Appendix A The Grassmannian
A.2 Quantum Cohomology Ring onGr(M, N) specific example of the Grassmannian Gr(2,4) to highlight the correspondence between 2d gauge theory and quantum cohomology.
Example: Gr(2,4)
For the Grassmannian Gr(2,4) the generators of the cohomology ring are the Schur polynomials in two variables 1, 2 ∈H2(Gr(2,4)). Note that we use for them identical terminology as for the Chern roots of the dual universal subbundle. Classically, they generate the symmetric polynomial ring C[1, 2]S2 and obeying,
σν·σµ = σν⊗µ , (A.11)
in terms of the tensor product ⊗of Young tableaux of the permutation groupS2. The kernel of the ring homomorphism from Schubert cycles to corresponding Schur polynomials is given by the two relations,
σ3(1, 2) = 31+212+122+32 = 0, σ4(1, 2) = 41+312+2122+132+42 = 0, (A.12) Here the subscripts of Schur polynomialsσ refer to the integer partition of a Young diagram.
The cohomology ring thus becomes, H∗(Gr(2,4),C) ' C[1, 2]S2.
hσ3, σ4i ' C[N1, N2]
D
N1N2,−N414 +N12N2+ N422
E . (A.13) In the last step the ideal hσ3, σ4i of the kernel of symmetric polynomials is expressed in terms of theNewton polynomials N` =x`1+x`2,`= 1,2, which in turn generate the symmetric polynomial ring C[1, 2]S2.
When the quantum deformation of is applied to the Grassmannian Gr(2,4) it yields the quantum cohomology ring [69],
H?∗(Gr(2,4),C) ' C[N1, N2][[Q]]
D
N1N2,−N414 +N12N2+N422 +QE
, (A.14)
i.e., the latter relation generating the classical cohomology relations is deformed by a factor of Q. This is reminiscent of the deformation of the classical to the quantum cohomology ring for projective spaces. The presented formulation relates directly to the gauge theory correlators and its correlator relations, as discussed in Chapter 3. We first note that the gauge invariant insertions are canonically identified with the Newton polynomials Nr=xr1+xr2 according to,
tr(σr) ←→ Nr . (A.15)
The Schubert cycles are multiplied with theQ-dependent quantum product and then integrated over the Grassmannian Gr(2,4). Here, we have used the following relations among Newton and Schur polynomials,
N1 = σ1 , N2 = σ2−σ1,1 . (A.16)
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