[Paper Review] D4-branes on Complete Intersection in Toric Variety

Davide Gaiotto, Lisa Huang|ArXiv.org|Dec 29, 2006

Black Holes and Theoretical Physics17 references3 citations

TL;DR

This paper proposes that the worldvolume of a D4-brane wrapping a holomorphic surface in a toric Calabi-Yau threefold is described by a determinantal variety defined by the vanishing of the determinant of a matrix of holomorphic sections. The construction arises from tachyon condensation in a D6–D̄6 brane system, with the Higgs branch of the resulting quiver gauge theory encoding the D4-brane moduli space. The key result is a geometric realization of D4-brane configurations via determinantal equations derived from open string gauge linear sigma models.

ABSTRACT

We consider D4-branes on toric Calabi-Yau spaces. The quiver gauge theory that describes several D4-branes on the Calabi-Yau has a Higgs branch, that describes configurations of a single large D4-brane with the same charges. We propose that the world volume of such a D4-brane is described by a determinantal variety. We discuss a description of the Higgs branch of the moduli space in terms of a quiver with twice as many nodes and only bifundamental fields, arising from a $D6-\bar{D}6$ system. We recast the tachyon condensation of the $D6-\bar{D}6$ system in the language of open string gauge linear sigma model.

Motivation & Objective

• To generalize the description of D4-brane moduli spaces from the quintic Calabi-Yau to general toric Calabi-Yau threefolds via complete intersections.
• To establish a correspondence between the Higgs branch of a quiver gauge theory for N D4-branes and the worldvolume geometry of a single large D4-brane.
• To derive the D4-brane worldvolume as the zero locus of a tachyon matrix in a D6–D̄6 system, using tachyon condensation.
• To reformulate the D4-brane geometry in the language of the open string gauged linear sigma model, recovering the determinantal condition from F-term constraints.

Proposed method

• Construct a quiver gauge theory with U(1)^N × U(1)^N gauge group and bifundamental fields T_IJ transforming in the line bundle ⊗_α L_α^{(n_α^I + n_α^J)/2 + k_α^I - k_α^J}.
• Realize the D4-brane as the sheaf of zero eigenvectors of the tachyon matrix T, supported on the locus det(T) = 0.
• Use the derived category description of D-branes to build the quiver theory from a D6–D̄6 system with fluxes encoded in line bundle twists.
• Apply the open string gauged linear sigma model with boundary couplings ∫ dθ ρ^I T_IJ β^J, where ρ and β are boundary fields with specific U(1) charges.
• Impose F-term constraints ρ^I T_IJ = 0 and require ρ ≠ 0 to enforce det(T) = 0, yielding the D4-brane worldvolume.
• Identify the D4-brane flux F as dual to the curve C defined by ∑_I v_I(z) ̃A_IJ(z) = 0, where ̃A_IJ are minors of A(z).

Experimental results

Research questions

• RQ1 How can the moduli space of N D4-branes on a complete intersection Calabi-Yau be described geometrically in terms of a single large D4-brane?
• RQ2 What is the precise role of tachyon condensation in the D6–D̄6 system in realizing D4-brane configurations?
• RQ3 How does the open string gauged linear sigma model reproduce the determinantal equation for the D4-brane worldvolume?
• RQ4 What is the relation between the D4-brane flux and the algebraic curve defined by the minors of the tachyon matrix?
• RQ5 How do the intersection numbers C·J^α and C·C compute the D2- and D0-brane charges carried by the D4-brane?

Key findings

• The worldvolume of a D4-brane wrapping a holomorphic surface in a toric Calabi-Yau threefold is described by the determinantal variety det(A_IJ(z)) = 0, where A_IJ are holomorphic sections of appropriate line bundles.
• The Higgs branch of the quiver gauge theory for N D4-branes is isomorphic to the moduli space of a single D4-brane with the same charges, confirming the proposal in [10] for the quintic.
• The D4-brane arises as the kernel of the tachyon matrix T_IJ in a D6–D̄6 system, with the support locus given by det(T) = 0.
• The D4-brane flux F is given by C - ½∑_I n_α^I J^α, where C is a non-complete-intersection curve defined by minors of the tachyon matrix.
• The open string gauged linear sigma model reproduces the determinantal condition via F-term constraints ρ^I T_IJ = 0 and the requirement that ρ ≠ 0, leading to det(T) = 0.
• The intersection numbers C·J^α and C·C are computed in terms of triple intersection numbers C^{αβγ} and the flux data, matching the D2- and D0-brane charges.