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Register a new userConstraints on 2d CFT partition functions
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Modular invariance is known to constrain the spectrum of 2d conformal field theories. We investigate this constraint systematically, using the linear functional method to put new improved upper bounds on the lowest gap in the spectrum. We also consider generalized partition functions of N = (2,2) superconformal theories and discuss the application of our results to Calabi-Yau compactifications. For Calabi-Yau threefolds with no enhanced symmetry we find that there must always be non-BPS primary states of weight 0.6 or less.
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We apply the theory of harmonic analysis on the fundamental domain of $SL(2,mathbb{Z})$ to partition functions of two-dimensional conformal field theories. We decompose the partition function of $c$ free bosons on a Narain lattice into eigenfunctions of the Laplacian of worldsheet moduli space $mathbb H/SL(2,mathbb Z)$, and of target space moduli space $O(c,c;mathbb Z)backslash O(c,c;mathbb R)/O(c)times O(c)$. This decomposition manifests certain properties of Narain theories and ensemble averages thereof. We extend the application of spectral theory to partition functions of general two-dimensional conformal field theories, and explore its meaning in connection to AdS$_3$ gravity. An implication of harmonic analysis is that the local operator spectrum is fully determined by a certain subset of degeneracies.
We observe that the partition function of the set of all free massless higher spins s=0,1,2,3,... in flat space is equal to one: the ghost determinants cancel against the physical ones or, equivalently, the (regularized) total number of degrees of freedom vanishes. This reflects large underlying gauge symmetry and suggests analogy with supersymmetric or topological theory. The Z=1 property extends also to the AdS background, i.e. the 1-loop vacuum partition function of Vasiliev theory is equal to 1 (assuming a particular regularization of the sum over spins); this was noticed earlier as a consistency requirement for the vectorial AdS/CFT duality. We find that Z=1 is also true in the conformal higher spin theory (with higher-derivative d^{2s} kinetic terms) expanded near flat or conformally flat S^4 background. We also consider the partition function of free conformal theory of symmetric traceless rank s tensor field which has 2-derivative kinetic term but only scalar gauge invariance in flat 4d space. This non-unitary theory has a Weyl-invariant action in curved background and corresponds to partially massless field in AdS_5. We discuss in detail the special case of s=2 (or conformal graviton), compute the corresponding conformal anomaly coefficients and compare them with previously found expressions for generic representations of conformal group in 4 dimensions.
Localization methods have produced explicit expressions for the sphere partition functions of (2,2) superconformal field theories. The mirror symmetry conjecture predicts an IR duality between pairs of Abelian gauged linear sigma models, a class of which describe families of Calabi-Yau manifolds realizable as complete intersections in toric varieties. We investigate this prediction for the sphere partition functions and find agreement between that of a model and its mirror up to the scheme-dependent ambiguities inherent in the definitions of these quantities.
We compute partition functions describing multiplicities and charges of massless and first massive string states of pure-spinor superstrings in 3,4,6,10 dimensions. At the massless level we find a spin-one gauge multiplet of minimal supersymmetry in d dimensions. At the first massive string level we find a massive spin-two multiplet. The result is confirmed by a direct analysis of the BRST cohomology at ghost number one. The central charges of the pure spinor systems are derived in a manifestly SO(d) covariant way confirming that the resulting string theories are critical. A critical string model with N=(2,0) supersymmetry in d=2 is also described.
We find a simple relation between a free higher spin field partition function on thermal quotient of AdS(d+1) and the partition function of the associated d-dimensional conformal higher spin field on thermal quotient of AdS(d). Starting with a conformal higher spin field defined on AdS(d) one may also associate to it another conformal field in d-1 dimensions, thus iterating AdS/CFT. We observe that in the case of d=4 this iteration leads to a trivial 3d higher spin conformal theory with parity-even non-local action: it describes zero total number of dynamical degrees of freedom and the corresponding partition function on thermal AdS(3) is equal to 1.
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