Quantum Integrable Systems Via Quantum K-theory - LSU Math
Anton ZeitlinOutlineQuantum IntegrabilityQuantum Integrable Systems via Quantum K-theoryNekrasov-ShatashviliideasQuantum K-theoryAnton M. ZeitlinLouisiana State University, Department of MathematicsUniversity of PennsylvaniaPhiladelphiaMarch, 2019Further Directions
IntroductionWe will talk about the relationship between two seemingly independentareas of mathematics:Anton ZeitlinOutlineQuantum IntegrabilityNekrasov-ShatashviliideasIQuantum Integrable SystemsExactly solvable models of statistical physics: spin chains, vertexmodels1930s: Hans Bethe: Bethe ansatz solution of Heisenberg model1960-70s: R.J. Baxter, C.N. Young: Yang-Baxter equation,Baxter operator1980s: Development of ”QISM” by Leningrad school leading to thediscovery of quantum groups by Drinfeld and JimboSince 1990s: textbook subject and an established area ofmathematics and physics.IEnumerative geometry: quantum K-theoryGeneralization of quantum cohomology in the early 2000s by A.Givental, Y.P. Lee and collaborators. Recently big progress in thisdirection by A. Okounkov and his school.Quantum K-theoryFurther Directions
IntroductionWe will talk about the relationship between two seemingly independentareas of mathematics:Anton ZeitlinOutlineQuantum IntegrabilityNekrasov-ShatashviliideasIQuantum Integrable SystemsExactly solvable models of statistical physics: spin chains, vertexmodels1930s: Hans Bethe: Bethe ansatz solution of Heisenberg model1960-70s: R.J. Baxter, C.N. Young: Yang-Baxter equation,Baxter operator1980s: Development of ”QISM” by Leningrad school leading to thediscovery of quantum groups by Drinfeld and JimboSince 1990s: textbook subject and an established area ofmathematics and physics.IEnumerative geometry: quantum K-theoryGeneralization of quantum cohomology in the early 2000s by A.Givental, Y.P. Lee and collaborators. Recently big progress in thisdirection by A. Okounkov and his school.Quantum K-theoryFurther Directions
IntroductionWe will talk about the relationship between two seemingly independentareas of mathematics:Anton ZeitlinOutlineQuantum IntegrabilityNekrasov-ShatashviliideasIQuantum Integrable SystemsExactly solvable models of statistical physics: spin chains, vertexmodels1930s: Hans Bethe: Bethe ansatz solution of Heisenberg model1960-70s: R.J. Baxter, C.N. Young: Yang-Baxter equation,Baxter operator1980s: Development of ”QISM” by Leningrad school leading to thediscovery of quantum groups by Drinfeld and JimboSince 1990s: textbook subject and an established area ofmathematics and physics.IEnumerative geometry: quantum K-theoryGeneralization of quantum cohomology in the early 2000s by A.Givental, Y.P. Lee and collaborators. Recently big progress in thisdirection by A. Okounkov and his school.Quantum K-theoryFurther Directions
Anton ZeitlinPath to this relationship:IFirst hints: work of Nekrasov and Shatashvili on 3-dimensionalgauge theories, now known as Gauge-Bethe correspondence:OutlineQuantum IntegrabilityNekrasov-ShatashviliideasQuantum K-theoryFurther DirectionsN. Nekrasov, S. Shatashvili, Supersymmetric vacua and Betheansatz, arXiv:0901.4744N. Nekrasov, S. Shatashvili, Quantum integrability andsupersymmetric vacua, arXiv:0901.4748ISubsequent work in geometric representation theory:A. Braverman, D. Maulik, A. Okounkov, Quantum cohomology ofthe Springer resolution, Adv. Math. 227 (2011) 421-458D. Maulik, A. Okounkov, Quantum Groups and QuantumCohomology, arXiv:1211.1287A. Okounkov, Lectures on K-theoretic computations inenumerative geometry, arXiv: arXiv:1512.07363
