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BPS states in type II string compactified on a Calabi-Yau threefold can typically be decomposed as moduli-dependent bound states of absolutely stable constituents, with a hierarchical structure labelled by attractor flow trees. This decomposition is best understood from the scattering diagram, an arrangement of real codimension-one loci (or rays) in the space of stability conditions where BPS states of given electromagnetic charge and fixed phase of the central charge exist. The consistency of the diagram when rays intersect determines all BPS indices in terms of the `attractor indices' carried by the initial rays. In this work we study the scattering diagram for a non-compact toric CY threefold known as local $\mathbb{F}_0$, namely the total space of the canonical bundle over $\mathbb{P}^1\times \mathbb{P}^1$. We first construct the scattering diagram for the quiver, valid near the orbifold point, and for the large volume slice, valid when both $\mathbb{P}^1$'s have large (and nearly equal) area. We then combine the insights gained from these simple limits to construct the scattering diagram along the physical slice of $\Pi$-stability conditions, which carries an action of a $\mathbb{Z}^4$ extension of the modular group $\Gamma_0(4)$. We sketch a proof of the Split Attractor Flow Tree Conjecture in this example, albeit for a restricted range of the central charge phase. Most arguments are similar to our early study of local $\mathbb{P}^2$ [arXiv:2210.10712], but complicated by the occurence of an extra mass parameter and ramification points on the $\Pi$-stability slice., Comment: 64 pages, 36 colorful figures

Motivated in part by the modular properties of enumerative invariants of K3-fibered Calabi-Yau threefolds, we introduce a family of 39 Calabi-Yau mirror pairs $(X,Y)$ with $h_{1,1}(X)=h_{2,1}(Y)=2$, labelled by certain integer quadruples $(m,i,j,s)$ with $m\leq 11$. On the A-model side, $X$ arises as a complete intersection in a projective bundle over a Fano fourfold $V_m^{[i,j]}$, and admits a Tyurin degeneration into a pair of degree $m$ Fano threefolds $F_m^{[i]}\cup F_m^{[j]}$ intersecting on an anticanonical K3 divisor of degree $2m$. On the B-model side, $Y$ is fibered by $M_{m}$-polarized K3-surfaces of Picard rank 19, and determined by a branched covering of $\mathbb{P}^1$, consistent with the Doran-Harder-Thompson mirror conjecture. When $s=0$, $Y$ itself acquires a Tyurin degeneration, and correspondingly $X$ acquires a fibration by degree $2m$ K3 surfaces, such that the two K\"ahler moduli control the size of the K3-fiber and base $\mathbb{P}^1$. While the mirror pairs with $m\leq 4$ can be realized as complete intersections in products of projective spaces or as hypersurfaces in toric varieties, the examples with $m\geq 5$ are intrinsically non-toric. We obtain uniform formulae for the genus 0 and 1 topological free energies near the Tyurin degeneration (mirror to the large base limit), exhibiting modular properties under the Fricke-extended congruence group $\Gamma_0(m)^+$. We use these results to compute the vertical Gopakumar-Vafa and Noether-Lefschetz invariants and check that their generating functions satisfy the expected modular properties. We also compute generating series of Gopakumar-Vafa invariants with fixed non-zero base degree and exhibit their modular properties., Comment: 46+36 pages, 3 figures, data available from https://github.com/bpioline/TwoParameterK3

In previous work, we used new mathematical relations between Gopakumar-Vafa (GV) invariants and rank 0 Donaldson-Thomas (DT) invariants to determine the first few terms in the generating series of Abelian D4-D2-D0 indices for a class of compact one-parameter Calabi-Yau threefolds. This allowed us to obtain striking checks of S-duality, namely the prediction that these series should be vector-valued weakly holomorphic modular forms under $SL(2,\mathbb{Z})$. In this work, we extend this analysis to the case of D4-D2-D0 indices with two units of D4-brane charge, where S-duality instead predicts that the corresponding generating series should be mock modular with a specific shadow. For the degree 10 hypersurface in weighted projective space $\mathbb{P}_{5,2,1,1,1}$, and the degree 8 hypersurface in $\\mathbb{P}_{4,1,1,1,1}$, where GV invariants can be computed to sufficiently high genus, we find that the first few terms indeed match a unique mock modular form with the required properties, which we determine explicitly. Turning the argument around, we obtain new boundary conditions on the holomorphic ambiguity of the topological string amplitude, which in principle allow to determine it completely up to genus 95 and 112, respectively, i.e. almost twice the maximal genus obtainable using gap and ordinary Castelnuovo vanishing conditions., Comment: 34 pages

We study the resurgent structure of the refined topological string partition function on a non-compact Calabi-Yau threefold, at large orders in the string coupling constant $g_s$ and fixed refinement parameter $\mathsf{b}$. For $\mathsf{b}\neq 1$, the Borel transform admits two families of simple poles, corresponding to integral periods rescaled by $\mathsf{b}$ and $1/\mathsf{b}$. We show that the corresponding Stokes automorphism is expressed in terms of a generalization of the non-compact quantum dilogarithm, and we conjecture that the Stokes constants are determined by the refined Donaldson-Thomas invariants counting spin-$j$ BPS states. This jump in the refined topological string partition function is a special case (unit five-brane charge) of a more general transformation property of wave functions on quantum twisted tori introduced in earlier work by two of the authors. We show that this property follows from the transformation of a suitable refined dual partition function across BPS rays, defined by extending the Moyal star product to the realm of contact geometry.

