arXiv subjects

Benjamin A. Burrington

Publications and source records attributed to Benjamin A. Burrington.

At least 19 recordsLinked to original sources

arXiv2025

Covering space maps for $n$-point functions with three long twists

We consider correlation functions in symmetric product orbifold CFTs on the sphere, focusing on the case where all operators are single-cycle twists, and the covering surface is also a sphere. We directly construct the general class of covering space maps where there are three twists of arbitrary lengths, along with any number of twist-2 insertions. These are written as a ratio of sums of Jacobi polynomials with $\Delta N+1$ coefficients $b_N$. These coefficients have a scaling symmetry $b_N\rightarrow \lambda b_N$, making them naturally valued in $\mathbb{CP}^{\Delta N}$. We explore limits where various ramified points on the cover approach each other, which are understood as crossing channel specific OPE limits, and find that these limits are defined by algebraic varieties of $\mathbb{CP}^{\Delta N}$. We compute the expressions needed to calculate the group element representative correlation functions for bare twists. Specializing to the cases $\Delta N=1,2$, we find closed form for these expressions which define four- and five-point functions of bare twists.

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arXiv2023

Conformal Perturbation Theory for $n$-Point Functions: Structure Constant Deformation

We consider conformal perturbation theory for $n$-point functions on the sphere in general 2D CFTs to first order in coupling constant. We regulate perturbation integrals using canonical hard disk excisions of size $ε$ around the fixed operator insertions, and identify the full set of counter terms which are sufficient to regulate all such integrated $n$-point functions. We further explore the integrated 4-point function which computes changes to the structure constants of the theory. Using an $sl(2)$ map, the three fixed locations of operators are mapped to $0$, $1$, and $\infty$. We show that approximating the mapped excised regions to leading order in $ε$ does not lead to the same perturbative shift to the structure constant as the exact in $ε$ region. We explicitly compute the correction back to the exact in $ε$ region of integration in terms of the CFT data. We consider the compact boson, and show that one must use the exact in $ε$ region to obtain agreement with the exact results for structure constants in this theory.

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arXiv2022

Larger Twists and Higher $n$-Point Functions with Fractional Conformal Descendants in $S_N$ Orbifold CFTs at Large $N$

We consider correlation functions in symmetric product ($S_N$) orbifold CFTs at large $N$ with arbitrary seed CFT, expanding on our earlier work arXiv:2211.04633 . Using covering space techniques, we calculate descent relations using fractional Virasoro generators in correlators, writing correlators of descendants in terms of correlators of ancestors. We first consider the case three-point functions of the form ($m$-cycle)-($n$-cycle)-($q$-cycle) which lift to arbitrary primaries on the cover, and descendants thereof. In these examples we show that the final descent relations do not depend on the covering space data, nor on the specific details of the seed CFT. This makes these descent relations universal in all $S_N$ orbifold CFTs. Next, we explore four-point functions of the form (2-cycle)-($n$-cycle)-($n$-cycle)-(2-cycle) which lift to arbitrary primaries on the cover, and descendants thereof. In such cases a single parameter in the map $s$ parameterizes both the base space cross ratio $ζ_z$ and the covering space cross ratio $ζ_t$. We find that the descent relations for the four point functions depend only on base space data and the parameter $s$, which we argue is tantamount to writing the descent relations in terms of the base space data and the base space cross ratio. These descent relations again do not depend on the covering space data, nor the specifics of the seed CFT, making these universal as well.

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arXiv2022

Fractional Conformal Descendants and Correlators in General 2D $S_N$ Orbifold CFTs at Large $N$

We consider correlation functions in symmetric product ($S_N$) orbifold CFTs at large $N$ with arbitrary seed CFT. Specifically, we consider correlators of descendant operators constructed using both the full Virasoro generators $L_{m}$ and fractional Virasoro generators $\ell_{m/n_i}$. Using covering space techniques, we show that correlators of descendants may be written entirely in terms of correlators of ancestors, and further that the appropriate set of ancestors are those operators that lift to conformal primaries on the cover. We argue that the covering space data should cancel out in such calculations. To back this claim, we provide some example calculations by considering a three-point function of the form (4-cycle)-(2-cycle)-(5-cycle) that lifts to a three-point function of arbitrary primaries on the cover, and descendants thereof. In these examples we show that while the covering space is used for the calculation, the final descent relations do not depend on covering space data, nor on the details of which seed CFT is used to construct the orbifold, making these results universal.