Anton ZeitlinPath to this relationship:IFirst hints: work of Nekrasov and Shatashvili on 3-dimensionalgauge theories, now known as Gauge-Bethe correspondence:OutlineQuantum IntegrabilityNekrasov-ShatashviliideasQuantum K-theoryFurther DirectionsN. Nekrasov, S. Shatashvili, Supersymmetric vacua and Betheansatz, arXiv:0901.4744N. Nekrasov, S. Shatashvili, Quantum integrability andsupersymmetric vacua, arXiv:0901.4748ISubsequent work in geometric representation theory:A. Braverman, D. Maulik, A. Okounkov, Quantum cohomology ofthe Springer resolution, Adv. Math. 227 (2011) 421-458D. Maulik, A. Okounkov, Quantum Groups and QuantumCohomology, arXiv:1211.1287A. Okounkov, Lectures on K-theoretic computations inenumerative geometry, arXiv: arXiv:1512.07363
Anton ZeitlinPath to this relationship:IFirst hints: work of Nekrasov and Shatashvili on 3-dimensionalgauge theories, now known as Gauge-Bethe correspondence:OutlineQuantum IntegrabilityNekrasov-ShatashviliideasQuantum K-theoryFurther DirectionsN. Nekrasov, S. Shatashvili, Supersymmetric vacua and Betheansatz, arXiv:0901.4744N. Nekrasov, S. Shatashvili, Quantum integrability andsupersymmetric vacua, arXiv:0901.4748ISubsequent work in geometric representation theory:A. Braverman, D. Maulik, A. Okounkov, Quantum cohomology ofthe Springer resolution, Adv. Math. 227 (2011) 421-458D. Maulik, A. Okounkov, Quantum Groups and QuantumCohomology, arXiv:1211.1287A. Okounkov, Lectures on K-theoretic computations inenumerative geometry, arXiv: arXiv:1512.07363
Anton ZeitlinOutlineQuantum IntegrabilityIn fact, this is a part of an ambitious program:Understanding (enumerative) geometry of symplectic resolutions:”Lie algebras of XXI century” (A. Okounkov’ 2012)Important examples: Springer resolution, Hilbert scheme of points inthe plane, Hypertoric varieties,.A large class of symplectic resolutions is provided by Nakajima quivervarieties (simplest subclass: T Gr (k, n))In this talk our main example will be T Gr (k, n) and more generally,cotangent bundles to (partial) flag varieties.Nekrasov-ShatashviliideasQuantum K-theoryFurther Directions
Anton ZeitlinOutlineQuantum IntegrabilityIn fact, this is a part of an ambitious program:Understanding (enumerative) geometry of symplectic resolutions:”Lie algebras of XXI century” (A. Okounkov’ 2012)Important examples: Springer resolution, Hilbert scheme of points inthe plane, Hypertoric varieties,.A large class of symplectic resolutions is provided by Nakajima quivervarieties (simplest subclass: T Gr (k, n))In this talk our main example will be T Gr (k, n) and more generally,cotangent bundles to (partial) flag varieties.Nekrasov-ShatashviliideasQuantum K-theoryFurther Directions
Anton ZeitlinOutlineQuantum IntegrabilityIn fact, this is a part of an ambitious program:Understanding (enumerative) geometry of symplectic resolutions:”Lie algebras of XXI century” (A. Okounkov’ 2012)Important examples: Springer resolution, Hilbert scheme of points inthe plane, Hypertoric varieties,.A large class of symplectic resolutions is provided by Nakajima quivervarieties (simplest subclass: T Gr (k, n))In this talk our main example will be T Gr (k, n) and more generally,cotangent bundles to (partial) flag varieties.Nekrasov-ShatashviliideasQuantum K-theoryFurther Directions
Anton ZeitlinOutlineQuantum IntegrabilityIn fact, this is a part of an ambitious program:Understanding (enumerative) geometry of symplectic resolutions:”Lie algebras of XXI century” (A. Okounkov’ 2012)Important examples: Springer resolution, Hilbert scheme of points inthe plane, Hypertoric varieties,.A large class of symplectic resolutions is provided by Nakajima quivervarieties (simplest subclass: T Gr (k, n))In this talk our main example will be T Gr (k, n) and more generally,cotangent bundles to (partial) flag varieties.Nekrasov-ShatashviliideasQuantum K-theoryFurther Directions
Anton ZeitlinOutlineQuantum IntegrabilityIn fact, this is a part of an ambitious program:Understanding (enumerative) geometry of symplectic resolutions:”Lie algebras of XXI century” (A. Okounkov’ 2012)Important examples: Springer resolution, Hilbert scheme of points inthe plane, Hypertoric varieties,.A large class of symplectic resolutions is provided by Nakajima quivervarieties (simplest subclass: T Gr (k, n))In this talk our main example will be T Gr (k, n) and more generally,cotangent bundles to (partial) flag varieties.Nekrasov-ShatashviliideasQuantum K-theoryFurther Directions
