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) Sklyanin 91, 92 In separated variables the wavefunction factorizes Powerful method, many aplications: spin chains, sigma models, AdS/CFT, …
This talk • Part 1 [Guica, FLM, Zarembo 17] 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 [Gromov, FLM, Sizov 16] SoV for higher rank SU(N) spin chains Leads to new and compact construction of eigenstates
Part 1 Dipole CFTs and integrability
AdS/CFT and dipole theories superstring theory in Yang-Mills theory in four dimensions , Schrödinger background Dipole CFT Nonlocal along null direction Nonrelativistic conf. symm. Son 08; Balasubramanian, McGreevy 08 Integrability preserved, hope for complete solution
Motivation • Non-AdS holography (cond- mat, …) Son 08; Balasubramanian, McGreevy 08 • Lower-dim versions (CFT 2 ) related to Kerr/CFT Guica, Hartman, Song, Strominger 08 extremal black holes • No susy • Integrable structure deformed nontrivially Usual Bethe ansatz not applicable even at 1 loop Solvable via Baxter equation / separation of variables
Non-relativistic holography Schrödinger background: Son 08; Balasubramanian, McGreevy 08 Scale-invariant but time and space scale differently CFT length scale, AdS length deformation parameter scale
Global symmetry Natural to consider states with fixed lightcone momentum Unbroken symmetry (Schrödinger algebra) : part of commuting with Galilean NR dilatation Susy is completely broken
Schr ö dinger from TsT Obtained from original theory by TsT transformation, Herzog, Rangamani, Ross 08 Maldacena, Martelli, preserves classical integrability Tachikawa 08 Alishahiha, Ganor 03 T-duality angle null direction on AdS boundary shift T-duality angle Alternatively: twisted boundary conditions, Frolov 05; Frolov, Roiban, Tseytlin 05 undeformed background total ang. momentum total light-cone momemtum
Example: BMN string Twisted b.c.’s : Undeformed model Dual operator is nonlocal, has anomalous dimension Can we match this result with gauge theory ?
AdS/CFT triality AdS 5 x S 5 N=4 SYM Spin chain TsT Star Product Drinfeld-Reshetikhin twist commuting charges commuting isometries Cartan generators Lunin, Maldacena 05 Matsumoto, Yoshida 14 Beisert, Roiban 05 van Tongeren 15, 16 Frolov 05 Araujo, Bakhmatov, Colgain, Sakamoto, Sheikh-Jabbari, Yoshida 17 …
Dipole CFT Star product (noncommutative field theory) : Seiberg, Witten 99 Bergman, Ganor 00 Dipole length : Intermediate place between • deformation (local CFT) • generic AdS deformations
Operators and spin chains • twist Hamiltonian is the same as in N=4 SYM • Boundary conditions are twisted • Hamiltonian is deformed • Boundary conditions are periodic Related by Seiberg-Witten map
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 Drinfeld 90 Reshetikhin 90 Deformed spin chain R-matrix satisfies YBE Cartan generators In our case Monodromy matrix encodes the twisted Hamiltonian Beisert, Roiban 05 Ahn, Bajnok, Bombardelli, Nepomechie 11
The sl(2) sector spectrum
Algebraic Bethe ansatz Infinite-dim spin ½ representation of sl(2) at each site want to diagonalize Undeformed case Eigenstates: Spectrum:
Twisted spin chain twist is a Jordan cell We want to diagonalize Need i.e. eigenvector of the two diagonal elements No such state no Bethe ansatz ! Ground state with nonzero momentum becomes unprotected
Baxter equation and SoV differential equation in variables, hopeless to solve directly In Sklyanin’s separated variables the wavefunction factorizes Sklyanin 91, 92 Baxter equation in undeformed case: Then
Baxter equation We expect Baxter equation can be recast as the “quantum characteristic equation” Quantum version of classical spectral curve Sklyanin 92 Chrevov,Falqui,Talalaev 06 For twisted case Baxter equation is the same q-det is unchanged
The Q-function But now Surprising asymptotics Guica, FLM, Zarembo 17 No longer polynomial similar to quark-antiquark potential Gromov, FLM 16 Baxter + regularity of fixes both and ! Energy still given by (conjecture) Gives full 1-loop spectrum in sl(2) sector