By exploiting new mathematical relations between Pandharipande-Thomas (PT) invariants, closely related to Gopakumar-Vafa (GV) invariants, and rank 0 Donaldson-Thomas (DT) invariants counting D4-D2-D0 BPS bound states, we rigorously compute the first few terms in the generating series of Abelian D4-D2-D0 indices for compact one-parameter Calabi-Yau threefolds of hypergeometric type. In all cases where GV invariants can be computed to sufficiently high genus, we find striking confirmation that the generating series is modular, and predict infinite series of Abelian D4-D2-D0 indices. Conversely, we use these results to provide new constraints for the direct integration method, which allows to compute GV invariants (and therefore the topological string partition function) to higher genus than hitherto possible. The triangle of relations between GV/PT/DT invariants is powered by a new explicit formula relating PT and rank 0 DT invariants, which is proven in an Appendix by the second named author. As a corollary, we obtain rigorous Castelnuovo-type bounds for PT and GV invariants for CY threefolds with Picard rank one., Comment: 39+24 pages, 6 figures, 13 tables of GV and PT invariants, further data available from http://www.th.physik.uni-bonn.de/Groups/Klemm/data.php; v2: minor corrections, version to appear in Comm. Num. Theo. Phys

The spectrum of BPS states in type IIA string theory compactified on a Calabi-Yau threefold famously jumps across codimension-one walls in complexified K\"ahler moduli space, leading to an intricate chamber structure. The Split Attractor Flow Conjecture posits that the BPS index $\Omega_z(\gamma)$ for given charge $\gamma$ and moduli $z$ can be reconstructed from the attractor indices $\Omega_*(\gamma_i)$ counting BPS states of charge $\gamma_i$ in their respective attractor chamber, by summing over a finite set of decorated rooted flow trees known as attractor flow trees. If correct, this provides a classification (or dendroscopy) of the BPS spectrum into different topologies of nested BPS bound states, each having a simple chamber structure. Here we investigate this conjecture for the simplest, albeit non-compact, Calabi-Yau threefold, namely the canonical bundle over the projective plane $P^2$. Since the K\"ahler moduli space has complex dimension one and the attractor flow preserves the argument of the central charge, attractor flow trees coincide with scattering sequences of rays in a two-dimensional slice of the scattering diagram in the space of stability conditions on the derived category of compactly supported coherent sheaves on $K_{P^2}$. We combine previous results on the scattering diagram of $K_{P^2}$ in the large volume slice with new results near the orbifold point $\mathbb{C}^3/\mathbb{Z}_3$, and prove that the Split Attractor Flow Conjecture holds true on the physical slice of $\Pi$-stability conditions. In particular, while there is an infinite set of initial rays related by the group $\Gamma_1(3)$ of auto-equivalences, only a finite number of possible decompositions $\gamma=\sum_i\gamma_i$ contribute to the index $\Omega_z(\gamma)$ for any $\gamma$ and $z$, with constituents $\gamma_i$ related by spectral flow to the fractional branes at the orbifold point., Comment: 61+23 pages, 32 figures; v3: final version to appear in Comm. Math. Phys

We investigate the modularity constraints on the generating series $h_r(\tau)$ of BPS indices counting D4-D2-D0 bound states with fixed D4-brane charge $r$ in type IIA string theory compactified on complete intersection Calabi-Yau threefolds with $b_2 = 1$. For unit D4-brane, $h_1$ transforms as a (vector-valued) modular form under the action of $SL(2,Z)$ and thus is completely determined by its polar terms. We propose an Ansatz for these terms in terms of rank 1 Donaldson-Thomas invariants, which incorporates contributions from a single D6-anti-D6 pair. Using an explicit overcomplete basis of the relevant space of weakly holomorphic modular forms (valid for any $r$), we find that for 10 of the 13 allowed threefolds, the Ansatz leads to a solution for $h_1$ with integer Fourier coefficients, thereby predicting an infinite series of DT invariants.For $r > 1$, $h_r$ is mock modular and determined by its polar part together with its shadow. Restricting to $r = 2$, we use the generating series of Hurwitz class numbers to construct a series $h^{an}_2$ with exactly the same modular anomaly as $h_2$, so that the difference $h_{2}-h^{an}_2$ is an ordinary modular form fixed by its polar terms. For lack of a satisfactory Ansatz, we leave the determination of these polar terms as an open problem., Comment: 44 pages; v4: minor corrections, version to appear in ATMP, see [arXiv:2301.08066] for a rigorous determination of the polar terms when $r=1$

For suitable charges of the constituents, the phase space of multi-centered BPS black holes in N=2 four-dimensional supergravity famously exhibits scaling regions where the distances between the centers can be made arbitrarily small, so that the bound state becomes indistinguishable from a single-centered black hole. In this note we establish necessary conditions on the Dirac product of charges for the existence of such regions for any number of centers, generalizing the standard triangular inequalities in the three-center case. Interestingly, the same conditions are necessary for the existence of multi-centered solutions at the attractor point. We prove that similar conditions are also necessary for the existence of self-stable representations of the corresponding quiver, as suggested by the duality between the Coulomb and Higgs branches of supersymmetric quantum mechanics., Comment: In this v3 we have changed a bit the presentation according to our reviewer's suggestions, and have extended the results on the Higgs branch to a wider generality

We revisit the Riemann-Hilbert problem determined by Donaldson-Thomas invariants for the resolved conifold and for other small crepant resolutions. While this problem can be recast as a system of TBA-type equations in the conformal limit, solutions are ill-defined due to divergences in the sum over infinite trajectories in the spectrum of D2-D0-brane bound states. We explore various prescriptions to make the sum well-defined, show that one of them reproduces the existing solution in the literature, and identify an alternative solution which is better behaved in a certain limit. Furthermore, we show that a suitable asymptotic expansion of the $\tau$ function reproduces the genus expansion of the topological string partition function for any small crepant resolution. As a by-product, we conjecture new integral representations for the triple sine function, similar to Woronowicz' integral representation for Faddeev's quantum dilogarithm., Comment: 25+14 pages; added proof for integral representation of the double sine function and a similar conjecture for the triple sine; version accepted for publication in Annales Henri Poincar\'e