Anton ZeitlinOutlineBased on:IPetr P. Pushkar, Andrey Smirnov, A.Z., Baxter Q-operator fromquantum K-theory, arXiv:1612.08723IPeter Koroteev, Petr P. Pushkar, Andrey Smirnov, A.Z., QuantumK-theory of Quiver Varieties and Many-Body Systems,arXiv:1705.10419IPeter Koroteev, Anton M. Zeitlin, Difference Equations forK-theoretic Vertex Functions of Type-A Nakajima VarietiesarXiv:1802.04463and to some extent onIPeter Koroteev, Daniel S. Sage, Anton M. Zeitlin, (SL(N),q)-opers,the q-Langlands correspondence, and quantum/classical dualityarXiv:1811.09937Quantum IntegrabilityNekrasov-ShatashviliideasQuantum K-theoryFurther Directions
OutlineAnton ZeitlinOutlineQuantum IntegrabilityNekrasov-ShatashviliideasQuantum groups and quantum integrabilityQuantum K-theoryFurther DirectionsNekrasov-Shatashvili ideasQuantum K-theory and integrabilityBack to Givental’s ideas further directions
Loop algebras and evaluation modulesAnton ZeitlinOutlineQuantum IntegrabilityLet us consider Lie algebra g.Nekrasov-ShatashviliideasQuantum K-theoryThe associated loop algebra is ĝ g[t, t 1 ] and t is known as spectralparameter.The following representations, known as evaluation modules form atensor category of ĝ:V1 (a1 ) V2 (a2 ) · · · Vn (an ),whereIVi are representations of gIai are values for tFurther Directions
Loop algebras and evaluation modulesAnton ZeitlinOutlineQuantum IntegrabilityLet us consider Lie algebra g.Nekrasov-ShatashviliideasQuantum K-theoryThe associated loop algebra is ĝ g[t, t 1 ] and t is known as spectralparameter.The following representations, known as evaluation modules form atensor category of ĝ:V1 (a1 ) V2 (a2 ) · · · Vn (an ),whereIVi are representations of gIai are values for tFurther Directions
Loop algebras and evaluation modulesAnton ZeitlinOutlineQuantum IntegrabilityLet us consider Lie algebra g.Nekrasov-ShatashviliideasQuantum K-theoryThe associated loop algebra is ĝ g[t, t 1 ] and t is known as spectralparameter.The following representations, known as evaluation modules form atensor category of ĝ:V1 (a1 ) V2 (a2 ) · · · Vn (an ),whereIVi are representations of gIai are values for tFurther Directions
Quantum groupsAnton ZeitlinOutlineQuantum groupQuantum IntegrabilityU (ĝ)is a deformation of U(ĝ), with a nontrivial intertwiner R V1 ,V2 (a1 /a2 ):V1 (a1 ) V2 (a2 )V2 (a2 ) V1 (a1 )which is a rational function of a1 , a2 , satisfying Yang-Baxter equation:The generators of U (ĝ) emerge as matrix elements of R-matrices (theso-called FRT construction).Nekrasov-ShatashviliideasQuantum K-theoryFurther Directions
Integrability and Baxter algebraAnton ZeitlinOutlineSource of integrability: commuting transfer matrices, generating Baxteralgebra which are weighted traces ofR̃ W (u),Hphys : W (u) Hphys W (u) HphysQuantum IntegrabilityNekrasov-ShatashviliideasQuantum K-theoryFurther Directions
Baxter algebra and IntegrabilityAnton ZeitlinOutlineQuantum IntegrabilitySource of integrability: commuting transfer matrices, generating Baxteralgebra which are weighted traces ofR̃ W (u),Hphys : W (u) Hphys W (u) Hphysover auxiliary W (u) space: T W (u) TrW (u) (Z 1) R̃ W (u),HphysHere Z e h , where h g are diagonal matrices.Nekrasov-ShatashviliideasQuantum K-theoryFurther Directions
Anton ZeitlinOutlineQuantum y:Quantum K-theoryFurther Directions0[T W 0 (u ), T W (u)] 0There are special transfer matrices is called Baxter Q-operators. Suchoperators generate all Baxter algebra.Primary goal for physicists is to diagonalize {T W (u)} simultaneously.
Anton ZeitlinOutlineQuantum y:Quantum K-theoryFurther Directions0[T W 0 (u ), T W (u)] 0There are special transfer matrices is called Baxter Q-operators. Suchoperators generate all Baxter algebra.Primary goal for physicists is to diagonalize {T W (u)} simultaneously.
