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SL ( 2 , C ) spin magnet Yang-Baxter equation R -operators, Q -operators and Baxter equation Construction of eigenfunctions Iterative construction of eigenfunctions of the matrix elements of the monodromy matrix S. Derkachov PDMI, St.Petersburg RAQIS’16 Recent Advances in Quantum Integrable Systems 22-26 August 2016

SL ( 2 , C ) spin magnet Yang-Baxter equation R -operators, Q -operators and Baxter equation Construction of eigenfunctions Plan • SL ( 2 , C ) spin magnet • principal series representations of SL ( 2 , C ) I. M. Gelfand, M. I. Graev, N. Ya. Vilenkin (1966) Generalized functions. Vol. 5: Integral geometry and representation theory • monodromy matrix • operators A ( u ) and B ( u ) • Yang-Baxter equation • Construction of eigenfunctions • R-matrices and Q-operators • Iterative construction • Eigenfunctions and Q-operators based on S. Derkachov, G. Korchemsky, A. Manashov Nucl.Phys. B617 (2001) 375-440 Noncompact Heisenberg spin magnets from high-energy QCD: 1. Baxter Q operator and separation of variables S. Derkachov, A.Manashov J.Phys. A47 (2014) 305204 Iterative construction of eigenfunctions of the monodromy matrix for SL(2,C) magnet

SL ( 2 , C ) spin magnet Yang-Baxter equation R -operators, Q -operators and Baxter equation Construction of eigenfunctions Motivation • SL ( 2 , C ) spin magnet • principal series representations of SL ( 2 , C ) I. M. Gelfand, M. A. Naimark, Unitary representations of the classical groups, Trudy Mat. Inst. Steklov., vol. 36, Izdat. Nauk SSSR, Moscow - Leningrad, 1950; German transl.: Academie - Verlag, Berlin, 1957. "In some sense infinite-dimensional representations in many respects are more simple in comparison with finite-dimensional representations." • the model describes high-energy behaviour in Yang-Mills theory L. N. Lipatov, High-energy asymptotics of multicolor QCD and two-dimensional conformal field theories, Phys. Lett. B 309 (1993) 394. High-energy asymptotics of multicolor QCD and exactly solvable lattice models, hep-th/9311037. L. D. Faddeev and G. P. Korchemsky, High-energy QCD as a completely integrable model, Phys. Lett. B 342 (1995) 311. • diagonalization of B ( u ) ↔ Sklyanin SOV representation E. K. Sklyanin, Quantum Inverse Scattering Method.Selected Topics , in Quantum Groups and Quantum Integrable Systems , (Nankai lectures), ed. Mo-Lin Ge, pp. 63-97, World Scientific Publ., Singapore 1992, [hep-th/9211111] Separation of variables - new trends, Prog.Theor.Phys.Suppl. 118 (1995) 35-60, [solv-int/9504001] • diagonalization of A ( u ) L. N. Lipatov, Integrability of scattering amplitudes in N=4 SUSY, J. Phys. A 42 (2009) 304020.

SL ( 2 , C ) spin magnet Yang-Baxter equation R -operators, Q -operators and Baxter equation Construction of eigenfunctions SL ( 2 , C ) spin magnet The quantum SL ( 2 , C ) spin magnet is a straightforward generalization of the standard XXX s spin chain. XXX s spin chain The Hilbert space of the XXX s model is given by the tensor product of the ( 2 s + 1 ) -dimensional representations of the SU ( 2 ) group V k = C 2 s + 1 , H N = V 1 ⊗ V 2 ⊗ · · · ⊗ V N , k = 1 , . . . , N . To each site k we associate the quantum L -operators with subscript k acting nontrivially on the k − th space in the tensor product � � iS ( k ) u + iS ( k ) − ] = 2 S ( k ) , [ S ( k ) , S ( k ) ; [ S ( k ) + , S ( k ) ± ] = ± S ( k ) − L k ( u ) = iS ( k ) ± u − iS ( k ) + The monodromy matrix is defined as a product of L − operators � � A N ( u ) B N ( u ) T ( u ) = L 1 ( u ) L 2 ( u ) . . . L N ( u ) = C N ( u ) D N ( u ) SL ( 2 , C ) spin magnet ( 2 s + 1 ) -dimensional representations of the SU ( 2 ) group − → unitary principal series representations of the SL ( 2 , C ) group V k = C 2 s + 1 → V k = L 2 ( C )

