Abstract
We study the relation between the instanton counting on ALE spaces and the BPS state counting on a toric Calabi-Yau three-fold. We put a single D4-brane on a divisor isomorphic to A N −1-ALE space in the Calabi-Yau three-fold, and evaluate the discrete changes of BPS partition function of D4-D2-D0 states in the wall-crossing phenomena. In particular, we find that the character of affine SU(N) algebra naturally arises in wall-crossings of D4-D2-D0 states. Our analysis is completely based on the wall-crossing formula for the d = 4, $$ \mathcal{N} $$ = 2 supersymmetric theory obtained by dimensionally reducing the Calabi-Yau three-fold.
Highlights
Where Ri denotes the radius of i-th compact two-cycle in the Calabi-Yau three-fold
The radii of compact cycles belong to Kähler moduli of the Calabi-Yau, and changing the Kähler moduli in general modifies the BPS conditions of the D4-D2-D0 bound states. This implies that some BPS bound states of D4-D2-D0 branes might be unstable or newly appear in the spectrum when we change the radii of the cycles, which gives rise to discontinuous changes in the BPS partition function
Recall that the divisor wrapped by the D4-brane is isomorphic to C2. In this limit, the BPS partition function of our D4-D2-D0 states should be equal to the instanton partition function on C2
Summary
In order to introduce non-vanishing effects of wall-crossings, we here “add” an additional dummy two-cycle at an edge of the toric diagram, which drastically changes the situation. This dummy cycle β0 gives us non-vanishing wall-crossing phenomena due to the non-zero intersection product (3.2). We will study such wall-crossing phenomena in subsection 3.3 Note that this modified Calabi-Yau three-fold can be reduced to the original one if we take the non-compact limit Im z0 → +∞ for the dummy cycle. The flop transition occurs at Im z0 = 0 and changes the topology of the Calabi-Yau three-fold as well as that of the divisor wrapped by the D4-brane. Which implies that, in the large radius limit, ze0 = −z0 and ze1 = z1 + z0 stand for the (complexified) areas of the modified two-cycles βe0 and βe , respectively. The corresponding BPS partition function should be equivalent to the instanton partition function of the Vafa-Witten theory on O(−1) → P1 , which was evaluated in [26]
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