Christoph A Keller

Interests

No activities entered.

Courses

Scholarly Contributions

Journals/Publications

• , N. B., , C. A., & , I. G. (2021). Lifting 1/4-BPS States in $AdS_3\times S^3 \times T^4$.
  More info

  We establish a framework for doing second order conformal perturbation theoryfor the symmetric orbifold Sym$^N(T^4)$ to all orders in $N$. This allows us tocompute how 1/4-BPS states of the D1-D5 system on $AdS_3\times S^3\times T^4$are lifted as we move away from the orbifold point. As an application weconfirm a previous observation that in the large $N$ limit not all 1/4-BPSstates that can be lifted do get lifted. This provides evidence that thesupersymmetric index actually undercounts the number of 1/4-BPS states at ageneric point in the moduli space.[Journal_ref: ]
• , N. B., , C. A., , H. O., & , I. G. (2021). Narain to Narnia.
  More info

  We generalize the holographic correspondence between topological gravitycoupled to an abelian Chern-Simons theory in three dimensions and an ensembleaverage of Narain's family of massless free bosons in two dimensions,discovered by Afkhami-Jeddi et al. and by Maloney and Witten. We find that thecorrespondence also works for toroidal orbifolds but not for K3 or Calabi-Yausigma-models and not always for the minimal models. We conjecture that thecorrespondence requires that the central charge is equal to the criticalcentral charge defined by the asymptotic density of states of the chiralalgebra. For toroidal orbifolds, we extend the holographic correspondence tocorrelation functions of twist operators by using topological properties ofrational tangles in the three-dimensional ball, which represent configurationsof vortices associated to a discrete gauge symmetry.[Journal_ref: ]
• , C. A., & , J. M. (2020). On the Space of Slow Growing Weak Jacobi Forms.
  More info

  Weak Jacobi forms of weight $0$ and index $m$ can be exponentially lifted tomeromorphic Siegel paramodular forms. It was recently observed that the Fouriercoefficients of such lifts are then either fast growing or slow growing. Inthis note we investigate the space of weak Jacobi forms that lead to slowgrowth. We provide analytic and numerical evidence for the conjecture thatthere are such slow growing forms for any index $m$.[Journal_ref: ]
• , N. B., , C. A., , H. O., & , I. G. (2020). On Rational Points in CFT Moduli Spaces.
  More info

  Motivated by the search for rational points in moduli spaces oftwo-dimensional conformal field theories, we investigate how points withenhanced symmetry algebras are distributed there. We first study the bosonicsigma-model with $S^1$ target space in detail and uncover hitherto unknownfeatures. We find for instance that the vanishing of the twist gap, though truefor the $S^1$ example, does not automatically follow from enhanced symmetrypoints being dense in the moduli space. We then explore the supersymmetricsigma-model on K3 by perturbing away from the torus orbifold locus. Though wedo not reach a definite conclusion on the distribution of enhanced symmetrypoints in the K3 moduli space, we make several observations on how chiralcurrents can emerge and disappear under conformal perturbation theory.[Journal_ref: ]
• Keller, C. A. (2020). $\mathcal{N}=2$ Minimal Models: A Holographic Needle in a Symmetric Orbifold Haystack. SciPost Phys..
  More info

  We explore large-$N$ symmetric orbifolds of the $\mathcal N=2$ minimalmodels, and find evidence that their moduli spaces each contain a supergravitypoint. We identify single-trace exactly marginal operators that deform themaway from the symmetric orbifold locus. We also show that their elliptic generaexhibit slow growth consistent with supergravity spectra in AdS$_3$. We thuspropose an infinite family of new holographic CFTs.[Journal_ref: SciPost Phys. 8, 084 (2020)]
• Keller, C., Keller, C., Belin, A., Belin, A., Benjamin, N., Benjamin, N., Castro, A., Castro, A., Harrison, S. M., & Harrison, S. M. (2020). $\mathcal{N}=2$ minimal models: A holographic needle in a symmetric orbifold haystack. SciPost Physics, 8(6). doi:10.21468/scipostphys.8.6.084
• , A. B., , A. C., , C. A., & , B. J. (2019). Siegel Paramodular Forms from Exponential Lifts: Slow versus Fast Growth.
  More info

  We investigate the growth of Fourier coefficients of Siegel paramodular formsbuilt by exponentially lifting weak Jacobi forms, focusing on terms with largenegative discriminant. To this end we implement a method based on deformingcontours that expresses the coefficients of all such terms as residues. We findthat there are two types of weak Jacobi forms, leading to two different growthbehaviors: the more common type leads to fast, exponential growth, whereas asecond type leads to slower growth, akin to the growth seen in ratios of thetafunctions. We give a simple criterion to distinguish between the two types, andgive a simple closed form expression for the coefficients in the slow growingcase. In a companion article [1], we provide physical applications of theseresults to symmetric product orbifolds and holography.[Journal_ref: ]
• , T. G., & , C. A. (2019). Non-Abelian Orbifolds of Lattice Vertex Operator Algebras.
  More info

  We construct orbifolds of holomorphic lattice Vertex Operator Algebras fornon-Abelian finite automorphism groups $G$. To this end, we construct twistedmodules for automorphisms $g$ together with the projective representation ofthe centralizer of $g$ on the twisted module. This allows us to extract theirreducible modules of the fixed point VOA $V^G$, and to compute theircharacters and modular transformation properties. We then construct holomorphicVOAs by adjoining such modules to $V^G$. Applying these methods to extremallattices in $d=48$ and $d=72$, we construct more than fifty new holomorphicVOAs of central charge 48 and 72, many of which have a very small number oflight states.[Journal_ref: ]
• , T. G., & , C. A. (2019). The Large $N$ Limit of Orbifold Vertex Operator Algebras.
  More info