Anton ZeitlinOutlineQuantum y:Quantum K-theoryFurther Directions0[T W 0 (u ), T W (u)] 0There are special transfer matrices is called Baxter Q-operators. Suchoperators generate all Baxter algebra.Primary goal for physicists is to diagonalize {T W (u)} simultaneously.
g sl(2): XXZ spin chainAnton ZeitlinOutlineQuantum IntegrabilityTextbook example (and main example in this talk) is XXZ Heisenbergspin chain:Nekrasov-ShatashviliideasQuantum K-theoryFurther DirectionsHXXZ C2 (a1 ) C2 (a2 ) · · · C2 (an )States: b 2 ).Here C2 stands for 2-dimensional representation of U (slAlgebraic method to diagonalize transfer matrices:Algebraic Bethe ansatzas a part of Quantum Inverse Scattering Method developed in the1980s.
g sl(2): XXZ spin chainAnton ZeitlinOutlineQuantum IntegrabilityTextbook example (and main example in this talk) is XXZ Heisenbergspin chain:Nekrasov-ShatashviliideasQuantum K-theoryFurther DirectionsHXXZ C2 (a1 ) C2 (a2 ) · · · C2 (an )States: b 2 ).Here C2 stands for 2-dimensional representation of U (slAlgebraic method to diagonalize transfer matrices:Algebraic Bethe ansatzas a part of Quantum Inverse Scattering Method developed in the1980s.
g sl(2): XXZ spin chainAnton ZeitlinOutlineQuantum IntegrabilityTextbook example (and main example in this talk) is XXZ Heisenbergspin chain:Nekrasov-ShatashviliideasQuantum K-theoryFurther DirectionsHXXZ C2 (a1 ) C2 (a2 ) · · · C2 (an )States: b 2 ).Here C2 stands for 2-dimensional representation of U (slAlgebraic method to diagonalize transfer matrices:Algebraic Bethe ansatzas a part of Quantum Inverse Scattering Method developed in the1980s.
Bethe equations and Q-operatorAnton ZeitlinOutlineQuantum IntegrabilityThe eigenvalues are generated by symmetric functions of Bethe roots{xi }:n x ak x xQQijij z n/2, i 1 · · · k,j 1 xi xj j 1 aj xij6 iso that the eigenvalues Λ(u) of the Q-operator are the generatingfunctions for the elementary symmetric functions of Bethe roots:Λ(u) kY(1 u · xi )i 1A real challenge is to describe representation-theoretic meaning ofQ-operator for general g (possibly ntum K-theoryFurther Directions
Bethe equations and Q-operatorAnton ZeitlinOutlineQuantum IntegrabilityThe eigenvalues are generated by symmetric functions of Bethe roots{xi }:n x ak x xQQijij z n/2, i 1 · · · k,j 1 xi xj j 1 aj xij6 iso that the eigenvalues Λ(u) of the Q-operator are the generatingfunctions for the elementary symmetric functions of Bethe roots:Λ(u) kY(1 u · xi )i 1A real challenge is to describe representation-theoretic meaning ofQ-operator for general g (possibly ntum K-theoryFurther Directions
Bethe equations and Q-operatorAnton ZeitlinOutlineQuantum IntegrabilityThe eigenvalues are generated by symmetric functions of Bethe roots{xi }:n x ak x xQQijij z n/2, i 1 · · · k,j 1 xi xj j 1 aj xij6 iso that the eigenvalues Λ(u) of the Q-operator are the generatingfunctions for the elementary symmetric functions of Bethe roots:Λ(u) kY(1 u · xi )i 1A real challenge is to describe representation-theoretic meaning ofQ-operator for general g (possibly ntum K-theoryFurther Directions
q-difference equationModern way of looking at Bethe ansatz: solving q-difference equationsforΨ(z1 , . . . , zk ; a1 , . . . , an ) V1 (a1 ) · · · Vn (an )[[z1 , . . . , zk ]]Anton ZeitlinOutlineQuantum IntegrabilityNekrasov-ShatashviliideasQuantum K-theoryFurther Directionsknown asQuantum Knizhnik-Zamolodchikov (aka Frenkel-Reshetikhin) equations:Ψ(qa1 , . . . , an , {zi }) (Z 1 · · · 1)R V1 ,Vn . . . R V1 ,V2 Ψ commuting difference equations in z variablesHere {zi } are the components of twist variable Z .The latter series of equations are known as dynamical equations,studied by Etingof, Felder, Tarasov, Varchenko, . . .In q 1 limit we arrive to an eigenvalue problem. Studying theasymptotics of the corresponding solutions we arrive to Bethe equationsand eigenvectors.