Recently T. Bridgeland defined a complex hyperk\"ahler metric on the tangent bundle over the space of stability conditions of a triangulated category, based on a Riemann-Hilbert problem determined by the Donaldson-Thomas invariants. This metric is encoded in a function $W(z,\theta)$ satisfying a heavenly equation, or a potential $F(z,\theta)$ satisfying an isomonodromy equation. After recasting the RH problem into a system of TBA-type equations, we obtain integral expressions for both $W$ and $F$ in terms of solutions of that system. These expressions are recognized as conformal limits of the `instanton generating potential' and `contact potential' appearing in studies of D-instantons and BPS black holes. By solving the TBA equations iteratively, we reproduce Joyce's original construction of $F$ as a formal series in the rational DT invariants. Furthermore, we produce similar solutions to deformed versions of the heavenly and isomonodromy equations involving a non-commutative star-product. In the case of a finite uncoupled BPS structure, we rederive the results previously obtained by Bridgeland and obtain the so-called $\tau$ function for arbitrary values of the fiber coordinates $\theta$, in terms of a suitable two-variable generalization of Barnes' $G$ function., Comment: 21+16 pages; version accepted for publication in Letters in Mathematical Physics

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 modified 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., Comment: 45 pages; v2: added subsection 3.4.2 on Abelian quivers with multiple oriented cycles, plus various clarifications and several references. Converted to SciPost style

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) $\gamma\in H^*(X)$ and a stability condition (or central 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., Comment: 58 pages, 19 figures

In an earlier paper, we constructed the genus-two amplitudes for five external massless states in Type II and Heterotic string theory, and showed that the alpha' expansion of the Type II amplitude reproduces the corresponding supergravity amplitude to leading order. In this paper, we analyze the effective interactions induced by Type IIB superstrings beyond supergravity, both for U(1)_R-preserving amplitudes such as for five gravitons, and for U(1)_R-violating amplitudes such as for one dilaton and four gravitons. At each order in alpha', the coefficients of the effective interactions are given by integrals over moduli space of genus-two modular graph functions, generalizing those already encountered for four external massless states. To leading and sub-leading orders, the coefficients of the effective interactions D^2 R^5 and D^4 R^5 are found to match those of D^4 R^4 and D^6 R^4, respectively, as required by non-linear supersymmetry. To the next order, a D^6 R^5 effective interaction arises, which is independent of the supersymmetric completion of D^8 R^4, and already arose at genus one. A novel identity on genus-two modular graph functions, which we prove, ensures that up to order D^6 R^5, the five-point amplitudes require only a single new modular graph function in addition to those needed for the four-point amplitude. We check that the supergravity limit of U(1)_R-violating amplitudes is free of UV divergences to this order, consistently with the known structure of divergences in Type IIB supergravity. Our results give strong consistency tests on the full five-point amplitude, and pave the way for understanding S-duality beyond the BPS-protected sector., Comment: 91 pages, 3 figures, dedicated to the life, science, and art of Professor Jean-Loup Gervais, v2: minor improvements throughout the paper, published version, v3: corrected normalization factors of parity-even and/or parity-odd terms in equations (2.30), (4.34), (4.44), (4.46) and footnote 20

The full two-loop amplitudes for five massless states in Type~II and Heterotic superstrings are constructed in terms of convergent integrals over the genus-two moduli space of compact Riemann surfaces and integrals of Green functions and Abelian differentials on the surface. The construction combines elements from the BRST cohomology of the pure spinor formulation and from chiral splitting with the help of loop momenta and homology invariance. The $\alpha' \to 0$ limit of the resulting superstring amplitude is shown to be in perfect agreement with the previously known amplitude computed in Type~II supergravity. Investigations of the $\alpha'$ expansion of the Type~II amplitude and comparisons with predictions from S-duality are relegated to a first companion paper. A construction from first principles in the RNS formulation of the genus-two amplitude with five external NS states is relegated to a second companion paper., Comment: 97 pages; minor typos corrected and minor edits in version 2

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 generalized 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 $\vec\zeta$ 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., Comment: 88+40 pages, one figure, quivers aplenty; v5: various misprints corrected

Unlike the $\mathcal{R}^4$ and $\nabla^4\mathcal{R}^4$ couplings, whose coefficients are Langlands-Eisenstein series of the U-duality group, the coefficient $\mathcal{E}_{(0,1)}^{(d)}$ of the $\nabla^6\mathcal{R}^4$ interaction in the low-energy effective action of type II strings compactified on a torus $T^d$ belongs to a more general class of automorphic functions, which satisfy Poisson rather than Laplace-type equations. In earlier work, it was proposed that the exact coefficient is given by a two-loop integral in exceptional field theory, with the full spectrum of mutually 1/2-BPS states running in the loops, up to the addition of a particular Langlands-Eisenstein series. Here we compute the weak coupling and large radius expansions of these automorphic functions for any $d$. We find perfect agreement with perturbative string theory up to genus three, along with non-perturbative corrections which have the expected form for 1/8-BPS instantons and bound states of 1/2-BPS instantons and anti-instantons. The additional Langlands-Eisenstein series arises from a subtle cancellation between the two-loop amplitude with 1/4-BPS states running in the loops, and the three-loop amplitude with mutually 1/2-BPS states in the loops. For $d=4$, the result is shown to coincide with an alternative proposal in terms of a covariantised genus-two string amplitude, due to interesting identities between the Kawazumi-Zhang invariant of genus-two curves and its tropical limit, and between double lattice sums for the particle and string multiplets, which may be of independent mathematical interest., Comment: Minor corrections

