| Week | Material covered | Exercises listed in class |
| 20 | Implementable unitaries; Segal's quantization criterion. Hilbert-Schmidt operators. Fermionic construction of some Vir- and LSU(n)-modules. The Wightman fields L and J for the Virasoro and LSU(n) CFTs. The commutation relations for L(f) and for J(f). | • Check the covariance axiom for the chiral Dirac fermion. • Prove the covariance axiom for the Wightman fields JX in the CFT associated to LSU(n) (level 1). • Show that the quantun field L associated to the Virasoro CFT fails the covariance axiom (this is related to the fact that L is not a primary field). • Prove that the definition of the Hilbert-Schmidt norm on the space of finite rank operators from H to K is independent on the choice of orthonormal basis of H. |
| 19 | The embedding of SU(1,1) into the double cover of Diff+(S1). The Spinor bundle on ℂP1 and its action of SL(2,ℂ). The fact that SU(1,1) preserves the polarization of L2(S1;Spinors), and hence acts on the Fock space. | • Check that there exists a holomorphic line bundle (the spinor bundle) on ℂP1 whose sheaf of sections is given by U↦{f(z)√dz : f holomorphic on U} if ∞∉U, and U↦{f(z)√dz : f holomorphic on U\∞ and limz→∞z f(z) exists} if ∞∈U. • Check that the group SL(2,ℂ) has a natural action on the spinor bundle over ℂP1. • Check that if Σn≥0anzn√dz is in H+, then the power series Σn≥0anzn has a radius of convergence that is ≥1, and thus defines a holomorphic function in the unit disc. • Check that the action of SU(1,1) on F presereves the ℤ/2-grading, and descends to an action of PSU(1,1) on Feven. • Check that the action of the double cover Diff+(2)(S1) on H:=L2(S1;Spinors) is unitary. |
| 18 | The fermionic one-particle space L2(S1;Spinors) and its polarization. The way spinors transform under the double cover of Diff+(S1). The Fermionic Fock space for the "Dirac fermion" and the way to index its standard base. Creation and annihilation operators. The creation and annihilation operators as Wightman fields. The Dirac free fermion satisfies the locality axiom. | • Check that, formally, if we apply the change of variable z=eix
to the expression zn√dz, we get (up to a constant)
ei(n+1/2)x√dx.
• Check that the wedge product is not globally defined on the fermionic Fock space.
• Check that [a(f),a*(g)]+ = <f,g>1 and that
[a(f),a(g)]+=[a*(f),a*(g)]+=0.
• Check that a(f) and a*(f) are bounded operators, and that their operator norm agrees with the
L2 norm of f.
• Check that the fields c(f) := a(f) + a*(f) satisfy
[c(f),c(g)]+ = 2Re<f,g>1.
• [hand in |
| 17 | The definition of full CFT, and of chiral CFT (in Wightmann's formalism). The Fermionic Fock space. | • Show that there is an isomorphism of Lie groups SO+(2,2)/{±1} ≅ PSL2(ℝ)×PSL2(ℝ). • Show that there are isomorphisms of Lie groups PSL2(ℝ) ≅ PSU(1,1) ≅ SO+(2,1). • Show that the second Lie algebra cohomolog of sl2(ℝ) with coefficients in ℝ vanishes. Hint: Use the fact that sl2(ℝ) and su(2) have the same complexification, and that SU(2) is compact. |
| 16 | Distributions: Schwartz space S(ℝd), tempered distributions. Unbounded operators: closed operators, closeable operators, the adjoint of a densely defined operator. Direct integrals of Hilbert spaces. Wightman's axioms for QFT. | • Try to make sense of "1/x2" as a distibution (hint: take the deriative of -1/x). • Check that d/dx : S(ℝ)&rarr L2(ℝ) is a closeable operator from L2(ℝ) to L2(ℝ) (hint: apply Fourier transform to turn it into the operator of multiplication by ix). • Check that the adjoint of an operator is always closed • If A is an abelian group and H carries an irreducible unitary representation of A, then H is one dimensional. • Let V be a real vector space. Then there is a natural bijection between its dual V* and the set isomorphism calsses of (continuous) unitary irreps of V. |