Exact solution for J=2 After Mellin transform the Baxter equation becomes Guica, FLM, Zarembo 17 Solved by spheroidal wavefunctions It’s also the wavefunction of the spin chain
Spectrum for any J – results Guica, FLM, Zarembo 17 Solving perturbatively we get At large this gives Quantitative test of holography Matches BMN string energy ! Got same result from large effective action for spin chain (Landau-Lifschitz model)
Part 2 SoV for higher rank spin chains
SoV should give access to 3-point functions in dipole CFT and original N=4 SYM So far used only in Kazama, Komatsu 11, 12 Kazama, Komatsu, Nishimura 13 - 16 rank-1 sectors like SU(2) Sobko 13 Jiang, Komatsu, Kostov, Serban 15 Q’s are known to all loops in N=4 SYM from the Quantum Spectral Curve Gromov, Kazakov, Leurent, Volin 13 Could give a formulation for 3-pt functions alternative to [Basso, Komatsu, Vieira 2015]
• Need to extend SoV to full • Need better descriptions for spin chain states
Our results New and simple construction of SU(N) spin chain eigenstates Explicitly describe SoV for these spin chains Interesting not only for AdS/CFT (condensed matter, pure mathematics, …)
Integrable SU(N) spin chains At each site we have a space Hamiltonian: (+ boundary twisted terms) Related to rational R-matrix Spectrum is captured by nested Bethe ansatz equations Sutherland 68; Kulish, Reshetikhin 83; …
The monodromy matrix We take generic inhomogeneities and diagonal twist Transfer matrix gives commuting integrals of motion We want to build their common eigenstates
Construction of eigenstates use a creation operator evaluated on the Bethe roots no simple analog despite Sutherland, Kulish, Reshetikhin, Slavnov, many efforts over 30 years Ragoucy, Pakouliak, Belliard, Mukhin, Tarasov, Varchenko, … We conjecture for any Surprisingly simple ! is an explicit polynomial in the monodromy matrix entries The same operator also provides separated variables
SU(2) case
Separated variables for SU(2) are the separated coordinates In their eigenbasis the wavefunction factorizes we normalize
Removing degeneracy nilpotent, cannot be diagonalized is a constant 2 x 2 matrix Gromov, FLM, Sizov 16 All comm. rels are preserved, trace of is unchanged Now we can diagonalize separated variables Jiang, Komatsu, See also Kostov, Serban 15
Eigenstates from B good Trace of is unchanged We can also build the states with Surprisingly it’s true even for generic Gromov, FLM, Sizov 16
SU(3)
Eigenstates for SU(3) is a 3 x 3 matrix The operator which should provide separated variables is [Sklyanin 92] Again cannot be diagonalized Replacing we get e.g. It generates the eigenstates ! Gromov, FLM, Sizov 16 Conjecture supported by many tests
Eigenstates for SU(3) Exactly like in SU(2) are the momentum-carrying Bethe roots fixed by usual nested Bethe equations Separated variables are found from Factorization of wavefunction follows at once
Comparison with known constructions Usual nested algebraic Bethe Ansatz gives Sutherland, Kulish, Reshetikhin 83 wavefunction of auxiliary SU(2) chain Our conjecture • Only a single operator • No recursion • Complexity in # of roots is linear, not exponential
Comparison with known constructions Large literature on the construction of eigenstates • representation as sum over partitions of roots Belliard, Pakouliak, Ragoucy, Slavnov 12, 14 Tarasov, Varchenko 94 • trace formulas Mukhin, Tarasov, Varchenko 06 Mukhin, Tarasov, Varchenko 07 … • Drinfeld current construction Khoroshkin, Pakouliak 06 Khoroshkin, Pakouliak, Tarasov 06 Frappat, Khoroshkin, Pakouliak, Ragoucy 08 Belliard, Pakouliak, Ragoucy 10 • … Pakouliak, Ragoucy, Slavnov 14 … 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) Scott 94 How to quantize this expression? Gekhtman 95 SU(3) result: quantum minors SU(4): make an ansatz, all coefficients fixed to 0 or 1 !
Results for SU(N) We propose for any SU(N) Gromov, FLM, Sizov 16 • Matches classical limit Highly nontrivial checks ! • Commutativity We conjecture that generates the states and separated variables
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