I'm studying the Nishinaka-Yoshida crystal models that encode the generating function of $D4$-$D2$-$D0$ BPS bound states on a Calabi-Yau divisor.

The case of conifold at its singular point is developed in Statistical model and BPS D4-D2-D0 counting and the case $A_{N-1} \times \mathbb{C}$ is discussed in A note on statistical model for BPS D4-D2-D0 states.

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The paper that dicuss the conifold case explicitly states that the completely "frozen" crystal (the triangular partition without "triangular atoms" removed) is identifyed as a $D4$-wrapping the conifold. The picture in the above shows the frozen crystal.

A general crystal melting configuration must have atoms removed and should represent a general $D2$-$D0$ bound state "sticked" to the $D4$ brane. The following picture shows an example.

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The authors show that if one identify blue triangles as one unit of $D2$-brane charge and pairs of red and blue triangles as white boxes representing one unit of $D0$ brane charge, then his model is able to compute the desired BPS index.

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I'm unable to see the rationality behind those choices, they seem totally arbitrary to me (beyond the fact that the identifications are needed to reproduce the desired partition function).


How can I derive those choices from first principles?

Is there any direct relation with the two dimensional Calabi-Yau crystals constructed in Crystals, instantons and quantum toric geometry?

Thanks in advance.


1 Answer 1


I recently found a paper that answer all my questions from first principles and in great detail.

Two-dimensional crystal melting and D4-D2-D0 on toric Calabi-Yau singularities

Further useful references:

Multiple D4-D2-D0 on the Conifold and Wall-crossing with the Flop

Vertical D4-D2-D0 bound states on K3 fibrations and modularity

Wall-crossing of D4-branes using flow trees


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