arXiv subjects

Scott Collier

Publications and source records attributed to Scott Collier.

29 records · Page 2Linked to original sources

arXiv2021

Wormholes and Spectral Statistics in the Narain Ensemble

We study the spectral statistics of primary operators in the recently formulated ensemble average of Narain's family of free boson conformal field theories, which provides an explicit (though exotic) example of an averaged holographic duality. In particular we study moments of the partition function by explicit computation of higher-degree Eisenstein series. This describes the analog of wormhole contributions coming from a sum of over geometries in the dual theory of "U(1) gravity" in AdS$_3$. We give an exact formula for the two-point correlation function of the density of primary states. We compute the spectral form factor and show that the wormhole sum reproduces precisely the late time plateau behaviour related to the discreteness of the spectrum. The spectral form factor does not exhibit a linear ramp.

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arXiv2020

Strings in Ramond-Ramond Backgrounds from the Neveu-Schwarz-Ramond Formalism

We treat RR flux backgrounds of type II string theory in the framework of closed superstring field theory based on the NSR formalism, focusing on two examples: (1) the pp-wave background supported by 5-form flux, and (2) $AdS_3\times S^3\times M_4$ supported by mixed 3-form fluxes. In both cases, we analyze the classical string field solution perturbatively, and compute the correction to the dispersion relation of string states to quadratic order in the RR flux. In the first example, our result is in a delicate way consistent with that obtained from lightcone quantization of the Green-Schwarz string. In the second example, we will obtain numerically the mass corrections to pulsating type IIB strings in $AdS_3\times S^3\times M_4$. Our results, valid at finite $AdS$ radius, agree with previously known answers in the semiclassical limit and in the BMN limit respectively.

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arXiv2020

Pure Gravity and Conical Defects

We revisit the spectrum of pure quantum gravity in AdS$_3$. The computation of the torus partition function will -- if computed using a gravitational path integral that includes only smooth saddle points -- lead to a density of states which is not physically sensible, as it has a negative degeneracy of states for some energies and spins. We consider a minimal cure for this non-unitarity of the pure gravity partition function, which involves the inclusion of additional states below the black hole threshold. We propose a geometric interpretation for these extra states: they are conical defects with deficit angle $2π(1-1/N)$, where $N$ is a positive integer. That only integer values of $N$ should be included can be seen from a modular bootstrap argument, and leads us to propose a modest extension of the set of saddle-point configurations that contribute to the gravitational path integral: one should sum over orbifolds in addition to smooth manifolds. These orbifold states are below the black hole threshold and are regarded as massive particles in AdS, but they are not perturbative states: they are too heavy to form multi-particle bound states. We compute the one-loop determinant for gravitons in these orbifold backgrounds, which confirms that the orbifold states are Virasoro primaries. We compute the gravitational partition function including the sum over these orbifolds and find a finite, modular invariant result; this finiteness involves a delicate cancellation between the infinite tower of orbifold states and an infinite number of instantons associated with $PSL(2,{\mathbb Z})$ images.

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arXiv2020

Universal Dynamics of Heavy Operators in CFT$_2$

We obtain an asymptotic formula for the average value of the operator product expansion coefficients of any unitary, compact two dimensional CFT with $c>1$. This formula is valid when one or more of the operators has large dimension or -- in the presence of a twist gap -- has large spin. Our formula is universal in the sense that it depends only on the central charge and not on any other details of the theory. This result unifies all previous asymptotic formulas for CFT$_2$ structure constants, including those derived from crossing symmetry of four point functions, modular covariance of torus correlation functions, and higher genus modular invariance. We determine this formula at finite central charge by deriving crossing kernels for higher genus crossing equations, which give analytic control over the structure constants even in the absence of exact knowledge of the conformal blocks. The higher genus modular kernels are obtained by sewing together the elementary kernels for four-point crossing and modular transforms of torus one-point functions. Our asymptotic formula is related to the DOZZ formula for the structure constants of Liouville theory, and makes precise the sense in which Liouville theory governs the universal dynamics of heavy operators in any CFT. The large central charge limit provides a link with 3D gravity, where the averaging over heavy states corresponds to a coarse-graining over black hole microstates in holographic theories. Our formula also provides an improved understanding of the Eigenstate Thermalization Hypothesis (ETH) in CFT$_2$, and suggests that ETH can be generalized to other kinematic regimes in two dimensional CFTs.