Whenever available, refined BPS indices provide considerably more information on the spectrum of BPS states than their unrefined version. Extending earlier work on the modularity of generalized Donaldson-Thomas invariants counting D4-D2-D0 brane bound states in type IIA strings on a Calabi-Yau threefold $\mathfrak{Y}$, we construct the modular completion of generating functions of refined BPS indices supported on a divisor class. Although for compact $\mathfrak{Y}$ the refined indices are not protected, switching on the refinement considerably simplifies the construction of the modular completion. Furthermore, it leads to a non-commutative analogue of the TBA equations, which suggests a quantization of the moduli space consistent with S-duality. In contrast, for a local CY threefold given by the total space of the canonical bundle over a complex surface $S$, refined BPS indices are well-defined, and equal to Vafa-Witten invariants of $S$. Our construction provides a modular completion of the generating function of these refined invariants for arbitrary rank. In cases where all reducible components of the divisor class are collinear (which occurs e.g. when $b_2(\mathfrak{Y})=1$, or in the local case), we show that the holomorphic anomaly equation satisfied by the completed generating function truncates at quadratic order. In the local case, it agrees with an earlier proposal by Minahan et al for unrefined invariants, and extends it to the refined level using the afore-mentioned non-commutative structure. Finally, we show that these general predictions reproduce known results for $U(2)$ and $U(3)$ Vafa-Witten theory on $\mathbb{P}^2$, and make them explicit for $U(4)$., Comment: 34+24 pages, 6 figures; misprints fixed

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., Comment: 40 pages; v2: few misprints corrected, title changed to match published version in JHEP. Mathematica code CoulombHiggs.m available from http://www.lpthe.jussieu.fr/~pioline/computing.html

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 series of higher-derivative vertices, including the much studied $\nabla^{4p+6q} \mathcal{R}^4$ 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 $\Lambda\to 0$ 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., Comment: 40 pages; v2: after fixing a factor of 2 mistake in Eq. (2.41), all divergent terms now agree with SUGRA predictions. Added a note at end of Sec 1 on the definition of the truncated moduli space M_{h,n}(\Lambda)

By enforcing invariance under S-duality in type IIB string theory compactified on a Calabi-Yau threefold, we derive modular properties of the generating function of BPS degeneracies of D4-D2-D0 black holes in type IIA string theory compactified on the same space. Mathematically, these BPS degeneracies are the generalized Donaldson-Thomas invariants counting coherent sheaves with support on a divisor $\cal D$, at the large volume attractor point. For $\cal D$ irreducible, this function is closely related to the elliptic genus of the superconformal field theory obtained by wrapping M5-brane on $\cal D$ and is therefore known to be modular. Instead, when $\cal D$ is the sum of $n$ irreducible divisors ${\cal D}_i$, we show that the generating function acquires a modular anomaly. We characterize this anomaly for arbitrary $n$ by providing an explicit expression for a non-holomorphic modular completion in terms of generalized error functions. As a result, the generating function turns out to be a (mixed) mock modular form of depth $n-1$., Comment: 43+35 pages, 11 figures; v2: various cosmetic changes, index of notations added to aid the reader; v3: minor corrections and clarifications, added refs and new prop. 9 on the shadow of the completion

The exact degeneracies of quarter-BPS dyons in Type II string theory on $K3 \times T^2$ are given by Fourier coefficients of the inverse of the Igusa cusp form. For a fixed magnetic charge invariant $m$, the generating function of these degeneracies naturally decomposes as a sum of two parts, which are supposed to account for single-centered black holes, and two-centered black hole bound states, respectively. The decomposition is such that each part is separately modular covariant but neither is holomorphic, calling for a physical interpretation of the non-holomorphy. We resolve this puzzle by computing the supersymmetric index of the quantum mechanics of two-centered half-BPS black-holes, which we model by geodesic motion on Taub-NUT space subject to a certain potential. We compute a suitable index using localization methods, and find that it includes both a temperature-independent contribution from BPS bound states, as well as a temperature-dependent contribution due to a spectral asymmetry in the continuum of scattering states. The continuum contribution agrees precisely with the non-holomorphic completion term required for the modularity of the generating function of two-centered black hole bound states.

Motivated by precision counting of BPS black holes, we analyze six-derivative couplings in the low energy effective action of three-dimensional string vacua with 16 supercharges. Based on perturbative computations up to two-loop, supersymmetry and duality arguments, we conjecture that the exact coefficient of the $\nabla^2(\nabla\phi)^4$ effective interaction is given by a genus-two modular integral of a Siegel theta series for the non-perturbative Narain lattice times a specific meromorphic Siegel modular form. The latter is familiar from the Dijkgraaf-Verlinde-Verlinde (DVV) conjecture on exact degeneracies of 1/4-BPS dyons. We show that this Ansatz reproduces the known perturbative corrections at weak heterotic coupling, including tree-level, one- and two-loop corrections, plus non-perturbative effects of order $e^{-1/g_3^2}$. We also examine the weak coupling expansions in type I and type II string duals and find agreement with known perturbative results, as well as new predictions for higher genus perturbative contributions. In the limit where a circle in the internal torus decompactifies, our Ansatz predicts the exact $\nabla^2 F^4$ effective interaction in four-dimensional CHL string vacua, along with infinite series of exponentially suppressed corrections of order $e^{-R}$ from Euclideanized BPS black holes winding around the circle, and further suppressed corrections of order $e^{-R^2}$ from Taub-NUT instantons. We show that instanton corrections from 1/4-BPS black holes are precisely weighted by the BPS index predicted from the DVV formula, including the detailed moduli dependence. We also extract two-instanton corrections from pairs of 1/2-BPS black holes, demonstrating consistency with supersymmetry and wall-crossing, and estimate the size of instanton-anti-instanton contributions., Comment: 77+78 pages, 1 figure; v2: cosmetic changes, references added

