Integrating out geometry in 2D dilaton gravity coupled to fermions

Integrating out geometry in 2D dilaton gravity coupled to fermions

René Meyer

<rene.meyer@itp.uni-leipzig.de>

Outline

1. Introduction 2. The Action

3. Constraint algebra 4. BRST gauge xing

5. Integrating out Geometry

6. Outlook

1. Introduction

• GOAL: Use Path Integral to quantize gravity in D=2 + scalar eld X (Dilaton) + Fermions

S ^(dil) = Z

d ² x √

−g

X R

2 − U (X )

2 (∇X ) ² + V (X )

+ S ^m

• Arise from compactications, spherical reduction and string theory

• 1980s: Jackiw, Teitelboim, D'Hoker, Katanaev, Volovich & others

• Exact path integration of geometry: Pioneering work by W. Kummer, H. Liebl

& D. Vassilevich 1996-1998

Literature

• Quantum gravity: S. Carlip gr-qc/0108040

• Review on Dilaton Gravity: D. Grumiller, W. Kummer, D. Vassilevich hep-th/0204253

• W. Kummer, H. Liebl, D. Vassilevich: gr-qc/9612012, hep-th/9707115, hep-th/9809168

• D. Grumiller's PhD Thesis: gr-qc/0105078

2. Action

S = S _{F OG} + S _kin + S _SI S _{F OG} =

Z

M

X ^a (De) _a + Xdω + (U (X )X ⁺ X ⁻ + V (X ))

S _kin = i Z

M

f (X ) e ^a ∧ (χγ _a dχ) + h.c.c.

S _SI = 1 2

Z

M

e ^b ∧ e ^a _ab h(X )g(χχ)

= Z

dx ² (e)h(X )g(χχ)

Remarks

• f (X ) , h(X ) and g(χχ) are real functions (hermiticity of S )

• Functions U , V (and f , h , g ) specify the model, e.g. D=4 SRG:

U (X ) = −1/2X , V (X ) = −λ ²

• χ _i , χ ^† _i are anticommuting, thus the general form of a polynomial self interaction is

g(χχ) = c + mχχ + e(χχ) ²

and c can be absorbed into V(X).

3. Constraint Algebra - Primary constraints

1st class constraints: p _i := ^{∂ L}

∂ q ˙

= (0, 0, 0) The P _i := ^∂

^L

∂ Q ˙

give rise to 2nd class constraints:

Φ ₀ = P ₀ + i

√ 2 f (p ₁ )q ³ Q ² Φ ₁ = P ₁ − i

√ 2 f (p ₁ )q ² Q ³ Φ ₂ = P ₂ + i

√ 2 f (p ₁ )q ³ Q ⁰ Φ ₃ = P ₃ − i

√ f (p ₁ )q ² Q ¹

Secondary Constraints

We pass to the Dirac bracket formalism. The time evolution of the primary 1st class constraints G _i := ˙ p _i = {p _i , H ⁰ } ^∗ = {p _i , H ⁰ } _s ≈ 0 gives rise to secondary 1st class constraints

G ₁ = G ^g ₁

G ₂ = G ^g ₂ + i

√ 2 f (X )(χ ^† ₁ ← →

∂ ₁ χ ₁ ) + e ⁺ ₁ h(X )g(χχ) G ₃ = G ^g ₃ − i

√ 2 f (X )(χ ^† ₀ ← →

∂ ₁ χ ₀ ) − e ⁻ ₁ h(X )g(χχ) G ^g ₁ = ∂ ₁ X + X ⁻ e ⁺ ₁ − X ⁺ e ⁻ ₁

G ^g ₂ = ∂ ₁ X ⁺ + ω ₁ X ⁺ − e ⁺ ₁ V

G ^g ₃ = ∂ ₁ X ⁻ − ω ₁ X ⁻ + e ⁻ ₁ V

The Hamiltonian

The Hamiltonian can be written as a sum over the secondary constraints

H = −q ⁱ G _i ≈ 0

and thus to check the absence of ternary constraints one has to explore the algebra of the G _i

{G _i , H ⁰ } ^∗ = − q ^{j 0} {G _i , G ⁰ _j } ^∗ ≈ 0

Algebra of secondary constraints

{G _i , G ⁰ _i } ^∗ = 0 i = 1, 2, 3 {G ₁ , G ⁰ ₂ } ^∗ = −G ₂ δ

{G ₁ , G ⁰ ₃ } ^∗ = G ₃ δ {G ₂ , G ⁰ ₃ } ^∗ =

"

−

3

X

i=1

dV

dp _i G _i +

gh ⁰ − h

f f ⁰ g ⁰ · (χχ)

G ₁

#

δ

Remarks on the additional term

• Closes with δ -functions → Lie algebra with structure functions

• It vanishes for minimal coupling, i.e. for h = f = const.

