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Perturbative topological field theory with Segal-likegluing

Pavel Mnev

Max Planck Institute for Mathematics, Bonn

ICMP, Santiago de Chile, July 27, 2015

Joint work with Alberto S. Cattaneo and Nikolai Reshetikhin

Introduction BV-BFV formalism, outline Examples

Plan

1 Introduction: calculating partition functions by cut/paste.

2 BV-BFV formalism for gauge theories on manifolds with boundary:an outline.

3 Abelian BF theory in BV-BFV formalism.

4 Further examples: Poisson sigma model, cellular models.


Plan






Plan






Plan






Cut/paste philosophy

Introduction: calculating partition functions by cut/paste.Idea:

Z

( )=

⟨Z

( ), Z

( )⟩

Functorial description (Atiyah-Segal):Closed (n− 1)-manifold Σ HΣ

n-cobordism M Partition function

ZM : HΣin → HΣout

Gluing Composition

ZMI∪MII= ZMII

ZMI

Atiyah: TQFT is a functor of monoidal categories(Cobn,t)→ (VectC,⊗).




Z

( )=

⟨Z

( ), Z

( )⟩Functorial description (Atiyah-Segal):

Closed (n− 1)-manifold Σ HΣ



Gluing Composition

ZMI∪MII= ZMII

ZMI





Z

( )=

⟨Z

( ), Z

( )⟩Functorial description (Atiyah-Segal):

Closed (n− 1)-manifold Σ HΣ



Gluing Composition

ZMI∪MII= ZMII

ZMI




Example: 2D TQFT

Z

can be expressed in terms of building blocks:

1 Z

( ): C→ HS1

2 Z

( ): HS1 → C

3 Z

: HS1 ⊗HS1 → HS1

4 Z

: HS1 → HS1 ⊗HS1

– Universal local building blocks for 2D TQFT!


Corners

For n > 2 we want to glue along pieces of boundary/ glue-cut withcorners.Building blocks: balls with stratified boundary (cells)

Extension of Atiyah’s axioms to gluing with corners: extended TQFT(Baez-Dolan-Lurie).Example: Turaev-Viro 3D state-sum model.

building block - 3-simplex

q6j-symbol

gluing sum over spins on edges


Corners


Extension of Atiyah’s axioms to gluing with corners: extended TQFT(Baez-Dolan-Lurie).

Example: Turaev-Viro 3D state-sum model.building block - 3-simplex

q6j-symbol



Corners


Extension of Atiyah’s axioms to gluing with corners: extended TQFT(Baez-Dolan-Lurie).Example: Turaev-Viro 3D state-sum model.

building block - 3-simplex

q6j-symbol



Goal

Problems:

Very few examples!

Some natural examples do not fit into Atiyah axiomatics.

Goal:

Construct TQFT with corners and gluing out of perturbative pathintegrals for diffeomorphism-invariant action functionals.

Investigate interesting examples.


Classical BV-BFV theories

Reminder. A classical BV theory on a closed spacetime manifold M :

Fω ∈ Ω2(F) odd-symplectic

Q ∈ X(F), odd, Q2 = 0

S ∈ C∞(F), ιQω = δS




F

ω ∈ Ω2(F) odd-symplectic

Q ∈ X(F), odd, Q2 = 0






Q ∈ X(F), odd, Q2 = 0






Q ∈ X(F), odd, Q2 = 0






Q ∈ X(F), odd, Q2 = 0






Q ∈ X(F), odd, Q2 = 0


Note: S, Sω = 0.



BV-BFV formalism for gauge theories on manifolds with boundaryReference: A. S. Cattaneo, P. Mnev, N. Reshetikhin, Classical BVtheories on manifolds with boundary, Comm. Math. Phys. 332 2 (2014)535–603.

For M with boundary:

M −−−−→ (F , ω, Q, S) – space of fieldsyπ yπ∗

∂M −−−−→ (F∂ , ω∂ = δα∂ , Q∂ , S∂) – phase space

Relations: Q2∂ = 0, ιQ∂ω∂ = δS∂ ; Q2 = 0, ιQω = δS + π∗α∂ .

⇒CME: 12 ιQιQω = π∗S∂

Gluing:MI ∪Σ MII → FMI

×FΣFMII

This picture extends to higher-codimension strata!





