Abstract
We compute the elliptic genus of the D1/D7 brane system in flat space, finding a non-trivial dependence on the number of D7 branes, and provide an F-theory interpretation of the result. We show that the JK-residues contributing to the elliptic genus are in one-to-one correspondence with coloured plane partitions and that the elliptic genus can be written as a chiral correlator of vertex operators on the torus. We also study the quantum mechanical system describing D0/D6 bound states on a circle, which leads to a plethystic exponential formula that can be connected to the M-theory graviton index on a multi-Taub-NUT background. The formula is a conjectural expression for higher-rank equivariant K-theoretic Donaldson-Thomas invariants on ℂ3.
Highlights
Properties of the moduli spaces of genus-zero pseudo-holomorphic maps to the target, and represent a convenient way to extract Gromov-Witten invariants [3]
We study the quantum mechanical system describing D0/D6 bound states on a circle, which leads to a plethystic exponential formula that can be connected to the M-theory graviton index on a multi-Taub-NUT background
The effective dynamics of the D1-branes is captured by a two-dimensional N = (2, 2) supersymmetric GLSM living on the elliptic curve, and whose classical vacua describe the moduli space of rank-N sheaves on C3, where N is the number of D7-branes
Summary
To study (equivariant) Donaldson-Thomas invariants [19] of a three-fold, one can employ a string theory brane construction [20, 21]. 2d N = (2, 2) quiver gauge theory with a U(k) vector multiplet; Q, Ba=1,2,3 chiral multiplets; SU(N ) flavour symmetry (in section 2 we take N = 1). We can study a D7-brane wrapped on the threefold and k D1-branes on its worldvolume: the two-dimensional theory on the D1-branes allows us to define “elliptic DT invariants” of the three-fold. While we study the D1/D7 system with a single D7-brane, in section 3 we will move to higher-rank DT invariants They are captured by the D1/D7 system with N multiple D7-branes wrapping the three-fold (here C3). Notice that fugacities are invariant under shift of the chemical potentials by 1, because of ’t Hooft anomalies, partition functions in general are not. For generic values of , instead, the integrand picks up a phase e2πib More details and examples can be found in [30, 31]
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