Christoph A Keller
 Associate Professor
 Member of the Graduate Faculty
Contact
 (315) 871
 Mathematics, Rm. 321
 Tucson, AZ 85721
 cakeller@arizona.edu
Awards
 Journal of Physics A Best Paper Prize 2021
 Journal of Physics A, Fall 2021
Interests
No activities entered.
Courses
202425 Courses

Dissertation
MATH 920 (Fall 2024)
202324 Courses

Dissertation
MATH 920 (Spring 2024) 
Independent Study
MATH 599 (Spring 2024) 
Complex Analysis
MATH 520A (Fall 2023) 
Dissertation
MATH 920 (Fall 2023) 
Independent Study
MATH 599 (Fall 2023)
202223 Courses

Dissertation
MATH 920 (Spring 2023) 
Independent Study
MATH 599 (Spring 2023) 
Theory of Probability
MATH 464 (Spring 2023) 
Theory of Statistics
MATH 466 (Spring 2023) 
Directed Research
PHYS 492 (Fall 2022) 
Dissertation
MATH 920 (Fall 2022) 
Theory of Probability
MATH 464 (Fall 2022)
202122 Courses

2nd Crs Abstract Algebra
MATH 415B (Spring 2022) 
2nd Crs Abstract Algebra
MATH 515B (Spring 2022) 
Independent Study
MATH 599 (Spring 2022) 
Independent Study
MATH 599 (Fall 2021) 
Intro Abstract Algebra
MATH 415A (Fall 2021) 
Intro Abstract Algebra
MATH 515A (Fall 2021)
202021 Courses

Math Analysis Engineers
MATH 322 (Spring 2021) 
Math Analysis Engineers
MATH 322 (Fall 2020)
201920 Courses

Dissertation
MATH 920 (Spring 2020) 
Intro To Cryptography
MATH 445 (Spring 2020) 
Dissertation
MATH 920 (Fall 2019) 
Intro Math Physics
MATH 541 (Fall 2019) 
Intro Math Physics
PHYS 541 (Fall 2019)
201819 Courses

Linear Algebra
MATH 413 (Spring 2019) 
Linear Algebra
MATH 513 (Spring 2019) 
Math Analysis Engineers
MATH 322 (Fall 2018)
201718 Courses

Calculus II
MATH 129 (Spring 2018)
Scholarly Contributions
Journals/Publications
 , N. B., , C. A., & , I. G. (2021). Lifting 1/4BPS States in $AdS_3\times S^3 \times T^4$.More infoWe 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 1/4BPS states of the D1D5 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 1/4BPSstates that can be lifted do get lifted. This provides evidence that thesupersymmetric index actually undercounts the number of 1/4BPS states at ageneric point in the moduli space.[Journal_ref: ]
 , N. B., , C. A., , H. O., & , I. G. (2021). Narain to Narnia.More infoWe generalize the holographic correspondence between topological gravitycoupled to an abelian ChernSimons theory in three dimensions and an ensembleaverage of Narain's family of massless free bosons in two dimensions,discovered by AfkhamiJeddi et al. and by Maloney and Witten. We find that thecorrespondence also works for toroidal orbifolds but not for K3 or CalabiYausigmamodels 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 threedimensional 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 infoWeak 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 infoMotivated by the search for rational points in moduli spaces oftwodimensional conformal field theories, we investigate how points withenhanced symmetry algebras are distributed there. We first study the bosonicsigmamodel 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 supersymmetricsigmamodel 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 infoWe explore large$N$ symmetric orbifolds of the $\mathcal N=2$ minimalmodels, and find evidence that their moduli spaces each contain a supergravitypoint. We identify singletrace 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 infoWe 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). NonAbelian Orbifolds of Lattice Vertex Operator Algebras.More infoWe construct orbifolds of holomorphic lattice Vertex Operator Algebras fornonAbelian 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 infoWe 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 infoWe 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 1/4BPS States on K3 and Mathieu Moonshine. Commun.Math.Phys.More infoThe elliptic genus of K3 is an index for the 1/4BPS 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 1/4BPSstates 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)111More infoWe 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 suchsupergravitylike growth, indicating the possible existence of new realizationsof AdS$_3$/CFT$_2$ where the bulk is a semiclassical 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 infoMotivated by the notion of extremal vertex operator algebras, we investigatecyclic orbifolds of vertex operator algebras coming from extremal evenselfdual 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., & MÃ¼hlmann, B. J. (2019). The Spectrum of Permutation Orbifolds. Lett. Math. Phys., 109(7), 15591572.
 Keller, C. A., Mathys, G., & Zadeh, I. G. (2018). Bootstrapping Chiral CFTs at Genus Two. Adv. Theor. Math. Phys., 22, 14471487.
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.