M −−−−→ (F , ω−1, Q

1, S

0) – space of fieldsyπ yπ∗

∂M −−−−→ (F∂ , ω∂ = δα∂0

, Q∂1, S∂

1) – phase space

Subscripts =“ghost numbers”.




×FΣ FMII











×FΣFMII











×FΣFMII











×FΣFMII




Example: abelian Chern-Simons theory, dimM = 3.

M −−−−→ (F , ω, Q, S)yπ yπ∗

∂M −−−−→ (F∂ , ω∂ = δα∂ , Q∂ , S∂)

Euler-Lagrange moduli spaces:

M −−−−→ H•(M)[1]

ι∗y

∂M −−−−→ H•(∂M)[1]




M −−−−→ (Ω•(M)[1], ω, Q, S)yπ:A7→A|∂yπ∗

∂M −−−−→ (Ω•(∂M)[1], ω∂ = δα∂ , Q∂ , S∂)

Superfield A = c︸︷︷︸ghost,1

+ A︸︷︷︸classical field,0

+A+

−1+ c+−2︸︷︷︸

antifields


M −−−−→ H•(M)[1]

ι∗y

∂M −−−−→ H•(∂M)[1]




M −−−−→ (Ω•(M)[1], 12

∫MδA ∧ δA, Q, S)yπ:A7→A|∂

yπ∗

∂M −−−−→ (Ω•(∂M)[1], 12

∫∂δA ∧ δA, Q∂ , S∂)



+A+

−1+ c+−2︸︷︷︸

antifields


M −−−−→ H•(M)[1]

ι∗y

∂M −−−−→ H•(∂M)[1]




M −−−−→ (Ω•(M)[1], 12

∫MδA ∧ δA,

∫MdA δ

δA , S)yπ:A7→A|∂yπ∗

∂M −−−−→ (Ω•(∂M)[1], 12

∫∂δA ∧ δA,

∫∂dA δ

δA , S∂)



+A+

−1+ c+−2︸︷︷︸

antifields


M −−−−→ H•(M)[1]

ι∗y

∂M −−−−→ H•(∂M)[1]




M −−−−→ (Ω•(M)[1], 12

∫MδA ∧ δA,

∫MdA δ

δA ,12

∫MA ∧ dA)yπ:A7→A|∂

yπ∗

∂M −−−−→ (Ω•(∂M)[1], 12

∫∂δA ∧ δA,

∫∂dA δ

δA ,12

∫∂A ∧ dA)



+A+

−1+ c+−2︸︷︷︸

antifields


M −−−−→ H•(M)[1]

ι∗y

∂M −−−−→ H•(∂M)[1]




M −−−−→ (Ω•(M)[1], 12

∫MδA ∧ δA,

∫MdA δ

δA ,12

∫MA ∧ dA)yπ:A7→A|∂

yπ∗

∂M −−−−→ (Ω•(∂M)[1], 12

∫∂δA ∧ δA,

∫∂dA δ

δA ,12

∫∂A ∧ dA)



+A+

−1+ c+−2︸︷︷︸

antifields


M −−−−→ H•(M)[1]

ι∗y

∂M −−−−→ H•(∂M)[1]


Quantum BV-BFV theories

Quantum BV-BFV formalism.

Σ closed, dim Σ = n− 1 7→ (H•Σ,ΩΣ)

M , dimM = n 7→(Fres, ωres)

ZM ∈ Dens12 (Fres)⊗H∂M satisfying mQME:(

i

~Ω∂M − i~∆res

)ZM = 0

Gauge-fixing ambiguity ⇒ ZM ∼ ZM +(i~Ω∂M − i~∆res

)(· · · ).

Gluing:

ZMI∪ΣMII= P∗(ZMI

∗Σ ZMII)

∗Σ — pairing of states in HΣ,P∗ — BV pushforward (fiber BV integral) for

FMIres ×FMII

resP−→ FMI∪ΣMII

res







i


)ZM = 0


)(· · · ).

Gluing:


∗Σ ZMII)


FMIres ×FMII


res






ZM ∈ Dens12 (Fres)⊗H∂M

satisfying mQME:(i


)ZM = 0


)(· · · ).

