Deformation quantization and Morita
equivalence
Henrique Bursztyn (IMPA) and Stefan Waldmann (Freiburg)
Buenos Aires, 2010
Lecture 2 - Outline:
1. Deformation quantization (lecture 1 reminder)
2. Morita equivalence reminder
3. Morita equivalence of star products
1. Deformation quantization reminder (lecture 1)
M smooth manifold, C ∞(M ) algebra of complex-valued functions
1. Deformation quantization reminder (lecture 1)
M smooth manifold, C ∞(M ) algebra of complex-valued functions
Star product: associative product ? on C ∞(M )[[~]],
f ? g = fg +
∞
X
~r Cr (f, g),
r=1
where Cr : C ∞(M ) × C ∞(M ) → C ∞(M ) bidifferential
(Bayen, Flato, Fronsdal, Lichnerowicz, Sternheimer, ’1978)
1. Deformation quantization reminder (lecture 1)
M smooth manifold, C ∞(M ) algebra of complex-valued functions
Star product: associative product ? on C ∞(M )[[~]],
f ? g = fg +
∞
X
~r Cr (f, g),
r=1
where Cr : C ∞(M ) × C ∞(M ) → C ∞(M ) bidifferential
(Bayen, Flato, Fronsdal, Lichnerowicz, Sternheimer, ’1978)
Poisson bracket: {f, g} =
1
[f, g]∗~=0
~
= C1(f, g) − C1(g, f )
1. Deformation quantization reminder (lecture 1)
M smooth manifold, C ∞(M ) algebra of complex-valued functions
Star product: associative product ? on C ∞(M )[[~]],
f ? g = fg +
∞
X
~r Cr (f, g),
r=1
where Cr : C ∞(M ) × C ∞(M ) → C ∞(M ) bidifferential
(Bayen, Flato, Fronsdal, Lichnerowicz, Sternheimer, ’1978)
Poisson bracket: {f, g} =
1
[f, g]∗~=0
~
= C1(f, g) − C1(g, f )
P∞
Equivalence: T (f ? g) = T (f ) ?0 T (g), where T = Id + r=1 ~r Tr
1. Deformation quantization reminder (lecture 1)
M smooth manifold, C ∞(M ) algebra of complex-valued functions
Star product: associative product ? on C ∞(M )[[~]],
f ? g = fg +
∞
X
~r Cr (f, g),
r=1
where Cr : C ∞(M ) × C ∞(M ) → C ∞(M ) bidifferential
(Bayen, Flato, Fronsdal, Lichnerowicz, Sternheimer, ’1978)
Poisson bracket: {f, g} =
1
[f, g]∗~=0
~
= C1(f, g) − C1(g, f )
P∞
Equivalence: T (f ? g) = T (f ) ?0 T (g), where T = Id + r=1 ~r Tr
Moduli of star products: Def(M ) = {?}/ ∼
Formal Poisson structures: π~ ∈ ~X 2(M )[[~]],
π~ = ~π1 + ~2π2 + . . . , such that [π~, π~] = 0.
Formal Poisson structures: π~ ∈ ~X 2(M )[[~]],
π~ = ~π1 + ~2π2 + . . . , such that [π~, π~] = 0.
P∞ r
0
Equivalence: T {f, g}~ = {T f, T g}~, T = exp( r=1 ~ Xr )
Formal Poisson structures: π~ ∈ ~X 2(M )[[~]],
π~ = ~π1 + ~2π2 + . . . , such that [π~, π~] = 0.
P∞ r
0
Equivalence: T {f, g}~ = {T f, T g}~, T = exp( r=1 ~ Xr )
Moduli:
FPois(M) = {π~, [π~, π~] = 0}/ ∼
Formal Poisson structures: π~ ∈ ~X 2(M )[[~]],
π~ = ~π1 + ~2π2 + . . . , such that [π~, π~] = 0.
P∞ r
0
Equivalence: T {f, g}~ = {T f, T g}~, T = exp( r=1 ~ Xr )
Moduli:
FPois(M) = {π~, [π~, π~] = 0}/ ∼
Consequence of Kontsevich’s formality (’97):
There is 1-1 correspondence
∼
K∗ : FPois(M ) → Def(M ), [π~ = ~π1 + . . .] 7→ [?]
Formal Poisson structures: π~ ∈ ~X 2(M )[[~]],
π~ = ~π1 + ~2π2 + . . . , such that [π~, π~] = 0.
