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The Segal–Sugawara Construction

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The Segal–Sugawara construction Peter J. Haine January 5, 2020 Contents 1 Introduction 1 1.1 Reminders from previous talks ...................... 1 1.2 Talk overview ................................ 2 2 Loop algebras & Kac–Moody algebras 2 2.1 Loop algebras ................................ 3 2.2 Recollection on bilinear forms & semisimplicity ............. 3 2.3 Kac–Moody algebras ............................ 4 3 The Segal–Sugawara construction 5 3.1 Motivating case: the Heisenberg algebra ................. 6 3.2 The Casimir element ............................ 8 3.3 The general case .............................. 9 References 11 1 Introduction 1.1 Reminders from previous talks 1.1.1 Definition. The (complex) Witt algebra is the complex Lie algebra Witt of poly- 1 푖푚휃 d nomial vector fields on S . Explicitly, Witt has generators 퐿푚 ≔ 푖푒 d휃 for 푚 ∈ 퐙 with Lie bracket [퐿푚, 퐿푛] ≔ (푚 − 푛)퐿푚+푛 for all 푚, 푛 ∈ 퐙. 1.1.2. Ignoring regularity issues, the Witt algebra is the complexification of the Lie al- gebra of the group Diff +(S1) of orientation-preserving diffeomorphisms of the circle. 2 1.1.3. We have that HLie(Witt; 퐂) ≅ 퐂, so there is a 1-dimensional space of central extensions of the Witt algebra. 1 1.1.4 Definition. The (complex) Virasoro algebra Vir is the central extension 1 퐂chg Vir Witt 1 of Witt with generators 퐿푚 for 푚 ∈ 퐙 and a central element chg, and nontrivial Lie bracket given by 푚3 − 푚 [퐿 , 퐿 ] ≔ (푚 − 푛)퐿 + 훿 chg 푚 푛 푚+푛 푚,−푛 12 for all 푚, 푛 ∈ 퐙. We call the central element chg ∈ Vir the central charge. 1.1.5. Again, ignoring regularity issues, the Virasoro algebra is the complexification of the Lie algebra of the Virasoro group Diff̃ +(S1). 1.2 Talk overview 1.2.1. Let 퐺 be a simply-connected, simple, compact Lie group with Lie algebra 픤. Last time we looked at central extensions 1 S1 L̃퐺 L퐺 1 of the loop group L퐺 ≔ C∞(S1, 퐺). The group Diff +(S1) of orientation-preserving diffeomorphisms of the circle acts on L퐺 by precomposition. So we might expect an action of the Virasoro group Diff̃ +(S1) on L̃퐺. Last time we saw that even though there is not an action of Diff̃ +(S1) on L̃퐺, roughly, the Virasoro group acts on any positive energy representation of L̃퐺. However, the Virasoro action on positive energy representation of L̃퐺 is very inexplicit, and we can only guarantee the existence of the Virasoro action up to ‘essential equivalence’, which is not actually an equivalence relation. In particular, the Pressley–Segal Theorem [8, Theorem 13.4.3] does not explicitly explain how the central circle S1 ⊂ L̃퐺 acts. 1.2.2 Goal. The goal of this talk is to explain the Lie algebra version of the Pressley– Segal Theorem, which gives an explicit representation of the Virasoro algebra on any ‘positive energy’ representation of the Kac–Moody algebra L̃픤 associated to a simple Lie algebra 픤 (over the complex numbers). We’ll be able to do this by writing down explicit universal formulas for ‘elements’ of the universal enveloping algebra U(L̃픤) that satisfy the Virasoro relations. The catch is that these universal formulas involve infinite sums, so they do not actually make sense as elements of U(L̃픤), but they do make sense whenever we act on a representation where only finitely many of the terms don’t act by zero (this is the ‘positive energy’ condition). 2 Loop algebras & Kac–Moody algebras The first thing we need to explain in order to state the Segal–Sugawara construction is what the Kac–Moody algebra L̃픤 is. As the notation suggests, L̃픤 is the Lie algebra analog of the central extension L̃퐺 of the loop group L퐺 (with suitable finiteness hypotheses). Before talking about Kac–Moody algebras, we need to talk about loop algebras. 2 2.1 Loop algebras 2.1.1 Recollection. Let 픤 be a Lie algebra over a ring 푅, and let 푆 be an 푅-algebra. The basechange 픤 ⊗푅 푆 of 픤 to 푆 is the Lie algebra over 푆 with underlying 푆-module the basechange 픤 ⊗푅 푆 of the underlying 푅-module of 픤 to 푆 with Lie bracket extended from pure tensors from the formula [푋 ⊗ 푠 , 푋 ⊗ 푠 ] ≔ [푋 , 푋 ] ⊗ 푠 푠 . 