Hamilton-Jacobi counterterms in quantum mechanical and black hole partition functions

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Hamilton-Jacobi counterterms in quantum mechanical and black hole partition functions

Daniel Grumiller

Center for Theoretical Physics Massachusetts Institute of Technology

7^th Workshop on Quantization, Dualities and Integrable Systems, Anadolu University,April 2008

hep-th/0703230, 0711.4115

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Appetizer: Action Principles And... Action!

I = Z

M

d^d+1x

(∂φ)²−V(φ)

First variation δI= 0 yields bulk EOM and boundary conditions: boundary :π δφ= 0 → chooseφ= const.at∂M

Problem: If π→ ∞ or φ→ ∞ at boundary thenπ δφ6= 0 possible Solution: subtractboundary counterterm

Γ =I−I_CT , I_CT = Z

∂M

d^dxL(φ, ∂_kφ) Γ has same EOM and same boundary value problem asI

Question: How to construct the countertermI_CT?

Another question: Why is this relevant?

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I = Z

M

d^d+1x

(∂φ)²−V(φ)

First variation δI= 0 yields bulk EOM and boundary conditions:

boundary :π δφ= 0 → chooseφ= const.at∂M

∂M

D. Grumiller — Hamilton-Jacobi counterterms 2/26

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I = Z

M

d^d+1x

(∂φ)²−V(φ)

boundary :π δφ= 0 → chooseφ= const.at∂M Problem: If π→ ∞ or φ→ ∞ at boundary thenπ δφ6= 0 possible

Solution: subtractboundary counterterm Γ =I−I_CT , I_CT =

Z

∂M

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I = Z

M

d^d+1x

(∂φ)²−V(φ)

∂M

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I = Z

M

d^d+1x

(∂φ)²−V(φ)

∂M

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I = Z

M

d^d+1x

(∂φ)²−V(φ)

∂M

Question: How to construct the countertermI_CT? Another question: Why is this relevant?

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Outline

Statement of problem: divergences and variational principle

Solution: Hamilton-Jacobi counterterm

Applications: Black Hole Thermodynamics, AdS/CFT, ...

Summary and Literature

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Outline

D. Grumiller — Hamilton-Jacobi counterterms Statement of problem: divergences and variational principle 4/26

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Quantum Mechanical Toy Model

Step 1: Boundary value problem

Let us start with an “extraordinary” Hamiltonianaction I_EH =

tf

Z

ti

dt[−pq˙ −H(q, p)]

Want to set up a Dirichlet boundary value problemq= fixed att_i, t_f Problem:

δIEH = 0 requiresq δp= 0 at boundary Solution: add“great, helpful (yellowish)”boundary term

I_E =I_EH +I_GHY , I_GHY = pq|^t_t^f

i

As expectedI_E =

t_f

R

ti

[pq˙−H(q, p)] is standard Hamiltonian action

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tf

Z

ti

Want to set up a Dirichlet boundary value problemq= fixed att_i, t_f

Problem:

i

As expectedI_E =

t_f

R

ti

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tf

Z

ti

δIEH = 0 requiresq δp= 0 at boundary

Solution: add“great, helpful (yellowish)”boundary term I_E =I_EH +I_GHY , I_GHY = pq|^t_t^f

i

As expectedI_E =

t_f

R

ti

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tf

Z

ti

i

As expectedI_E =

t_f

R

ti

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tf

Z

ti

i

As expectedI_E =

t_f

R[pq˙−H(q, p)] is standard Hamiltonian action

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Half-binding potentials

Step 2: Impose asymptotic boundary conditions

Specific Hamiltonian with half-binding potential:

H(q, p) = p²

2 +V(q), V(q) = 1 q²

1 2 3 4 5 q

0.2 0.4 0.6 0.8 1 V

I Of interest is asymptotic limitt_f → ∞

I In that case almost free propagation

I Appropriate boundary condition: q_f =∞

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H(q, p) = p²

2 +V(q), V(q) = 1 q²

1 2 3 4 5 q

0.2 0.4 0.6 0.8 1 V

I Of interest is asymptotic limittf → ∞

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H(q, p) = p²

2 +V(q), V(q) = 1 q²

1 2 3 4 5 q

0.2 0.4 0.6 0.8 1 V

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H(q, p) = p²

2 +V(q), V(q) = 1 q²

1 2 3 4 5 q

0.2 0.4 0.6 0.8 1 V

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A Problem Arises...