| 15 | Every 2-cocycle on Lg is cohomologous to a G-invariant one. Two G-invariant 2-cocycles are cohomologous iff they are equal. Canonical identifications between H2(Lg, ℝ) and the space of G-invariant 2-cocycles. Every G-invariant 2-cocycle on Lg is a multiple of the basic cocycle. | • For G compact, simple (= gℂ is an irreducible G-rep) and simply connected, show that [gℂ, gℂ]=gℂ. •Assume that G is compact and simple. Show that the space of G-invariant bilinear forms gℂ×gℂ &rarr ℂ is one dimensional. • [hand in on week 16] Prove that the space of continuous Diff(S1)-invariant 2-cocycles on the abelian Lie algebra Lℝ=C∞(S1,ℝ) is one dimensional, and that it is spanned by the basic cocycle ω(ξ,η)=1/2π∫S1ξ(θ)η'(θ)dθ. • [hand in on week 16] Prove that the space of continuous S1-invariant 2-cocycles on Vect(S1) is two dimensional, with basis ω1(ξ,η) := ∫ ξ'''(θ)η(θ)dθ and ω2(ξ,η) := ∫ ξ'(θ)η(θ)dθ. |
| 14 | The second cohomology groups of the lie algebras Vect(S1), and Lg:=C∞(S1,g) with values in ℝ is one dimensional. (Here, g is the Lie algebra of a simple, compact, simply connected Lie group). The definition of simple Lie group. The basic cocycle ω(ξ,η)=1/2π∫S1<ξ,dη> on Lg. Lemma: The action of a compact group G on the set of cocycles on Lg becomes trivial at the level of cohomology. | • Let G be a Lie group, with corresponding Lie algebra g. Prove that the loop Lie algebra Lg:=C∞(S1,g) is the Lie algebra of the loop group LG:=MapC∞(S1,G). • Prove that Vect(S1) is the Lie algebra of Diff+(S1). • [hand in on week 15] Let g be a Lie algebra defined over a field k with the property that H2(g, k) = k. Prove that, up to isomorphism, there is only one non-trivial central extension of g by k. Note: you are allowed to use the classification (∗), but all other arguments should be justified. |
| 13 | Lie algebra cohomology. The chain complexes computing group cohomology, and Lie algebra cohomology. The classification of Lie algebra central extensions of g by a by H2(g, a). (∗) | 
• Finish the proof that there is a canonical bijection between the set of Lie algebra central extensions of g by a, and the second Lie algebra cohomology group of g with coefficients in a. |
| 12 | The S1-valued 2-cocycle associated to a homomorphism G→PU(H). The A-valued 2-cocycle on G of a central extension of G by A. The central extension associated to a 2-cocycle. Definition of H2(G, A). Bijection between iso classes of central extensions of G by A and H2(G, A). The Heisenberg group and the corresponding ℝ-valued 2-cocycle on ℝ2n. | • Given a homomorphism G→PU(H) and a lift τ:G→U(H), show that τ(x)τ(y)τ(xy)-1 is a 2-cocycle. • Given an A-valued 2-cocycle ω on G, check that ∗ω defined by (x,a)∗ω(y,b) := (xy, a+b+&omega(x,y)) is a unital operation. • Given cohomologous 2-cocycles ω1 and ω2, check that the corresponding central extensions (G × A, ∗ω1) and (G × A, ∗ω2) are isomoprhic (as central extensions, not just as groups). • Check that there are two non-isomorphic central extensions of ℤ/3 by ℤ/3 that both look like ℤ/3→ℤ/9→ℤ/3. • [hand in on week 13. all sources should be mentionned.] Do a literature search and find the definition of the Bott-Virasoro cocycle. This is an ℝ-valued 2-cocycle on the group Diff+(S1) of orientation preserving diffeomorphisms of the circle. Check that it is indeed a 2-cocycle. |
| 11 | The projection SU(2)→SO(3) is not topologically split. The definition of homotopy groups. The projection U(H)→PU(H) is not topologically split. | • Prove that a central extension is trivial (i.e. split) if and only it is isomorphic to a direct product. • Check that the projection SL(2,ℂ)→SO+(1,3) is not topologically split. |
| 10 | The norm and weak* topologies on H. These topologies agree on the unit sphere S(H) ⊂ H. The strong and weak topologies on the algebra B(H) of bounded operators on H. These topologies agree on U(H) ⊂ B(H). Proof that (U(H), strong topology) is a topological group. | • Check that the two descriptions of the weak* topology on H given in class are equivalent to each other (one is given in terms of open subsets; the other is given in terms of convergent nets). • Let H be an infinite dimensional Hilbert space. Check that the norm, strong, and weak topologies on B(H) are all distinct. • Check that the map that sends a to a* is discontinuous for the strong topology on B(H). • Let H be separable Hilbert space. Check that U(H) is then separable for the strong topology. • Let H=L2[0,1], and let Ts:H→H be the map given by (Tsf)(x)=(1/√s)f(x/s) if x≤s, and 0 otherwise. Let Fs(U):=TsUTs*+(1-TsTs*). Prove that the map F:(0,1]× U(H) → U(H) sending (s,U) to Fs(U) extends to a contracting homotopy [0,1]× U(H) → U(H) that is continuous for the strong topology on U(H). |