SL ( 2 , C ) spin magnet Yang-Baxter equation R -operators, Q -operators and Baxter equation Construction of eigenfunctions unitary principal series representations of SL ( 2 , C ) SL ( 2 , C ) spin magnet � � a b g → T ( s , ¯ s ) ( g ) : L 2 ( C ) → L 2 ( C ) g = ∈ SL ( 2 , C ) ; c d � � − c + az d − bz , − ¯ c + ¯ a ¯ s φ z � s ) ( g ) φ � z ) = ( d − bz ) − 2 s � ¯ z � − 2 ¯ T ( s , ¯ d − ¯ ( z , ¯ b ¯ ¯ d − ¯ b ¯ z � d 2 z φ ( z , ¯ � T ( s , ¯ s ) ( g ) φ | T ( s , ¯ s ) ( g ) ψ � = � φ | ψ � � φ | ψ � = z ) ψ ( z , ¯ z ) ; The spins s and ¯ s are parameterized as follows ( n s ∈ Z , ν s ∈ R ) s = 1 + n s s = 1 − n s + i ν s ; ¯ + i ν s 2 2 generators: S + = z 2 ∂ z + 2 s z S − = − ∂ z , S = z ∂ z + s , ¯ ¯ ¯ z 2 ∂ ¯ S − = − ∂ ¯ , S = ¯ z ∂ ¯ z + ¯ s , S + = ¯ z + 2 ¯ s ¯ z z commutation relations: [¯ S + , ¯ S − ] = 2 ¯ [¯ S , ¯ S ± ] = ± ¯ [ S + , S − ] = 2 S , [ S , S ± ] = ± S ± , S , S ± conjugation: S † = − ¯ S † ± = − ¯ S ± , S

SL ( 2 , C ) spin magnet Yang-Baxter equation R -operators, Q -operators and Baxter equation Construction of eigenfunctions Intertwining operator There exists an operator W which intertwines a pair of principal series s ) and T ( 1 − s , 1 − ¯ s ) at generic complex s and ¯ representations T ( s , ¯ s s ) T ( s , ¯ s ) ( g ) = T ( 1 − s , 1 − ¯ s ) ( g ) W ( s , ¯ W ( s , ¯ s ) Integral operator � Φ( x , ¯ x ) d 2 x [ W ( s , ¯ s )Φ ] ( z , ¯ z ) = const ( z − x ) 2 − 2 s (¯ z − ¯ x ) 2 − 2 ¯ s C • The operator W is well-defined at generic s , ¯ s and the problems emerge for the discrete set of points 2 s = − n , 2 ¯ s = − ¯ n at n , ¯ n ∈ Z ≥ 0 • At these special values of spins an ( n + 1 )(¯ n + 1 ) -dimensional representation decouples from the general infinite-dimensional case � � � � n Φ − c + az d − bz , − ¯ c + ¯ a ¯ z z ) = ( d − bz ) n � ¯ z � ¯ T ( − n 2 , − ¯ n 2 ) ( g ) Φ d − ¯ ( z , ¯ b ¯ d − ¯ ¯ b ¯ z ¯ k , n + 1 ) basis vectors z k ¯ The space of polynomials spanned by ( n + 1 )(¯ z where k = 0 , 1 , · · · , n and ¯ k = 0 , 1 , · · · , ¯ n is invariant subspace.