Step 3: Variational principle

Consider the first variation of the action δI_E =δI_EH +δI_GHY =

tf

Z

ti

dt[ ˙q δp+p δq˙−δH(q, p)]

I Bulk: Hamilton EOM

I Boundary (recallq|^t^f =∞):

δIE =p δq|^t^f −p δq|^tⁱ =p δq

|{z}

finite!

|^t^f 6= 0.

Not all variations preserving the boundary conditionqf =∞ lead to vanishing first variation of the actionIE!

SERIOUS PROBLEM!

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A Problem Arises...

tf

Z

ti

|{z}

finite!

|^t^f 6= 0.

SERIOUS PROBLEM!

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A Problem Arises...

tf

Z

ti

|{z}

finite!

|^t^f 6= 0.

SERIOUS PROBLEM!

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A Problem Arises...

tf

Z

ti

|{z}

finite!

|^t^f 6= 0.

SERIOUS PROBLEM!

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A Problem Arises...

tf

Z

ti

|{z}

finite!

|^t^f 6= 0.

SERIOUS PROBLEM!

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Another Problem Arises...

Step 4: Check on-shell action

Classical approximation to partition function:

Z = Z

DqDp exp

−1

~I_E[q, p]

∼exp

−1

~I_E[q_cl, p_cl]

To leading order determined by on-shell action I_E[q_cl, p_cl] :=I_E|_EOM.

In our case (p(t_f → ∞)→v= const.): IE|_EOM=

∞

Z

ti

dt

pq˙−p² 2 − 1

q²

= v² 2

∞

Z

ti

dt+O(1)→ ∞

The on-shell action I_E|_EOM diverges!

PROBLEM IN THERMODYNAMICAL APPLICATIONS!

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Z = Z

DqDp exp

−1

~I_E[q, p]

∼exp

−1

~I_E[q_cl, p_cl]

To leading order determined by on-shell action I_E[q_cl, p_cl] :=I_E|_EOM. In our case (p(tf → ∞)→v= const.):

IE|_EOM=

∞

Z

ti

dt

pq˙−p² 2 − 1

q²

= v² 2

∞

Z

ti

dt+O(1)→ ∞

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Z = Z

DqDp exp

−1

~I_E[q, p]

∼exp

−1

~I_E[q_cl, p_cl]

IE|_EOM=

∞

Z

ti

dt

pq˙−p² 2 − 1

q²

= v² 2

∞

Z

ti

dt+O(1)→ ∞

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Z = Z

DqDp exp

−1

~I_E[q, p]

∼exp

−1

~I_E[q_cl, p_cl]

IE|_EOM=

∞

Z

ti

dt

pq˙−p² 2 − 1

q²

= v² 2

∞

Z

ti

dt+O(1)→ ∞

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Outline

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Exploit Freedom to Subtract Further Boundary Terms

Step 5: Ansatz for counterterm

Key observation: Dirichlet boundary problem not changed under I_E →Γ =I_E−I_CT =I_EH+I_GHY −I_CT with

ICT = S(q, t)|^t^f

Improved action: Γ =

tf

Z

ti

dt

−pq˙ −p² 2 − 1

q²

+ pq|^t_t^f

i − S(q, t)|^t^f

First variation:

δΓ =

p−∂S(q, t)

∂q

δq

tf

= 0?

D. Grumiller — Hamilton-Jacobi counterterms Solution: Hamilton-Jacobi counterterm 10/26

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ICT = S(q, t)|^t^f Improved action:

Γ =

tf

Z

ti

dt

−pq˙ −p² 2 − 1

q²

+ pq|^t_t^f

i −S(q, t)|^t^f

First variation:

δΓ =

p−∂S(q, t)

∂q

δq

tf

= 0?

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ICT = S(q, t)|^t^f Improved action:

Γ =

tf

Z

ti

dt

−pq˙ −p² 2 − 1

q²

+ pq|^t_t^f

i −S(q, t)|^t^f

First variation:

δΓ =

p−∂S(q, t)

∂q

δq

tf

= 0?

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Constructing the Counterterm

Step 6: Solve Hamilton-Jacobi equation

Need a function S(q, t) of the boundary data qf,tf that obeys asymptotically

p= ∂S(q, t)

∂q

Hamilton’s principal function in the Hamilton-Jacobi equation

∂S(q, t)

∂t +H

q,∂S(q, t)

∂q

= 0 has precisely this property!

