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Register a new userNoncommutative spectral geometry, dissipation and the origin of quantization
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Mairi Sakellariadou
Publication date
2012
fields
Physics
and research's language is
English
Authors
Mairi Sakellariadou
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We present a physical interpretation of the doubling of the algebra, which is the basic ingredient of the noncommutative spectral geometry, developed by Connes and collaborators as an approach to unification. We discuss its connection to dissipation and to the gauge structure of the theory. We then argue, following t Hoofts conjecture, that noncommutative spectral geometry classical construction carries implicit in its feature of the doubling of the algebra the seeds of quantization.
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We introduce a framework in noncommutative geometry consisting of a $*$-algebra $mathcal A$, a bimodule $Omega^1$ endowed with a derivation $mathcal Ato Omega^1$ and with a Hermitian structure $Omega^1otimes bar{Omega}^1to mathcal A$ (a noncommutative Kahler form), and a cyclic 1-cochain $mathcal Ato mathbb C$ whose coboundary is determined by the previous structures. These data give moment map equations on the space of connections on an arbitrary finitely-generated projective $mathcal A$-module. As particular cases, we obtain a large class of equations in algebra (Kings equations for representations of quivers, including ADHM equations), in classical gauge theory (Hermitian Yang-Mills equations, Hitchin equations, Bogomolny and Nahm equations, etc.), as well as in noncommutative gauge theory by Connes, Douglas and Schwarz. We also discuss Nekrasovs beautiful proposal for re-interpreting noncommutative instantons on $mathbb{C}^nsimeq mathbb{R}^{2n}$ as infinite-dimensional solutions of Kings equation $$sum_{i=1}^n [T_i^dagger, T_i]=hbarcdot ncdotmathrm{Id}_{mathcal H}$$ where $mathcal H$ is a Hilbert space completion of a finitely-generated $mathbb C[T_1,dots,T_n]$-module (e.g. an ideal of finite codimension).
A fuzzy geometry is a certain type of spectral triple whose Dirac operator crucially turns out to be a finite matrix. This notion was introduced in [J. Barrett, J. Math. Phys. 56, 082301 (2015)] and accommodates familiar fuzzy spaces like spheres and tori. In the framework of random noncommutative geometry, we use Barretts characterization of Dirac operators of fuzzy geometries in order to systematically compute the spectral action $S(D)= mathrm{Tr} f(D)$ for $2n$-dimensional fuzzy geometries. In contrast to the original Chamseddine-Connes spectral action, we take a polynomial $f$ with $f(x)to infty$ as $ |x|toinfty$ in order to obtain a well-defined path integral that can be stated as a random matrix model with action of the type $S(D)=N cdot mathrm{tr}, F+textstylesum_i mathrm{tr},A_i cdot mathrm{tr} ,B_i $, being $F,A_i $ and $B_i $ noncommutative polynomials in $2^{2n-1}$ complex $Ntimes N$ matrices that parametrize the Dirac operator $D$. For arbitrary signature---thus for any admissible KO-dimension---formulas for 2-dimensional fuzzy geometries are given up to a sextic polynomial, and up to a quartic polynomial for 4-dimensional ones, with focus on the octo-matrix models for Lorentzian and Riemannian signatures. The noncommutative polynomials $F,A_i $ and $B_i$ are obtained via chord diagrams and satisfy: independence of $N$; self-adjointness of the main polynomial $F$ (modulo cyclic reordering of each monomial); also up to cyclicity, either self-adjointness or anti-self-adjointness of $A_i $ and $B_i $ simultaneously, for fixed $i$. Collectively, this favors a free probabilistic perspective for the large-$N$ limit we elaborate on.
In this series of lectures, we (re)view the geometric method that reconstructs, from a geometric object: the spectral curve, an integrable system, and in particular its Tau function, Baker-Akhiezer functions and current amplitudes, all having an interpretation as CFT conformal blocks. The construction identifies Hamiltonians with cycles on the curve, and times with periods (integrals of forms over cycles). All the integrable structure is formulated in terms of homology of contours, the phase space is a space of cycles where the symplectic form is the intersection, the Hirota operator is a degree 2 second-kind cycle, a Sato shift is an addition of a 3rd kind cycle, the Hirota equations amount to saying that merging 3rd kind cycles (monopoles) yields a 2nd kind cycle (dipole). The lecture is divided into 3 parts: 1) classical case, perturbative: the spectral curve is a ramified cover of a base Riemann surface -- with some additional structure -- and the integrable system is defined as a formal power series of a small dispersion parameter $epsilon$. 2) dispersive classical case, non perturbative: the spectral curve is defined not as a ramified cover (which would be a bundle with discrete fiber), but as a vector bundle -- whose dispersionless limit consists in chosing a finite set of vectors in each fiber. 3) non-commutative case, and perturbative. The spectral curve is here a non-commutative surface, whose geometry will be defined in lecture III. 4) the full non-commutative dispersionless theory is under development is not presented in these lectures.
These are lecture notes for the course Poisson geometry and deformation quantization given by the author during the fall semester 2020 at the University of Zurich. The first chapter is an introduction to differential geometry, where we cover manifolds, tensor fields, integration on manifolds, Stokes theorem, de Rhams theorem and Frobenius theorem. The second chapter covers the most important notions of symplectic geometry such as Lagrangian submanifolds, Weinsteins tubular neighborhood theorem, Hamiltonian mechanics, moment maps and symplectic reduction. The third chapter gives an introduction to Poisson geometry where we also cover Courant structures, Dirac structures, the local splitting theorem, symplectic foliations and Poisson maps. The fourth chapter is about deformation quantization where we cover the Moyal product, $L_infty$-algebras, Kontsevichs formality theorem, Kontsevichs star product construction through graphs, the globalization approach to Kontsevichs star product and the operadic approach to formality. The fifth chapter is about the quantum field theoretic approach to Kontsevichs deformation quantization where we cover functional integral methods, the Moyal product as a path integral quantization, the Faddeev-Popov and BRST method for gauge theories, infinite-dimensional extensions, the Poisson sigma model, the construction of Kontsevichs star product through a perturbative expansion of the functional integral quantization for the Poisson sigma model for affine Poisson structures and the general construction.
Random noncommutative geometry can be seen as a Euclidean path-integral approach to the quantization of the theory defined by the Spectral Action in noncommutative geometry (NCG). With the aim of investigating phase transitions in random NCG of arbitrary dimension, we study the non-perturbative Functional Renormalization Group for multimatrix models whose action consists of noncommutative polynomials in Hermitian and anti-Hermitian matrices. Such structure is dictated by the Spectral Action for the Dirac operator in Barretts spectral triple formulation of fuzzy spaces.The present mathematically rigorous treatment puts forward coordinate-free language that might be useful also elsewhere, all the more so because our approach holds for general multimatrix models. The toolkit is a noncommutative calculus on the free algebra that allows to describe the generator of the renormalization group flow -- a noncommutative Laplacian introduced here -- in terms of Voiculescus cyclic gradient and Rota-Sagan-Stein noncommutative derivative. We explore the algebraic structure of the Functional Renormalization Group Equation and, as an application of this formalism, we find the $beta$-functions, identify the fixed points in the large-$N$ limit and obtain the critical exponents of $2$-dimensional geometries in two different signatures.
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