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Dirac Operator in Matrix Geometry New Mexico Tech (January 2005) Dirac Operator in Matrix Geometry Ivan G. Avramidi Department of Mathematics New Mexico Institute of Mining and Technology Socorro, NM 87801, USA E-mail: [email protected] Dedicated to the Memory of Vladimir Fock and Dmitri Ivanenko We review the construction of the Dirac operator and its proper- ties in Riemannian geometry and show how the asymptotic expansion of the trace of the heat kernel determines the spectral invariants of the Dirac operator and its index. We also point out that the Einstein- Hilbert functional can be obtained as a linear combination of the first two spectral invariants of the Dirac operator. Next, we report on arXiv:math-ph/0502001v1 31 Jan 2005 our previous attempts to generalize the notion of the Dirac opera- tor to the case of Matrix Geometry, when instead of a Riemannian metric there is a matrix valued self-adjoint symmetric two-tensor that plays a role of a “non-commutative” metric. We construct invariant first-order and second-order self-adjoint elliptic partial differential op- erators, that can be called “non-commutative” Dirac operator and non-commutative Laplace operator. We construct the corresponding heat kernel for the non-commutative Laplace type operator and com- pute its first two spectral invariants. A linear combination of these two spectral invariants gives a functional that can be considered as a non-commutative generalization of the Einstein-Hilbert action. 1 1 Introduction Dirac operator was discovered by Dirac in 1928 as a “square root” of the D’Alambert operator in a flat Minkowskian space in an attempt to develop a relativistic theory of the electron. In 1929, almost immediately after Dirac’s paper, Fock and Ivanenko [1, 2] showed how to generalize the Dirac’s equation for the case of General Relativity. Fock completed the geometrization of the theory of spinors on Riemannian manifolds in [3, 4, 5]. This development was purely local. Only much later, at the end of forties in a global setting it was understood that there are topological obstructions to the existence of the spinor structure and spinor fields and Dirac operators cannot be introduced on every Riemannian manifold (see, for example, [6, 7, 8]). The construction of a square root of the Laplacian naturally leads to the study of complex representations of the Clifford algebra. The spinors are then introduced as the elements of the corresponding vector space. It turns out that there are no non-trivial representations of the orthogonal group in the vector space of spinors compatible with Clifford multiplication. There- fore, spinors on a Riemannian manifold cannot be introduced as sections of a vector bundle associated with the frame bundle of the manifold. Instead of the special orthogonal group one considers its double covering group, so called spin group. Contrary to the orthogonal group the spin group has a representation in the vector space of spinors compatible with Clifford multi- plication. Now, if the frame bundle of the manifold allows a reduction to the spin group, then the manifold is said to admit a spin structure and one can define the spinor bundle, which is a vector bundle associated with this reduc- tion via the representation of the spin group. The spinors are the sections of the spinor bundle. The spinor bundle inherits a connection from the canonical Levi-Civita connection, which enables one to define the Dirac operator. The Dirac op- erator on a spinor bundle is now called the Dirac operator in the narrow sense, while the general (also called twisted) Dirac operator (or Dirac type operator) is any self-adjoint first-order operator whose square is equal to the D’Alambert operator (or the Laplace operator). The Dirac operator in Riemannian geometry is defined with the help of a Riemannian metric and the spin connection. Following the ideas of our papers [9, 10, 11] we are going to generalize this formalism to the case of Matrix Geometry, when instead of a single Riemannian metric there is a matrix-valued symmetric 2-tensor, which we call a “non-commutative met- 2 ric”. Matrix Geometry is motivated by the relativistic interpretation of gauge theories and is intimately related to Finsler geometry (rather a collection of Finsler geometries) (see [9, 10, 11]). In the present paper we will not discuss the origin of Matrix Geometry, but simply assume the existense of such a structure. We will not be concerned about the global issues as well. We will simply assume that there are no topological obstructions to all the structures