The Coulomb Branch Formula conjecturally expresses the refined Witten index for $N=4$ Quiver Quantum Mechanics as a sum over multi-centered collinear black hole solutions, weighted by so-called `single-centered or `pure-Higgs indices, and suitably mo
dified when the quiver has oriented cycles. On the other hand, localization expresses the same index as an integral over the complexified Cartan torus and auxiliary fields, which by Stokes theorem leads to the famous Jeffrey-Kirwan residue formula. Here, by evaluating the same integral using steepest descent methods, we show the index is in fact given by a sum over deformed multi-centered collinear solutions, which encompasses both regular and scaling collinear solutions. As a result, we confirm the Coulomb Branch Formula for Abelian quivers in the presence of oriented cycles, and identify the origin of the pure-Higgs and minimal modification terms as coming from collinear scaling solutions. For cyclic Abelian quivers, we observe that part of the scaling contributions reproduce the stacky invariants for trivial stability, a mathematically well-defined notion whose physics significance had remained obscure.
Supersymmetric D-brane bound states on a Calabi-Yau threefold $X$ are counted by generalized Donaldsdon-Thomas invariants $Omega_Z(gamma)$, depending on a Chern character (or electromagnetic charge) $gammain H^*(X)$ and a stability condition (or cent
ral charge) $Z$. Attractor invariants $Omega_*(gamma)$ are special instances of DT invariants, where $Z$ is the attractor stability condition $Z_gamma$ (a generic perturbation of self-stability), from which DT invariants for any other stability condition can be deduced. While difficult to compute in general, these invariants become tractable when $X$ is a crepant resolution of a singular toric Calabi-Yau threefold associated to a brane tiling, and hence to a quiver with potential. We survey some known results and conjectures about framed and unframed refined DT invariants in this context, and compute attractor invariants explicitly for a variety of toric Calabi-Yau threefolds, in particular when $X$ is the total space of the canonical bundle of a smooth projective surface, or when $X$ is a crepant resolution of $C^3/G$. We check that in all these cases, $Omega_*(gamma)=0$ unless $gamma$ is the dimension vector of a simple representation or belongs to the kernel of the skew-symmetrized Euler form. Based on computations in small dimensions, we predict the values of all attractor invariants, thus potentially solving the problem of counting DT invariants of these threefolds in all stability chambers. We also compute the non-commutative refined DT invariants and verify that they agree with the counting of molten crystals in the unrefined limit.
Supersymmetric D-branes supported on the complex two-dimensional base $S$ of the local Calabi-Yau threefold $K_S$ are described by semi-stable coherent sheaves on $S$. Under suitable conditions, the BPS indices counting these objects (known as genera
lized Donaldson-Thomas invariants) coincide with the Vafa-Witten invariants of $S$ (which encode the Betti numbers of the moduli space of semi-stable sheaves). For surfaces which admit a strong collection of exceptional sheaves, we develop a general method for computing these invariants by exploiting the isomorphism between the derived category of coherent sheaves and the derived category of representations of a suitable quiver with potential $(Q,W)$ constructed from the exceptional collection. We spell out the dictionary between the Chern class $gamma$ and polarization $J$ on $S$ vs. the dimension vector $vec N$ and stability parameters $veczeta$ on the quiver side. For all examples that we consider, which include all del Pezzo and Hirzebruch surfaces, we find that the BPS indices $Omega_star(gamma)$ at the attractor point (or self-stability condition) vanish, except for dimension vectors corresponding to simple representations and pure D0-branes. This opens up the possibility to compute the BPS indices in any chamber using either the flow tree or the Coulomb branch formula. In all cases we find precise agreement with independent computations of Vafa-Witten invariants based on wall-crossing and blow-up formulae. This agreement suggests that i) generating functions of DT invariants for a large class of quivers coming from strong exceptional collections are mock modular functions of higher depth and ii) non-trivial single-centered black holes and scaling solutions do not exist quantum mechanically in such local Calabi-Yau geometries.
In quiver quantum mechanics with 4 supercharges, supersymmetric ground states are known to be in one-to-one correspondence with Dolbeault cohomology classes on the moduli space of stable quiver representations. Using supersymmetric localization, the
refined Witten index can be expressed as a residue integral with a specific contour prescription, originally due to Jeffrey and Kirwan, depending on the stability parameters. On the other hand, the physical picture of quiver quantum mechanics describing interactions of BPS black holes predicts that the refined Witten index of a non-Abelian quiver can be expressed as a sum of indices for Abelian quivers, weighted by `single-centered invariants. In the case of quivers without oriented loops, we show that this decomposition naturally arises from the residue formula, as a consequence of applying the Cauchy-Bose identity to the vector multiplet contributions. For quivers with loops, the same procedure produces a natural decomposition of the single-centered invariants, which remains to be elucidated. In the process, we clarify some under-appreciated aspects of the localization formula. Part of the results reported herein have been obtained by implementing the Jeffrey-Kirwan residue formula in a public Mathematica code.
At low energies, interactions of massless particles in type II strings compactified on a torus $T^d$ are described by an effective Wilsonian action $mathcal{S}(Lambda)$, consisting of the usual supergravity Lagrangian supplemented by an infinite seri
es of higher-derivative vertices, including the much studied $ abla^{4p+6q} mathcal{R}^4$ gravitational interactions. Using recent results on the asymptotics of the integrands governing four-graviton scattering at genus one and two, I determine the $Lambda$-dependence of the coefficient of the above interaction, and show that the logarithmic terms appearing in the limit $Lambdato 0$ are related to UV divergences in supergravity amplitudes, augmented by stringy interactions. This provides a strong consistency check on the expansion of the integrand near the boundaries of moduli space, in particular it elucidates the appearance of odd zeta values in these expansions. I briefly discuss how these logarithms are reflected in non-analytic terms in the low energy expansion of the string scattering amplitude.