1 1 2 2 픤⊗푅푆 1 2 픤 1 2 2.1.2 Definition. Let 픤 be a complex Lie algebra. The loop algebra L픤 of 픤 is the Lie algebra ±1 L픤 ≔ 픤 ⊗퐂 퐂[푡 ] , regarded as a Lie algebra over 퐂 (rather than 퐂[푡±1]). 2.1.3 Notation. Let 픤 be a complex Lie algebra, 푋 ∈ 픤, and 푚 an integer. We write 푋⟨푚⟩ ≔ 푋 ⊗ 푡푚 ∈ L픤 . 2.1.4. If {푢푖}푖∈퐼 is a Lie algebra basis for 픤, then {푢푖⟨푚⟩}(푖,푚)∈퐼×퐙 is a basis for L픤. 2.1.5 Remark. The loop algebra functor L∶ Lie퐂 → Lie퐂 preserves finite products. 2.1.6 Recollection. A finite dimensional Lie algebra 픤 is simple if 픤 is not abelian and the only ideals of 픤 are 픤 and 0. 2.1.7 Theorem (Garland [3, §§1 & 2]). If 픤 is a simple Lie algebra over 퐂, then 2 HLie(L픤; 퐂) ≅ 퐂 . In particular, if 픤 is simple there is a 1-dimensional space of central extensions of L픤. 2.2 Recollection on bilinear forms & semisimplicity 2.2.1 Notation. Let 픤 be a complex Lie algebra. We write ad∶ 픤 → End퐂(픤) for the adjoint representation, defined by ad(푋) ≔ [푋, −] . 2.2.2 Example. A Lie algebra 픤 is abelian if and only if the adjoint representation of 픤 is trivial. 3 2.2.3 Recollection (Killing form). Let 픤 be a finite dimensional Lie algebra. The Killing form on 픤 is the bilinear form K픤 ∶ 픤 × 픤 → 퐂 (푋, 푌) ↦ tr(ad(푋) ∘ ad(푌)) . The Killing form is symmetric and invariant in the sense that K픤([푋, 푌], 푍) = K픤(푋, [푌, 푍]) for all 푋, 푌, 푍 ∈ 픤. 2.2.4 Example. If 픤 is a simple Lie algebra, then every invariant symmetric bilinear form on 픤 is a 퐂-multiple of the Killing form K픤. See [2] for a nice exposition of this fact. 2.2.5 Example. Let 픞 be a finite dimensional abelian Lie algebra over 퐂. Then since the adjoint representation of 픞 is trivial, the Killing form of 픞 is identically zero. Also note that every bilinear form on the underlying vector space of 픞 is an invariant bilinear form on 픞. 2.2.6 Proposition ([9, Chapter ii, Theorems 2 & 4]). Let 픤 be a finite dimensional com- plex Lie algebra. The following conditions are equivalent: (2.2.6.1) The center of 픤 is trivial. (2.2.6.2) The only abelian ideal in 픤 is 0. (2.2.6.3) The Lie algebra 픤 is isomorphic to a product of simple Lie algebras. (2.2.6.4) Cartan–Killing criterion: the Killing form of 픤 is nondegenerate. If the equivalent conditions (2.2.6.2)–(2.2.6.4) are satisfied, we say that 픤 is semisimple. 2.3 Kac–Moody algebras Now we define the Lie algebra analogue of the central extension L̃퐺 of the loop group L퐺. In addition to the Lie algebra 픤, we need to input an invariant symmetric bilinear form on 픤; the Killing form provides a canonical choice. 2.3.1 Definition. Let 픤 be a Lie algebra over 퐂 with invariant symmetric bilinear form 퐵∶ 픤 × 픤 → 퐂. The Kac–Moody algebra of 픤 with respect to the form 퐵 is the central extension ̃ 1 퐂푐 L퐵픤 L픤 1 with central element 푐 and with Lie bracket extended from the relation [푋⟨푚⟩, 푌⟨푛⟩]̃ ≔ [푋⟨푚⟩, 푌⟨푛⟩] + 훿 푚퐵(푋, 푌)푐 L퐵픤 L픤 푚,−푛 = [푋, 푌]픤⟨푚 + 푛⟩ + 훿푚,−푛푚퐵(푋, 푌)푐 for all 푋, 푌 ∈ 픤. 4 2.3.2. If {푢푖}푖∈퐼 is a Lie algebra basis for 픤, then {푢푖⟨푚⟩}(푖,푚)∈퐼×퐙 ∪ {푐} ̃ is a basis for L퐵픤. 2.3.3 Remark. The Kac–Moody algebra L̃픤 is usually denoted by 픤̂ and is also known as the affine Lie algebra of 픤. 2.3.4 Remark. Let 픤1 and 픤2 be complex Lie algebras equipped with invariant symmet- ric bilinear forms 퐵1 ∶ 픤1 × 픤1 → 퐂 and 퐵2 ∶ 픤2 × 픤2 → 퐂 . Write 퐵 for the bilinear form on the product Lie algebra 픤1 × 픤2 defined by 퐵((푥1, 푥2), (푦1, 푦2)) ≔ 퐵1(푥1, 푦1) + 퐵(푥2, 푦2) . Then we have a canonical isomorphism L̃ (픤 × 픤 ) ≅ L̃ 픤 × L̃ 픤 . 퐵 1 2 퐵1 1 퐵2 2 3 The Segal–Sugawara construction We now have enough of of the background on Lie algebras to give a vague statement of the Segal–Sugawara construction. 3.0.1 Definition. Let 픤 be a Lie algebra over 퐂 and 퐵 an invariant symmetric bilinear ̃ form on 픤. A representation 휌∶ L퐵픤 → End퐂(푉) has positive energy if for all 푣 ∈ 푉 and 푋 ∈ 픤 there exists an integer 푚 > 0 such that 휌(푋⟨푚⟩)푣 = 0 . 3.0.2 Remark. In the theory of Kac–Moody algebras, positive energy representations are more often called admissible. We have chosen the term ‘positive energy’ to align with the loop group terminology. 3.0.3 Theorem (Segal–Sugawara construction, vague formulation). Let 픤 be an abelian or simple Lie algebra over 퐂 and let 퐵∶ 픤 × 픤 → 퐂 be a nondegenerate invariant symmetric bilinear form on 픤.