introduced below. Our “non-commutative” Dirac operator is a first-order el- liptic partial differential operator such that its square is a second-order self- adjoint elliptic operator with positive definite leading symbol (not necessarily of Laplace type). The outline of the paper is as follows. First, we review the construction of the Dirac operator in Riemannian geometry. In section 2.1 we introduce the Clifford algebra and describe its properties. In section 2.2 we introduce the spin group and show that it is a double cover of the orthogonal group. In section 2.3 the spin representation of the spin group is introduced and the spinors are defined. In section 2.4 the derivation of the spin connection and its curvature is described. In section 2.5 we define the Dirac operator and its index. In section 2.6 we introduce the heat kernel. It is explained how the asymptotic expansion of the trace of the heat kernel generates the spectral invariants of the Dirac operator, in particular, its index. In section 3.1 we introduce non-commutative (or matrix) generalization of the Riemannian metric and the Dirac matrices as a deformation of the commutative limit. In section 3.2 we introduce non-commutative versions of the Hodge star operator acting on space of matrix valued p-forms. In section 3.3 the relation of the matrix geometry to Finsler geometry is ex- plored. In section 3.4 we promote the vector spaces introduced above to vector bundles and define the corresponding Hilbert spaces. Since we do not have a Riemannian volume element, we work here not with tensors but rather with densities of various weight. In section 3.5 we develop an exterior calculus for matrix-valued densities and in section 3.6 we introduce a “non- commutative connection”. Sections 3.7 and 3.8 discuss the construction of the non-commutative versions of the Dirac operator and the Laplacian. In sections 3.9 and 3.10 we discuss the spectral asymptotics of the operators introduced before and compute the first two spectral invariants. In section 4 we construct a non-commutative deformation of the Einstein- Hilbert functional. 3 2 Dirac Operators in Riemannian Geometry 2.1 Clifford Algebra Let M be an n-dimensional manifold, x be a point in M and V = TxM be the tangent space at the point x (which is isomorphic to Rn) equipped with a positive-definite scalar product , . Let m = n so that for even h i 2 dimension n = 2m and for odd dimension n = 2m +1. Let also N = 2m. The real Clifford algebra Cliff(n) is the universal associative algebra with unit generated multiplicatively by the range γ(V ) of a linear map γ : V Cliff(n) (1) → satisfying γ(u)γ(v)+ γ(v)γ(u)=2 u, v I , (2) h i where I is the unit in the algebra. In particular, γ(u)γ(u)= u 2I , (3) | | where u 2 = u,u , and, therefore, for any unit vector u, u = 1, the element | | h i | | γ(u) is invertible and [γ(u)]−1 = γ(u) . (4) Let V be the tensor algebra ⊗ ∞ V = kV , (5) ⊗ ⊗ Mk=0 where 0V = R, and be the ideal generated by the set ⊗ I u v + v u 2 u, v u, v V . { ⊗ ⊗ − h i| ∈ } Then the Clifford algebra can be identified with the quotient Cliff(n)= V/ . (6) ⊗ I The corresponding complex Clifford algebra is obtained by tensoring the real algebra with the complex numbers CliffC(n) = Cliff(n) R C . (7) ⊗ 4 The tensor algebra V has a natural N-grading, which after reduction ⊗ mod 2 leads to a natural Z2-grading of the Clifford algebra Cliff(n) = Cliff (n) Cliff−(n) , (8) + ⊕ where Cliff+(n) and Cliff−(n) are the even and the odd parts respectively consisting of the sums of products of even and odd number of elements from γ(V ) with Cliff (n)Cliff (n) Cliff (n) , (9) k j ⊂ jk where k, j = 1. Therefore, the even part Cliff+(n) is a subalgebra of the Clifford algebra± Cliff(n). Let F (n)= R and for 1 k n 0 ≤ ≤ F (n) = Span γ(u ) γ(u ) 1 j k, u V (10) k { 1 ··· j | ≤ ≤ i ∈ } be the subspace of Cliff(n) consisting of the sums of the products of at most k elements from γ(V ). Then the Clifford algebra has a natural increasing filtration F (n) F (n) F (n) = Cliff(n) (11) 0 ⊂ 1 ⊂···⊂ n such that F (n)F (n) F (n) , (12) j k ⊂ k+j where, by definition Fj(n)= Fn(n) if j > n. Further, let C (n)= R , C (n)= F (n)/F − (n), 1 k n . (13) 0 k k k 1 ≤ ≤ Then k F2k(n)= C2k−2j(n) (14) Mj=0 and k F2k+1(n)= C2k−2j+1(n) . (15) Mj=0 k The space Ck(n) is isomorphic to V and the Clifford algebra is a graded algebra, which, as a vector space, has∧ the form n Cliff(n)= Ck(n) , (16) Mk=0 5 with Cj(n)Ck(n)= Ck+j−2l(n) , (17) 0≤M2l≤k+j and is naturally isomorphic to the exterior algebra V ∧ n V = kV , (18) ∧ ∧ Mk=0 where 0V = R.
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