q-difference equationModern way of looking at Bethe ansatz: solving q-difference equationsforΨ(z1 , . . . , zk ; a1 , . . . , an ) V1 (a1 ) · · · Vn (an )[[z1 , . . . , zk ]]Anton ZeitlinOutlineQuantum IntegrabilityNekrasov-ShatashviliideasQuantum K-theoryFurther Directionsknown asQuantum Knizhnik-Zamolodchikov (aka Frenkel-Reshetikhin) equations:Ψ(qa1 , . . . , an , {zi }) (Z 1 · · · 1)R V1 ,Vn . . . R V1 ,V2 Ψ commuting difference equations in z variablesHere {zi } are the components of twist variable Z .The latter series of equations are known as dynamical equations,studied by Etingof, Felder, Tarasov, Varchenko, . . .In q 1 limit we arrive to an eigenvalue problem. Studying theasymptotics of the corresponding solutions we arrive to Bethe equationsand eigenvectors.
q-difference equationModern way of looking at Bethe ansatz: solving q-difference equationsforΨ(z1 , . . . , zk ; a1 , . . . , an ) V1 (a1 ) · · · Vn (an )[[z1 , . . . , zk ]]Anton ZeitlinOutlineQuantum IntegrabilityNekrasov-ShatashviliideasQuantum K-theoryFurther Directionsknown asQuantum Knizhnik-Zamolodchikov (aka Frenkel-Reshetikhin) equations:Ψ(qa1 , . . . , an , {zi }) (Z 1 · · · 1)R V1 ,Vn . . . R V1 ,V2 Ψ commuting difference equations in z variablesHere {zi } are the components of twist variable Z .The latter series of equations are known as dynamical equations,studied by Etingof, Felder, Tarasov, Varchenko, . . .In q 1 limit we arrive to an eigenvalue problem. Studying theasymptotics of the corresponding solutions we arrive to Bethe equationsand eigenvectors.
q-difference equationModern way of looking at Bethe ansatz: solving q-difference equationsforΨ(z1 , . . . , zk ; a1 , . . . , an ) V1 (a1 ) · · · Vn (an )[[z1 , . . . , zk ]]Anton ZeitlinOutlineQuantum IntegrabilityNekrasov-ShatashviliideasQuantum K-theoryFurther Directionsknown asQuantum Knizhnik-Zamolodchikov (aka Frenkel-Reshetikhin) equations:Ψ(qa1 , . . . , an , {zi }) (Z 1 · · · 1)R V1 ,Vn . . . R V1 ,V2 Ψ commuting difference equations in z variablesHere {zi } are the components of twist variable Z .The latter series of equations are known as dynamical equations,studied by Etingof, Felder, Tarasov, Varchenko, . . .In q 1 limit we arrive to an eigenvalue problem. Studying theasymptotics of the corresponding solutions we arrive to Bethe equationsand eigenvectors.
Nekrasov-Shatashvili ideasAnton ZeitlinOutlineIn 2009 Nekrasov and Shatashvili looked at 3d SUSY gauge theories onC S 1:Quantum IntegrabilityNekrasov-ShatashviliideasQuantum K-theoryFurther Directionswith gauge groupG U(v1 ) U(v2 ) . . . U(vrankg ),and some ”matter fields” (sections of associated vector G -bundles), tobe specified below.The collection {vi } determines the weights of the correspondingsubspace in H.In the simplest case of g sl(2) we just have one U(v ) and , and # v
Nekrasov-Shatashvili ideasAnton ZeitlinOutlineIn 2009 Nekrasov and Shatashvili looked at 3d SUSY gauge theories onC S 1:Quantum IntegrabilityNekrasov-ShatashviliideasQuantum K-theoryFurther Directionswith gauge groupG U(v1 ) U(v2 ) . . . U(vrankg ),and some ”matter fields” (sections of associated vector G -bundles), tobe specified below.The collection {vi } determines the weights of the correspondingsubspace in H.In the simplest case of g sl(2) we just have one U(v ) and , and # v
Nekrasov-Shatashvili ideasAnton ZeitlinOutlineIn 2009 Nekrasov and Shatashvili looked at 3d SUSY gauge theories onC S 1:Quantum IntegrabilityNekrasov-ShatashviliideasQuantum K-theoryFurther Directionswith gauge groupG U(v1 ) U(v2 ) . . . U(vrankg ),and some ”matter fields” (sections of associated vector G -bundles), tobe specified below.The collection {vi } determines the weights of the correspondingsubspace in H.In the simplest case of g sl(2) we just have one U(v ) and , and # v
Nekrasov-Shatashvili ideasAnton ZeitlinOutlineIn 2009 Nekrasov and Shatashvili looked at 3d SUSY gauge theories onC S 1:Quantum IntegrabilityNekrasov-ShatashviliideasQuantum K-theoryFurther Directionswith gauge groupG U(v1 ) U(v2 ) . . . U(vrankg ),and some ”matter fields” (sections of associated vector G -bundles), tobe specified below.The collection {vi } determines the weights of the correspondingsubspace in H.In the simplest case of g sl(2) we just have one U(v ) and , and # v
Full Gauge/Bethe correspondence dictionaryAnton ZeitlinOutlineQuantum IntegrabilityGauge group G : U(v1 ) U(v2 ) . . . U(vrankg )Nekrasov-ShatashviliideasThe set {vi } determines the weight (e.g. number of inverted spins)Quantum K-theoryFurther DirectionsMaximal torus: {xi1 , . . . , xivi } — these are Bethe roots variables.Matter Fields: affine space MI Standard matter fields: rankgi 1 Vi Wi , s.t. dim(Vi ) vi ; Wi is a framing (“flavor”) space, where C a1 Ca2 . . . act.