| 9 | An example of quantum state space: P(ℂ2) = S3/S1 = ℂP1 = S2. The central extensions ±1→SU(2)→SO(3) and ±1→SL(2,ℂ)→SO+(1,3). The central extension S1→U(H)→PU(H). The norm topology on U(H), and the fact that the rotation action of S1 on L2(S1) is not norm continuous. | • [hand in on week 10] The transition probability between two elements [ξ], [η] of P(H) is defined by the formula δ([ξ],[η]) = |<ξ,η>|2. Let H:= ℂ2, and let φ and ψ be points of P(H) = S2. Viewing them as vectors in ℝ3, we may consider the angle θ formed by φ and ψ. Prove that δ(φ,ψ) = cos2(θ/2). • Write down the formula for the natural isomorphism ℂP1 → S2. • The map A ↦ Ad(A) defines a group homomorphism SU(2) → SO(3). Write down the corresponding Lie algebra homomorphism su(2)→so(3) explicitely, and check that it is an isomorphism. • Check that the map ℂP1 → S2 is su(2)-equivariant. • Check that ±1 → Q8 → V is a central extension. Here, V stands for the group ℤ/2ℤ × ℤ/2ℤ. • Check that (U(H), norm topology) is a topological group. Namely, check that multiplication and inversion are continuous operations in that topology. • Let H be a separable Hilbert space. Find an uncountable collection of elements of U(H) whose pairwise distance is 2 in the norm metric. |
| 8 | The pseudogroup of conformal transformations of (subsets of) ℂ. Comparison between the complexifications of the Lie algebras of conformal Killing fields in signatures (2,0) and (1,1). What it means for a CFT to be chiral. The basic anzatz of quantum mechanics: quantum state space is P(H). Transition probabilities. Quantization of symmetries. | • Check that the Lie bracket [f(z)∂z,g(z)∂z] := (f(z)g'(z)-f'(z)g(z))∂z on the vector space of holomorphic vector fields is the restriction of the Lie bracket of real vector fields on ℝ2 under the map sending f(z)∂z to Re(f)∂x + Im(f)∂y. • Check that the group of (globally defined) conformal automorphisms of ℝ1,1 generates the pseudogroup of conformal transformations (of subsets) of ℝ1,1 • Read and understand the following proof of Wigner's theorem. |
| 7 | Conformal Killing vector fields. Proof that, if n+m > 2, the Lie algebra of conformal Killing vector fields on ℝn,m is isomorphic to so(n+1,m+1). Classification of conformal Killing fields on ℂ. | • Prove that the only conformal automorphisms of ℂ are the maps z → az+b and z → a zbar + b. • [hand in on week 8] Given a holomorphic function f:ℂ→ℂ, consider the vector field X := Re(f)∂x + Im(f)∂y. Its flow-at-time-t is a map φt:Ut→ℂ defined on an open subset Ut⊂ℂ. Given a vector field v on Ut, check that the equation dφt(Jv) = Jdφt(v) holds on φt(Ut). Here, J stands for the standard complex structure on ℂ. |
| 6 | Inversions. Special conformal transformations. Construction of a Lie algebra map so(n+1,m+1) → {conformal Killing vector fields on ℝn,m}. Classification of conformal transformations of ℝn,m in the cases (n,m) = (2,0) and (n,m) = (1,1). | • Check that the diffeomorphism ℝn \ {0} → ℝn \ {0} sending x to x / ||x||2 is conformal. • Proposition 2.5. of the book claims that the map τ : ℝn,m → Sn×Sm is conformal. Finish the proof of that proposition and find the typo. • Check that the restriction of the flat semi-Riemannian metric of ℝn,m to the manifold of non-zero null-vectors has a kernel that is everywhere 1-dimensional. • Check that, on an orientable 2-manifold, a conformal structure is equivalent to the data of a complex structure that is well defined up to complex conjugation. • Prove that any diffeomorhpism ℝ1,1 → ℝ1,1 that preserves the two null-foliations is conformal. |
| 5 | Semi-Riemannian manifolds. Eucledian, Minkowskian manifolds. Conformal equivalence of semi-Riemannian metrics. Conformal transformations. Translations, dilations, rotations of ℝn,m. The group O(n,m). The stereographic projection ℝn → Sn is a conformal transformation. | • Any non-degenerate symmetric bilinear form on ℝn is congugate to one of the form < , >ℝr - < , >ℝn-r . • Check the formula of the stereographic projection. • Check that, under the assumtion that ε2=0, we have 1/(a + ε) = 1/a - ε / a2. |