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arXiv2019

The large $N$ limit of OPEs in symmetric orbifold CFTs with $\mathcal{N}=(4,4)$ supersymmetry

We explore the OPE of certain twist operators in symmetric product ($S_N$) orbifold CFTs, extending our previous work arXiv:1804.01562 to the case of $\mathcal{N}=(4,4)$ supersymmetry. We consider a class of twist operators related to the chiral primaries by spectral flow parallel to the twist. We conjecture that at large $N$, the OPE of two such operators contains only fields in this class, along with excitations by fractional modes of the superconformal currents. We provide evidence for this by studying the coincidence limits of two 4-point functions to several non-trivial orders. We show how the fractional excitations of the twist operators in our restricted class fully reproduce the crossing channels appearing in the coincidence limits of the 4-point functions.

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arXiv2018

The OPE of bare twist operators in bosonic $S_N$ orbifold CFTs at large $N$

In this work, we explore the twist operator OPEs of a generic bosonic symmetric product ($S_N$) orbifold CFT. We conjecture that at large $N$ the OPE of bare twist operators contains only bare twists and excitations of bare twists with fractional Virasoro modes. These fractionally excited operators are the only ones that depend exclusively on the lengths of the twists and the central charge, agreeing with the general structure of correlators of bare twists found in the literature. To provide evidence for this, we study the coincidence limit of a four point function of bare twist operators to several non-leading orders. We show how the coefficients of these powers can be reproduced by considering bare twist operators excited by fractional Virasoro modes in the exchange channels.

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arXiv2017

Operator mixing in deformed D1D5 CFT and the OPE on the cover

We consider the D1D5 CFT near the orbifold point and develop methods for computing the mixing of untwisted operators to first order by using the OPE on the covering surface. We argue that the OPE on the cover encodes both the structure constants for the orbifold CFT and the explicit form of the mixing operators. We show this explicitly for some example operators. We start by considering a family of operators dual to supergravity modes, and show that the OPE implies that there is no shift in the anomalous dimension to first order, as expected. We specialize to the operator dual to the dilaton, and show that the leading order singularity in the OPE reproduces the correct structure constant. Finally, we consider an unprotected operator of conformal dimension (2,2), and show that the leading order singularity and one of the subleading singularies both reproduce the correct structure constant. We check that the operator produced at subleading order using the OPE method is correct by calculating a number of three point functions using a Mathematica package we developed. Further development of this OPE technique should lead to more efficient calculations for the D1D5 CFT perturbed away from the orbifold point.

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arXiv2015

Bosonization, cocycles, and the D1-D5 CFT on the covering surface

We consider the D1-D5 CFT near the orbifold point, specifically the computation of correlators involving twist sector fields using covering surface techniques. As is well known, certain twists introduce spin fields on the cover. Here we consider the bosonization of fermions to facilitate computations involving the spin fields. We find a set of cocycle operators that satisfy constraints coming from various $SU(2)$ symmetries, including the $SU(2)_L\times SU(2)_R$ R-symmetry. Using these cocycles, we consider the correlator of four spin fields on the cover, and show that it is invariant under all of the $SU(2)$ symmetries of the theory. We consider the mutual locality of operators, and compute several three-point functions. These computations lead us to a notion of radial ordering on the cover that is inherited from the original computation before lifting. Further, we note that summing over orbifold images sets certain branch-cut ambiguous correlators to zero.

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arXiv2014

Analyzing the squeezed state generated by a twist deformation

The D1D5 CFT has provided a useful microscopic model for studying black holes. The coupling in this theory is a twist deformation whose action on the vacuum generates a squeezed state. We give a new derivation of the expression for this squeezed state using the conformal Ward identity; this derivation provides an insight into several features of the state. We also examine the squeezed state in a continuum limit where we describe it in terms of position space correlations created by the twist.

