Algebraic geometric properties of spectral surfaces of quantum - PowerPoint PPT Presentation

Algebraic geometric properties of spectral surfaces of quantum integrable systems and their isospectral deformations Alexander Zheglov Moscow State University XXXVIII Workshop on Geometric Methods in Physics, 2019 Alexander Zheglov (Moscow) Algebraic geometric properties of spectral surfaces of quantum integrable systems and their XXXVIII 1 / 29

Quantum completely integrable systems The quantum completely integrable system (QCIS) (with n degrees of freedom) over a n -dimensional algebraic variety X is a pair (Λ , θ ) , where Λ is an irreducible n -dimensional affine algebraic variety; O Λ is the ring of regular functions on Λ D ( X ) is the ring of differential operators on a variety X (without loss of generality X may be taken to be a formal polydisc Spec ( k [[ x 1 , . . . , x n ]] ) θ : O Λ → D ( X ) is an embedding. Recall that for a commutative K -algebra R the filtered ring D ( R ) is generated by Der K ( R ) and R inside the ring End K ( R ) . D 0 ( R ) ⊂ D 1 ( R ) ⊂ D 2 ( R ) ⊂ . . . , D i ( R ) D j ( R ) ⊂ D i + j ( R ) , where D i ( R ) are defined inductively. In particular, the usual function ord is defined on D ( R ) . Alexander Zheglov (Moscow) Algebraic geometric properties of spectral surfaces of quantum integrable systems and their XXXVIII 2 / 29

From now on we introduce the following notation: ˆ R := K [[ x 1 , . . . , x n ]] , where char K = 0 — the ring of regular functions on a formal polydisc. D n := ˆ R [ ∂ 1 , . . . , ∂ n ] — the ring of differential operators on a formal polydisc. Then QCIS are just subrings of commuting operators in D n . We’ll study such subrings and their isospectral deformations. Alexander Zheglov (Moscow) Algebraic geometric properties of spectral surfaces of quantum integrable systems and their XXXVIII 3 / 29

Case n = 1 : commuting ordinary differential operators Definition An ordinary differential operator P = a n ∂ n + a n − 1 ∂ n − 1 + · · · + a 0 ∈ D 1 of positive order n is called (formally) elliptic if a n ∈ K ∗ . A ring B ⊂ D 1 containing an elliptic element is called elliptic . (reduction to the elliptic case) If P = a n ∂ n + a n − 1 ∂ n − 1 + · · · + a 0 ∈ D 1 , where a n (0) � = 0 , then there is a change of variables ϕ ∈ Aut ( D 1 ) such that Q := ϕ ( P ) = ∂ n + b n − 2 ∂ n − 2 + · · · + b 0 (1) for some b 0 , . . . , b n − 2 ∈ K [[ x ]] . Let B be a commutative subalgebra of D 1 containing an elliptic element P . Then all elements of B are elliptic. Alexander Zheglov (Moscow) Algebraic geometric properties of spectral surfaces of quantum integrable systems and their XXXVIII 4 / 29

Krichever-Mumford classification in n = 1 case. Theorem There is a one-to-one correspondence [ B ⊂ D 1 of rank r ] ← → [( C, p, F , z, φ ) of rank r ] / ≃ [ B ⊂ D 1 of rank 1] / ∼ ← → [( C, p, F ) of rank 1] / ≃ where [ B ] means a class of equivalent commutative elliptic subrings, where B ∼ B ′ iff B = f − 1 B ′ f , f ∈ D ∗ 1 . ∼ means ”up to linear changes of variables” ( C, p, F , z, φ ) means the algebraic-geometric spectral data of rank r Here the rank of B is rk( B ) := GCD { ord( P ) , P ∈ B } . Alexander Zheglov (Moscow) Algebraic geometric properties of spectral surfaces of quantum integrable systems and their XXXVIII 5 / 29

Spectral data Definition C is an integral projective curve over K ; p ∈ C is a closed regular K -point; F is a coherent torsion free sheaf of rank r on C with h 0 ( C, F ) = h 1 ( C, F ) = 0; z is a local coordinate at p ; F p ≃ ( K [[ z ]]) ⊕ r is a trivialisation (i.e. an ˆ φ : ˆ O p ≃ K [[ z ]] -module isomorphism). Alexander Zheglov (Moscow) Algebraic geometric properties of spectral surfaces of quantum integrable systems and their XXXVIII 6 / 29

Recall: Isospectral deformations of rank one commutative rings of ODOs determine the KP flows on the compactified Jacobian of the spectral curve C . If C is smooth, K = C and rk( F ) = 1 , then there are explicit formulae due to Krichever. If C is singular and rational, then there are explicit formulae due to Wilson. The n > 1 cases are much more complicated. To explain the corresponding results we need to introduce new notation. Alexander Zheglov (Moscow) Algebraic geometric properties of spectral surfaces of quantum integrable systems and their XXXVIII 7 / 29

The ring ˆ D sym and its order function n Consider the K –vector space:    a k ∂ k �  M := ˆ � � a k ∈ ˆ R for all k ∈ N n R [[ ∂ 1 , . . . , ∂ n ]] =  . � 0  k ≥ 0 Definition a k ∂ k ∈ M we define its order to be For any 0 � = P := � k ≥ 0 � � ord ( P ) := sup | k | − υ ( a k ) ∈ Z ∪ {∞} , a k ∈ m n , m = ( x 1 , . . . , x n ) } , and where υ ( a k ) := max { n | | k | = k 1 + . . . + k n . Alexander Zheglov (Moscow) Algebraic geometric properties of spectral surfaces of quantum integrable systems and their XXXVIII 8 / 29

