D6R4D^6 R^4 amplitudes in various dimensions

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 JHE

Four ways across the wall

An important question in the study of N=2 supersymmetric string or field theories is to compute the jump of the BPS spectrum across walls of marginal stability in the space of parameters or vacua. I survey four apparently different answers for this problem, two of which are based on the mathematics of generalized Donaldson-Thomas invariants (the Kontsevich-Soibelman and the Joyce-Song formulae), while the other two are based on the physics of multi-centered black hole solutions (the Coulomb branch and the Higgs branch formulae, discovered in joint work with Jan Manschot and Ashoke Sen). Explicit computations indicate that these formulae are equivalent, though a combinatorial proof is currently lacking.Comment: 17 pages, 2 figures, proceedings of the workshop "Algebra, Geometry and Mathematical Physics", Tj\"arn\"o, Sweden, 25-30 October 201

Rankin-Selberg methods for closed string amplitudes

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 reference

String theory integrands and supergravity divergences

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

A Theta lift representation for the Kawazumi-Zhang and Faltings invariants of genus-two Riemann surfaces

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 versio

Topological wave functions and the 4D-5D lift

We revisit the holomorphic anomaly equations satisfied by the topological string amplitude from the perspective of the 4D-5D lift, in the context of ''magic'' N=2 supergravity theories. In particular, we interpret the Gopakumar-Vafa relation between 5D black hole degeneracies and the topological string amplitude as the result of a canonical transformation from 4D to 5D charges. Moreover we use the known Bekenstein-Hawking entropy of 5D black holes to constrain the asymptotic behavior of the topological wave function at finite topological coupling but large K\"ahler classes. In the process, some subtleties in the relation between 5D black hole degeneracies and the topological string amplitude are uncovered, but not resolved. Finally we extend these considerations to the putative one-parameter generalization of the topological string amplitude, and identify the canonical transformation as a Weyl reflection inside the 3D duality group.Comment: minor corrections, version to appear in JHE

Theta series, wall-crossing and quantum dilogarithm identities

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 correcte

On the Rankin-Selberg method for higher genus string amplitudes

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

Corfu lectures on wall-crossing, multi-centered black holes, and quiver invariants

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, 201