• If f ∝ h , a mass term mχχ doesn't contribute, but a Thirring term does

∝ h ⁰ (g − g ⁰ · (χχ)) = −h ⁰ e(χχ) ²

• The new contribution does not contain derivatives of χ , opposing to the scalar

case, where ∝ ^f _f

L _scalar .

4. BRST gauge xing

• BRST charge: Ω ² . := {Ω, {Ω, .} ^∗ } ^∗ = 0 ⇐⇒ {Ω, ^J.I. Ω} ^∗ = 0

• Ghosts for the G _i : (c ⁱ , p ^c _i ) s.t. {c ⁱ (x), p ^c _j (x ⁰ )} _s = −δ _j ⁱ δ(x − x ⁰ )

• Result:

Ω = Ω ⁽¹⁾ + Ω ⁽²⁾ Ω ⁽¹⁾ = c ⁱ G _i

Ω ⁽²⁾ = 1

2 c ⁱ c ^j C _ij ^k p ^c _k

Gauge xed Hamiltonian

Gauge xing fermion: Ψ = p ^c ₂ Gauge xed Hamiltonian:

H _gf =

=0

z }| {

H _BRST +{Ω, Ψ} ^∗

= −G ₂ − C _2i ^k c ⁱ p ^c _k

= −G ₂ − c ¹ p ^c ₂ − C ₂₃ ^k c ³ p ^c _k

⇒ Eddington-Finkelstein gauge (ω ₀ , e ⁻ ₀ , e ⁺ ₀ ) = (0, 1, 0)

5. Integrating out Geometry

Path integral: Z [J, j, S ] = N R

Dµ[Q, P, q, p, c, p ^c ]e ⁱ

R d

x(L

+L

)

Measure: Dµ[Q, P, q, p, c, p ^c ] =

3

Q

i=0

DQ ⁱ Dµ ⁰ [q, p]

3

Q

i=0

DP _i δ(Φ _i )

3

Q

j=1

Dc ^j Dp ^c _j

Lagrangian:

L _gf = p _i q ˙ ⁱ + ˙ Q ⁱ P _i + p ^c _i c ⁱ − H _gf

= p _i q ˙ ⁱ + ˙ Q ⁱ P _i + G ₂ + p ^c _k M _l ^k c ^l

L _src = J ⁱ p _i + j _i q ⁱ + S _i Q ⁱ

Ghost integration

M =









∂ ₀ 0 _∂p ^∂V

1

−

gh ⁰ − ^h _f f ⁰ g ⁰ · (χχ)

−1 ∂ _{0 ∂p} ^∂V

2

0 0 ∂ ₀ + _∂p ^{∂ V}

3









Z ³ Y

j =1

Dc ^j Dp ^c _j exp{i Z

d ² xp ^c _k M _l ^k c ^l } ∝ DetM = Det(∂ ₀ ² (∂ ₀ + U (p ₁ )p ₂ ))

will be cancelled later

Matter Momenta

Φ _i linear in P , Q & δ(Φ _i ) in the path integral ⇒ P _i -Integration trivial

⇒ L ⁽¹⁾ _{ef f} = p _i q ˙ ⁱ + G ₂ + i

√ 2 f (p ₁ ) h

q ³ (Q ⁰ ← →

∂ ₀ Q ² ) − q ² (Q ¹ ← →

∂ ₀ Q ³ ) i

+ L _src G ₂ = G ^g ₂ + i

√ 2 f (p ₁ )(Q ³ ← →

∂ ₁ Q ³ ) + q ³ h(p ₁ )g(χχ) G ^g ₂ = ∂ ₁ p ₂ + q ¹ p ₂ − q ³ V

→ linear in q ⁱ , thus their integration is trivial too, giving Delta functionals

Covariant path integral measure for fermions (1)

2nd class constraints → local measure in path integral [Henneaux/Teitelboim Ch. 16;

Henneaux/Slavnov hep-th/9406161]

q

sdet{Φ _i , Φ _j } _s = 2f ² (p ₁ )|q ² q ³ |

Dirac fermions on a curved, but xed background (in D=2) [D.J. Toms Phys. Rev. D 35, 3796 (1987); M. Basler Fortsch.Phys.41:1-43,1993]

Y

i=0,1

Dχ ^† _i Dχ _i Y

x

[−g(x)] ⁻¹ = Y

i=0,1

Dχ ^† _i Dχ _i Y

x

q ^{3 −2}

→ change the measure by hand [Kummer, Liebl, Vassilevich hep-th/9707115 for scalar elds]

Covariant path integral measure for fermions (2)

→ Trick to get rid of the measure factor [Kummer, Liebl, Vassilevich hep-th/9809168]

