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1. What the question is asking (in plain language)

The problem is about minimal string theory (a minimal conformal model coupled to Liouville gravity) and its description as a topological B‑model on the non‑compact Calabi–Yau three‑fold

[ uv-H(x,y)=0 ,\qquad H(x,y)=y^{p}+x^{q}\; . ]

In Liouville theory there are two basic types of D‑branes

  • FZZT branes – non‑compact branes whose boundary condition is labelled by a point ((x,y)) on the spectral curve (H(x,y)=0).
  • ZZ branes – compact branes that sit at the critical (or double‑point) loci of the same curve.

The question asks:

What objects in the B‑model play the rôle of the FZZT and ZZ open‑string partition functions?

In other words, we must translate the two Liouville brane partition functions into the language of the B‑model (or, equivalently, of the matrix model that reproduces the B‑model after the Dijkgraaf–Vafa construction).

2. Step‑by‑step answer

2.1 Review of the Dijkgraaf–Vafa correspondence

  1. Matrix model
    Consider the one‑matrix model

    [ Z_{\text{MM}}=\int!dM\,\exp!\Big[-\frac{1}{g_s}\, \operatorname{Tr}V(M)\Big] , \qquad V’(x)=W’(x)=y(x) . ]

    In the large‑(N) (planar) limit the eigenvalue density (\rho(x)) is supported on cuts of the complex (x)‑plane. The resolvent

    [ \omega(x)=\Big\langle\operatorname{Tr}\frac{1}{x-M}\Big\rangle =\frac{1}{g_s}y(x) , ]

    satisfies the spectral curve equation

    [ H(x,y)=0,\qquad H(x,y)=y^{p}+x^{q} ]

    (after the double‑scaling limit that zooms onto a critical point of order ((p,q))).

  2. B‑model mirror geometry
    The same curve appears as the reduction of the Calabi–Yau three‑fold

    [ X:\; uv-H(x,y)=0 . ]

    The holomorphic three‑form reduces to the one‑form

    [ \lambda = y\,dx . ]

    The B‑model closed‑string free energy (F_g) is reproduced by the matrix‑model genus‑(g) free energies.

2.2 B‑model open strings = “branes on the curve”

In the B‑model a B‑brane is a holomorphic sub‑manifold of complex dimension one (a curve) together with a holomorphic line bundle.
Because the Calabi–Yau is a fibration over the ((x,y))‑plane with fibre (uv=H(x,y)), a B‑brane that wraps the fibre and sits at a point ((x_0,y_0)) on the base is completely characterised by that point.

Thus:

Liouville brane B‑model description
FZZT (non‑compact) Non‑compact B‑brane localized at an arbitrary point ((x_0,y_0)) on the spectral curve.
ZZ (compact) Compact B‑brane that wraps a compact cycle of the curve, i.e. a cycle that collapses at a branch (double) point of the curve.

2.3 The open‑string partition functions

The open‑string partition function of a B‑brane is the wavefunction (\Psi) obtained by integrating the holomorphic three‑form along a path that ends on the brane. Because the fibre part (uv) is trivial, the integral reduces to an integral of (\lambda=y\,dx) on the spectral curve:

[ \boxed{\;\Psi_{\text{brane}}(x)=\exp!\Big[\frac{1}{g_s}\int^{x} y(x’)\,dx’ \Big]\times\big(\text{quantum corrections}\big)\;} \tag{2.1} ]

The quantum corrections are generated by the topological recursion of Eynard‑Orantin, which is precisely the all‑order (1/N) (or (g_s)) expansion of the matrix model.

