The spectral problem of the modular oscillator in the strongly coupled regime Rinat Kashaev University of Geneva Joint work with Sergey Sergeev (University of Canberra, Australia) Quantum fields, knots, and strings, University of Warsaw, September 24–28, 2018 Rinat Kashaev The spectral problem of the modular oscillator. . .
Motivation: topological strings Mirror Symmetry Toric CY 3-fold M (trace class operator) ρ M The spectrum of ρ M is expected to be related to enumerative invariants of M through the topological string partition functions. Suggested by Aganagic–Dijkgraaf–Klemm–Mari˜ no–Vafa (2006) and materialized by Grassi–Hatsuda–Mari˜ no (2016). Example: the local P 1 × P 1 or F 0 ρ − 1 = ρ − 1 F 0 , m = v + v − 1 + u + m u − 1 , m ∈ R > 0 , with positive self-adjoint operators u and v satisfying the Heisenberg–Weyl commutation relation uv = e i � vu , � ∈ R > 0 . Rinat Kashaev The spectral problem of the modular oscillator. . .
Implications of the Grassi–Hatsuda–Mari˜ no conjecture Fredholm determinant ∞ � Z ( N , � ) κ N det(1 + κ ρ ) = 1 + (convergent series) N =1 where the fermionic spectral traces Z ( N , � ) = e F ( N , � ) provide a non-perturbative definition of the topological string partition functions. with fixed λ := � � → ∞ , N → ∞ , (t’Hooft limit) N ∞ � F g ( λ ) � 2 − 2 g F ( N , � ) ≃ (asymptotic series) g =0 with the genus g standard topological string free energies F g ( λ ) in the conifold frame where λ is a flat coordinate for the CY moduli space vanishing at the conifold point. Rinat Kashaev The spectral problem of the modular oscillator. . .
Statement of the problem For b ∈ C � =0 , define operators in L 2 ( R ) v := e 2 π b − 1 p . u := e 2 π b − 1 x , u := e 2 π b x , v := e 2 π b p , ¯ ¯ with Heisenberg operators p ψ ( x ) = (2 π i) − 1 ψ ′ ( x ) . x ψ ( x ) = x ψ ( x ) , Spectral problem for two Hamiltonians H := v + v − 1 + u + u − 1 , v − 1 + ¯ ¯ u − 1 H := ¯ v + ¯ u + ¯ which (formally) commute ( Faddeev’s modular duality ). Strongly coupled regime b = e i θ , 0 < θ < π 2 ⇒ ¯ H = H ∗ (Hermitian conjugate) . Small b limit H = 4 + (2 π b) 2 ( p 2 + x 2 ) + O (b 4 ) (“modular oscillator”) . Rinat Kashaev The spectral problem of the modular oscillator. . .
Functional difference equations The common spectral problem for H and ¯ H is equivalent to constructing an element ψ ( x ) ∈ L 2 ( R ) admitting analytic continuation to a domain containing the strip |ℑ z | ≤ max( ℜ b , ℜ b − 1 ), satisfying the functional difference equations ψ ( x + ib) + ψ ( x − ib) = ( ε − 2 cosh(2 π b x )) ψ ( x ) , ψ ( x + ib − 1 ) + ψ ( x − ib − 1 ) = (¯ ε − 2 cosh(2 π b − 1 x )) ψ ( x ) , and the restrictions ψ ( x + i λ ) being elements of L 2 ( R ), where λ ∈ {ℜ b , −ℜ b , ℜ b − 1 , −ℜ b − 1 } . In the general case of Baxter’s T − Q equations, an approach for constructing the solution in the strongly coupled regime is suggested by S. Sergeev (2005). A different approach through auxiliary non-linear integral equations is developed by O. Babelon, K. Kozlowski, V. Pasquier (2018). Rinat Kashaev The spectral problem of the modular oscillator. . .
Behavior at infinity In the limit x → −∞ , equation ψ ( x + ib) + ψ ( x − ib) = ( ε − 2 cosh(2 π b x )) ψ ( x ) is approximated by the equation ψ ( x + ib) + ψ ( x − ib) = − e − 2 π b x ψ ( x ) , where, in the left hand side, any one of the two terms can be dominating giving rise to two possible asymtotics η := b + b − 1 ψ ( x ) | x →−∞ ∼ e ± i π x 2 +2 πη x , = cos θ. 2 Thus, there are two solutions of the form ψ ± ( x ) = e ± i π x 2 +2 πη x φ ± ( x ) , φ ± ( x ) | x →−∞ = O (1) . Thus, a general exponentially decaying at x → −∞ solution is of the form e i π x 2 φ + ( x ) + e − i π x 2 φ − ( x ) ψ ( x ) = e 2 πη x � � . Rinat Kashaev The spectral problem of the modular oscillator. . .
The factorization ansatz We look for solutions of the form ψ ± ( x ) = e ± i π x 2 +2 πη x φ ± ( x ) with � e π i b 2 , ε, e 2 π b x � e − π i b 2 , ε, e 2 π b x � � φ + ( x ) = f f , φ − ( x ) = αφ + ( x ) , where α ∈ C and ∞ � c n ( q , ε ) u n f ( q , ε, u ) = n =0 solves the functional equation q , ε, uq − 2 � + q 2 u 2 f ( q , ε, q 2 u ) = (1 − ε u + u 2 ) f ( q , ε, u ) . � f Rinat Kashaev The spectral problem of the modular oscillator. . .
