Quantum Groups and Quantum Field Theory

a historical perspective

Anna Lachowska

Institut de Math´ematiques EPFL, Lausanne Switzerland

August 2018 Models in Quantum Field Theory Dedicated to Professor A.N. Vasiliev

who was also my professor during my studies at the Physics Department of the St.Petersburg State University in 1987-93.

A. Lachowska (EPFL) Quantum Groups and QFT August 2018 2 / 26 A few memories from my college classmates

He always emphasized what was profound and signiﬁcant for our understanding of a topic... He would concentrate on the essential mathematical dependence when deriving a formula. The constant coeﬃcients would be relegated to a separate ’box for the constants’ in the corner of the blackboard These ears are what you should remember!

(radiation of a moving relativistic charged particle) He practiced the much-loved express-exams...

A. Lachowska (EPFL) Quantum Groups and QFT August 2018 3 / 26 That was a long time ago...

A. Lachowska (EPFL) Quantum Groups and QFT August 2018 4 / 26 ... but we still remember Professor A.N. Vasiliev for the joy of learning and studying physics that he impersonated.

One of us has dedicated his thesis "Habilitation `a diriger les recherches"to A.N. Vasiliev. Others continue to promote his ideas in far away lands where many of us now work.

I have another informal connection to Professor Vasiliev...

A. Lachowska (EPFL) Quantum Groups and QFT August 2018 5 / 26 Conferences dedicated to A.N. Vasiliev: starting from 2007 MQFT-2010

A. Lachowska (EPFL) Quantum Groups and QFT August 2018 6 / 26 Quantum Field Theory and Quantum Groups

inspiration −→ Physics language Mathematics ←−

Physical inspirations for the development of quantum groups:

1 Quantum integrable models. 2 Deformation quantization. (Commutative observables −→ non-commutative operators). 3 Topological quantum ﬁeld theory.

A. Lachowska (EPFL) Quantum Groups and QFT August 2018 7 / 26 Quantum Groups: an example

Let sl2 denote the simple complex Lie algebra of 2 × 2 traceless matrices over C. The simplest quantum group is denoted Uq(sl2). It has the same set of generators as U(sl2) and a “quantum"parameter q = e~. Compare the relations: U(sl2) Uq(sl2)

HE − EH = 2E qH E = q2E qH HF − FH = 2F qH F = q−2F qH qH −q−H EF − FE = H EF − FE = q−q−1

Note that at the limit ~ → 0 we recover the commutation relations for sl2.

A. Lachowska (EPFL) Quantum Groups and QFT August 2018 8 / 26 1. The discovery of quantum groups: the inverse scattering method for quantum integrable systems Classical sine-Gordon problem (Sklyanin-Takhtajan-Faddeev, 1979).

2 φtt − φxx + m sin φ = 0 ⇔ Ut − Vx + [U, V ] = 0 - the Lax equation, where i 1 π m sinh(λ + 2 φ) U(λ) = i , 4 m sinh(λ − 2 φ) −π i 1 φx m sinh(λ − 2 φ) V (λ) = i , 4 m sinh(λ + 2 φ) −φx φt = π. From the Poisson structure {π(x, t) , φ(x, t)} = δ(x − y) we have U(x, λ) ⊗{} U(y, µ) = [r(λ − µ), U(x, λ) ⊗ I ⊕ I ⊗ U(y, µ)]δ(x − y). Here r(λ) is the classical r-matrix. The Jacoby identity for { } ⇒ classical Yang-Baxter equation for r(λ). A. Lachowska (EPFL) Quantum Groups and QFT August 2018 9 / 26 Quantum sine-Gordon problem

Classical {π(x), φ(y)} = δ(x − y) → quantum [Πn, Φm] = i~δnm.

Classical U(x, λ) → quantum analog Ln(λ) on the lattice, written using 0 1 0 −i 1 0 the Pauli matrices σ = , σ = , σ = 1 1 0 2 i 0 3 0 −1

⇒ R12(λ − µ)L(λ) ⊗ L(µ) = L(µ) ⊗ L(λ)R12(λ − µ), where R12(λ) is the quantum R-matrix satisfying the QYBE: 2 ⊗2 R12(λ − µ)R13(λ)R23(µ) = R23(µ)R13(λ)R12(λ − µ). R ∈ (End(C )) ,

Solution (Kulish-Reshetikhin, 1981) : 1 If the Pauli matrices σ3, σ± = 2 (σ1 ± iσ2) in Ln(λ) are replaced with operators H, E, F acting in a vector space, then one obtains a solution satisfying the relations [H, E] = 2E, [H, F ] = −2F and [E, F ] = sinh(~H) . sinh ~ The relations are equivalent to those for the quantum group Uq(sl2) with q = e~.

A. Lachowska (EPFL) Quantum Groups and QFT August 2018 10 / 26 The dual construction

Uq(sl2) is a deformation of the universal enveloping algebra U(sl2).A deformation of its ’dual’, the algebra of functions on the Lie group SL2, was obtained by Faddeev-Takhtajan (Liouville model on the lattice, 1984).