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 genus-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., Comment: 50+44 pages, 13 figures; v2: minor corrections, including an overall factor of 2 in Eq. 5.18, published version; v3: more factors of 2 corrected, see footnote 7

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., Comment: 36+17 pages, 9 figures; v2: figure 4 fixed, to appear in ATMP

The concept and the construction of modular graph functions are generalized from genus-one to higher genus surfaces. The integrand of the four-graviton superstring amplitude at genus-two provides a generating function for a special class of such functions. A general method is developed for analyzing the behavior of modular graph functions under non-separating degenerations in terms of a natural real parameter $t$. For arbitrary genus, the Arakelov Green function and the Kawazumi-Zhang invariant degenerate to a Laurent polynomial in $t$ of degree $(1,1)$ in the limit $t\to\infty$. For genus two, each coefficient of the low energy expansion of the string amplitude degenerates to a Laurent polynomial of degree $(w,w)$ in $t$, where $w+2$ is the degree of homogeneity in the kinematic invariants. These results are exact to all orders in $t$, up to exponentially suppressed corrections. The non-separating degeneration of a general class of modular graph functions at arbitrary genus is sketched and similarly results in a Laurent polynomial in $t$ of bounded degree. The coefficients in the Laurent polynomial are generalized modular graph functions for a punctured Riemann surface of lower genus., Comment: 63 pages, 8 figures; minor typos corrected in new version

We analyze the modular properties of D3-brane instanton corrections to the hypermultiplet moduli space in type IIB string theory compactified on a Calabi-Yau threefold. In Part I, we found a necessary condition for the existence of an isometric action of S-duality on this moduli space: the generating function of DT invariants in the large volume attractor chamber must be a vector-valued mock modular form with specified modular properties. In this work, we prove that this condition is also sufficient at two-instanton order. This is achieved by producing a holomorphic action of SL(2,Z) on the twistor space which preserves the holomorphic contact structure. The key step is to cancel the anomalous modular variation of the Darboux coordinates by a local holomorphic contact transformation, which is generated by a suitable indefinite theta series. For this purpose we introduce a new family of theta series of signature (2,n-2), find their modular completion, and conjecture sufficient conditions for their convergence, which may be of independent mathematical interest., Comment: 24+24 pages

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 scalar couplings of the form $F(\Phi) (\nabla\Phi)^4$ 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., Comment: 42+19 pages; v3: references added, few misprints corrected, several clarifications in section 2.1 and 2.2

In type II string theory compactified on a $d$-dimensional torus $T^d$ down to $D=10-d$ dimensions, the $R^4$ and $\nabla^4 R^4$ four-graviton couplings are known exactly, for all values of the moduli, in terms of certain Eisenstein series of the U-duality group $E_{d}(\mathbb{Z})$. In the limit where one circle in the torus becomes large, these couplings are expected to reduce to their counterpart in dimension $D+1$, plus threshold effects and exponentially suppressed corrections corresponding to BPS black holes in dimension $D+1$ whose worldline winds around the circle. By combining the weak coupling and large radius limits, we determine these exponentially suppressed corrections exactly, and demonstrate that the contributions of 1/4-BPS black holes to the $\nabla^4 R^4$ coupling are proportional to the appropriate helicity supertrace. Mathematically, our results provide the complete Fourier expansion of the next-to-minimal theta series of $E_{d+1}(\mathbb{Z})$ with respect to the maximal parabolic subgroup with Levi component $E_{d}$ for $d\leq 6$, and the complete Abelian part of the Fourier expansion of the same for $d=7$., Comment: 40 pages

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 $(\nabla\Phi)^4$ coupling, we propose that the $\nabla^2(\nabla\Phi)^4$ coupling 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 $\nabla^2(\nabla\Phi)^4$ 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., Comment: 6 pages

Theta series for lattices with indefinite signature $(n_+,n_-)$ arise in many areas of mathematics including representation theory and enumerative algebraic geometry. Their modular properties are well understood in the Lorentzian case ($n_+=1$), but have remained obscure when $n_+\geq 2$. Using a higher-dimensional generalization of the usual (complementary) error function, discovered in an independent physics project, we construct the modular completion of a class of `conformal' holomorphic theta series ($n_+=2$). As an application, we determine the modular properties of a generalized Appell-Lerch sum attached to the lattice $A_2$, which arose in the study of rank 3 vector bundles on $\mathbb{P}^2$. The extension of our method to $n_+>2$ is outlined., Comment: 32 pages, 2 figures; v2: discussed $\Delta_{12}=0$ case at end of section 3, added subsection 4.4 on $C_1=C_2$ case (relevant for signature (2,1)), and added several references; v3: published version in Selecta Mathematica (with apologies for not uploading it earlier)

