where P(t) = Li2(e−t) + t2

4 .

K not invariant under Zi → Zi + δαi. T-duality does not preserve the K¨ ahler property (otherwise T: chiral ↔ twisted chiral [Rocek, Verlinde 1991]). The metric satisfies the (simple) Ricci flow equation −

dgij dτ

= 4 Rij with s = eNτ.

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