Quantum Cohomology

Euler Vector Field

Great snippet to zoom in on. Here’s what that paragraph is saying, piece by piece.

What is the “Euler vector field” here?

On the big quantum cohomology of \(X\) (a Frobenius manifold with coordinates \(t=(t_i)\) along a basis \(\{T_i\}\subset H^\bullet(X)\)), the Euler vector field is the canonical vector field encoding the grading and the first Chern class:

\[ E \;=\; c_1(TX)\;+\;\sum_i\Bigl(1-\tfrac{\deg T_i}{2}\Bigr)\,t_i\,T_i\;. \]

It enters the quantum connection by the standard formula

\[ \boxed{\;\nabla_{\partial_z}=\partial_z-\frac{1}{z^{2}}\,(E\star_t)\ +\ \frac{1}{z}\,\mu\;} \qquad\text{equivalently}\qquad \nabla_{z\partial_z}=z\partial_z-\frac{1}{z}(E\star_t)+\mu, \]

where \(\star_t\) is quantum product at the point \(t\) and \(\mu\) is the Hodge-grading operator. So “Euler vector field action” just means the endomorphism

\[ K_t \;:=\; E\star_t\ :\ H^\bullet(X)\longrightarrow H^\bullet(X) \]

(“quantum multiplication by \(E\)”). At the small point \(t=0\), \(E=c_1(TX)\), so this is the familiar operator “quantum cup by \(c_1(TX)\)”. (math.kyoto-u.ac.jp)

In the TE/TEP/F-bundle language used in the paper, the “Euler action” is the same thing encoded in the \(u\)-direction of the meromorphic connection:

\[ \nabla_{\partial_u}=\frac{1}{2u}\,Gr\;-\;\frac{1}{u}\,\nabla_E, \]

so residues at \(u=0\) recover the endomorphism \(E\star_t\) on the special fibre. That’s why they speak of the action of the Euler vector field on the fibre \(H|_{u=0}\) (which is identified with \(H^\bullet(X)\)). (SpringerLink)

Where does it act, and how?

• Where: on the state space \(H^\bullet(X)\) (the fibre of the quantum/F-bundle at the base point), or equivalently on the tangent space of the Frobenius manifold after the standard identification.
• How: by left quantum multiplication with \(E\): \(v\mapsto E\star_t v\). The spectrum (eigenvalues/eigenspaces) of this operator is what the paper studies. (math.kyoto-u.ac.jp)

Why the blow-up decomposition matches an Euler-spectrum decomposition

For a blow-up \(\pi:\widehat X\to X\) along a smooth center \(Z\subset X\) of codimension \(r\ge2\), the paper recalls the classical cohomology splitting

\[ H^\bullet(\widehat X,\mathbb Q)\;\cong\;H^\bullet(X,\mathbb Q)\ \oplus\ \bigoplus_{i=1}^{r-1}H^\bullet(Z,\mathbb Q)[-2i]. \]

They then show that at a specific “limiting” base point \(b_\infty\) (large radius; pullback of an ample class), this is exactly the eigenspace decomposition of the Euler action \(K_{b_\infty}\):

• one eigenvalue \(0\) with eigenspace \(H^\bullet(X)\);

• and \(r-1\) other eigenvalues

  \[ (r-1)\,e^{\pi i (2j-1)/(r-1)}\quad (j=1,\ldots,r-1), \]

  each with eigenspace isomorphic to \(H^\bullet(Z)\). Moving away from \(b_\infty\), those eigenvalues may split further, producing the spectral “clusters” pictured in the paper. (arXiv)

This behaviour is a manifestation of Iritani’s blow-up theorem for the quantum \(D\)-module: the quantum connection of \(\widehat X\) decomposes into the piece from \(X\) plus \(r-1\) copies of the piece from \(Z\) (after a standard transform), and the Euler operator \(E\star\) respects this decomposition—hence the eigenvalue pattern above. (arXiv)

Takeaway. “Euler vector field action” = the endomorphism \(E\star\) on the quantum cohomology state space. In the F-bundle formulation it’s read off from the \(u\)-part of the connection and acts on the fibre \(H|_{u=0}\cong H^\bullet\). For a blow-up, its spectrum at the large-radius point splits into one \(X\)-block (eigenvalue 0) and \(r-1\) \(Z\)-blocks (eigenvalues arranged as the \((2j{-}1)\)-th roots of \(-1\) scaled by \(r-1\)), matching the classical cohomology splitting and Iritani’s \(D\)-module decomposition. (math.kyoto-u.ac.jp)