We study D3-instanton corrections to the hypermultiplet moduli space in type IIB string theory compactified on a Calabi-Yau threefold. In a previous work, consistency of D3-instantons with S-duality was established at first order in the instanton expansion, using the modular properties of the M5-brane elliptic genus. We extend this analysis to the two-instanton level, where wall-crossing phenomena start playing a role. We focus on the contact potential, an analogue of the Kahler potential which must transform as a modular form under S-duality. We show that it can be expressed in terms of a suitable modification of the partition function of D4-D2-D0 BPS black holes, constructed out of the generating function of MSW invariants (the latter coincide with Donaldson-Thomas invariants in a particular chamber). Modular invariance of the contact potential then requires that, in case where the D3-brane wraps a reducible divisor, the generating function of MSW invariants must transform as a vector-valued mock modular form, with a specific modular completion built from the MSW invariants of the constituents. Physically, this gives a powerful constraint on the degeneracies of BPS black holes. Mathematically, our result gives a universal prediction for the modular properties of Donaldson-Thomas invariants of pure two-dimensional sheaves., Comment: 33 pages; v2: minor corrections and clarifications

Closed string amplitudes at genus $h\leq 3$ are given by integrals of Siegel modular functions on a fundamental domain of the Siegel upper half-plane. When the integrand is of rapid decay near the cusps, the integral can be computed by the Rankin-Selberg method, which consists of inserting an Eisenstein series $E_h(s)$ in the integrand, computing the integral by the orbit method, and finally extracting the residue at a suitable value of $s$. String amplitudes, however, typically involve integrands with polynomial or even exponential growth at the cusps, and a renormalization scheme is required to treat infrared divergences. Generalizing Zagier's extension of the Rankin-Selberg method at genus one, we develop the Rankin-Selberg method for Siegel modular functions of degree 2 and 3 with polynomial growth near the cusps. In particular, we show that the renormalized modular integral of the Siegel-Narain partition function of an even self-dual lattice of signature $(d,d)$ is proportional to a residue of the Langlands-Eisenstein series attached to the $h$-th antisymmetric tensor representation of the T-duality group $O(d,d,Z)$., Comment: 53 pages, 3 figures; v2: various clarifications and cosmetic changes, new appendix B on the Rankin-Selberg transform of the lattice partition function in arbitrary degree, small correction to Figure 1

Motivated by mathematical structures which arise in string vacua and gauge theories with N=2 supersymmetry, we study the properties of certain generalized theta series which appear as Fourier coefficients of functions on a twisted torus. In Calabi-Yau string vacua, such theta series encode instanton corrections from $k$ Neveu-Schwarz five-branes. The theta series are determined by vector-valued wave-functions, and in this work we obtain the transformation of these wave-functions induced by Kontsevich-Soibelman symplectomorphisms. This effectively provides a quantum version of these transformations, where the quantization parameter is inversely proportional to the five-brane charge $k$. Consistency with wall-crossing implies a new five-term relation for Faddeev's quantum dilogarithm $\Phi_b$ at $b=1$, which we prove. By allowing the torus to be non-commutative, we obtain a more general five-term relation valid for arbitrary $b$ and $k$, which may be relevant for the physics of five-branes at finite chemical potential for angular momentum., Comment: 26 pages; v2: added discussion on relation to complex Chern-Simons, misprints corrected

We analyze the pertubative contributions to the $D^4 R^4$ and $D^6 R^4$ couplings in the low-energy effective action of type II string theory compactified on a torus $T^d$, with particular emphasis on two-loop corrections. In general, it is necessary to introduce an infrared cut-off $\Lambda$ to separate local interactions from non-local effects due to the exchange of massless states. We identify the degenerations of the genus-two Riemann surface which are responsible for power-like dependence on $\Lambda$, and give an explicit prescription for extracting the $\Lambda$-independent effective couplings. These renormalized couplings are then shown to be eigenmodes of the Laplace operator with respect to the torus moduli, up to computable anomalous source terms arising in the presence of logarithmic divergences, in precise agreement with predictions from U-duality. Our results for the two-loop $D^6 R^4$ contribution also probe essential properties of the Kawazumi-Zhang invariant, Comment: 23 pages

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 of $\varphi$ 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., Comment: 16 pages; v2: many improvements: the main conjecture is now a theorem, numerical checks and applications are performed, connections to Gromov-Witten invariants are discussed, various clarifications throughout, 3 extra pages, 10 extra references; v3: cosmetic changes, added details on the proof of (78), one new reference; v4: journal version

Four-graviton couplings in the low energy effective action of type II string vacua compactified on tori are strongly constrained by supersymmetry and U-duality. While the $R^4$ and $D^4 R^4$ couplings are known exactly in terms of Langlands-Eisenstein series of the U-duality group, the $D^6 R^4$ couplings are not nearly as well understood. Exploiting the coincidence of the U-duality group in $D=6$ with the T-duality group in $D=5$, we propose an exact formula for the $D^6 R^4$ couplings in type II string theory compactified on $T^4$, in terms of a genus-two modular integral plus a suitable Eisenstein series. The same modular integral computes the two-loop correction to $D^6 R^4$ in 5 dimensions, but here provides the non-perturbative completion of the known perturbative terms in $D=6$. This proposal hinges on a systematic re-analysis of the weak coupling and large radius of the $D^6 R^4$ in all dimensions $D\geq 3$, which fills in some gaps and resolves some inconsistencies in earlier studies., Comment: 18 pages (+ 26 page appendix), three Mathematica files included with submission; v2: minor corrections, version to appear in JHEP

We continue our study of one-loop integrals associated to BPS-saturated amplitudes in $\mathcal{N}=2$ heterotic vacua. We compute their large-volume behaviour, and express them as Fourier series in the complexified volume, with Fourier coefficients given in terms of Niebur-Poincar\'e series in the complex structure modulus. The closure of Niebur-Poincar\'e series under modular derivatives implies that such integrals derive from holomorphic prepotentials $f_n$, generalising the familiar prepotential of $\mathcal{N}=2$ supergravity. These holomorphic prepotentials transform anomalously under T-duality, in a way characteristic of Eichler integrals. We use this observation to compute their quantum monodromies under the duality group. We extend the analysis to modular integrals with respect to Hecke congruence subgroups, which naturally arise in compactifications on non-factorisable tori and freely-acting orbifolds. In this case, we derive new explicit results including closed-form expressions for integrals involving the ${\varGamma}_0(N)$ Hauptmodul, a full characterisation of holomorphic prepotentials including their quantum monodromies, as well as concrete formulae for holomorphic Yukawa couplings., Comment: 46 pages, eqs 4.38, 4.40-4.43 and 4.47 corrected

