Introduction to 2D dilaton gravity

Dmitri Vassilevich

UFABC

Verão Quântico, Ubu, February 2019

Dmitri Vassilevich 2D gravity

Contents

Why dilaton?General model and particular casesEven more general: Poisson sigma modelsAll classical solutionsAdS2/CFT1

Review: Grumiller, Kummer and D.V, Phys.Rept. (2002).

Support: CNPq, FAPESP

Dmitri Vassilevich 2D gravity

Why dilaton?

The Einstein-Hilbert action in 2D∫d2x√−gR

has vanishing local variations and thus describes no dynamics.A remedy: include a scalar field X . For example:

S =

∫d2x√−g X (R − 2Λ).

This is the first 2D dilaton gravity - the Jackiw-Teitelboim (JT)model (1984). Classical solutions to this model are locally dS orAdS spacetimes.

Dmitri Vassilevich 2D gravity

Generalizations

Let us add to the action a kinetic term for X and a potential

[Banks, ..., I. Shapiro, ..., around 1990-1992]:

L =

∫d2x√−g[1

2RX −12U(X )(∇X )2 + V (X )

]Important particular cases:Spherical reduction from D dimensions:

V = −(D − 2)(D − 3)XD−4D−2 U = − D − 3

(D − 2)X.

String gravity (CGHS):

V = −2X U = − 1X.

Dmitri Vassilevich 2D gravity

First order formulation

The action above is equivalent to

L =

∫[X aDea + Xdω + εV(XaX

a,X )] ,

where ω is the spin-connection, ea is zweibein, a = ±,Dea = dea + aω ∧ ea, ε is the volume 2-form, X a are two auxiliaryfields that generate torsion constraints.

V = 12U(X )X aXa + V (X )

To prove the equivalence: solve the torsion constraints to express ωthrough e. The resulting action will depend on the metric ratherthan on the zweibein.

Dmitri Vassilevich 2D gravity

Poisson sigma models

These are 2D sigma models with a target space being a Poissonmanifold with local coordinates X I (scalars from the 2D point ofview), gauge fields AI (one-forms from the 2D point of view) and aPoisson tensor P IJ(X ) satisfying the Jacobi identity

P IJ∂JPKL + cycl(IKL) = 0

The action [Schaller & Strobl, 1994]

L =

∫ [dX I ∧ AI + 1

2PIJAJ ∧ AI

]is invariant under the gauge transformations

δX I = P IJλJ

δAI = −dλI − (∂IPJK )λKAJ

Dmitri Vassilevich 2D gravity

With the choice X I = (X ,X a), AI = (ω, ea) and

Pab = Vεab, PaX = X bε ab

one recovers 2D dilaton gravities in the 1st order formalism. Thegauge parameters λa correspond to diffeomorphisms, while λX

describes local Lorentz rotations.

Dmitri Vassilevich 2D gravity

All classical solutions

[Bergamin, Grumiller, Kummer, DV, 2005]Consider a 1st order Euclidean dilaton gravity in complex variables∫ [

Y De + Y De + Xdω + εV(2Y Y ,X )]

whereY = (X 1 + iX 2)/

√2, e = (e1 + ie2)/

√2

Though the bar means a complex conjugation, it is convenient toview all fields as independent complex variables.Equations of motion:

dX − i Y e + iY e = 0, DY + ieV = 0

dω + ε ∂V∂X = 0, De + ε∂V∂Y

= 0

Dmitri Vassilevich 2D gravity

By combining equations on the 1st line, we obtain

d(Y Y ) + VdX = 0

that is integrated to

dC = 0, C = w(X ) + eQ(X )Y Y

with

Q(X ) =

∫ X

U(z)dz , w(X ) =

∫ X

eQ(z)V (z)

This allows to express Y Y through X .The existence and the form of absolutely conserved quantity followsalso from the Poisson sigma model arguments.

Dmitri Vassilevich 2D gravity

Similarly,d(e/Y ) = dX ∧ (e/Y )U(X ),

which yieldsd(e−Q/Y ) = 0

So thate−Q/Y = df

with some complex zero-form f . That is it,

e = YeQdf , e = idX

Y+ Y eQdf

ω = VeQdf − idY

Y

By imposing the reality condition one obtains a general classicalsolution depending of 3 arbitrary functions (gauge degrees offreedom) and one integration constant C (mass of the solution).

Dmitri Vassilevich 2D gravity

AdS/CFT

According to the AdS/CFT conjecture gravity theories inasymptotically AdS spaces correspond to conformal theories at theasymptotic boundary. To implement this conjecture in 2D at theclassical level, one has to

define asymptotic conditions asymptotic symmetry algebrasdefine asymptotic degrees of freedomcompute the action for these degrees of freedom

Let us see, how the first step can be done [Grumiller, Salzer, DV,2015].

Dmitri Vassilevich 2D gravity

Let us take the JT gravity, denote by ρ the radial coordinate, s.t.ρ→∞ corresponds to the asymptotic region of AdS2. Let ϕ be asecond coordinate. We require that at the asymptotics all fieldsbehave as listed below plus subleading terms

X 0 = XReρ − XLe

−ρ eϕ0 = 12eρ − 1

2Me−ρ

X 1 = X 1(ϕ) eρ1 = 1X = XRe

ρ + XLe−ρ ωϕ = −1

2eρ − 1

2Me−ρ

Other fields vanish. XR , XL and M are some functions of ϕ. Thiscomes from partially fixing the gauge, partially solving theequations of motion and by an educated guess. By equations ofmotion, XR,L and X 1 may be expressed through M.

Dmitri Vassilevich 2D gravity

The asymptotic symmetry algebra is generated by bulk gaugetransformations that

Preserve the form of asymptotic conditionsChange M.

Parameters of these transformations depend on a single arbitraryfunction λ(ϕ)

λ0 = 12λe

ρ − 12(Mλ+ 2λ′′)e−ρ

λ1 = −λ′

λX = −12λe

ρ − 12(Mλ+ 2λ′′)e−ρ

M is transformed as

δM = −M ′λ− 2Mλ′ − 2λ′′′

This is the Virasoro algebra with a non-zero central charge.

Dmitri Vassilevich 2D gravity

Q & A

Q: How can one make sure that M represents a true asymptoticdegree of freedom?A: It is enough to compute canonical boundary charges that haveto depend on asymptotic degrees of freedom and be finite.

Q: How can one obtain an action for asymptotic degrees offreedom?A: By substituting the solutions in JT action with a suitableboundary term. Depending on details, the resulting model may be ade Alfaro-Fubini-Furlan conformal quantum mechanics or aSchwarzian action [Maldacena et al, 2016].

Dmitri Vassilevich 2D gravity

Further developments

By imposing looser asymptotic conditions in the JT model onecan get larger asymptotic symmetry algebras, as, e.g., the loopalgebra of sl(2), Virasoro plus an u(1)k , see [Grumiller et al,2017].There are higher spin and supersymmetric extensions.Extensions to generic dilaton gravities: work in progress.

Dmitri Vassilevich 2D gravity

Dmitri Vassilevich 2D gravity