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    Symmetry, Integrability and Geometry: Methods and Applications SIGMA 17 (2021), 007, 38 pages Harmonic Analysis in d-Dimensional Superconformal Field Theory Ilija BURIC´ DESY, Notkestraße 85, D-22607 Hamburg, Germany E-mail: [email protected] Received September 02, 2020, in final form January 15, 2021; Published online January 25, 2021 https://doi.org/10.3842/SIGMA.2021.007 Abstract. Superconformal blocks and crossing symmetry equations are among central in- gredients in any superconformal field theory. We review the approach to these objects rooted in harmonic analysis on the superconformal group that was put forward in [J. High Energy Phys. 2020 (2020), no. 1, 159, 40 pages, arXiv:1904.04852] and [J. High Energy Phys. 2020 (2020), no. 10, 147, 44 pages, arXiv:2005.13547]. After lifting conformal four-point functions to functions on the superconformal group, we explain how to obtain compact expressions for crossing constraints and Casimir equations. The later allow to write superconformal blocks as finite sums of spinning bosonic blocks. Key words: conformal blocks; crossing equations; Calogero{Sutherland models 2020 Mathematics Subject Classification: 81R05; 81R12 1 Introduction Conformal field theories (CFTs) are a class of quantum field theories that are interesting for several reasons. On the one hand, they describe the critical behaviour of statistical mechanics systems such as the Ising model. Indeed, the identification of two-dimensional statistical systems with CFT minimal models, first suggested in [2], was a celebrated early achievement in the field. For similar reasons, conformal theories classify universality classes of quantum field theories in the Wilsonian renormalisation group paradigm. On the other hand, CFTs also play a role in the description of physical systems that do not posses scale invariance, through certain \dualities".
  • Quantum Mechanics 2 Tutorial 9: the Lorentz Group

    Quantum Mechanics 2 Tutorial 9: the Lorentz Group

    Quantum Mechanics 2 Tutorial 9: The Lorentz Group Yehonatan Viernik December 27, 2020 1 Remarks on Lie Theory and Representations 1.1 Casimir elements and the Cartan subalgebra Let us first give loose definitions, and then discuss their implications. A Cartan subalgebra for semi-simple1 Lie algebras is an abelian subalgebra with the nice property that op- erators in the adjoint representation of the Cartan can be diagonalized simultaneously. Often it is claimed to be the largest commuting subalgebra, but this is actually slightly wrong. A correct version of this statement is that the Cartan is the maximal subalgebra that is both commuting and consisting of simultaneously diagonalizable operators in the adjoint. Next, a Casimir element is something that is constructed from the algebra, but is not actually part of the algebra. Its main property is that it commutes with all the generators of the algebra. The most commonly used one is the quadratic Casimir, which is typically just the sum of squares of the generators of the algebra 2 2 2 2 (e.g. L = Lx + Ly + Lz is a quadratic Casimir of so(3)). We now need to say what we mean by \square" 2 and \commutes", because the Lie algebra itself doesn't have these notions. For example, Lx is meaningless in terms of elements of so(3). As a matrix, once a representation is specified, it of course gains meaning, because matrices are endowed with multiplication. But the Lie algebra only has the Lie bracket and linear combinations to work with. So instead, the Casimir is living inside a \larger" algebra called the universal enveloping algebra, where the Lie bracket is realized as an explicit commutator.