In $D=4,N=2$ theories on $R^{3,1}$, the index receives contributions not only from single-particle BPS states, counted by the BPS indices, but also from multi-particle states made of BPS constituents. In a recent work [arXiv:1406.2360], a general formula expressing the index in terms of the BPS indices was proposed, which is smooth across walls of marginal stability and reproduces the expected single-particle contributions. In this note, I analyze the two-particle contributions predicted by this formula, and show agreement with the spectral asymmetry of the continuum of scattering states in the supersymmetric quantum mechanics of two non-relativistic, mutually non-local dyons. This provides a physical justification for the error function profile used in the mathematics literature on indefinite theta series, and in the physics literature on black hole partition functions., Comment: 14 pages, 1 figure

In theories with $N=2$ supersymmetry on $R^{3,1}$, BPS bound states can decay across walls of marginal stability in the space of Coulomb branch parameters, leading to discontinuities in the BPS indices $\Omega(\gamma,u)$. We consider a supersymmetric index $I$ which receives contributions from 1/2-BPS states, generalizing the familiar Witten index $Tr (-1)^F e^{-\beta H}$. We expect $I$ to be smooth away from loci where massless particles appear, thanks to contributions from the continuum of multi-particle states. Taking inspiration from a similar phenomenon in the hypermultiplet moduli space of $N=2$ string vacua, we conjecture a formula expressing $I$ in terms of the BPS indices $\Omega(\gamma,u)$, which is continuous across the walls and exhibits the expected contributions from single particle states at large $\beta$. This gives a universal prediction for the contributions of multi-particle states to the index $I$. This index is naturally a function on the moduli space after reduction on a circle, closely related to the canonical hyperk\"ahler metric and hyperholomorphic connection on this space., Comment: 7 pages; v2: introduction expanded, minor corrections, differs from published version in PRL in that supplemental material is included as an Appendix

The coefficient of the $D^6 {\cal R}^4$ interaction in the low energy expansion of the two-loop four-graviton amplitude in type II superstring theory is known to be proportional to the integral of the Zhang-Kawazumi (ZK) invariant over the moduli space of genus-two Riemann surfaces. We demonstrate that the ZK invariant is an eigenfunction with eigenvalue 5 of the Laplace-Beltrami operator in the interior of moduli space. Exploiting this result, we evaluate the integral of the ZK invariant explicitly, finding agreement with the value of the two-loop $D^6 {\cal R}^4$ interaction predicted on the basis of S-duality and supersymmetry. A review of the current understanding of the $D^{2p} {\cal R}^4$ interactions in type II superstring theory compactified on a torus $T^d$ with $p \leq 3$ and $d \leq 4$ is included., Comment: 40 pages, various typos and coefficients corrected in version 2

Heterotic string compactifications on a $K3$ surface $\mathfrak{S}$ depend on a choice of hyperk\"ahler metric, anti-self-dual gauge connection and Kalb-Ramond flux, parametrized by hypermultiplet scalars. The metric on hypermultiplet moduli space is in principle computable within the $(0,2)$ superconformal field theory on the heterotic string worldsheet, although little is known about it in practice. Using duality with type II strings compactified on a Calabi-Yau threefold, we predict the form of the quaternion-K\"ahler metric on hypermultiplet moduli space when $\mathfrak{S}$ is elliptically fibered, in the limit of a large fiber and even larger base. The result is in general agreement with expectations from Kaluza-Klein reduction, in particular the metric has a two-stage fibration structure, where the $B$-field moduli are fibered over bundle and metric moduli, while bundle moduli are themselves fibered over metric moduli. A more precise match must await a detailed analysis of $R^2$-corrected ten-dimensional supergravity., Comment: 20 pages

In recent series of works, by translating properties of multi-centered supersymmetric black holes into the language of quiver representations, we proposed a formula that expresses the Hodge numbers of the moduli space of semi-stable representations of quivers with generic superpotential in terms of a set of invariants associated to `single-centered' or `pure-Higgs' states. The distinguishing feature of these invariants is that they are independent of the choice of stability condition. Furthermore they are uniquely determined by the $\chi_y$-genus of the moduli space. Here, we provide a self-contained summary of the Coulomb branch formula, spelling out mathematical details but leaving out proofs and physical motivations., Comment: 24 pages. v2: final version; minor changes, including a new diagram

After integrating over supermoduli and vertex operator positions, scattering amplitudes in superstring theory at genus $h\leq 3$ are reduced to an integral of a Siegel modular function of degree $h$ on a fundamental domain of the Siegel upper half plane. A direct computation is in general unwieldy, but becomes feasible if the integrand can be expressed as a sum over images under a suitable subgroup of the Siegel modular group: if so, the integration domain can be extended to a simpler domain at the expense of keeping a single term in each orbit -- a technique known as the Rankin-Selberg method. Motivated by applications to BPS-saturated amplitudes, Angelantonj, Florakis and I have applied this technique to one-loop modular integrals where the integrand is the product of a Siegel-Narain theta function times a weakly, almost holomorphic modular form. I survey our main results, and take some steps in extending this method to genus greater than one., Comment: 24 pages, contribution to the proceedings of String-math 2013; v2: minor corrections and improvements, especially in section 4, more references