Full Gauge/Bethe correspondence dictionaryAnton ZeitlinOutlineQuantum IntegrabilityGauge group G : U(v1 ) U(v2 ) . . . U(vrankg )Nekrasov-ShatashviliideasThe set {vi } determines the weight (e.g. number of inverted spins)Quantum K-theoryFurther DirectionsMaximal torus: {xi1 , . . . , xivi } — these are Bethe roots variables.Matter Fields: affine space MI Standard matter fields: rankgi 1 Vi Wi , s.t. dim(Vi ) vi ; Wi is a framing (“flavor”) space, where C a1 Ca2 . . . act.
Full Gauge/Bethe correspondence dictionaryAnton ZeitlinOutlineQuantum IntegrabilityGauge group G : U(v1 ) U(v2 ) . . . U(vrankg )Nekrasov-ShatashviliideasThe set {vi } determines the weight (e.g. number of inverted spins)Quantum K-theoryFurther DirectionsMaximal torus: {xi1 , . . . , xivi } — these are Bethe roots variables.Matter Fields: affine space MI Standard matter fields: rankgi 1 Vi Wi , s.t. dim(Vi ) vi ; Wi is a framing (“flavor”) space, where C a1 Ca2 . . . act.
Full Gauge/Bethe correspondence dictionaryAnton ZeitlinOutlineQuantum IntegrabilityGauge group G : U(v1 ) U(v2 ) . . . U(vrankg )Nekrasov-ShatashviliideasThe set {vi } determines the weight (e.g. number of inverted spins)Quantum K-theoryFurther DirectionsMaximal torus: {xi1 , . . . , xivi } — these are Bethe roots variables.Matter Fields: affine space MI Standard matter fields: rankgi 1 Vi Wi , s.t. dim(Vi ) vi ; Wi is a framing (“flavor”) space, where C a1 Ca2 . . . act.
Full Gauge/Bethe correspondence dictionaryAnton ZeitlinOutlineQuantum IntegrabilityGauge group G : U(v1 ) U(v2 ) . . . U(vrankg )Nekrasov-ShatashviliideasThe set {vi } determines the weight (e.g. number of inverted spins)Quantum K-theoryFurther DirectionsMaximal torus: {xi1 , . . . , xivi } — these are Bethe roots variables.Matter Fields: affine space MI Standard matter fields: rankgi 1 Vi Wi , s.t. dim(Vi ) vi ; Wi is a framing (“flavor”) space, where C a1 Ca2 . . . act.