Gluing:


∗Σ ZMII)


FMIres ×FMII


res







i


)ZM = 0


)(· · · ).

Gluing:


∗Σ ZMII)


FMIres ×FMII


res







i


)ZM = 0


)(· · · ).

Gluing:


∗Σ ZMII)


FMIres ×FMII


res







i


)ZM = 0


)(· · · ).

Gluing:


∗Σ ZMII)


FMIres ×FMII


res







i


)ZM = 0


)(· · · ).

Gluing:


∗Σ ZMII)

∗Σ — pairing of states in HΣ,

P∗ — BV pushforward (fiber BV integral) for

FMIres ×FMII


res







i


)ZM = 0


)(· · · ).

Gluing:


∗Σ ZMII)


FMIres ×FMII


res


Quantization

QuantizationChoose p : F∂ → B Lagrangian fibration, α∂ |p−1(b) = 0.

H∂ = Dens12 (B) , Ω∂ = S∂ ∈ End(H∂)1.

F

π

yF∂p

yB


Quantization


H∂ = Dens12 (B) , Ω∂ = S∂ ∈ End(H∂)1.

F

π

yF∂p

yB


Quantization


H∂ = Dens12 (B) , Ω∂ = S∂ ∈ End(H∂)1.

F ⊃ Fb = π−1p−1b

π

yF∂p

yB 3 b boundary condition

Partition function:

ZM (b) =

∫L⊂Fb

ei~S , ZM ∈ Dens

12 (B)

L ⊂ Fb gauge-fixing Lagrangian.Problem: ZM may be ill-defined due to zero-modes.


Quantization


H∂ = Dens12 (B) , Ω∂ = S∂ ∈ End(H∂)1.

F ⊃ Fb = π−1p−1b

π

yF∂p


Partition function:

ZM (b) =

∫L⊂Fb

ei~S , ZM ∈ Dens

12 (B)

L ⊂ Fb gauge-fixing Lagrangian.Problem: ZM may be ill-defined due to zero-modes.


Quantization


H∂ = Dens12 (B) , Ω∂ = S∂ ∈ End(H∂)1.

F ⊃ Fb = π−1p−1b

π

yF∂p


Solution: Split Fb = Fres × F 3 (φres, φ). Partition function:

ZM (b, φres) =

∫L⊂F

ei~S(b,φres,φ), ZM ∈ Dens

12 (B)⊗Dens

12 (Fres)

L ⊂ F gauge-fixing Lagrangian.

FresP−→ F ′res ⇒ Z ′M = P∗ZM


Quantization


H∂ = Dens12 (B) , Ω∂ = S∂ ∈ End(H∂)1.

F ⊃ Fb = π−1p−1b

π

yF∂p


Solution: Split Fb = Fres × F 3 (φres, φ). Partition function:

ZM (b, φres) =

∫L⊂F

ei~S(b,φres,φ), ZM ∈ Dens

12 (B)⊗Dens

12 (Fres)

L ⊂ F gauge-fixing Lagrangian.

FresP−→ F ′res ⇒ Z ′M = P∗ZM


Abelian BF theory

Abelian BF theory: the continuum model.Input:

M a closed oriented n-manifold M .

E an SL(m)-local system.

Space of BV fields: F = Ω•(M,E)[1]⊕ Ω•(M,E∗)[n− 2] 3 (A,B)Action: S =

∫M〈B, dEA〉.

Reference: A. S. Schwarz, The partition function of degeneratequadratic functional and Ray-Singer invariants, Lett. Math. Phys. 2, 3(1978) 247–252.

A. S. Schwarz: For M closed and E acyclic, the partition function is theR-torsion τ(M,E) ∈ R.


Abelian BF theory

Abelian BF theory: the continuum model.Input:

M a closed oriented n-manifold M .

E an SL(m)-local system.

Space of BV fields: F = Ω•(M,E)[1]⊕ Ω•(M,E∗)[n− 2] 3 (A,B)Action: S =

∫M〈B, dEA〉.

Reference: A. S. Schwarz, The partition function of degeneratequadratic functional and Ray-Singer invariants, Lett. Math. Phys. 2, 3(1978) 247–252.