Quiver quantum mechanics is invariant under Seiberg duality. A mathematical consequence is that the cohomology of the Higgs branch moduli space is invariant under mutations of the quiver. The Coulomb branch formula, on the other hand, conjecturally expresses the Poincar\'e / Dolbeault polynomial of the Higgs branch moduli space in terms of certain quantities known as single-centered indices. In this work we determine the transformations of these single-centered indices under mutations. Moreover, we generalize these mutations to quivers whose nodes carry single-centered indices different from unity. Although the Higgs branch description of these generalized quivers is currently unknown, the Coulomb branch formula is conjectured to be invariant under generalized mutations., Comment: 33 pages, 1 figure; a mathematica notebook using an updated version of the CoulombHiggs.m package released along with our previous work arXiv:1302.5498 is included as ancillary file; v2: refs added, one extra paragraph in sec 2.2

The BPS state spectrum in four-dimensional gauge theories or string vacua with N=2 supersymmetries is well known to depend on the values of the parameters or moduli at spatial infinity. The BPS index is locally constant, but discontinuous across real codimension-one walls where some of the BPS states decay. By postulating that BPS states are bound states of more elementary constituents carrying their own degrees of freedom and interacting via supersymmetric quantum mechanics, we provide a physically transparent derivation of the universal wall-crossing formula which governs the jump of the index. The same physical picture suggests that at any point in moduli space, the total index can be written as a sum of contributions from all possible bound states of elementary, absolutely stable constituents with the same total charge. For D-brane bound states described by quivers, this `Coulomb branch formula' predicts that the cohomology of quiver moduli spaces is uniquely determined by certain `pure-Higgs' invariants, which are the microscopic analogues of single-centered black holes. These lectures are based on joint work with J. Manschot and A. Sen., Comment: 19 pages, 1 figure; prepared for the Proceedings of the Corfu Summer Institute 2012 "School and Workshops on Elementary Particle Physics and Gravity", September 8-27, 2012

In recent work we have developed a new unfolding method for computing one-loop modular integrals in string theory involving the Narain partition function and, possibly, a weak almost holomorphic elliptic genus. Unlike the traditional approach, the Narain lattice does not play any role in the unfolding procedure, T-duality is kept manifest at all steps, a choice of Weyl chamber is not required and the analytic structure of the amplitude is transparent. In the present paper, we generalise this procedure to the case of Abelian Z_N orbifolds, where the integrand decomposes into a sum of orbifold blocks that can be organised into orbits of the Hecke congruence subgroup {\Gamma}_0(N). As a result, the original modular integral reduces to an integral over the fundamental domain of {\Gamma}_0(N), which we then evaluate by extending our previous techniques. Our method is applicable, for instance, to the evaluation of one-loop corrections to BPS-saturated couplings in the low energy effective action of closed string models, of quantum corrections to the K\"ahler metric and, in principle, of the free-energy of superstring vacua., Comment: 47 pages, 1 figure

The hypermultiplet moduli space M_H in type II string theories compactified on a Calabi-Yau threefold X is largely constrained by supersymmetry (which demands quaternion-K\"ahlerity), S-duality (which requires an isometric action of SL(2, Z)) and regularity. Mathematically, M_H ought to encode all generalized Donaldson-Thomas invariants on X consistently with wall-crossing, modularity and homological mirror symmetry. We review recent progress towards computing the exact metric on M_H, or rather the exact complex contact structure on its twistor space., Comment: 31 pages; Contribution to the Proceedings of String Math 2012; v2: references added, misprints corrected, published version

In previous work we have shown that the equivariant index of multi-centered N=2 black holes localizes on collinear configurations along a fixed axis. Here we provide a general algorithm for enumerating such collinear configurations and computing their contribution to the index. We apply this machinery to the case of black holes described by quiver quantum mechanics, and give a systematic prescription -- the Coulomb branch formula -- for computing the cohomology of the moduli space of quiver representations. For quivers without oriented loops, the Coulomb branch formula is shown to agree with the Higgs branch formula based on Reineke's result for stack invariants, even when the dimension vector is not primitive. For quivers with oriented loops, the Coulomb branch formula parametrizes the Poincar\'e polynomial of the quiver moduli space in terms of single-centered (or pure-Higgs) BPS invariants, which are conjecturally independent of the stability condition (i.e. the choice of Fayet-Iliopoulos parameters) and angular-momentum free. To facilitate further investigation we provide a Mathematica package "CoulombHiggs.m" implementing the Coulomb and Higgs branch formulae., Comment: 45 pages, 1 figure; v2 (after publication in JHEP): New recursion scheme introduced in Note Added, 4.31-4.34; Appendix updated to document new features of the Mathematica package "CoulombHiggs.m" v2.0, available along with example files from the submission source, or from http://www.lpthe.jussieu.fr/~pioline/computing.html

Heterotic string theory compactified on a K3 surface times T^2 is believed to be equivalent to type II string theory on a suitable Calabi-Yau threefold. In particular, it must share the same hypermultiplet moduli space. Building on the known twistorial description on the type II side, and on recent progress on the map between type II and heterotic moduli in the `double scaling' limit (where both the type II and heterotic strings become classical), we provide a new twistorial construction of the hypermultiplet moduli space in this limit which is adapted to the symmetries of the heterotic string. We also take steps towards understanding the twistorial description for heterotic worldsheet instanton corrections away from the double scaling limit. As a spin-off, we obtain a twistorial description of a class of automorphic forms of SO(4,n,Z) obtained by Borcherds' lift., Comment: 29 pages; the duality map is restricted to heterotic vacua without bundle moduli, for reasons explained in footnote 1