An Introduction to Geometric Quantization and Witten’s Quantum Invariant Kadri İlker Berktav ∗ Department of Mathematics, Middle East Technical University, 06800 Ankara, Turkey Abstract This note, in a rather expository manner, serves as a conceptional introduction to the certain underlying mathematical structures encoding the geometric quantization formalism and the construction of Witten’s quantum invariants, which is in fact organized in the language topological quantum field theory. Contents 1 Introduction 1 2 Quantization in What Sense and GQ Formalism2 3 The Construction of Prequantization4 4 A Review on Chern-Simons Theory6 5 TQFT and Category Theory8 6 The Path Integral Formalism 10 7 The Construction of Witten’s Quantum Invariants 12 1 Introduction A number of remarkable techniques arising from particular gauge theories in physics have long been incarnated into different branches of mathematics. They have been notably employed to study low dimensional topology and geometry in a rather sophisticated way, such as Donaldson theory on four-manifolds [6], the work of Floer on the topology of 3-manifolds and Yang-Mills instantons that serves as a Morse-theoretic interpretation of Chern-Simons gauge theory (and hence an infinite-dimensional counterpart of the classical smooth Morse theory [8], [7], [12]), and Witten’s knot invariants [13] arising from a certain three-dimensional Chern-Simons theory. Main arXiv:1902.10813v1 [math-ph] 26 Feb 2019 motivations of this current discussion are as follows: (i) to provide a brief introduction to the notion of quantization, (ii) to introduce the geometric quantization formalism (GQ) and try to understand how the notion of quantization boils down to the study of representation theory of classical observables in the sense that one can construct the quantum Hilbert space H and a certain Lie algebra homomorphism, and (iii) to elaborate in a rather intuitive manner the quantization of Chern-Simons theory together with a brief discussion of a TQFT in the sense of Atiyah [1] and the language of category theory (cf. [16], [17]) that manifestly captures the essence of TQFT. With this formalism in hand, we shall investigate Witten’s construction of quantum invariants [13] in three-dimensions, and where geometric quantization formalism comes into play. Acknowledgments. This is an extended version of the talk given by the author at the Workshop on Mathematical Topics in Quantization, Galatasaray University, Istanbul, Turkey in 2018. The shorter version, on the other hand, will appear in the proceedings of this workshop. This note con- sists of introductory materials to the notion of geometric quantization based on a series of lectures, ∗E-mail: [email protected] 1 namely Geometric Quantization and Its Applications, delivered by the author as a weekly semi- nar/lecture at theoretical physics group meetings organized by Bayram Tekin at the Department of Physics, METU, Spring 2016-2017. Throughout the note, we do not intend to provide neither original nor new results related to subject that are not known to the experts. The references, on the other hand, are not meant to be complete either. But we hope that the material we present herein provides a brief introduction and a naïve guideline to the existing literature for non-experts who may wish to learn the subject. For a quick and accessible treatment to the geometric quan- tization formalism, including a short introduction to symplectic geometry, see [3], [4] or [10]. [5], on the other hand, provides pedagogically-oriented complete treatment to symplectic geometry. The full story with a more systematic formulation is available in [14] and [9]. Finally, I’d like to thank Özgür Kişisel and Bayram Tekin for their comments and corrections on this note and I am also very grateful to them for their enlightening, fruitful and enjoyable conversations during our regular research meetings. Also, I would like to thank organizers and all people who make the event possible and give me such an opportunity to be the part of it. 2 Quantization in What Sense and GQ Formalism We would like to elaborate the notion of geometric quantization in the case of quantization of classical mechanics. Recall that observables in classical mechanics with a phase space (X; !), a finite dimensional symplectic manifold, form a Poisson algebra with respect to the Poisson bracket {·; ·} on C1(X) given by 1 ff; gg := −w(Xf ;Xg) = Xf (g) for all f; g 2 C (X); (2.1) where Xf is the Hamiltonian vector field associated to f defined implicitly as {Xf ! = df: (2.2) Here, {Xf ! denotes the contraction of a 2-form ! with the vector field Xf in the sense that {Xf ! (·) := !