RT1T2 = T2T1R, T1 = T ⊗ I , T2 = I ⊗ T ,

 q 0 0 0  a b  0 1 0 0  T = , R =   c d  0 q − q−1 1 0  0 0 0 q ab = qba, ac = qca, dc = q−1cd, db = q−1bd, ad − da = (q − q−1)dc, bc = cb, and detqT = ad − qbc is central.

In the limit ~ → 0 the matrix entries a, b, c, d commute and detT = 1, just like the matrix elements of the group SL2.

A. Lachowska (EPFL) Quantum Groups and QFT August 2018 11 / 26 Generalization to any semisimple Lie algebra g

Quantization of Lie groups and Lie algebras (Faddeev-Reshetikhin-Takhtajan, 1987 ) – the FRT formalism.

The FRT deﬁnition of quantum groups :

Quantum −→ integrable U (g), A (G) ←− q q models

– Both algebras are non-commutative. – New object: R-matrix, R ∈ (End(V 2))2, V a representation of g.

A. Lachowska (EPFL) Quantum Groups and QFT August 2018 12 / 26 2. Deformation quantization Let M be a manifold and consider the algebra of functions on M.

A(M) commutative −→ Aq(M) non − commutative, Aq/~Aq ' A

The deformation quantization problem (Bayen, Flato, Fronsdal, Lichnerowicz, Steinheimer, 1977) :

Let M be a Poisson manifold. Construct a series of bidiﬀerential operators ∞ ∞ {Bn}n=1 on C (M) such that

1 P∞ n The star product f ? g = fg + n=1 ~ Bn(f ⊗ g) is associative 1 2 (f ? g − g ? f ) = {f , g} + O( ) ~ ~

If G = M is a Lie group with the semisimple Lie algebra g, then G has a natural Poisson structure induced from the Lie algebra structure on g∗ via the Killing form.

A. Lachowska (EPFL) Quantum Groups and QFT August 2018 13 / 26 Drinfeld: A quantum group is a solution to the dual problem

duality U(g) ←→ A(G)

Both U(g) and A(G) are Hopf algebras: ∆x = 1 ⊗ x + x ⊗ 1, S(x) = −x for x ∈ g; −1 ∆f (g1, g2) = f (g1g2) and S(f )(g) = f (g ) for g, g1, g2 ∈ G. Pairing: A(G) interpreted as matrix elements of U(g)-modules and

hf1f2, ai = hf1 ⊗ f2, ∆ai where a ∈ U(g), f1, f2 ∈ A(G). U(g) ←→ A(G) cocommutative Hopf duality commutative

↓q ↓q

Uq(g) ←→ Aq(G) non−cocommutative Hopf duality non−commutative

A. Lachowska (EPFL) Quantum Groups and QFT August 2018 14 / 26 The Poisson structure on A(G) is determined by the element r ∈ g ⊗ g ("classical r-matrix") In general the consistency conditions of the Poisson and the Lie structure in g lead to the CYBE for r:

[r 12, r 13] + [r 12, r 23] + [r 13, r 23] = 0.

Quantization of U(g) is a q-deformation of the Hopf algebra U(g) with the given r. Therefore the comultiplication is q-deformed: - modulo ~ it stays the same, - in the ﬁrst order on ~ it is determined by r.

The resulting Hopf algebra Uq(g) has a non-cocommutative ∆ described by the universal R-matrix - an operator acting in Uq(g) ⊗ Uq(g) such that

−1 op R∆R = ∆ , R12R13R23 = R23R13R12.

The Hopf dual Aq(g) of Uq(g) solves the deformation quantization problem for the algebra of functions on G with the natural Poisson structure. The FRT formalism leads roughly to the same object.

A. Lachowska (EPFL) Quantum Groups and QFT August 2018 15 / 26 General deﬁnition (Drinfeld-Jimbo, 1989) Let g be a semisimple complex Lie algebra of rank r, and Π = {α1, . . . αr } the simple roots for g. Then the quantum group Uq(g) is generated by the ±1 r Hi elements {Ei , Fi , Ki }i=1 satisfying the relations (Ki = qi ): −1 Ki Ki = 1 Ki Kj = Kj Ki

(αi ,αj ) (αi ,αj ) Ki Ej = q Ej Ki Ki Fj = q Fj Ki K − K 1 i i (αi ,αi )/2 Ei Fj − Fj Ei = δij −1 , qi = q . qi − qi 1−aij X 1 − a 1−a −s m [n]i ! (−1)s ij X ij X X s = 0, = s i j i n [m] ![n − m] ! s=0 i i i i n −n qi −qi Here [n]i = −1 is the q-integer, and X stands for E or F . qi −qi −1 ∆(Ei ) = Ei ⊗ 1 + Ki ⊗ Ei , ∆(Fi ) = Fi ⊗ K + 1 ⊗ Fi .

A. Lachowska (EPFL) Quantum Groups and QFT August 2018 16 / 26 Drinfeld’s derivation of quantum groups

Quantum Groups (Drinfeld, Lecture at ICM-86).

−→ Deformation quantization U (g), A (G) ←− q q

Important new concept: the universal R-matrix, R ∈ Uq(g)⊗ˆ Uq(g).

