SUSY in Silico: numerical D-brane bound state spectroscopy

We numerically construct the BPS and non-BPS wavefunctions of an $\mathcal{N}=4$ quiver quantum mechanics with two Abelian nodes and a single arrow. This model captures the dynamics of a pair of wrapped D-branes interacting via a single light string mode. A dimensionless parameter $\nu$, which is inversely proportional to the Fayet-Iliopoulos parameter, controls whether the bulk of the wavefunctions are supported on the Higgs branch or the Coulomb branch. We demonstrate how the BPS and excited states morph as $\nu$ is tuned. We also numerically compute the energy gap between the ground state and the first excited states as a function of $\nu$. An expression for the gap, computed on the Coulomb branch, matches nicely with our numerics at large $\nu$ but deviates at small $\nu$ where the Higgs branch becomes the relevant description of the physics. In the appendix, we provide the Schr\"{o}dinger equations fully reduced via symmetries which, in principle, allow for the numerical determination of the entire spectrum at any point in moduli space. For the ground states, this numerical determination of the spectrum can be thought of as the first \emph{in silico} check of various Witten index calculations.Comment: 23 pages, 4 figures, v2. slight modifications, v3. references added, typos correcte

Parity and the modular bootstrap

We consider unitary, modular invariant, two-dimensional CFTs which are invariant under the parity transformation $P$. Combining $P$ with modular inversion $S$ leads to a continuous family of fixed points of the $SP$ transformation. A particular subset of this locus of fixed points exists along the line of positive left- and right-moving temperatures satisfying $\beta_L \beta_R = 4\pi^2$. We use this fixed locus to prove a conjecture of Hartman, Keller, and Stoica that the free energy of a large-$c$ CFT$_2$ with a suitably sparse low-lying spectrum matches that of AdS$_3$ gravity at all temperatures and all angular potentials. We also use the fixed locus to generalize the modular bootstrap equations, obtaining novel constraints on the operator spectrum and providing a new proof of the statement that the twist gap is smaller than $(c-1)/12$ when $c>1$. At large $c$ we show that the operator dimension of the first excited primary lies in a region in the $(h,\overline{h})$-plane that is significantly smaller than $h+\overline{h}<c/6$. Our results for the free energy and constraints on the operator spectrum extend to theories without parity symmetry through the construction of an auxiliary parity-invariant partition function.Comment: 21 pages, 3 figures, v2 reference and equation added, v3 minor edits and figure 2 improve

Inside Out: Meet The Operators Inside The Horizon

Based on the work of Heemskerk, Marolf, Polchinski and Sully (HMPS), we study the reconstruction of operators behind causal horizons in time dependent geometries obtained by acting with shockwaves on pure states or thermal states. These geometries admit a natural basis of gauge invariant operators, namely those geodesically dressed to the boundary along geodesics which emanate from the bifurcate horizon at constant Rindler time. We outline a procedure for obtaining operators behind the causal horizon but inside the entanglement wedge by exploiting the equality between bulk and boundary time evolution, as well as the freedom to consider the operators evolved by distinct Hamiltonians. This requires we carefully keep track of how the operators are gravitationally dressed and that we address issues regarding background dependence. We compare this procedure to reconstruction using modular flow, and illustrate some formal points in simple cases such as AdS$_2$ and AdS$_3$.Comment: 48 pages, 14 figure

Marginal Deformations and Rotating Horizons

Motivated by the near-horizon geometry of four-dimensional extremal black holes, we study a disordered quantum mechanical system invariant under a global $SU(2)$ symmetry. As in the Sachdev-Ye-Kitaev model, this system exhibits an approximate $SL(2,\mathbb{R})$ symmetry at low energies, but also allows for a continuous family of $SU(2)$ breaking marginal deformations. Beyond a certain critical value for the marginal coupling, the model exhibits a quantum phase transition from the gapless phase to a gapped one and we calculate the critical exponents of this transition. We also show that charged, rotating extremal black holes exhibit a transition when the angular velocity of the horizon is tuned to a certain critical value. Where possible we draw parallels between the disordered quantum mechanics and charged, rotating black holes.Comment: 29 pages, 5 figure

From Conformal Blocks to Path Integrals in the Vaidya Geometry

Correlators in conformal field theory are naturally organized as a sum over conformal blocks. In holographic theories, this sum must reorganize into a path integral over bulk fields and geometries. We explore how these two sums are related in the case of a point particle moving in the background of a 3d collapsing black hole. The conformal block expansion is recast as a sum over paths of the first-quantized particle moving in the bulk geometry. Off-shell worldlines of the particle correspond to subdominant contributions in the Euclidean conformal block expansion, but these same operators must be included in order to correctly reproduce complex saddles in the Lorentzian theory. During thermalization, a complex saddle dominates under certain circumstances; in this case, the CFT correlator is not given by the Virasoro identity block in any channel, but can be recovered by summing heavy operators. This effectively converts the conformal block expansion in CFT from a sum over intermediate states to a sum over channels that mimics the bulk path integral.Comment: 23 pages, 8 figure

