Dipole CFTs, Bethe states and separation of variables Fedor - - PowerPoint PPT Presentation

Dipole CFTs, Bethe states and separation of variables

Fedor Levkovich-Maslyuk Nordita Stockholm

based on 1706.07957 [M. Guica, FLM, K. Zarembo] 1610.08032 [N. Gromov, FLM, G. Sizov]

I will present two results linked by a common theme: Sklyanin’s separation of variables in integrable systems (SoV) In separated variables the wavefunction factorizes Powerful method, many aplications: spin chains, sigma models, AdS/CFT, …

Sklyanin 91, 92

SoV for higher rank SU(N) spin chains Leads to new and compact construction of eigenstates

• Part 1

Integrable dipole deformation of N = 4 SYM, nonrelativistic CFT Bethe ansatz breaks down yet SoV gives access to spectrum We show that 1-loop spectrum matches string predictions

• Part 2

This talk

[Guica, FLM, Zarembo 17] [Gromov, FLM, Sizov 16]

Dipole CFTs and integrability Part 1

AdS/CFT and dipole theories

Integrability preserved, hope for complete solution

Nonlocal along null direction Dipole CFT in four dimensions superstring theory in Yang-Mills theory , Schrödinger background Nonrelativistic conf. symm.

Son 08; Balasubramanian, McGreevy 08

Motivation

Solvable via Baxter equation / separation of variables

• Non-AdS holography (cond-mat, …)
• Lower-dim versions (CFT2) related to Kerr/CFT

extremal black holes

• No susy
• Integrable structure deformed nontrivially

Usual Bethe ansatz not applicable even at 1 loop

Guica, Hartman, Song, Strominger 08 Son 08; Balasubramanian, McGreevy 08

Non-relativistic holography

AdS length scale CFT length scale, deformation parameter

Schrödinger background: Scale-invariant but time and space scale differently

Son 08; Balasubramanian, McGreevy 08

Unbroken symmetry (Schrödinger algebra) : part of commuting with

NR dilatation Galilean

Susy is completely broken

Global symmetry

Natural to consider states with fixed lightcone momentum

total

• ang. momentum

total light-cone momemtum Obtained from original theory by TsT transformation, preserves classical integrability Alternatively: twisted boundary conditions, undeformed background

Schrödinger from TsT

T-duality angle null direction on AdS boundary shift T-duality angle

Herzog, Rangamani, Ross 08 Maldacena, Martelli, Tachikawa 08 Alishahiha, Ganor 03 Frolov 05; Frolov, Roiban, Tseytlin 05

Undeformed model Twisted b.c.’s:

Can we match this result with gauge theory ? Dual operator is nonlocal, has anomalous dimension

Example: BMN string

AdS5 x S5 N=4 SYM Spin chain

TsT Star Product Drinfeld-Reshetikhin twist commuting charges commuting isometries Cartan generators

AdS/CFT triality

Lunin, Maldacena 05 Beisert, Roiban 05 Frolov 05 Matsumoto, Yoshida 14 van Tongeren 15, 16 Araujo, Bakhmatov, Colgain, Sakamoto, Sheikh-Jabbari, Yoshida 17 …

Star product (noncommutative field theory) : Dipole length :

Dipole CFT

Intermediate place between

• deformation (local CFT)
• generic AdS deformations

Seiberg, Witten 99 Bergman, Ganor 00

• Hamiltonian is deformed
• Boundary conditions are periodic
• Hamiltonian is the same as in N=4 SYM
• Boundary conditions are twisted

Related by Seiberg-Witten map

Operators and spin chains

twist

sl(2) sector

Dipole CFT: eigenstates are nonlocal, mixture of many operators 1-loop dilatation operator:

Balitsky, Braun 89 Belitsky, Derkachov, Korchemsky, Manashov 04

Drinfeld-Reshetikhin twist

In our case

Cartan generators

Monodromy matrix encodes the twisted Hamiltonian satisfies YBE Deformed spin chain R-matrix

Drinfeld 90 Reshetikhin 90 Beisert, Roiban 05 Ahn, Bajnok, Bombardelli, Nepomechie 11

The sl(2) sector spectrum

Eigenstates: Undeformed case

Algebraic Bethe ansatz

Infinite-dim spin ½ representation of sl(2) at each site Spectrum: want to diagonalize

no Bethe ansatz !

Twisted spin chain

No such state Need i.e. eigenvector of the two diagonal elements We want to diagonalize Ground state with nonzero momentum becomes unprotected twist is a Jordan cell

In Sklyanin’s separated variables the wavefunction factorizes

Baxter equation and SoV

differential equation in variables, hopeless to solve directly Baxter equation in undeformed case: Then

Sklyanin 91, 92

Baxter equation

We expect Baxter equation can be recast as the “quantum characteristic equation” For twisted case q-det is unchanged Baxter equation is the same Quantum version of classical spectral curve

Sklyanin 92 Chrevov,Falqui,Talalaev 06

The Q-function

Surprising asymptotics No longer polynomial Baxter + regularity of fixes both and ! Energy still given by