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arXiv2013

Operator mixing for string states in the D1-D5 CFT near the orbifold point

In the context of the fuzzball programme, we investigate deforming the microscopic string description of the D1-D5 system on T^4xS^1 away from the orbifold point. Using conformal perturbation theory and a generalization of Lunin-Mathur symmetric orbifold technology for computing twist-nontwist correlators developed in a companion work, we initiate a program to compute the anomalous dimensions of low-lying string states in the D1-D5 superconformal field theory. Our method entails finding four-point functions involving a string operator O of interest and the deformation operator, taking coincidence limits to identify which other operators mix with O, subtracting the identified conformal family to isolate other contributions to the four-point function, finding the mixing coefficients, and iterating. For the lowest-lying string modes, this procedure should truncate in a finite number of steps. We check our method by showing how the operator dual to the dilaton does not participate in mixing that would change its conformal dimension, as expected. Next we complete the first stage of the iteration procedure for a low-lying string state of the form \partial X \partial X \bar\partial X \bar\partial X and find its mixing coefficient. Our main qualitative result is evidence of operator mixing at first order in the deformation parameter, which means that the string state acquires an anomalous dimension. After diagonalization this will mean that anomalous dimensions of some string states in the D1-D5 SCFT must decrease away from the orbifold point while others increase.

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arXiv2013

Twist-nontwist correlators in M^N/S_N orbifold CFTs

We consider general 2D orbifold CFTs of the form M^N/S_N, with M a target space manifold and S_N the symmetric group, and generalize the Lunin-Mathur covering space technique in two ways. First, we consider excitations of twist operators by modes of fields that are not twisted by that operator, and show how to account for these excitations when computing correlation functions in the covering space. Second, we consider non-twist sector operators and show how to include the effects of these insertions in the covering space. We work two examples, one using a simple bosonic CFT, and one using the D1-D5 CFT at the orbifold point. We show that the resulting correlators have the correct form for a 2D CFT.

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arXiv2011

General Leznov-Savelev solutions for Pohlmeyer reduced AdS$_5$ minimal surfaces

We consider the Pohlmeyer reduced sigma model describing AdS$_5$ minimal surfaces. We show that, similar to the affine Toda models, there exists a conformal extension to this model which admits a Lax formulation. The Lax connection is shown to be valued in a ${\mathbb Z}_4$-invariant subalgebra of the affine Lie algebra $\widehat{su(4)}$. Using this, we perform a modified version of a Laznov-Savelev analysis, which allows us to write formal expressions for the general solutions for the Pohlmeyer reduced AdS$_5$ theory. This analysis relies on the a certain decomposition for the exponentiated algebra elements.

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arXiv2011

Lifshitz-like black brane thermodynamics in higher dimensions

Gravitational backgrounds in d+2 dimensions have been proposed as holographic duals to Lifshitz-like theories describing critical phenomena in d+1 dimensions with critical exponent z\geq 1. We numerically explore a dilaton-Einstein-Maxwell model admitting such backgrounds as solutions. Such backgrounds are characterized by a temperature T and chemical potential μ, and we find how to embed these solutions into AdS for a range of values of z and d. We find no thermal instability going from the (T\llμ) to the (T\ggμ) regimes, regardless of the dimension, and find that the solutions smoothly interpolate between the Lifshitz-like behaviour and the relativistic AdS-like behaviour. We exploit some conserved quantities to find a relationship between the energy density E, entropy density s, and number density n, E=\frac{d}{d+1}(Ts+nμ), as is required by the isometries of AdS_{d+2}. Finally, in the (T\llμ) regime the entropy density is found to satisfy a power law s \propto c T^{d/z} μ^{(z-1)d/z}, and we numerically explore the dependence of the constant c, a measure of the number of degrees of freedom, on d and z.