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arXiv2019

Quantum Regge Trajectories and the Virasoro Analytic Bootstrap

Every conformal field theory (CFT) above two dimensions contains an infinite set of Regge trajectories of local operators which, at large spin, asymptote to "double-twist" composites with vanishing anomalous dimension. In two dimensions, due to the existence of local conformal symmetry, this and other central results of the conformal bootstrap do not apply. We incorporate exact stress tensor dynamics into the CFT$_2$ analytic bootstrap, and extract several implications for AdS$_3$ quantum gravity. Our main tool is the Virasoro fusion kernel, which we newly analyze and interpret in the bootstrap context. The contribution to double-twist data from the Virasoro vacuum module defines a "Virasoro Mean Field Theory" (VMFT), its spectrum includes a finite number of discrete Regge trajectories, whose dimensions obey a simple formula exact in the central charge $c$ and external operator dimensions. We then show that VMFT provides a baseline for large spin universality in two dimensions: in every unitary compact CFT$_2$ with $c > 1$ and a twist gap above the vacuum, the double-twist data approaches that of VMFT at large spin $\ell$. Corrections to the large spin spectrum from individual non-vacuum primaries are exponentially small in $\sqrt{\ell}$ for fixed $c$. We analyze our results in various large $c$ limits. Further applications include a derivation of the late-time behavior of Virasoro blocks at generic $c$, a refined understanding and new derivation of heavy-light blocks, and the determination of the cross-channel limit of generic Virasoro blocks. We deduce non-perturbative results about the bound state spectrum and dynamics of quantum gravity in AdS$_3$.

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arXiv2018

Genus Two Modular Bootstrap

We study the Virasoro conformal block decomposition of the genus two partition function of a two-dimensional CFT by expanding around a Z3-invariant Riemann surface that is a three-fold cover of the Riemann sphere branched at four points, and explore constraints from genus two modular invariance and unitarity. In particular, we find 'critical surfaces' that constrain the structure constants of a CFT beyond what is accessible via the crossing equation on the sphere.

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arXiv2018

Recursive Representations of Arbitrary Virasoro Conformal Blocks

We derive recursive representations in the internal weights of N-point Virasoro conformal blocks in the sphere linear channel and the torus necklace channel, and recursive representations in the central charge of arbitrary Virasoro conformal blocks on the sphere, the torus, and higher genus Riemann surfaces in the plumbing frame.

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arXiv2017

Bootstrapping the Spectral Function: On the Uniqueness of Liouville and the Universality of BTZ

We introduce spectral functions that capture the distribution of OPE coefficients and density of states in two-dimensional conformal field theories, and show that nontrivial upper and lower bounds on the spectral function can be obtained from semidefinite programming. We find substantial numerical evidence indicating that OPEs involving only scalar Virasoro primaries in a c>1 CFT are necessarily governed by the structure constants of Liouville theory. Combining this with analytic results in modular bootstrap, we conjecture that Liouville theory is the unique unitary c>1 CFT whose primaries have bounded spins. We also use the spectral function method to study modular constraints on CFT spectra, and discuss some implications of our results on CFTs of large c and large gap, in particular, to what extent the BTZ spectral density is universal.

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arXiv2016

Modular Bootstrap Revisited

We constrain the spectrum of two-dimensional unitary, compact conformal field theories with central charge c > 1 using modular bootstrap. Upper bounds on the gap in the dimension of primary operators of any spin, as well as in the dimension of scalar primaries, are computed numerically as functions of the central charge using semi-definite programming. Our bounds refine those of Hellerman and Friedan-Keller, and are in some cases saturated by known CFTs. In particular, we show that unitary CFTs with c < 8 must admit relevant deformations, and that a nontrivial bound on the gap of scalar primaries exists for c < 25. We also study bounds on the dimension gap in the presence of twist gaps, bounds on the degeneracy of operators, and demonstrate how "extremal spectra" which maximize the degeneracy at the gap can be determined numerically.