(Xf ; ·): (2.3) Employing canonical/geometric quantization formalism (cf. [10], [3], [14], [9]), the notion of quan- tization boils down to the study of representation theory of classical observables in the sense that one can construct the quantum Hilbert space H and a Lie algebra homomorphism 1 1 Q : C (X); {·; ·} −! End(H); [·; ·] (2.4) together with the Dirac’s quantum condition: 8 f; g 2 C1(X) we have [Q(f); Q(g)] = −i~Q ff; gg (2.5) where [·; ·] denotes the usual commutator on End(H). A primary motivation of this part is to understand how to associate manifestly a suitable Hilbert space H to a given symplectic manifold (M; !) of dimension 2n together with its Poisson algebra 1 C (M); {·; ·} in accordance with a certain set of quantization axioms given as follows: Definition 2.1. (cf. [3], [4]) Let (M; !) be the classical phase space and A a subalgebra 1 1 of C (M). The quantum system H; Q associated to M; C (M) consists of the following data: 1. A complex separable Hilbert space H where its elements are called the quantum wave functions and the rays fλψ : λ 2 Cg are the quantum states. 2. For each f 2 A, Q(f) is a self-adjoint C-linear map on H such that Q sends the function f = 1 to the identity operator idH 2 End(H): 3. The quantum condition 2.5 for f; g 2 A. 1 A Lie algebra homomorphism β : g ! h is a linear map of vector spaces such that β([X; Y ]g) = [β(X); β(Y )]h. Keep in mind that, one can easily suppress the constant "-i~" in 2.5 into the definition of Q such that the quantum condition 2.5 becomes the usual compatibility condition that a Lie algebra homomorphism satisfies. 2 4. The irreducibility condition: If ff1; :::; fng is a complete set of observables in A, i.e. a function g 2 A commuting with all fi’s must be constant: fg; fig = 0 for all i , g = c for some c 2 C; (2.6) then so is the set fQ(f1); :::; Q(fn)g of corresponding operators. Geometric quantization (GQ) is a formalism that encodes the construction of the assignment H; Q in a well-established manner (cf. [3], [4], [14], [9]). In that respects, it enjoys the following properties: 1. GQ is available for any finite dimensional symplectic manifold (M; !). 2. If (M; !; G; µ) is a Hamiltonian G-space with the gauge group G and the moment map µ (cf. [5] ch.22), then GQ remembers the symmetries of classical system in the sense that the corresponding quantum states form an irreducible representation of G (this is in fact the representation-theoretic interpretation [15] of so-called the irreducibility condition stated above). GQ is a two-step process: (i) Pre-quantization, and (ii) the polarization. The fist step involves the construction of so-called a prequantum line bundle L on (M; !), the description of a pre-quantum Hilbert space Hpre as the space Γ(M; L) of smooth square-integrable sections of L , and a (pre-) assignment Qpre as a certain differential operator acting on such sections of L (cf. Theorem 3.1 and Definition 3.1). Note that even if the first step captures almost all necessary constructions related to the axioms in Definition 2.1, it satisfies all but one: the irreducibility condition. This is where the second step comes into play: In order to circumvent such a pathological assignment, which fails to satisfy the irreducibility condition, we need to restrict the space of smooth functions to be quantized in a certain subalgebra A that the irreducibility condition holds as well. This corresponds to a particular choice of a certain Lagrangian n-subbundle P of TM, called the polarization, and hence it leads to define the quantum Hilbert space H as the space ΓP (M; L) of sections of L which are covariantly constant along P ⊂ TM (aka the space of P-polarized sections of L). That is, ΓP (M; L) = fs 2 Γ(M; L): rX s = 0;X 2 Γ(M; P) ⊂ Γ(M; T M)g: (2.7) A motivational example. This example motivates the notion of polarization in a particular case without providing the formal definition of a polarization (for more detail see [9], [3]): Every Kähler manifold (M; !; J), where for all p 2 M, J : p 7! Jp 2 End(TpM) is an integrable almost complex structure compatible with the sypmlectic structure !, gives rise to a holomorphic Kähler polarization associated to (M; !) by setting P := T (0;1)(M), the (−i)-eigenspace subbundle of the complexified tangent bundle TM ⊗ C. Indeed, since the complex structure J is diagonizable, it defines the splitting of the complexified tangent bundle TM ⊗ C as follows: For each p 2 M, (0;1) (1;0) TpM ⊗ C = Tp (M) ⊕ Tp (M) (2.8) (1;0) (0;1) where Tp (M) = fv 2 TpM⊗C : Jv = ivg and Tp (M) = fv 2 TpM⊗C : Jv = −ivg, which are called J-holomorphic (anti-holomorphic resp.) tangent spaces of M; are both Lagrangian subspaces (0;1) (1;0) TpM ⊗ Tp (M)\Tp (M) = f0g (U; z1; z2; :::; zdim M ) of C such that .