2.3.1 FZZT brane ↔ non‑compact B‑brane

  • In the matrix model the insertion of an FZZT brane corresponds to the determinant operator

    [ \mathcal{O}_{\text{FZZT}}(x)=\det\big(x-M\big)=\exp!\Big[\operatorname{Tr}\ln(x-M)\Big] . ]

  • Its expectation value is the Baker‑Akhiezer (wave) function

    [ \Psi_{\text{FZZT}}(x)=\Big\langle\det(x-M)\Big\rangle . ]

  • In the double‑scaling limit the matrix model becomes the minimal string, and

    [ \boxed{\; Z_{\text{FZZT}}(x)\;=\;\Psi_{\text{FZZT}}(x) \;=\;\exp!\Big[\frac{1}{g_s}\int^{x} y\,dx\Big] \Big(1+\mathcal{O}(g_s)\Big) \;} \tag{2.2} ]

    The leading exponent (\int^{x}y\,dx) is exactly the disk amplitude of an FZZT brane; higher‑order terms encode annulus, genus‑(g) corrections.

  • Geometrically, the B‑model brane is non‑compact: it stretches to infinity along the fibre direction (u) (or (v)) while being pinned at the chosen point ((x,y)) on the curve.

2.3.2 ZZ brane ↔ compact B‑brane

  • A ZZ brane sits at a branch point where two sheets of the curve meet, i.e. a point where

    [ H(x,y)=0,\qquad \frac{\partial H}{\partial y}=0 \quad\Longleftrightarrow\quad y=0,\; x=x_i\;(i=1,\dots ,#\text{cuts}) . ]

  • In the matrix model this corresponds to an eigenvalue tunnelling (instanton) from one cut to another. The instanton weight is the integral of the one‑form (\lambda) along a contour that connects the two sheets:

    [ S_i=\int_{a_i}^{b_i} y(x)\,dx \qquad\text{(A‑period of the curve)} . ]

  • The ZZ partition function is therefore a sum over instanton sectors

    [ \boxed{\; Z_{\text{ZZ}}=\sum_{i} n_i\, \exp!\Big[-\frac{S_i}{g_s}\Big]\; \big(1+\mathcal{O}(g_s)\big) \;} \tag{2.3} ]

    where (n_i\in\mathbb Z) are the (integer) D‑brane charges. The exponent (-S_i/g_s) is the disk amplitude of a compact B‑brane wrapping the vanishing cycle that collapses at the branch point.

  • In the B‑model language the compact brane is a holomorphic curve that is the preimage of the compact 1‑cycle (the A‑cycle) on the spectral curve. Its open‑string wavefunction can be written as a difference of two FZZT wavefunctions evaluated at the two sheets meeting at the branch point:

    [ \Psi_{\text{ZZ}}^{(i)}\;\propto\; \Psi_{\text{FZZT}}(x_i^{+})-\Psi_{\text{FZZT}}(x_i^{-}) \;\sim\; \exp!\Big[-\frac{S_i}{g_s}\Big]. ]

  • The full minimal‑string ZZ amplitude (including all genera) is reproduced by the topological recursion applied to the compact B‑brane; equivalently it is the non‑perturbative completion of the B‑model closed‑string partition function.

2.4 Summary of the dictionary

Minimal‑string object B‑model (or matrix‑model) object Partition function
FZZT brane (labelled by a point (x) on (H=0)) Non‑compact B‑brane at ((x,y)) on the spectral curve; insertion of (\det(x-M)) in the matrix model. (\displaystyle Z_{\text{FZZT}}(x)=\big\langle\det(x-M)\big\rangle = \exp!\big[\frac{1}{g_s}!\int^{x}! y\,dx\big]\big(1+\mathcal O(g_s)\big))
ZZ brane (labelled by a branch point / integer charge) Compact B‑brane wrapping the vanishing A‑cycle of the curve; instanton (eigenvalue tunnelling) sector of the matrix model. (\displaystyle Z_{\text{ZZ}} = \sum_i n_i \exp!\big[-\frac{1}{g_s}!\int_{a_i}^{b_i}! y\,dx\big]\big(1+\mathcal O(g_s)\big))

The closed‑string B‑model partition function (\exp\big(\sum_{g}g_s^{2g-2}F_g\big)) is reproduced by the matrix‑model planar and higher‑genus free energies; the open‑string pieces above are simply the wavefunctions of B‑branes living on the same spectral curve.

3. Final answer (concise statement)

  • The FZZT open‑string partition function is the matrix‑model expectation value of a determinant operator (the Baker‑Akhiezer wavefunction). In the B‑model it is

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