By Martin A. Guest

ISBN-10: 0198565992

ISBN-13: 9780198565994

Quantum cohomology has its origins in symplectic geometry and algebraic geometry, yet is deeply with regards to differential equations and integrable structures. this article explains what's in the back of the intense luck of quantum cohomology, resulting in its connections with many current parts of arithmetic in addition to its visual appeal in new components reminiscent of reflect symmetry. sure types of differential equations (or D-modules) give you the key hyperlinks among quantum cohomology and conventional arithmetic; those hyperlinks are the main target of the booklet, and quantum cohomology and different integrable PDEs similar to the KdV equation and the harmonic map equation are mentioned inside of this unified framework. geared toward graduate scholars in arithmetic who are looking to know about quantum cohomology in a extensive context, and theoretical physicists who're drawn to the mathematical atmosphere, the textual content assumes uncomplicated familiarity with differential equations and cohomology.

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**Additional resources for From Quantum Cohomology to Integrable Systems **

**Example text**

As B is a group, we have canonical isomorphisms TB ∼ = B × H 2 (M; C), T ∗B ∼ = B × H 2 (M; C)∗ . Via this identiﬁcation, the natural (complex) symplectic form of T ∗ B is dλ, where r λ= i=1 dqi ∧ pi . qi With respect to this symplectic structure, the Poisson bracket of two functions f , g : T ∗ B → C is given by r {f , g} = qi i=1 ∂f ∂g ∂g ∂f − ∂pi ∂qi ∂pi ∂qi . Let us consider next the algebraic variety VM deﬁned by VM = {(q1 , . . , qr , p1 , . . , pr ) ∈ T ∗ B | R1 = · · · = Ru = 0}. It is shown in [55] and [7] that, under certain conditions, VM is a Lagrangian subvariety of T ∗ B.

We conclude that α, X1 = 1 and α, X4 = 0, which forces α = x2 . Next, we have β = X1 |X1 |Z 0,1 , where Z is the generator of H0 ( 1 ; Z). Let us choose X1 , X3 , and any point of 1 as representatives of the three homology classes. e. each ﬁbre) fails to intersect all three representatives. Hence β = 0. Finally, γ = X1 |X1 |Z 2,0 and this is zero because a holomorphic map of degree (2,0) must be a map CP1 → X2 of degree two. 10 x1 ◦ x4 = x1 x4 . Proof Let us write x1 ◦ x4 = x1 x4 + αq1 + βq2 + γ q21 .

6 Let F3 = F1,2 (C3 ) = {(L, V) ∈ Gr1 (C3 ) × Gr2 (C3 ) | L ⊆ V} ∼ = U3 /U1 × U1 × U1 be the (full) ﬂag manifold of the unitary group U3 . The ‘Borel description’ of H ∗ (F3 ; Z) is H ∗ (F3 ; Z) ∼ = Z[x1 , x2 , x3 ]/(σ1 , σ2 , σ3 ) where σ1 , σ2 , σ3 are the elementary symmetric functions of x1 , x2 , x3 . Geometrically, xi = −c1 (Li ), where L1 , L2 , L3 are the complex line bundles on F3 whose ﬁbres over (L, V) are L, L⊥ ∩ V, V ⊥ respectively. On the other hand, the ‘Schubert description’ of H∗ (F3 ; Z) involves speciﬁc cycles, the Schubert varieties, which generalize the subvarieties CPi of CPn .

### From Quantum Cohomology to Integrable Systems by Martin A. Guest

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