Thus we postulate:

The counterterm S(q, t) is a solution of the Hamilton-Jacobi equation!

Does it work?

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p= ∂S(q, t)

∂q

∂S(q, t)

∂t +H

q,∂S(q, t)

∂q

Thus we postulate:

Does it work?

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p= ∂S(q, t)

∂q

∂S(q, t)

∂t +H

q,∂S(q, t)

∂q

Thus we postulate:

Does it work?

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p= ∂S(q, t)

∂q

∂S(q, t)

∂t +H

q,∂S(q, t)

∂q

Thus we postulate:

Does it work?

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Final Result for the Improved Action

Step 7: Check that everything works

For our exampleH(q, p) = ^p₂² +_q¹2 we obtain (∆+:= ¹₂(1 +p

1−8t²/q⁴))

S(q, t) = q² 2t

p4∆+−8t²/q⁴−∆+

+√

2 arctan 1

pq⁴∆₊/(2t²)−1

Asymptotic expansion: S(q, t) = q²

2t +O(t/q²) = v²

2 t+O(1/t) Check on-shell action:

Γ|_EOM= lim

tf→∞

v² 2

tf

Z

ti

dt− v²

2 t_f +O(1) =O(1)

→OK!

Check first variation: δΓ|_EOM =

p−∂S(q, t)

∂q

δq

tf

= O(1/t_f)δq|^t^f = 0

→OK!

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1−8t²/q⁴))

S(q, t) = q² 2t

p4∆+−8t²/q⁴−∆+

+√

2 arctan 1

pq⁴∆₊/(2t²)−1 Asymptotic expansion:

S(q, t) = q²

2t +O(t/q²) = v²

2 t+O(1/t)

Check on-shell action: Γ|_EOM= lim

tf→∞

v² 2

tf

Z

ti

dt− v²

2 t_f +O(1) =O(1)

→OK!

p−∂S(q, t)

∂q

δq

tf

= O(1/t_f)δq|^t^f = 0

→OK!

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1−8t²/q⁴))

S(q, t) = q² 2t

p4∆+−8t²/q⁴−∆+

+√

2 arctan 1

S(q, t) = q²

2t +O(t/q²) = v²

Γ|_EOM= lim

tf→∞

v² 2

tf

Z

ti

dt−v²

2 t_f +O(1) =O(1)

→OK!

p−∂S(q, t)

∂q

δq

tf

= O(1/t_f)δq|^t^f = 0

→OK!

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1−8t²/q⁴))

S(q, t) = q² 2t

p4∆+−8t²/q⁴−∆+

+√

2 arctan 1

S(q, t) = q²

2t +O(t/q²) = v²

Γ|_EOM= lim

tf→∞

v² 2

tf

Z

ti

dt−v²

2 t_f +O(1) =O(1)→OK!

p−∂S(q, t)

∂q

δq

tf

= O(1/t_f)δq|^t^f = 0

→OK!

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1−8t²/q⁴))

S(q, t) = q² 2t

p4∆+−8t²/q⁴−∆+

+√

2 arctan 1

S(q, t) = q²

2t +O(t/q²) = v²

Γ|_EOM= lim

tf→∞

v² 2

tf

Z

ti

dt−v²

2 t_f +O(1) =O(1)→OK!

Check first variation:

δΓ|_EOM =

p−∂S(q, t) δq

tf

= O(1/t_f)δq|^t^f = 0

→OK!

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1−8t²/q⁴))

S(q, t) = q² 2t

p4∆+−8t²/q⁴−∆+

+√

2 arctan 1

S(q, t) = q²

2t +O(t/q²) = v²

Γ|_EOM= lim

tf→∞

v² 2

tf

Z

ti

dt−v²

2 t_f +O(1) =O(1)→OK!

Check first variation:

δΓ|_EOM =

p−∂S(q, t)

∂q

δq

tf

= O(1/t_f)δq|^t^f = 0→OK!

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Summary and Recipe

What we have learned so far...

I Start with bulk action IEH

→ Einstein-Hilbert

I Check consistency of boundary value problem

I If necessary, add boundary term IGHY

→ Gibbons-Hawking-York

I Check consistency of variational principle

I If necessary, subtract Hamilton-Jacobi counterterm ICT

→ ???

I Use improved action

Γ =I_EH +I_GHY −I_CT for applications!

Examples where this is of relevance?