Gives full 1-loop spectrum in sl(2) sector

(conjecture) But now

similar to quark-antiquark potential

Gromov, FLM 16 Guica, FLM, Zarembo 17

Solved by spheroidal wavefunctions

Exact solution for J=2

After Mellin transform the Baxter equation becomes It’s also the wavefunction

• f the spin chain

Guica, FLM, Zarembo 17

Spectrum for any J – results

Solving perturbatively we get At large this gives Matches BMN string energy ! Got same result from large effective action for spin chain (Landau-Lifschitz model) Quantitative test of holography

Guica, FLM, Zarembo 17

SoV for higher rank spin chains Part 2

SoV should give access to 3-point functions in dipole CFT and original N=4 SYM Q’s are known to all loops in N=4 SYM from the Quantum Spectral Curve Could give a formulation for 3-pt functions alternative to [Basso, Komatsu, Vieira 2015] So far used only in rank-1 sectors like SU(2)

Kazama, Komatsu 11, 12 Kazama, Komatsu, Nishimura 13 - 16 Sobko 13 Jiang, Komatsu, Kostov, Serban 15 Gromov, Kazakov, Leurent, Volin 13

• Need to extend SoV to full
• Need better descriptions for spin chain states

Our results

Explicitly describe SoV for these spin chains New and simple construction of SU(N) spin chain eigenstates Interesting not only for AdS/CFT (condensed matter, pure mathematics, …)

Integrable SU(N) spin chains

Hamiltonian: At each site we have a space Spectrum is captured by nested Bethe ansatz equations

Sutherland 68; Kulish, Reshetikhin 83; …

(+ boundary twisted terms)

Related to rational R-matrix

The monodromy matrix

We take generic inhomogeneities and diagonal twist gives commuting integrals of motion We want to build their common eigenstates Transfer matrix

Construction of eigenstates

is an explicit polynomial in the monodromy matrix entries no simple analog despite many efforts over 30 years We conjecture for any use a creation operator evaluated on the Bethe roots Surprisingly simple ! The same operator also provides separated variables

Sutherland, Kulish, Reshetikhin, Slavnov, Ragoucy, Pakouliak, Belliard, Mukhin, Tarasov, Varchenko, …

SU(2) case

Separated variables for SU(2)

In their eigenbasis the wavefunction factorizes

we normalize

are the separated coordinates

Removing degeneracy

nilpotent, cannot be diagonalized is a constant 2 x 2 matrix All comm. rels are preserved, trace of is unchanged Now we can diagonalize

Jiang, Komatsu, Kostov, Serban 15

See also

separated variables

Gromov, FLM, Sizov 16

Eigenstates from Bgood

We can also build the states with Surprisingly it’s true even for generic Trace of is unchanged

Gromov, FLM, Sizov 16

SU(3)

Eigenstates for SU(3)

The operator which should provide separated variables is [Sklyanin 92] Again cannot be diagonalized Replacing we get It generates the eigenstates !

Gromov, FLM, Sizov 16

Conjecture supported by many tests

e.g.

is a 3 x 3 matrix

are the momentum-carrying Bethe roots fixed by usual nested Bethe equations Separated variables are found from Exactly like in SU(2) Factorization of wavefunction follows at once

Eigenstates for SU(3)

Comparison with known constructions

Usual nested algebraic Bethe Ansatz gives

wavefunction of auxiliary SU(2) chain

• Only a single operator
• No recursion
• Complexity in # of roots

is linear, not exponential Our conjecture

Sutherland, Kulish, Reshetikhin 83

Comparison with known constructions

Large literature on the construction of eigenstates

• representation as sum over partitions of roots
• trace formulas
• Drinfeld current construction
• …

Belliard, Pakouliak, Ragoucy, Slavnov 12, 14 Khoroshkin, Pakouliak 06 Khoroshkin, Pakouliak, Tarasov 06 Frappat, Khoroshkin, Pakouliak, Ragoucy 08 Belliard, Pakouliak, Ragoucy 10 Pakouliak, Ragoucy, Slavnov 14 … Tarasov, Varchenko 94 Mukhin, Tarasov, Varchenko 06 Mukhin, Tarasov, Varchenko 07 … Albert, Boos, Frume, Ruhling 00 …

Our proposal seems much more compact

We focussed on spin chains with fundamental representation at each site Many of the results should apply more generally

Extension to SU(N)

Classical B and quantization

In the classical limit for SU(N) How to quantize this expression? SU(3) result: quantum minors

Scott 94 Gekhtman 95

SU(4): make an ansatz, all coefficients fixed to 0 or 1 !

Results for SU(N)

We propose for any SU(N) We conjecture that generates the states and separated variables

• Matches classical limit
• Commutativity

Highly nontrivial checks !

Gromov, FLM, Sizov 16

Tests

No need for recursion ! Numerically: many states with , all states with for : analytic proof for up to 2 magnons, any Huge matrices, e.g. entries

For quantum which provides SoV was also proposed without construction of eigenstates in:

• [F. Smirnov 2001], would be interesting to compare
• [Chervov, Falqui, Talalaev 2006-7], fails for

Based on Manin matrices, perhaps can be improved

Supersymmetric spin chains

[work in progress] SOV is not known even for Our construction of eigenstates generalizes to at least some

Future

• Rigorous proof, algebraic origins
• Can one extend Slavnov’s determinant

(on-shell/off-shell scalar product) beyond SU(2) ?

• Higher-loop guesses in AdS/CFT
• SoV for fishnet theory, see talk of J. Caetano
• All-loop spectrum of dipole CFT from Quantum Spectral Curve