A. S. Schwarz: For M closed and E acyclic, the partition function is theR-torsion τ(M,E) ∈ R.


Abelian BF theory

Result, C-M-R arXiv:1507.01221

For M closed, E possibly non-acyclic,Fres = H•(M,E)[1]⊕H•(M,E∗)[n− 2] and

ZM = ξ · τ(M,E)

where τ(M,E) ∈ DetH•(M,E) = Dens12 (Fres) is the R-torsion and

ξ = (2π~)∑nk=0(− 1

4− 1

2k(−1)k)·dim Hk(M,E)·(e−

πi2 ~)

∑nk=0( 1

4− 1

2k(−1)k)·dim Hk(M,E)

In particular ZM contains a mod16 phase e2πi16 s with

s =∑nk=0(−1 + 2k(−1)k) · dimHk(M,E).


Abelian BF theory



ZM = ξ · τ(M,E)

where τ(M,E) ∈ DetH•(M,E) = Dens12 (Fres) is the R-torsion

and

ξ = (2π~)∑nk=0(− 1

4− 1


πi2 ~)

∑nk=0( 1

4− 1



s =∑nk=0(−1 + 2k(−1)k) · dimHk(M,E).


Abelian BF theory



ZM = ξ · τ(M,E)


ξ = (2π~)∑nk=0(− 1

4− 1


πi2 ~)

∑nk=0( 1

4− 1



s =∑nk=0(−1 + 2k(−1)k) · dimHk(M,E).


Abelian BF theory



ZM = ξ · τ(M,E)


ξ = (2π~)∑nk=0(− 1

4− 1


πi2 ~)

∑nk=0( 1

4− 1



s =∑nk=0(−1 + 2k(−1)k) · dimHk(M,E).


Abelian BF theory


For M with boundary, E possibly non-acyclic,

ZM = ξ · τ(M,Σin;E)·

· expi

~

(∫Σout

Ba +

∫Σin

bA−∫

Σout×Σin 3(x,y)

B(x)η(x, y)A(y)

)

This result satisfies:

gluing

mQME

change of η shifts ZM by(i~Ω∂ − i~∆res

)-exact term.

BFV operator: Ω∂ = −i~(∫

ΣoutdEB δ

δB +∫

ΣindEA δ

δA

)


Abelian BF theory




· expi

~

(∫Σout

Ba +

∫Σin

bA−∫

Σout×Σin 3(x,y)

B(x)η(x, y)A(y)

)


gluing

mQME


)-exact term.


ΣoutdEB δ

δB +∫

ΣindEA δ

δA

)


Abelian BF theory




· expi

~

(∫Σout

Ba +

∫Σin

bA−∫

Σout×Σin 3(x,y)

B(x)η(x, y)A(y)

)

Where: Fres = H•(M,Σin;E)[1]⊕H•(M,Σout;E∗)[n− 2] 3 (a, b)


gluing

mQME


)-exact term.


ΣoutdEB δ

δB +∫

ΣindEA δ

δA

)


Abelian BF theory




· expi

~

(∫Σout

Ba +

∫Σin

bA−∫

Σout×Σin 3(x,y)

B(x)η(x, y)A(y)

)

Where: B = Ω•(Σin)[1]⊕ Ω•(Σout)[n− 2] 3 (A,B)


gluing

mQME


)-exact term.


ΣoutdEB δ

δB +∫

ΣindEA δ

δA

)


Abelian BF theory




· expi

~

(∫Σout

Ba +

∫Σin

bA−∫

Σout×Σin 3(x,y)

B(x)η(x, y)A(y)

)

Where: ξ as before (but for relative cohomology),


gluing

mQME


)-exact term.


ΣoutdEB δ

δB +∫

ΣindEA δ

δA

)


Abelian BF theory




· expi

~

(∫Σout

Ba +

∫Σin

bA−∫

Σout×Σin 3(x,y)

B(x)η(x, y)A(y)

)

Where: τ - relative R-torsion,


gluing

mQME


)-exact term.


ΣoutdEB δ

δB +∫

ΣindEA δ

δA

)


Abelian BF theory




· expi

~

(∫Σout

Ba +

∫Σin

bA−∫

Σout×Σin 3(x,y)

B(x)η(x, y)A(y)

)

Where: η ∈ Ωn−1(Conf2(M), E E∗) – propagator.


gluing

mQME


)-exact term.