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arXiv2010

Phase transitions in Wilson loop correlator from integrability in global AdS

We directly compute Wilson loop/Wilson loop correlators on ${\mathbb R}\times $S$^3$ in AdS/CFT by constructing space-like minimal surfaces that connect two space-like circular contours on the boundary of global AdS that are separated by a space-like interval. We compare these minimal surfaces to the disconnected "double cap" solutions both to regulate the area, and show when the connected/disconnected solution is preferred. We find that for sufficiently large Wilson loops no transition occurs because the Wilson loops cannot be sufficiently separated on the sphere. This may be considered an effect similar to the Hawking-Page transition: the size of the sphere introduces a new scale into the problem, and so one can expect phase transitions to depend on this data. To construct the minimal area solutions, we employ a reduction a la Arutyunov-Russo-Tseytlin (used by them for spinning strings), and rely on the integrability of the reduced set of equations to write explicit results.

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arXiv2010

Thermal behavior of charged dilatonic black branes in AdS and UV completions of Lifshitz-like geometries

Several classes of gravitational backgrounds in $3+1$ dimensions have been proposed as holographic duals to Lifshitz-like theories describing critical phenomena in $2+1$ dimensions with critical exponent $z\geq 1$. We numerically explore one such model, characterized by a temperature $T$ and chemical potential $μ$, and find how to embed these solutions into AdS for a range of values of $z$. We find no phase transition going from the $T\llμ$ to the $T\gg μ$ regimes, and find that the solutions smoothly interpolate between the Lifshitz-like behavior and the relativistic AdS-like behavior. Finally, we exploit some conserved quantities to find a relationship between the energy density $\mc E$, entropy density $s$, and number density $n$, $\mc E=\frac{2}{3} \left(Ts+μn\right)$. We show that this result is expected from general scaling arguments, and generalizes to $\mc E= \frac{d}{d+1}\left(Ts+μn\right)$ for a theory dual to AdS$_{d+2}$ (Poincaré patch) asymptotics with a local $U(1)$ gauge invariance.

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arXiv2010

Minimal surfaces in AdS space and Integrable systems

We consider the Pohlmeyer reduction for spacelike minimal area worldsheets in AdS$_5$. The Lax pair for the reduced theory is found, and written entirely in terms of the $A_3=D_3$ root system, generalizing the $B_2$ affine Toda system which appears for the AdS$_4$ string. For the $B_2$ affine Toda system, we show that the area of the worlsheet is obtainable from the moduli space Kähler potential of a related Hitchin system. We also explore the Saveliev-Leznov construction for solutions of the $B_2$ affine Toda system, and recover the rotationally symmetric solution associated to Painleve transcendent.

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arXiv2009

Black holes in asymptotically Lifshitz spacetimes with arbitrary critical exponent

Recently, a class of gravitational backgrounds in 3+1 dimensions have been proposed as holographic duals to a Lifshitz theory describing critical phenomena in 2+1 dimensions with critical exponent $z\geq 1$. We numerically explore black holes in these backgrounds for a range of values of $z$. We find drastically different behavior for $z>2$ and $z<2$. We find that for $z>2$ ($z<2$) the Lifshitz fixed point is repulsive (attractive) when going to larger radial parameter $r$. For the repulsive $z>2$ backgrounds, we find a continuous family of black holes satisfying a finite energy condition. However, for $z<2$ we find that the finite energy condition is more restrictive, and we expect only a discrete set of black hole solutions, unless some unexpected cancellations occur. For all black holes, we plot temperature $T$ as a function of horizon radius $r_0$. For $z\lessapprox 1.761$ we find that this curve develops a negative slope for certain values of $r_0$ possibly indicating a thermodynamic instability.

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arXiv2009

Thermodynamics of black branes in asymptotically Lifshitz spacetimes

Recently, a class of gravitational backgrounds in 3+1 dimensions have been proposed as holographic duals to a Lifshitz theory describing critical phenomena in 2+1 dimensions with critical exponent $z\geq 1$. We continue our earlier work \cite{Bertoldi:2009vn}, exploring the thermodynamic properties of the "black brane" solutions with horizon topology $\mathbb{R}^2$. We find that the black branes satisfy the relation $\mathcal{E}=\frac{2}{2+z}Ts$ where $\mathcal{E}$ is the energy density, $T$ is the temperature, and $s$ is the entropy density. This matches the expected behavior for a 2+1 dimensional theory with a scaling symmetry $(x_1,x_2)\to λ(x_1,x_2)$, $t\to λ^z t$.

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