Full Gauge/Bethe correspondence dictionaryAnton ZeitlinOutlineGauge group G : U(v1 ) U(v2 ) . . . U(vrankg )The set {vi } determines the weight (i.e. number of inverted spins)Quantum IntegrabilityNekrasov-ShatashviliideasQuantum K-theoryMaximal torus: {xi1 , . . . , xivi } — these are Bethe roots variables.Matter Fields: affine space MI Standard matter fields: rankgi 1 Vi Wi , s.t. dim(Vi ) vi ; Wi is a framing (“flavor”) space, where C a1 Ca2 . . . act.I“Bifundamental” quiver data: i j Vi VjThe quiver serves as a “kind of” Dynkin diagram for g.To have enough supersymmetries duals : T M.Further Directions
Full Gauge/Bethe correspondence dictionaryAnton ZeitlinOutlineGauge group G : U(v1 ) U(v2 ) . . . U(vrankg )The set {vi } determines the weight (i.e. number of inverted spins)Quantum IntegrabilityNekrasov-ShatashviliideasQuantum K-theoryMaximal torus: {xi1 , . . . , xivi } — these are Bethe roots variables.Matter Fields: affine space MI Standard matter fields: rankgi 1 Vi Wi , s.t. dim(Vi ) vi ; Wi is a framing (“flavor”) space, where C a1 Ca2 . . . act.I“Bifundamental” quiver data: i j Vi VjThe quiver serves as a “kind of” Dynkin diagram for g.To have enough supersymmetries duals : T M.Further Directions
Anton ZeitlinOutlineModuli of Higgs vacua Nakajima quiver variety:T M////G µ 1 (0)//G Nwhere µ 0 is a momentum map (low energy configuration) condition.In the case of quiver with one vertex and one framing:N T Gr (v , w ).Quantum IntegrabilityNekrasov-ShatashviliideasQuantum K-theoryFurther Directions
Anton ZeitlinOutlineModuli of Higgs vacua Nakajima quiver variety:T M////G µ 1 (0)//G Nwhere µ 0 is a momentum map (low energy configuration) condition.In the case of quiver with one vertex and one framing:N T Gr (v , w ).Quantum IntegrabilityNekrasov-ShatashviliideasQuantum K-theoryFurther Directions
Anton ZeitlinModuli of Higgs vacua Nakajima quiver variety:OutlineT M////G µ 1 (0)//G Nwhere µ 0 is low energy configuration condition.Quantum IntegrabilityNekrasov-ShatashviliideasQuantum K-theoryIn the case of quiver with one vertex and one framing:N T Gr (v , w ).Hilbert space of vacua H Wilson line operators equivariant K-theory of Nakajima variety.Known to be a module for the action of a quantum group U (ĝ) due toNakajima.Further Directions
Anton ZeitlinModuli of Higgs vacua Nakajima quiver variety:OutlineT M////G µ 1 (0)//G Nwhere µ 0 is low energy configuration condition.Quantum IntegrabilityNekrasov-ShatashviliideasQuantum K-theoryIn the case of quiver with one vertex and one framing:N T Gr (v , w ).Hilbert space of vacua H Wilson line operators equivariant K-theory of Nakajima variety.Known to be a module for the action of a quantum group U (ĝ) due toNakajima.Further Directions
Anton ZeitlinPhysicists interested in computing SUSY indices:OutlineQuantum Integrability/2str(e β D A) trKerD/even (A) trKerD/odd (A) strindexD/ (A)Nekrasov-ShatashviliideasQuantum K-theoryMathematically those correspond to (very similar to GW curvecounting!) weighted K-theoretic counts of quasimaps:quasimap fC Nakajima variety NThe weight (Kähler) parameter is Z deg(f) , which is exactly twistparameter Z we encountered before.Further Directions
Anton ZeitlinPhysicists interested in computing SUSY indices:OutlineQuantum Integrability/2str(e β D A) trKerD/even (A) trKerD/odd (A) strindexD/ (A)Nekrasov-ShatashviliideasQuantum K-theoryMathematically those correspond to (very similar to GW curvecounting!) weighted K-theoretic counts of quasimaps:quasimap fC Nakajima variety NThe weight (Kähler) parameter is Z deg(f) , which is exactly twistparameter Z we encountered before.Further Directions
Anton ZeitlinPhysicists interested in computing SUSY indices:OutlineQuantum Integrability/2str(e β D A) trKerD/even (A) trKerD/odd (A) strindexD/ (A)Nekrasov-ShatashviliideasQuantum K-theoryMathematically those correspond to (very similar to GW curvecounting!) weighted K-theoretic counts of quasimaps:quasimap fC Nakajima variety NThe weight (Kähler) parameter is Z deg(f) , which is exactly twistparameter Z we encountered before.Further Directions
Anton ZeitlinPhysicists interested in computing SUSY indices:OutlineQuantum Integrability/2str(e β D A) trKerD/even (A) trKerD/odd (A) strindexD/ (A)Nekrasov-ShatashviliideasQuantum K-theoryMathematically those correspond to (very similar to GW curvecounting!) weighted K-theoretic counts of quasimaps:quasimap fC Nakajima variety NThe weight (Kähler) parameter is Z deg(f) , which is exactly twistparameter Z we encountered before.Further Directions