The main functional equation q := e π i b 2 . u / q 2 � + q 2 u 2 f ( q 2 u ) = (1 − ε u + u 2 ) f ( u ) , � f Involution in the space of solutions: f ( u ) �→ ˇ f ( u ) := u − 1 f � u − 1 � . An equivalent first order difference matrix equation � f � 1 − ε u + u 2 u / q 2 � − q 2 u 2 � � � � � f ( u ) = L ( u ) , L ( u ) := f ( q 2 u ) f ( u ) 1 0 � f � f � u / q 2 � � q 2 n − 2 u � � � = M n ( u ) , ∀ n ∈ Z > 0 , f ( q 2 n u ) f ( u ) � χ q u / q 2 � � 0 � M n ( u ) := L ( u ) L ( q 2 u ) · · · L ( q 2( n − 1) u ) , M ∞ ( u ) = , χ q ( u ) 0 where χ q ( u ) = χ q ( u , ε ) is an entire function of u ∈ C normalised so that χ q (0) = 1 and which solves the main functional equation. χ q ( u ) := u − 1 χ q ( u − 1 ) leads to a non-zero The second solution ˇ q − 2 u q − 2 u � � � � Wronskian [ χ q , ˇ χ q ]( u ) := χ q χ q ( u ) − ˇ ˇ χ q ( u ) . χ q Rinat Kashaev The spectral problem of the modular oscillator. . .
Orthogonal polynomials associated to χ q ( u , ε ) χ q , n ( ε ) ( − 1) n q n ( n +1) χ q , n ( ε ) u n = u n . � � χ q ( u , ε ) = ( q − 2 ; q − 2 ) n ( q 2 ; q 2 ) n n ≥ 0 n ≥ 0 with polynomials χ q , n ( ε ) ∈ C [ ε ] satisfying the recurrence relation χ q , n +1 ( ε ) = εχ q , n ( ε ) + ( q n − q − n ) 2 χ q , n − 1 ( ε ) , χ q , 0 ( ε ) = 1 , with few first polynomials χ q , 2 ( ε ) = ε 2 + ( q − q − 1 ) 2 , χ q , 1 ( ε ) = ε, χ q , 3 ( ε ) = ε ( ε 2 + ( q 2 − q − 2 ) 2 + ( q − q − 1 ) 2 ) , . . . Multiplication rule χ q , m ( ε ) χ q , n ( ε ) min( m , n ) ( q 2 m ; q − 2 ) k ( q 2 n ; q − 2 ) k ( q 2( k − m − n ) ; q 2 ) k � = χ q , m + n − 2 k ( ε ) ( q 2 ; q 2 ) k k =0 Rinat Kashaev The spectral problem of the modular oscillator. . .
The main functional equation with q replaced by q − 1 � u f ( q 2 u ) + u 2 � = (1 − ε u + u 2 ) f ( u ) . q 2 f q 2 There is no solution regular at u = 0. The series χ q , n ( ε ) � u n χ q − 1 ( u , ε ) ≃ ( q 2 ; q 2 ) n n ≥ 0 does not converge, it is only an asymptotic expansion of the true solution χ q ( u , ε ) ˇ χ q − 1 ( u , ε ) := χ q ]( u ) . [ χ q , ˇ Rinat Kashaev The spectral problem of the modular oscillator. . .
Result for the eigenfunction ψ ( x ) := b − 1 e π i σ 2 − ξπ i / 4 e 2 πη x +i π x 2 ˇ χ q ( u ) χ q ( u ) + ξ χ q ( u )ˇ χ q ( u ) . θ 1 ( su , q ) θ 1 ( s − 1 u , q ) where χ q ]( u ) = ̺θ 1 ( su , q ) θ 1 ( s − 1 u , q ) , [ χ q , ˇ θ 1 ( u , q ) := 1 ( − 1) n q ( n +1 / 2) 2 u n +1 / 2 , � i n ∈ Z with certain functions s = s ( ε, q ), ̺ = ̺ ( ε, q ), s := e 2 π b σ , and the variable ξ ∈ {± 1 } is the parity of the eigenstate: ψ ( − x ) = ξψ ( x ) . The function is real ψ ( x ) = ψ ( x ) (thus modular invariant b ↔ b − 1 ) and exponentially decays at both infinities | ψ ( x ) | ∼ e − 2 πη | x | , x → ±∞ . Rinat Kashaev The spectral problem of the modular oscillator. . .
Quantization condition for the eigenvalues The quantization condition is the analyticity condition for ψ ( x ) with complex x in the strip z ∈ C | |ℑ z | < max( |ℜ b | , |ℜ b − 1 | ) � � S b := . Define G q ( u , ε ) := χ q ( u , ε ) χ q ( u , ε ) , G q ( u , ε ) G q (1 / u , ε ) = 1 , ∀ u ∈ C � =0 . ˇ Theorem χ q ]( u ) = ̺θ 1 ( su ) θ 1 ( s − 1 u ) for any Let ε = ε ( σ ) be such that [ χ q , ˇ u ∈ C , and assume that s �∈ ± q Z (recall that s = s ( σ ) = e 2 π b σ ). Then the eigenfunction ψ ( x ) does not have poles in the strip S b if the variable σ is such that G q ( s , ε ) = − ξ G q ( s , ε ) . Moreover, in that case, ψ ( x ) is an entire function on C . Rinat Kashaev The spectral problem of the modular oscillator. . .
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