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arXiv2013

Deconfinement in N=1 super Yang-Mills theory on R^3 x S^1 via dual-Coulomb gas and "affine" XY-model

We study finite-temperature N=1 SU(2) super Yang-Mills theory, compactified on a spatial circle of size L with supersymmetric boundary conditions. In the semiclassical small-L regime, a deconfinement transition occurs at T_c <<1/L. The transition is due to a competition between non-perturbative topological "molecules"---magnetic and neutral bion-instantons---and electrically charged W-bosons and superpartners. Compared to deconfinement in non-supersymmetric QCD(adj) arXiv:1112.6389, the novelty is the relevance of the light modulus scalar field. It mediates interactions between neutral bions (and W-bosons), serves as an order parameter for the Z_2^{L} center symmetry associated with the non-thermal circle, and explicitly breaks the electric-magnetic (Kramers-Wannier) duality enjoyed by non-supersymmetric QCD(adj) near T_c. We show that deconfinement can be studied using an effective two-dimensional gas of electric and magnetic charges with (dual) Coulomb and Aharonov-Bohm interactions, or, equivalently, via an XY-spin model with a symmetry-breaking perturbation, where each system couples to the scalar field. To study the realization of the discrete R-symmetry and the Z_2^{beta} thermal and Z_2^{L} non-thermal center symmetries, we perform Monte Carlo simulations of both systems. The dual-Coulomb gas simulations are a novel way to analyze deconfinement and provide a new venue to study the phase structure of a class of two-dimensional condensed matter models that can be mapped into dual-Coulomb gases. Our results indicate a continuous deconfinement transition, with Z_2^{L} remaining unbroken at the transition. Thus, the SYM transition appears similar to the one in SU(2) QCD(adj) arXiv:1112.6389 and is also likely to be characterized by continuously varying critical exponents.

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arXiv2012

The SU(3)/Z_3 QCD(adj) deconfinement transition via the gauge theory/"affine" XY-model duality

Earlier, two of us and M. Unsal [arXiv:1112.6389] showed that some 4d gauge theories, compactified on a small spatial circle of size L and considered at temperatures 1/beta near deconfinement, are dual to 2d "affine" XY-spin models. We use the duality to study deconfinement in SU(3)/Z_3 theories with n_f>1 massless adjoint Weyl fermions, QCD(adj) on R^2 x S^1_beta x S^1_L. The"affine" XY-model describes two "spins" - compact scalars taking values in the SU(3) root lattice, with nearest-neighbor interactions and subject to an "external field" preserving the topological Z_3^t and a discrete Z_3^chi subgroup of the chiral symmetry of the 4d gauge theory. The equivalent Coulomb gas representation of the theory exhibits electric-magnetic duality, which is also a high-/low-temperature duality. A renormalization group analysis suggests - but is not convincing, due to the onset of strong coupling - that the self-dual point is a fixed point, implying a continuous deconfinement transition. Here, we study the nature of the transition via Monte Carlo simulations. The Z_3^t x Z_3^chi order parameter, its susceptibility, the vortex density, the energy per spin, and the specific heat are measured over a range of volumes, temperatures, and "external field" strengths (in the gauge theory, these correspond to magnetic bion fugacities). The finite-size scaling of the susceptibility and specific heat we find is characteristic of a first-order transition. Furthermore, for sufficiently large but still smaller than unity bion fugacity (as can be achieved upon an increase of the S^1_L size), at the critical temperature we find two distinct peaks of the energy probability distribution, indicative of a first-order transition, as has been seen in earlier simulations of the full 4d QCD(adj) theory. We end with discussions of the global phase diagram in the beta-L plane for different numbers of flavors.

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