I Black hole thermodynamics from Euclidean path integral

I Holographic renormalization in AdS/CFT

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Summary and Recipe

→ ???

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Summary and Recipe

→ ???

(45)

Summary and Recipe

→ ???

(46)

Summary and Recipe

→ ???

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Summary and Recipe

→ ???

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Summary and Recipe

→ ???

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Summary and Recipe

I Start with bulk action IEH → Einstein-Hilbert

I If necessary, add boundary term IGHY → Gibbons-Hawking-York

I If necessary, subtract Hamilton-Jacobi counterterm ICT → ???

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Outline

(51)

Black Hole Thermodynamics

Consider Euclidean path integral (Gibbons, Hawking, 1977)

Z = Z

Dg exp

−1

~I_E[g]

I g: metric; I_E[g]: Euclidean action

I Semiclassical limit (~→0): dominated by classical solutions

I Exploit relationship between Z and Euclidean partition function Z ∼e^−β^Ω

I Ω: thermodynamic potential for appropriate ensemble

I β: periodicity in Euclidean time

I Derive thermodynamical quantities from Ω(β, . . .) Requires periodicity in Euclidean time and accessibility of (semi-)classical approximation

D. Grumiller — Hamilton-Jacobi counterterms Applications: Black Hole Thermodynamics, AdS/CFT, ... 15/26

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Z = Z

Dg exp

−1

~I_E[g]

∼exp

−1

~I_E[g_cl]

I g: metric; I_E[g]: Euclidean action

I Semiclassical limit (~→0): dominated by classical solutions

I Exploit relationship between Z and Euclidean partition function Z ∼e^−β^Ω

I Ω: thermodynamic potential for appropriate ensemble

I β: periodicity in Euclidean time

I Derive thermodynamical quantities from Ω(β, . . .) Requires periodicity in Euclidean time and

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Z = Z

Dg exp

−1

~IE[g]

∼exp

−1

~IE[gcl]

Semi-classical approximation typically is not accessible for bulk action!

First variation of action not zero for all field configurations contributing to path integral due to boundary terms

δI_E EOM∼

Z

∂M

d^dx√ γ h

π^abδγ_abi 6= 0

SAME PROBLEM AS IN QUANTUM MECHANICS EXAMPLE!

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Z = Z

Dg exp

−1

~IE[g]

∼exp

−1

~IE[g_cl]

In view of quantum mechanics example: Which action to take forI_E?

I Naive attempt: I_E[g] =I_EH

Problem: I_EH = 0 on classical solutions! ThusZ = 1 ???

I Less naive attempt: I_E[g] =I_EH +I_GHY

Problem: typicallyIGHY → −∞ asymptotically! ThusZ → ∞ ???

I More sophisticated attempt: IE[g] = Γ =IEH+IGHY −ICT, I_CT = “∞⁰⁰

Problem 1: Finite ambiguity remains!

Problem 2: Sometimes IE → ∞ correct physical result! Need systematic way to constructICT!

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Z = Z

Dg exp

−1

~IE[g]

∼exp

−1

~IE[g_cl]

Problem: typicallyIGHY → −∞ asymptotically! ThusZ → ∞ ???

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Z = Z

Dg exp

−1

~IE[g]

∼exp

−1

~IE[g_cl]

Problem: typicallyIGHY → −∞asymptotically! Thus Z → ∞ ???

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Z = Z

Dg exp

−1

~IE[g]

∼exp

−1

~IE[g_cl]

Problem 2: Sometimes IE → ∞ correct physical result!

Need systematic way to constructICT!

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Z = Z

Dg exp

−1

~IE[g]

Problem 2: Sometimes IE → ∞ correct physical result!

Need systematic way to constructI !

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Example: 2D dilaton gravity

Same problem – same solution!