A. Lachowska (EPFL) Quantum Groups and QFT August 2018 17 / 26 3. TQFT: the Chern-Simons theory

Let M be an oriented 3-manifold, G a compact simple group G, and E = G × M a G-bundle on M, G = M → G the gauge group. The space of connections on E can be viewed as A := Ω1(M, g). Then Witten (1989) suggested to consider the Chern-Simons action : k Z 2 CS(A) = 2 Tr(A ∧ dA + A ∧ A ∧ A) 8π M 3 for A ∈ A, and an integer k. The partition function is given by the Feynman path integral Z Z(M) = exp(2πiCS(A))DA A/G

which is a topological invariant of M (in the classical theory the critical points of the Chern-Simons functional are ﬂat connections – independent of the metric).

A. Lachowska (EPFL) Quantum Groups and QFT August 2018 18 / 26 Topological invariants of links

m Now let L = {Li }i=1 be a link embedded in M, and HolLi (A) ∈ G the holonomy of A around Li , and choose a representation Ri for each component of the link (gauge-invariant observables). Then

m ! Z Y Z(M, L) = TrRi HolLi (A) exp(2πiCS(A))DA A/G i=1 is a topological invariant of the link. Witten’s insight: this invariant is the same as the one that is constructed purely mathematically using the representations of quantum groups.

The deﬁnition of Z(M) (and Z(M, L)) can be extended to manifolds with boundary ∂M = Σ1 ∪ (−Σ2). Then Z(M) corresponds to the time evolution from the physical state on Σ1 to the physical state on Σ2. Mathematically this is formalized in topological quantum ﬁeld theory.

A. Lachowska (EPFL) Quantum Groups and QFT August 2018 19 / 26 Topological quantum ﬁeld theory

Z : {closed oriented surfaces} −→ {vector spaces}

Σ −→ HΣ Cobordisms M −→ vectors Z(M) satisfying the axioms :

∗ H∅ = C, H−Σ = HΣ, Z(Σ × I ) = IdHΣ , HΣ1∪Σ2 = HΣ1 ⊗ HΣ2

Z(M1 ∪Σ M2) = Z(M2) ◦ Z(M1). If ∂M = (−Σ ) ∪ Σ , then Z(M) ∈ H∗ ⊗ H = Hom(H , H ). 1 2 Σ1 Σ2 Σ1 Σ2

For the Chern-Simons theory HΣ are determined by the representations of the aﬃne Lie algebra gˆ of level k (inclusion of the link components leads to the addition of points in Σ marked with representations of gˆ).

A. Lachowska (EPFL) Quantum Groups and QFT August 2018 20 / 26 Computing the invariant

Split the manifold along a surface Σ. Then M1 gives Z(M1) ∈ HΣ, and M2 gives Z(M2) ∈ H−Σ. Their scalar product is the invariant Z(M) ∈ H∅ = C.

A. Lachowska (EPFL) Quantum Groups and QFT August 2018 21 / 26 Quantum surgical invariants of 3-manifolds

Passing from the representations of the aﬃne Lie algebra gˆ at integer level k to the representations of the quantum group Uq(g) with the value of the parameter q a root of unity: qk+hv = 1 (equivalence of categories).

Reshetikhin and Turaev (1991) proposed a construction of an invariant of 3-manifolds using the non-cocommutativity property of quantum groups. Any 3-manifold can be obtained by surgery along a link imbedded in S3; links lead to the same manifold if and only if they are related by Kirby (or Fenn-Rourke) moves (in addition to the homotopy transformations).

A. Lachowska (EPFL) Quantum Groups and QFT August 2018 22 / 26 At roots of unity Uq(g) gives rise to a category T (the fusion category) with only ﬁnitely many representations. Each link component is marked with a simple representation and the topology of the link induces the Uq(g)-homomorphisms:

Applied consecutively these homomorphisms give an invariant of a link. Moreover, a certain weighted sum over all simple modules from the fusion category upholds the invariance with respect to the Kirby moves and thus produces an invariant of the 3-manifold obtained by the surgery on this link. This is the mathematical deﬁnition of the Witten’s invariant inspired by the Chern-Simons QFT.

A. Lachowska (EPFL) Quantum Groups and QFT August 2018 23 / 26 Fusion category: future directions

−→ TQFT (Chern − Simons) Rep(U (g)), ql = 1 ←− q

l The fusion category exists over Uq(g) only if q is a root of unity, q = 1. It

is a semisimple tensor category with ﬁnitely many simple objects {Lλ}λ∈C0 , |C0| < ∞. It provides a mathematical foundation for the conformal and topological quantum ﬁeld theories.

However, from the representation theoretical viewpoint, T is only the tip of the iceberg. In fact, the representation theory over Uq(g) at a root of unity is non-semisimple and very rich. It provides hope for interesting applications in CFT and TQFT.

A. Lachowska (EPFL) Quantum Groups and QFT August 2018 24 / 26 – Ну ты понял?

A. Lachowska (EPFL) Quantum Groups and QFT August 2018 25 / 26 Thank you!

A. Lachowska (EPFL) Quantum Groups and QFT August 2018 26 / 26