P∞ r
0
Equivalence: T {f, g}~ = {T f, T g}~, T = exp( r=1 ~ Xr )
Moduli:
FPois(M) = {π~, [π~, π~] = 0}/ ∼
Consequence of Kontsevich’s formality (’97):
There is 1-1 correspondence
∼
K∗ : FPois(M ) → Def(M ), [π~ = ~π1 + . . .] 7→ [?]
1
such that ~ [f, g]?~=0 = π1(df, dg).
Formal Poisson structures: π~ ∈ ~X 2(M )[[~]],
π~ = ~π1 + ~2π2 + . . . , such that [π~, π~] = 0.
P∞ r
0
Equivalence: T {f, g}~ = {T f, T g}~, T = exp( r=1 ~ Xr )
Moduli:
FPois(M) = {π~, [π~, π~] = 0}/ ∼
Consequence of Kontsevich’s formality (’97):
There is 1-1 correspondence
∼
K∗ : FPois(M ) → Def(M ), [π~ = ~π1 + . . .] 7→ [?]
1
such that ~ [f, g]?~=0 = π1(df, dg).
Problem: When are two star products Morita equivalent ?
2. Morita equivalence reminder
2. Morita equivalence reminder
A, B unital k-algebras,
2. Morita equivalence reminder
A, B unital k-algebras,
A and B are Morita equivalent if exists invertible bimodule B XA
2. Morita equivalence reminder
A, B unital k-algebras,
A and B are Morita equivalent if exists invertible bimodule B XA
Equivalences of representations A−Mod ∼
= B−Mod
2. Morita equivalence reminder
A, B unital k-algebras,
A and B are Morita equivalent if exists invertible bimodule B XA
Equivalences of representations A−Mod ∼
= B−Mod
Morita self-equivalences = Pic(A)
2. Morita equivalence reminder
A, B unital k-algebras,
A and B are Morita equivalent if exists invertible bimodule B XA
Equivalences of representations A−Mod ∼
= B−Mod
Morita self-equivalences = Pic(A)
Morita theorem: characterization of equivalence bimodules.
2. Morita equivalence reminder
A, B unital k-algebras,
A and B are Morita equivalent if exists invertible bimodule B XA
Equivalences of representations A−Mod ∼
= B−Mod
Morita self-equivalences = Pic(A)
Morita theorem: characterization of equivalence bimodules.
When are two star products ?, ?0 Morita equivalent?
3. Morita equivalence of star products
3. Morita equivalence of star products
f.g.p. modules over (C ∞(M )[[~]], ?)?
3. Morita equivalence of star products
f.g.p. modules over (C ∞(M )[[~]], ?)?
There is a canonical action
(B., ’02)
Φ : Pic(M ) × Def(M ) → Def(M ), (L, ?) 7→ ΦL(?)
3. Morita equivalence of star products
f.g.p. modules over (C ∞(M )[[~]], ?)?
There is a canonical action
(B., ’02)
Φ : Pic(M ) × Def(M ) → Def(M ), (L, ?) 7→ ΦL(?)
Theorem
Star products ? and ?0 are Morita equivalent iff [?], [?0] lie in the
same Diff(M ) n Pic(M )-orbit:
[?0] = ΦL([?ϕ])
3. Morita equivalence of star products
f.g.p. modules over (C ∞(M )[[~]], ?)?
There is a canonical action
(B., ’02)
Φ : Pic(M ) × Def(M ) → Def(M ), (L, ?) 7→ ΦL(?)
Theorem
Star products ? and ?0 are Morita equivalent iff [?], [?0] lie in the
same Diff(M ) n Pic(M )-orbit:
[?0] = ΦL([?ϕ])
works also purely algebraically...
What comes next ?
What comes next ?
Ȟ 2 (M, Z) K
∗
FPois(M ) −→
Def(M )
Pic(M ) = Ȟ 2 (M, Z)
Description of ME star products via characteristic classes?
Poisson-geometric counterpart of algebraic ME of quantum algebras?
Some references:
B., Waldmann: Deformation quantization of Hermitian vector bundles, Lett. Math. Phys.
53.4 (2000), 349–365
B.: Semiclassical geometry of quantum line bundles and Morita equivalence of star products,
Intern. Math. Res. Notices , 16 (2002) 821–846.
B., Waldmann: Bimodule deformations, Picard groups and contravariant connections, KTheory 31 (2004), 1-37.