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Register a new userGeneralized q-deformed Correlation Functions as Spectral Functions of Hyperbolic Geometry
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We analyse the role of vertex operator algebra and 2d amplitudes from the point of view of the representation theory of infinite dimensional Lie algebras, MacMahon and Ruelle functions. A p-dimensional MacMahon function is the generating function of p-dimensional partitions of integers. These functions can be represented as amplitudes of a two-dimensional c=1 CFT. In this paper we show that p-dimensional MacMahon functions can be rewritten in terms of Ruelle spectral functions, whose spectrum is encoded in the Patterson-Selberg function of three dimensional hyperbolic geometry.
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We construct the wave functions in the q-deformed 2d Yang-Mills theory that compute torus correlation functions of affine currents in the VOA associated to a class of 4d $N = 2$ SCFTs. These wave functions are then shown to reduce to the topological correlators of a set of Coulomb branch operators in the $T[SU(N)]$ theory, from which those correlators in the 3d mirror dual of the 4d TN theories can be computed.
We discuss the homological aspects of the connection between quantum string generating function and the formal power series associated to the dimensions of chains and homologies of suitable Lie algebras. Our analysis can be considered as a new straightforward application of the machinery of modular forms and spectral functions (with values in the congruence subgroup of $SL(2,{mathbb Z})$) to the partition functions of Lagrangian branes, refined vertex and open string partition functions, represented by means of formal power series that encode Lie algebra properties. The common feature in our examples lies in the modular properties of the characters of certain representations of the pertinent affine Lie algebras and in the role of Selberg-type spectral functions of an hyperbolic three-geometry associated with $q$-series in the computation of the string amplitudes.
Gauged N=8 supergravity in four dimensions is now known to admit a deformation characterized by a real parameter $omega$ lying in the interval $0leomegale pi/8$. We analyse the fluctuations about its anti-de Sitter vacuum, and show that the full N=8 supersymmetry can be maintained by the boundary conditions only for $omega=0$. For non-vanishing $omega$, and requiring that there be no propagating spin s>1 fields on the boundary, we show that N=3 is the maximum degree of supersymmetry that can be preserved by the boundary conditions. We then construct in detail the consistent truncation of the N=8 theory to give $omega$-deformed SO(6) gauged N=6 supergravity, again with $omega$ in the range $0leomegale pi/8$. We show that this theory admits fully N=6 supersymmetry-preserving boundary conditions not only for $omega=0$, but also for $omega=pi/8$. These two theories are related by a U(1) electric-magnetic duality. We observe that the only three-point functions that depend on $omega$ involve the coupling of an SO(6) gauge field with the U(1) gauge field and a scalar or pseudo-scalar field. We compute these correlation functions and compare them with those of the undeformed N=6 theory. We find that the correlation functions in the $omega=pi/8$ theory holographically correspond to amplitudes in the U(N)_k x U(N)_{-k} ABJM model in which the U(1) Noether current is replaced by a dynamical U(1) gauge field. We also show that the $omega$-deformed N=6 gauged supergravities can be obtained via consistent reductions from the eleven-dimensional or ten-dimensional type IIA supergravities.
In this article we exploit Ruelle-type spectral functions and analyze the Verma module over Virasoro algebra, boson-fermion correspondence, the analytic torsion, the Chern-Simons and $eta$ invariants, as well as the generation function associated to dimensions of the Hochschild homology of the crossed product $mathbb{C}[S_n]ltimes mathcal{A}^{otimes n}$ ($mathcal{A}$ is the $q$-Weyl algebra). After analysing the Chern-Simons and $eta$ invariants of Dirac operators by using irreducible $SU(n)$-flat connections on locally symmetric manifolds of non-positive section curvature, we describe the exponential action for the Chern-Simons theory.
We study the evolution of correlation functions of local fields in a two-dimensional quantum field theory under the $lambda Tbar T$ deformation, suitably regularized. We show that this may be viewed in terms of the evolution of each field, with a Dirac-like string being attached at each infinitesimal step. The deformation then acts as a derivation on the whole operator algebra, satisfying the Leibniz rule. We derive an explicit equation which allows for the analysis of UV divergences, which may be absorbed into a non-local field renormalization to give correlation functions which are UV finite to all orders, satisfying a (deformed) operator product expansion and a Callan-Symanzik equation. We solve this in the case of a deformed CFT, showing that the Fourier-transformed renormalized two-point functions behave as $k^{2Delta+2lambda k^2}$, where $Delta$ is their IR conformal dimension. We discuss in detail deformed Noether currents, including the energy-momentum tensor, and show that, although they also become non-local, when suitably improved they remain finite, conserved and satisfy the expected Ward identities. Finally, we discuss how the equivalence of the $Tbar T$ deformation to a state-dependent coordinate transformation emerges in this picture.
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