Universal flops of length 1 and 2 from D2-branes at surface singularities

Abstract

We study families of deformed ADE surfaces by probing them with a D2-brane in Type IIA string theory. The geometry of the total space $X$ of such a family can be encoded in a scalar field $Φ$, which lives in the corresponding ADE algebra and depends on the deformation parameters. The superpotential of the probe three dimensional (3d) theory incorporates a term that depends on the field $Φ$. By varying the parameters on which $Φ$ depends, one generates a family of 3d theories whose moduli space always includes a geometric branch, isomorphic to the deformed surface. By fibering this geometric branch over the parameter space, the total space $X$ of the family of ADE surfaces is reconstructed. We explore various cases, including when $X$ is the universal flop of length $\ell=1,2$. The effective theory, obtained after the introduction of $Φ$, provides valuable insights into the geometric features of $X$, such as the loci in parameter space where the fiber becomes singular and, more notably, the conditions under which this induces a singularity in the total space. By analyzing the monopole operators in the 3d theory, we determine the charges of certain M2-brane states arising in M-theory compactifications on $X$.

Figures (11)

• Figure 1: ADE Dynkin diagrams and dual Coxeter labels of the nodes. The colored node corresponds to the root that is blown up in the (partial) simultaneous resolution in the universal flops. The label of the colored node is the length associated with the corresponding flop.
• Figure 2: $A_r$ theory. For each node $i$, $i=1,...,r+1$, there is a $\mathcal{N}=4$$U(1)$ vector multiplet $V_i$ containing a $\mathcal{N}=2$ vector multiplet and an adjoint chiral $\phi_i$. Pairs of oriented lines between adjacent nodes represent bifundamental hypermultiplets $(q_i, \tilde{q}_i)$.
• Figure 3: $A_1$ quiver.
• Figure 4: $A_3$ Dynkin diagram. The colored node corresponds to the blown up sphere in the partial simultaneous resolution of the $A_3$ family.
• Figure 5: $A_3$ quiver.