Definition ˆ D sym � � ord ( Q ) < ∞ � � := Q ∈ M . n Properties of ˆ D sym : n D sym ˆ is a ring (with natural operations · , + coming from D n ); n D sym ˆ ⊃ D n . n R has a natural structure of a left ˆ ˆ D n -module, which extends its natural structure of a left D n -module. Operators from ˆ D sym can realize arbitrary endomorphisms of the n K -algebra ˆ R which are continuous in the m -adic topology: e.g. for n = 1 the operator ∞ u k � k ! ∂ k , exp( u ∗ ∂ ) := u ∈ xK [[ x ]] k =0 acts as � � exp( u ∗ ∂ ) ◦ f ( x ) = f u + x . Alexander Zheglov (Moscow) Algebraic geometric properties of spectral surfaces of quantum integrable systems and their XXXVIII 9 / 29

There are Dirac delta functions: for δ i := exp(( − x i ) ∗ ∂ i ) � � δ i ◦ f ( x 1 , . . . , x i − 1 , x i , x i +1 , . . . , x n ) = f x 1 , . . . , x i − 1 , 0 , x i +1 , . . . , x n ; Operators of integration: ∞ x k +1 � · ∂ − 1 � ( k + 1)!( − ∂ ) k , � � := 1 − exp(( − x i ) ∗ ∂ i ) = i i k =0 i = x m +1 � ◦ x m i m + 1 i Difference operators ( n = 1 ): M f i ( n ) T i ֒ � → ˆ D n via T �→ x, n �→ − x∂, i =0 etc. Alexander Zheglov (Moscow) Algebraic geometric properties of spectral surfaces of quantum integrable systems and their XXXVIII 10 / 29

The ring ˆ D n and the order function ord n Definition We define ˆ D 1 = ˆ D sym and define ˆ D n = ˆ D sym n − 1 [ ∂ n ] . 1 Obviously, ˆ D n ⊂ ˆ D sym . n Definition We define the order function ord n on ˆ D n as l ˆ � p s ∂ s ord n ( P ) = l if D n ∋ P = n . s =0 The coefficient p l is called the highest term and will be denoted by HT n ( P ) (as the term naturally associated with the function ord n ). Alexander Zheglov (Moscow) Algebraic geometric properties of spectral surfaces of quantum integrable systems and their XXXVIII 11 / 29

The notion of Γ -order The Γ -order ord Γ is defined on some elements of the algebra ˆ D sym : n Let’s denote by ˆ D i 1 ,...,i q the subalgebra in ˆ D sym consisting of operators n n not depending on ∂ i 1 , . . . , ∂ i q . The Γ -order is defined recursively. Definition D 2 , 3 ,...n We say that ord Γ ( P ) = k 1 , where 0 � = P ∈ ˆ , if P = � k 1 s =0 p s ∂ s 1 , n where 0 � = p k 1 ∈ ˆ R . D i +1 ,i +2 ,...,n We say that ord Γ ( P ) = ( k 1 , . . . , k i ) , where P ∈ ˆ , if n i , where p s ∈ ˆ D i,i +1 ,...,n P = � k i s =0 p s ∂ s , and ord Γ ( p k i ) = ( k 1 , . . . , k i − 1 ) . n We say that ord Γ ( P ) = ( k 1 , . . . , k n ) , where P ∈ ˆ D sym n , where p s ∈ ˆ , if P = � k n s =0 p s ∂ s D n n , and n ord Γ ( p k n ) = ( k 1 , . . . , k n − 1 ) . In this situation we say that the operator P is monic if the highest coefficient p k 1 ,...,k n (defined recursively in analogous way) is 1 . Alexander Zheglov (Moscow) Algebraic geometric properties of spectral surfaces of quantum integrable systems and their XXXVIII 12 / 29

Quasi-elliptic algebras Now we define the algebras that admit an effective description in terms of its algebro-geometric spectral data (which will be defined below). Definition The subalgebra B ⊂ ˆ D n ⊂ ˆ D sym of commuting operators is called 1 -quasi n elliptic if there are n operators P 1 , . . . , P n such that For 1 ≤ i < n ord Γ ( P i ) = (0 , . . . 0 , 1 , 0 . . . 0 , l i ) , where 1 stands at the i -th place and l i ∈ Z + ; ord Γ ( P n ) = (0 , . . . , 0 , l n ) , where l n > 0 ; For 1 ≤ i ≤ n ord ( P i ) = | ord Γ ( P i ) | . P i are monic. Alexander Zheglov (Moscow) Algebraic geometric properties of spectral surfaces of quantum integrable systems and their XXXVIII 13 / 29

The notion of rank Definition The analytic rank of B ⊂ ˆ D n is An.rank ( B ) := rk F = dim( Q ( B ) ⊗ B F ) = dim { ψ | P ◦ ψ = χ ( P ) ψ ∀ P ∈ B, χ – generic point } . The algebraic rank is Alg.rank ( B ) = GCD { ord ( P ) | P ∈ B } . Fact: An.rank ( B ) ≥ Alg.rank ( B ) . We’ll say that B ⊂ ˆ D n is of rank r if An.rank ( B ) = Alg.rank ( B ) = r . Alexander Zheglov (Moscow) Algebraic geometric properties of spectral surfaces of quantum integrable systems and their XXXVIII 14 / 29