SL ( 2 , C ) spin magnet Yang-Baxter equation R -operators, Q -operators and Baxter equation Construction of eigenfunctions Normalized intertwining operator [ z ] α ≡ z α ¯ α – single valued function in the complex plane provided that z ¯ α − ¯ α ∈ Z Fourier transformation I � d 2 z e i ( pz +¯ p ¯ z ) 1 α a ( α ) = π i α − ¯ [ z ] α [ p ] 1 − α α ) = Γ( 1 − ¯ α ) a ( α ) ≡ a ( α, ¯ , a ( α, β, γ, . . . ) = a ( α ) a ( β ) a ( γ ) . . . Γ( α ) Fourier transformation II � e ipx + i ¯ p ¯ x 1 d 2 x α = p α ¯ p ¯ α π i α − ¯ α a ( 1 + α ) x 1 + α ¯ x 1 + ¯ p → i ∂ z , ¯ p → i ∂ ¯ z � 1 [ i ∂ z ] α Φ( z , ¯ d 2 x [ z − x ] − 1 − α Φ( x , ¯ z ) = x ) π i α − ¯ α a ( 1 + α ) normalized intertwining operator � i 2 s − 2 ¯ s d 2 x [ z − x ] 2 s − 2 Φ( x , ¯ [ W ( s , ¯ s )Φ ] ( z , ¯ z ) = x ) π a ( 2 − 2 s ) W ( s ) = [ i ∂ z ] 1 − 2 s

SL ( 2 , C ) spin magnet Yang-Baxter equation R -operators, Q -operators and Baxter equation Construction of eigenfunctions SL ( 2 , C ) spin magnet The Hilbert space of the SL ( 2 , C ) spin magnet is given by the tensor product of the unitary principal series representations of the SL ( 2 , C ) group H N = V 1 ⊗ V 2 ⊗ · · · ⊗ V N , V k = L 2 ( C ) , k = 1 , . . . , N . The space H N is the space of functions Ψ( z 1 , ¯ z 1 . . . , z N , ¯ z N ) . To each site k we associate the pair of quantum L -operators with subscript k acting nontrivially on the k − th space in the tensor product � � � � iS ( k ) u + i ¯ i ¯ S ( k ) u + iS ( k ) S ( k ) ¯ ¯ − − L k ( u ) = , L k (¯ u ) = , iS ( k ) i ¯ S ( k ) u − i ¯ u − iS ( k ) S ( k ) ¯ + + where ¯ u is complex conjugate to u . The monodromy matrices T N ( u ) and ¯ T N (¯ u ) are defined as a product of L operators ¯ u ) = ¯ u )¯ u ) . . . ¯ T ( u ) = L 1 ( u ) L 2 ( u ) . . . L N ( u ) , T (¯ L 1 (¯ L 2 (¯ L N (¯ u ) � ¯ � � � ¯ A N ( u ) B N ( u ) A N (¯ u ) B N (¯ u ) ¯ T ( u ) = , T (¯ u ) = ¯ ¯ C N ( u ) D N ( u ) C N (¯ u ) D N (¯ u ) The operators A N ( u ) , B N ( u ) are differential operators of N − th order in the variables z 1 , . . . , z N and ¯ u ) , ¯ A N (¯ B N (¯ u ) are differential operators of N − th order in the variables ¯ z 1 , . . . , ¯ z N .

SL ( 2 , C ) spin magnet Yang-Baxter equation R -operators, Q -operators and Baxter equation Construction of eigenfunctions Eigenfunctions of A N ( u ) and ¯ A N (¯ u ) A N ( u ) , ¯ A N (¯ u ) are polynomials of degree N in u and ¯ u correspondingly N N � � A N ( u ) = u N + iu N − 1 S + u N − k a k , S ( k ) S = k = 2 k = 1 N N � � u N + i ¯ u N − 1 ¯ ¯ ¯ ¯ u N − k ¯ S ( k ) A N (¯ u ) = ¯ S + ¯ a k , S = k = 2 k = 1 by construction [ A N ( u ) , ¯ v )] = 0 → [ S , ¯ A N (¯ S ] = [ a k , ¯ a j ] = 0 RTT → [ A N ( u ) , A N ( v )] = 0 , [¯ u ) , ¯ A N (¯ A N (¯ v )] = 0 → [ a k , a j ] = [¯ a k , ¯ a j ] = 0 u ) = ( A N ( u )) † → a † conjugation rules for generators → ¯ A N (¯ k = ¯ a k A N ( u ) , ¯ A N (¯ u ) generate the set of commuting self-adjoint operators � � i ( S + ¯ S ) , S − ¯ S , a k + ¯ a k , i ( a k − ¯ a k ) , k = 2 , . . . N and can be diagonalized simultaneously