We continue our investigation of the modular graph functions and string invariants that arise at genus-two as coefficients of low energy effective interactions in Type II superstring theory. In previous work, the non-separating degeneration of a genu
s-two modular graph function of weight $w$ was shown to be given by a Laurent polynomial in the degeneration parameter $t$ of degree $(w,w)$. The coefficients of this polynomial generalize genus-one modular graph functions, up to terms which are exponentially suppressed in $t$ as $t to infty$. In this paper, we evaluate this expansion explicitly for the modular graph functions associated with the $D^8 {cal R}^4$ effective interaction for which the Laurent polynomial has degree $(2,2)$. We also prove that the separating degeneration is given by a polynomial in the degeneration parameter $ln (|v|)$ up to contributions which are power-behaved in $v$ as $v to 0$. We further extract the complete, or tropical, degeneration and compare it with the independent calculation of the integrand of the sum of Feynman diagrams that contributes to two-loop type II supergravity expanded to the same order in the low energy expansion. We find that the tropical limit of the string theory integrand reproduces the supergravity integrand as its leading term, but also includes sub-leading terms proportional to odd zeta values that are absent in supergravity and can be ascribed to higher-derivative stringy interactions.
Inspired by the split attractor flow conjecture for multi-centered black hole solutions in N=2 supergravity, we propose a formula expressing the BPS index $Omega(gamma,z)$ in terms of `attractor indices $Omega_*(gamma_i)$. The latter count BPS states
in their respective attractor chamber. This formula expresses the index as a sum over stable flow trees weighted by products of attractor indices. We show how to compute the contribution of each tree directly in terms of asymptotic data, without having to integrate the attractor flow explicitly. Furthermore, we derive new representations for the index which make it manifest that discontinuities associated to distinct trees cancel in the sum, leaving only the discontinuities consistent with wall-crossing. We apply these results in the context of quiver quantum mechanics, providing a new way of computing the Betti numbers of quiver moduli spaces, and compare them with the Coulomb branch formula, clarifying the relation between attractor and single-centered indices.
Three-dimensional string models with half-maximal supersymmetry are believed to be invariant under a large U-duality group which unifies the S and T dualities in four dimensions. We propose an exact, U-duality invariant formula for four-derivative sc
alar couplings of the form $F(Phi) ( ablaPhi)^4$ in a class of string vacua known as CHL $mathbb{Z}_N$ heterotic orbifolds with $N$ prime, generalizing our previous work which dealt with the case of heterotic string on $T^6$. We derive the Ward identities that $F(Phi)$ must satisfy, and check that our formula obeys them. We analyze the weak coupling expansion of $F(Phi)$, and show that it reproduces the correct tree-level and one-loop contributions, plus an infinite series of non-perturbative contributions. Similarly, the large radius expansion reproduces the exact $F^4$ coupling in four dimensions, including both supersymmetric invariants, plus infinite series of instanton corrections from half-BPS dyons winding around the large circle, and from Taub-NUT instantons. The summation measure for dyonic instantons agrees with the helicity supertrace for half-BPS dyons in 4 dimensions in all charge sectors. In the process we clarify several subtleties about CHL models in $D=4$ and $D=3$, in particular we obtain the exact helicity supertraces for 1/2-BPS dyonic states in all duality orbits.
We analyze four- and six-derivative couplings in the low energy effective action of $D=3$ string vacua with half-maximal supersymmetry. In analogy with an earlier proposal for the $( ablaPhi)^4$ coupling, we propose that the $ abla^2( ablaPhi)^4$ cou
pling is given exactly by a manifestly U-duality invariant genus-two modular integral. In the limit where a circle in the internal torus decompactifies, the $ abla^2( ablaPhi)^4$ coupling reduces to the $D^2 F^4$ and $R^2 F^2$ couplings in $D=4$, along with an infinite series of corrections of order $e^{-R}$, from four-dimensional 1/4-BPS dyons whose wordline winds around the circle. Each of these contributions is weighted by a Fourier coefficient of a meromorphic Siegel modular form, explaining and extending standard results for the BPS index of 1/4-BPS dyons.
The Kawazumi-Zhang invariant $varphi$ for compact genus-two Riemann surfaces was recently shown to be a eigenmode of the Laplacian on the Siegel upper half-plane, away from the separating degeneration divisor. Using this fact and the known behavior o
f $varphi$ in the non-separating degeneration limit, it is shown that $varphi$ is equal to the Theta lift of the unique (up to normalization) weak Jacobi form of weight $-2$. This identification provides the complete Fourier-Jacobi expansion of $varphi$ near the non-separating node, gives full control on the asymptotics of $varphi$ in the various degeneration limits, and provides a efficient numerical procedure to evaluate $varphi$ to arbitrary accuracy. It also reveals a mock-type holomorphic Siegel modular form of weight $-2$ underlying $varphi$. From the general relation between the Faltings invariant, the Kawazumi-Zhang invariant and the discriminant for hyperelliptic Riemann surfaces, a Theta lift representation for the Faltings invariant in genus two readily follows.