Z [J, j, S ] = N Z

Df f ⁻² δ

f − 1 i

δ δj ₃

Z ˜ [J, j, S ]

Z ˜ [J, j, S ] =

Z Y

i=0,1

Dχ ^† _i χ _i

3

Y

i=1

Dp _i Dq ⁱ detM e ⁱ

R d

xL

ef f

Integrating the q ⁱ

→ Delta functionals give system of PDEs

0 = ∂ ₀ p ₁ − p ₂ − j ₁ (1)

0 = ∂ ₀ p ₂ − j ₂ + i

√ 2 f (p ₁ )(Q ¹ ← →

∂ ₀ Q ³ ) (2)

0 = (∂ ₀ + U (p ₁ )p ₂ )p ₃ − j ₃ − i

√ 2 f (p ₁ )(Q ⁰ ← →

∂ ₀ Q ² ) − h(p ₁ )g(χχ) + V (p ₁ ) (3)

→ det M cancels during p _i integration

Solving the PDEs (1)

Minimal coupling f (p ₁ ) ≡ f = const.

p ₁ = B ₁ = ∇ ⁻¹ ₀ (j ₁ + B ₂ ) + ˜ p ₁ (x ¹ ) p ₂ = B ₂ = ∇ ⁻¹ ₀

j ₂ − f i

√ 2 (Q ¹ ← →

∂ ₀ Q ³ )

+ ˜ p ₂ (x ¹ ) p ₃ = B ₃ = e ^−Q

∇ ⁻¹ ₀ e ^Q

j ₃ − V (B ₁ ) + h(B ₁ )g(χχ) + f i

√ 2 (Q ⁰ ← →

∂ ₀ Q ² )

+ ˜ p ₃ (x ¹ )

Q := ∇ ⁻¹ ₀ (U (B ₁ )B ₂ )

Solving the PDEs (2)

General case → Weak matter approximation, i.e. p _i =

∞

P

n=0

p ⁽ⁿ⁾ _i , i = 1, 2 and p ⁽ⁿ⁾ ₁ (Q ¹ ← →

∂ ₀ Q ³ )

| {z }

(n+1)th order

→ recursion for e.g. f (p ₁ ) = p ₁

p ⁽⁰⁾ ₂ = ∇ ⁻¹ ₀ j ₂ + ˜ p ⁽⁰⁾ ₂ (x ¹ )

p ⁽⁰⁾ ₁ = ∇ ⁻¹ ₀ (j ₁ + p ⁽⁰⁾ ₂ ) + ˜ p ⁽⁰⁾ ₁ (x ¹ ) p ⁽ⁿ⁾ ₂ = − i

√ 2 ∇ ⁻¹ ₀

p ⁽ⁿ⁻¹⁾ ₁ (Q ¹ ← →

∂ ₀ Q ³ )

+ ˜ p ⁽ⁿ⁾ ₂ (x ¹ ) n ≥ 1

p ⁽ⁿ⁾ ₁ = ∇ ⁻¹ ₀ p ⁽ⁿ⁾ ₂ + ˜ p ⁽ⁿ⁾ ₁ (x ¹ )

Eective action

L ⁽²⁾ _{ef f} = J ⁱ B ˆ _i + S _i Q ⁱ + i

√ 2 f ( ˆ B ₁ )(Q ³ ← →

∂ ₁ Q ¹ ) + L _amb L _amb = g ˜ ₃ e ^{Q ˆ}

j ₃ − V ( ˆ B ₁ ) + h( ˆ B ₁ )g(χχ) + i

√ 2 f ( ˆ B ₁ )(Q ⁰ ← →

∂ ₀ Q ² )

Z [J, j = 0; S ] = N R Q ³

i=0

DQ ⁱ [˜ g ₃ e ^{Q ˆ} ] ⁻² e ⁱ

R d

xL

ef f

J,j=0

→ propagators, vertices, perturbation theory, S-Matrix elements

6. Outlook

• "Bosonization": g = mχχ + e(χχ) ² , f ∝ h

→ {G ₂ , G ₃ } ^∗ = − _∂p ^{∂ V}

i

G _i − h ⁰ e(χχ) ² G ₁

→ {G ₂ , G ₂ } ^scalar _p = − _∂p ^∂V

i

G _i − 2h ⁰ G ₁ S ⁺ S ⁻ S ^± = ∗d(S ∧ e ^± ) (anti)selfdual

→ j ^± = χγ ^± χ ⇒ j ⁺ j ⁻ = −(χχ) ²

→ Map S ^± = p _e

2 e ^±iφ j ^±

→ Q: Relationship to Coleman's bosonization, Quantum level?

• Conformal and chiral anomalies → known on xed background & for scalar

matter in this formalism

Discussion