ΣoutdEB δ

δB +∫

ΣindEA δ

δA

)


Abelian BF theory




· expi

~

(∫Σout

Ba +

∫Σin

bA−∫

Σout×Σin 3(x,y)

B(x)η(x, y)A(y)

)


gluing

mQME


)-exact term.


ΣoutdEB δ

δB +∫

ΣindEA δ

δA

)


Abelian BF theory




· expi

~

(∫Σout

Ba +

∫Σin

bA−∫

Σout×Σin 3(x,y)

B(x)η(x, y)A(y)

)


gluing

mQME


)-exact term.


ΣoutdEB δ

δB +∫

ΣindEA δ

δA

)


Abelian BF theory




· expi

~

(∫Σout

Ba +

∫Σin

bA−∫

Σout×Σin 3(x,y)

B(x)η(x, y)A(y)

)


gluing

mQME


)-exact term.


ΣoutdEB δ

δB +∫

ΣindEA δ

δA

)


Abelian BF theory




· expi

~

(∫Σout

Ba +

∫Σin

bA−∫

Σout×Σin 3(x,y)

B(x)η(x, y)A(y)

)


gluing

mQME


)-exact term.


ΣoutdEB δ

δB +∫

ΣindEA δ

δA

)


Gluing of propagators


ηI , ηII – propagators on MI , MII .Assume H•(M,Σ1) = H•(MI ,Σ1)⊕H•(MII ,Σ2).Then the glued propagator on M is:

η(x, y) =

ηI(x, y) if x, y ∈MI

ηII(x, y) if x, y ∈MII

0 if x ∈MI , y ∈MII∫z∈Σ2

ηII(x, z)ηI(z, y) if x ∈MII , y ∈MI


Poisson sigma model

Example: Poisson sigma model, n = 2.Action: S =

∫M〈B, dA〉+ 1

2 〈π(B), A⊗A〉π =

∑ij π

ij(x) ∂∂xi ∧

∂∂xj Poisson bivector on Rm.


ZM = ξ · τ · expi

~∑

graphs

ZM satisfies:

gluing

mQME


)-exact term.

Ω∂ = standard-ordering quantization (B 7→ −i~ δδA on Σin, A 7→ −i~ δ

δB

on Σout) of

∫∂

BidAi +1

2Πij(B)AiAj where Πij(x) = xi∗xj−xj∗xi

i~ is

Kontsevich’s deformation of π.


Poisson sigma model


∫M〈B, dA〉+ 1

2 〈π(B), A⊗A〉π =

∑ij π

ij(x) ∂∂xi ∧




~∑

graphs

ZM satisfies:

gluing

mQME


)-exact term.


δB

on Σout) of

∫∂

BidAi +1


i~ is



Poisson sigma model


∫M〈B, dA〉+ 1

2 〈π(B), A⊗A〉π =

∑ij π

ij(x) ∂∂xi ∧




~∑

graphs

ZM satisfies:

gluing

mQME


)-exact term.


δB

on Σout) of

∫∂

BidAi +1


i~ is



Exact discretizations

Reference. Abelian and non-abelian BF :P. Mnev, Discrete BF theory, arXiv:0809.1160 (– for M closed),A. S. Cattaneo, P. Mnev, N. Reshetikhin, Cellular BV-BFV-BF theory.(– with gluing).1D Chern-Simons: A. Alekseev, P. Mnev, One-dimensional Chern-Simonstheory, Comm. Math. Phys. 307 1 (2011) 185–227.


Conclusion

1 → Corners.

2 Partition function for a “building block” (cell) in interestingexamples.

3 Compute cohomology of Ω∂ , e.g. in PSM.

4 More general polarizations, generalized Hitchin’s connection.

5 Observables supported on submanifolds.

Main references:

A. S. Cattaneo, P. Mnev, N. Reshetikhin, Classical BV theories onmanifolds with boundary, Comm. Math. Phys. 332 2 (2014)535–603.

A. S. Cattaneo, P. Mnev, N. Reshetikhin, Perturbative quantumgauge theories on manifolds with boundary, arXiv:1507.01221