Physicists interested in computing SUSY indices:Anton ZeitlinOutline/2str(e β D A) trKerD/even (A) trKerD/odd (A) strindexD/ (A)Mathematically those correspond to (very similar to GW curvecounting!) weighted K-theoretic counts of quasimaps:quasimap fC Nakajima variety NThe weight (Kähler) parameter is Z deg(f) , which is exactly twistparameter Z we encountered before.One can think of quantum K-theory ring:Quantum IntegrabilityNekrasov-ShatashviliideasQuantum K-theoryFurther Directions
Key IdeasAnton ZeitlinOutlineQuantum IntegrabilityNekrasov-ShatashviliideasQuantum K-theoryNekrasov and Shatashvili:Quantum K theory ring of Nakajima variety symmetric polynomials in xij / Bethe equationsFurther Directions
Key IdeasAnton ZeitlinOutlineQuantum IntegrabilityNekrasov and Shatashvili:Quantum K theory ring of Nakajima variety Nekrasov-ShatashviliideasQuantum K-theoryFurther Directionssymmetric polynomials in xij / Bethe equationsInput by Okounkov:q difference equations qKZ equations dynamical equations
Anton ZeitlinOutlineQuantum IntegrabilityNekrasov-ShatashviliideasQuantum K-theoryIn the following we will talk about this in the simplest case:INakajima variety: N T Gr (k, n)IQuantum Integrable System: sl(2) XXZ spin chain.Further Directions
NotationAnton ZeitlinOutline T Gr (k, n) Nk,n ,tk Nk,n N(n).Quantum IntegrabilityNekrasov-ShatashviliideasQuantum K-theoryAs a Nakajima variety:Further Directions Nk,n T M////GL(V ) µ 1(0)s /GL(V ),whereT M Hom(V , W ) Hom(W , V )Tautological bundles:V T M V ////GL(V ),W T M W ////GL(V )For any τ KGL(V ) (·) Λ(x1 1 , x2 1 , . . . xk 1 ) we introduce atautological bundle:τ T M τ (V )////GL(V )
NotationAnton ZeitlinOutline T Gr (k, n) Nk,n ,tk Nk,n N(n).Quantum IntegrabilityNekrasov-ShatashviliideasQuantum K-theoryAs a
Quantum Integrability Nekrasov-Shatashvili ideas Quantum K-theory . Algebraic method to diagonalize transfer matrices: Algebraic Bethe ansatz as a part of Quantum Inverse Scattering Method developed in the 1980s. Anton Zeitlin Outline Quantum Integrability Nekrasov-Shatashvili ideas Quantum K-theory Further Directions
MATHEMATICS 458 Integrable Systems and Random Matrices In honor of Percy Deift Conference on Integrable Systems, Random Matrices, and Applications in Honor of Percy Deift' s 60th Birthday May 22-26,2006 Courant Institute of Mathematical Sciences New York University, New York Jinho Baik Thomas Kriecherbauer Luen-Chau Li Kenneth D. T-R Mclaughlin
5 Integrable systems and Arnold-Liouville theorem De nition 5.1. A Hamiltonian system ((M;!);H) is (completely) integrable if it pos- . bining the two methods, one can construct the action variables, which are functions of the . the details of the construction in V.I Arnold’s Mathematical
1) develop an inverse scattering method on the half-line based on Sklyanin's approach in spirit of solving initial-boundary value problems, can be applied to a wide range of integrable models in contrast to Fokas' method. 2) develop the notion of hierarchy of integrable boundary conditions. 3) having the nonlinear method of re
of Quantum Statistical Physics has been gradually discovered, these develop- ments coming to a culminating point in the works of R. Baxter [B]. On the other hand, and quite independently of that, there was in the late 60 s and in the 70 s a major breakthrough in the study of classical integrable systems which led to creation of the Classical Inverse Scattering Method.
1. Quantum bits In quantum computing, a qubit or quantum bit is the basic unit of quantum information—the quantum version of the classical binary bit physically realized with a two-state device. A qubit is a two-state (or two-level) quantum-mechanical system, one of the simplest quantum systems displaying the peculiarity of quantum mechanics.
Inverse scattering method! UD Cellular automata Box-ball systems Ultradiscrete integrable system KKR bijection 0 q Lattice statistical models Solvable vertex models Quantum integrable system Bethe ansatz Kerov-Kirillov-Reshetikhin (KKR) bijection (1986) asserts "formal completeness"
According to the quantum model, an electron can be given a name with the use of quantum numbers. Four types of quantum numbers are used in this; Principle quantum number, n Angular momentum quantum number, I Magnetic quantum number, m l Spin quantum number, m s The principle quantum
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