  We investigate the large $N$ limit of permutation orbifolds of vertexoperator algebras. To this end, we introduce the notion of nested oligomorphicpermutation orbifolds and discuss under which conditions their fixed point VOAsconverge. We show that if this limit exists, then it has the structure of avertex algebra. Finally, we give an example based on $\mathrm{GL}(N,q)$ forwhich the fixed point VOA limit is also the limit of the full permutationorbifold VOA.[Journal_ref: ]
• Keller, C. A. (2020). Conformal Perturbation Theory for Twisted Fields. J.Phys.A.
  More info

  We investigate second order conformal perturbation theory for $\mathbb{Z}_2$orbifolds of conformal field theories in two dimensions. To evaluate thenecessary twisted sector correlation functions and their integrals, we map themfrom the sphere to its torus double cover. We discuss how this relates crossingsymmetry to the modular group, and introduce a regularization scheme on thecover that allows to evaluate the integrals numerically. These methods do notrequire supersymmetry. As an application, we show that in the torus orbifold of8 and 16 free bosons, $\mathbb{Z}_2$ twist fields are marginal at first order,but stop being marginal at second order.[Journal_ref: ]
• Keller, C. A. (2020). Lifting 1/4-BPS States on K3 and Mathieu Moonshine. Commun.Math.Phys.
  More info

  The elliptic genus of K3 is an index for the 1/4-BPS states of its sigmamodel. At the torus orbifold point there is an accidental degeneracy of suchstates. We blow up the orbifold fixed points using conformal perturbationtheory, and find that this fully lifts the accidental degeneracy of the 1/4-BPSstates with h=1. At a generic point near the Kummer surface the elliptic genusthus measures not just their index, but counts the actual number of these BPSstates. We comment on the implication of this for symmetry surfing and Mathieumoonshine.[Journal_ref: ]
• Keller, C. A. (2020). The Holographic Landscape of Symmetric Product Orbifolds. JHEP, 01, 111. doi:10.1007/JHEP01(2020)111
  More info

  We investigate the growth of coefficients in the elliptic genus of symmetricproduct orbifolds at large central charge. We find that this landscapedecomposes into two regions. In one region, the growth of the low energy statesis Hagedorn, which indicates a stringy dual. In the other, the growth is muchslower, and compatible with the spectrum of a supergravity theory on AdS$_3$.We provide a simple diagnostic which places any symmetric product orbifold ineither region. We construct a class of elliptic genera with suchsupergravity-like growth, indicating the possible existence of new realizationsof AdS$_3$/CFT$_2$ where the bulk is a semi-classical supergravity theory. Insuch cases, we give exact expressions for the BPS degeneracies, which could bematched with the spectrum of perturbative states in a dual supergravitydescription.[Journal_ref: ]
• "Belin, A., Castro, A., Gomes, J., & Keller, C. A. (2018). "{Siegel paramodular forms and sparseness in AdS$_3$/CFT$_2$}". "JHEP", "11", "037".
• Keller, C. A. (2019). Orbifolds of Lattice Vertex Operator Algebras at $d=48$ and $d=72$. Journal of Algebra.
  More info

  Motivated by the notion of extremal vertex operator algebras, we investigatecyclic orbifolds of vertex operator algebras coming from extremal evenself-dual lattices in $d=48$ and $d=72$. In this way we construct about onehundred new examples of holomorphic VOAs with a small number of low weightstates.[Journal_ref: ]
• Keller, C. A., & Mühlmann, B. J. (2019). The Spectrum of Permutation Orbifolds. Lett. Math. Phys., 109(7), 1559-1572.
• Keller, C. A., Mathys, G., & Zadeh, I. G. (2018). Bootstrapping Chiral CFTs at Genus Two. Adv. Theor. Math. Phys., 22, 1447-1487.

Presentations

• Keller, C. A. (2021, February). Holographic Families of VOAs. Rocky Mountain Representation Theory Seminar. online.
• Keller, C. A. (2020, May). Holographic Orbifold CFTs in Two Dimensions. invited seminar. UC Davis.
• Keller, C. A. (2019, February). Holographic Conformal Field Theories in Two Dimensions. Seminar, ASU. ASU, Tempe AZ: ASU.
• Keller, C. A. (2019, May). Holographic Conformal Field Theories in Two Dimensions. invited lecture series, KIAS. Seoul, Korea: KIAS.
• Keller, C. A. (2019, October). Fun with Modular Forms: Moonshine and Black Holes. invited seminar, KSU. Manhattan, Kansas: Kansas State University.
• Keller, C. A. (2018, April). Cyclic orbifolds of lattice vertex operator algebras. Mathematical Physics and Probability Seminar (Internal seminar). UA.
• Keller, C. A. (2018, February). Cyclic orbifolds of lattice vertex operator algebras. Algebra and Number Theory Seminar (internal seminar). UA.
• Keller, C. A. (2018, June). Holographic Orbifold CFTs. String Math 2018 (plenary speaker). Sendai, Japan.
• Keller, C. A. (2018, March). Conformal Field Theory, Modular Forms and Black Holes. Mathematics Colloquium Lehigh University. Bethlehem PA.
• Keller, C. A. (2018, May). Conformal Field Theories and the Entropy of Black Holes. Physics Colloquium (Lehigh University). Bethlehem PA.
• Keller, C. A. (2018, May). Holographic Orbifold CFTs. Caltech (invited seminar). Pasadena.