Bulk action of 2D dilaton gravity:

Γ =− 1 16πG₂

Z

M

d²x√ g

X R−U(X) (∇X)²−2V(X)

− 1 8πG₂

Z

∂M

dx√

γ X K− 1 8πG₂

Z

∂M

dx√

γ S(X,∇_kX, γ,∇_kγ)

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Γ =− 1 16πG₂

Z

M

d²x√ g

X R−U(X) (∇X)²−2V(X)

− 1 8πG₂

Z

∂M

dx√

γ X K− 1 8πG₂

Z

∂M

dx√

I Dilaton X defined via coupling to Ricci scalar R

I Model specified by kinetic and potential functions for dilaton

I Dilaton gravity analog of Gibbons-Hawking-Yorkboundary term: coupling of X to trace of extrinsic curvature K of (∂M, γ)

I Boundary value problem: fix X and induced metric γ at∂M For later: Q(X) :=RX

U, w(X) :=RX

e^QV

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Γ =− 1 16πG₂

Z

M

d²x√ g

X R−U(X)(∇X)²−2V(X)

− 1 8πG₂

Z

∂M

dx√

γ X K− 1 8πG₂

Z

∂M

dx√

I

I Model specified by kineticandpotential functions for dilaton

I Dilaton gravity analog of Gibbons-Hawking-Yorkboundary term: coupling of X to trace of extrinsic curvature K of (∂M, γ)

I Boundary value problem: fix X and induced metric γ at∂M

For later: Q(X) :=RX

U, w(X) :=RX

e^QV

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Classical action of 2D dilaton gravity:

Γ =− 1 16πG₂

Z

M

d²x√ g

X R−U(X) (∇X)²−2V(X)

− 1 8πG₂

Z

∂M

dx√ γX K

− 1 8πG₂

Z

∂M

dx√

I

I Dilaton gravity analog of Gibbons-Hawking-Yorkboundary term:

coupling of X to trace of extrinsic curvature K of (∂M, γ)

X X

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Improved action of 2D dilaton gravity:

Γ =− 1 16πG₂

Z

M

d²x√ g

X R−U(X) (∇X)²−2V(X)

− 1 8πG₂

Z

∂M

dx√

γ XK− 1 8πG₂

Z

∂M

dx√

I

I Boundary value problem: fix X and induced metric γ at∂M For later: Q(X) :=RX

U, w(X) :=RX

e^QV

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Γ =− 1 16πG₂

Z

M

d²x√ g

X R−U(X) (∇X)²−2V(X)

− 1 8πG₂

Z

∂M

dx√

γ X K− 1 8πG₂

Z

∂M

dx√

I

I Generic Ansatz for countertermS

X X

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Γ =− 1 16πG₂

Z

M

d²x√ g

X R−U(X) (∇X)²−2V(X)

− 1 8πG₂

Z

∂M

dx√

γ X K− 1 8πG₂

Z

∂M

dx√

γ S(X,∇_kX)

I Diffeomorphism invariance along boundary simplifies countertermS For later: Q(X) :=RX

U, w(X) :=RX

e^QV

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Γ =− 1 16πG₂

Z

M

d²x√ g

X R−U(X) (∇X)²−2V(X)

− 1 8πG₂

Z

∂M

dx√

γ X K− 1 8πG₂

Z

∂M

dx√ γ S(X)

I Assumption about boundary ∂Mfurther simplifies counterterm S

X X

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Γ =− 1 16πG₂

Z

M

d²x√ g

X R−U(X) (∇X)²−2V(X)

− 1 8πG₂

Z

∂M

dx√

γ X K+ 1 8πG₂

Z

∂M

dx√ γ

q

w(X)e^−Q(X⁾

I Solving Hamilton-Jacobi equation yields simple universal result!

For later: Q(X) :=RX

U, w(X) :=RX

e^QV

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Γ =− 1 16πG2

Z

M

d²x√ g

X R−U(X) (∇X)²−2V(X)

− 1 8πG2

Z

∂M

dx√

γ X K+ 1 8πG2

Z

∂M

dx√ γ

q

w(X)e^−Q(X⁾ Properties of improved action:

I On-shell actionΓ|_EOM finite

I δΓ = 0 for all variations that preserve boundary conditions

I Counter-termis (essentially) unique

I Counter-termis universal and applies to different asymptotics! Prime application: Black hole thermodynamics!

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Γ =− 1 16πG2

Z

M

d²x√ g

X R−U(X) (∇X)²−2V(X)

− 1 8πG2

Z

∂M

dx√

γ X K+ 1 8πG2

Z

∂M

dx√ γ

q

I Counter-termis universal and applies to different asymptotics!

Prime application: Black hole thermodynamics!

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Γ =− 1 16πG2

Z

M

d²x√ g

X R−U(X) (∇X)²−2V(X)

− 1 8πG2

Z

∂M

dx√

γ X K+ 1 8πG2

Z

∂M

dx√ γ

q

I Counter-termis universal and applies to different asymptotics!

Prime application: Black hole thermodynamics!