Black Hole Collapse in the 1/c Expansion

We present a first-principles CFT calculation corresponding to the spherical collapse of a shell of matter in three dimensional quantum gravity. In field theory terms, we describe the equilibration process, from early times to thermalization, of a CFT following a sudden injection of energy at time t=0. By formulating a continuum version of Zamolodchikov's monodromy method to calculate conformal blocks at large central charge c, we give a framework to compute a general class of probe observables in the collapse state, incorporating the full backreaction of matter fields on the dual geometry. This is illustrated by calculating a scalar field two-point function at time-like separation and the time-dependent entanglement entropy of an interval, both showing thermalization at late times. The results are in perfect agreement with previous gravity calculations in the AdS$_3$-Vaidya geometry. Information loss appears in the CFT as an explicit violation of unitarity in the 1/c expansion, restored by nonperturbative corrections.Comment: 39 pages, references added, corresponds with published versio

Glassy slowdown and replica-symmetry-breaking instantons

Glass-forming liquids exhibit a dramatic dynamical slowdown as the temperature is lowered. This can be attributed to relaxation proceeding via large structural rearrangements whose characteristic size increases as the system cools. These cooperative rearrangements are well modeled by instantons in a replica effective field theory, with the size of the dominant instanton encoding the liquid's cavity point-to-set correlation length. Varying the parameters of the effective theory corresponds to varying the statistics of the underlying free-energy landscape. We demonstrate that, for a wide range of parameters, replica-symmetry-breaking instantons dominate. The detailed structure of the dominant instanton provides a rich window into point-to-set correlations and glassy dynamics.Comment: 6 pages, 3 figures; v2: narrative revised to clarify our effective-theoretic viewpoint, results unchanged, added reference

Explorations in de Sitter Space and Amorphous Black Hole Bound States in String Theory

This dissertation is split into two distinct halves. The first covers various calculations done in order gain insights on holography in de Sitter space. The dispersion relation of linear perturbations of empty de Sitter space are numerically computed as a function of the location of a hypersurface on which conformal Dirichlet boundary conditions are imposed. When the hypersurface is near the south pole, the dispersion relation is linear, whereas for a hypersurface near the cosmological horizon, it satisfies that of the incompressible Navier-Stokes equation. This result is shown to hold for non-linear perturbations. We also compute the thermodynamic stability of rotating black holes in $dS_4$ as a function of their mass and angular momentum. We focus particularly on the rotating Nariai geometry, which is a near horizon limit of the rotating black hole as the outer and cosmological horizons tend towards each other. We study massless scalar fields in these backgrounds and obtain their quasinormal mode spectrum explicitly. We uncover an interesting structure in their two-point functions, namely that they resemble thermal Green's functions of a two-dimensional conformal field theory. The second half of this dissertation deals with the study of multicentered black holes in string theory and their finite temperature extensions. We show that there exist finite temperature single-centered solutions in $\mathcal{N}=2$ supergravity in asymptotically flat space that admit bound states with BPS probe particles. We compute the existence regions of these bound states as well as their dependence on temperature. We embed these solutions in Fayet-Illiopoulos gauged supergravity and show that bound states persist in asymptotically $AdS_4$ spacetimes. We make attempts to understand these disordered bound states as amorphous/glassy phases of the dual conformal field theory.Physic

Late-time Structure of the Bunch-Davies De Sitter Wavefunction

We examine the late time behavior of the Bunch-Davies wavefunction for interacting light fields in a de Sitter background. We use perturbative techniques developed in the framework of AdS/CFT, and analytically continue to compute tree and loop level contributions to the Bunch-Davies wavefunction. We consider self-interacting scalars of general mass, but focus especially on the massless and conformally coupled cases. We show that certain contributions grow logarithmically in conformal time both at tree and loop level. We also consider gauge fields and gravitons. The four-dimensional Fefferman-Graham expansion of classical asymptotically de Sitter solutions is used to show that the wavefunction contains no logarithmic growth in the pure graviton sector at tree level. Finally, assuming a holographic relation between the wavefunction and the partition function of a conformal field theory, we interpret the logarithmic growths in the language of conformal field theory.Comment: 41 pages, 1 figure, minor typos fixed, journal references adde