Weyl invariance of string theories
in the generalized supergravity backgrounds
Junichi Sakamoto (Kyoto Univ.)
arXiv : 1703.09213, 1811.10600

These works are a collaboration withJose J. Fernandez-Melgarejoa, Yuho Sakatani, and Kentaroh Yoshida
One-day workshop for QFT and string theory @Osaka City University 2018/12/14

Introduction
 Well-known fact

String theory

Supergravity

Low energy limit
 An important progress

Recently, a new class of low energy effective theories
has been discovered:Generalized Supergravity[ Arutyunov-Frolov-Hoare-Roiban-Tseytlin, 1511.05795 ] [ Tseytlin-Wulff, 1605.04884 ]2Introduction
There is a deep connection between the generalized SUGRA
and the -symmetry of the GS superstring.
 Old result[ Grisaru-Howe-Mezincescu-Nilsson-Townsend, 1985 ][ Bergshoeff-Sezgin-Townsend, 1986 ]A target spacetime solves the EOM of SUGRA
The-symmetry of the GS superstring

The inverse of the statement had been conjectured.
3

Introduction
 New Result
The[ Tseytlin-Wulff, 1605.04884 ]-symmetry of the GS action

The generalized supergravity equations
Consistency at the classical level
A long-standing problem had been resolved!
4

Introduction
Question

Consistency at the quantum level?

Our result
We showed that bosonic string theories defined on
generalized SUGRA backgrounds are Weyl invariant.

5

Talk plan

1. What is the generalized supergravity?

4 slides2. Weyl invariance of bosonic string theory 5 slides

3. Weyl invariance forgeneralized supergravity backgrounds

4 slides4. summary and discussions
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1. What is the generalized SUGRA?

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The generalized supergravity equations[ Arutyunov-Frolov-Hoare-Roiban-Tseytlin, 1511.05795 ] [ Tseytlin-Wulff, 1605.04884 ]The generalized supergravity equations ( GSE )Here, we have ignored R-R fields, dilatino, and gravitino.
The modifications are characterized by I and Z.
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The generalized supergravity equations[ Arutyunov-Frolov-Hoare-Roiban-Tseytlin, 1511.05795 ] [ Tseytlin-Wulff, 1605.04884 ]The generalized supergravity equations ( GSE )Here, we have ignored R-R fields, dilatino, and gravitino.
The modifications are characterized by I and Z.
9

The relation between I and Z( Killing equations )By fixing the gauge of B as,

The GSE can be characterized by the Killing vector I

10Weyl or Scale invariance?
Originally, the GSE was derived from the conditions forthe scale invariance condition of the string sigma model.
[ Arutyunov-Frolov-Hoare-Roiban-Tseytlin, 1511.05795 ]At that time, they could not find a local counterterm.
Inconsistency at the quantum level?
For this issue, it is useful to embed the GSE into
Double Field Theory (DFT).

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GSE from Double field theory
The GSE can be embedded into DFT by adding the dilaton
to a linear dual coordinate dependence. [ J.S.-Sakatani-Yoshida, 1703.09213 ]In particular, the linear dual coordinate dependence isconsistent with the section condition in DFT.
This modification enables us to construct
an appropriate counterterm.
[ J.S.-Sakatani-Yoshida, 1703.09213 ]12

2. Weyl invariance ofbosonic string theory

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Weyl invariance of bosonic string theory
The action of bosonic string theory ( D=26 ): The world-sheet metric: The anti-symmetric tensor

Here,
are the metric and the B-fieldof an arbitrary background.
 Classical Weyl invariance of the action

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Weyl invariance of bosonic string theory
The Weyl invariance of the model is broken
at the quantum level.
 The trace anomaly

 The β-functions at the one-loop level [Callan-Friedan-Martinec-Perry, 1985]

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Weyl invariance of bosonic string theory
Quantum scale invariance condition [Hull-Townsend, 1986]

where I and Z are arbitrary vector fields.

 In fact, the trace anomaly on the on-shell becomes

No trace anomaly
 The quantum scale invariance condition has the same form
as in the NS-NS sector of the GSE.
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Weyl invariance of bosonic string theory
How do we improve the EM tensor to obtainquantum Weyl invariant string sigma models?
Let us consider a special case

The Weyl anomaly reduces toWe can find the following local counterterm.
17Weyl invariance of bosonic string theory
The Fradkin-Tseytlin term [Fradkin-Tseytlin, 1985]

Indeed,Cancel !

1. The term is not classical Weyl invariant
2.

a quantum correction18

2. Weyl invariance forgeneralized supergravity backgrounds

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A generalization of the Fradkin-Tseytlin term
Our proposal [ J.S.-Sakatani-Yoshida, 1703.09213 ]We can obtain
Cancel !
Here, we have used the eom of the DSM:

is non-local[Hull, 0406102 ]Is the c.t. non-local as well?
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A generalization of the Fradkin-Tseytlin term
In 2d, the EH action is a total derivative,

vector density
By using the fact, the counterterm can be rewritten as

whereis the Noether current associated withWhen, this reduces to the FT term.

.

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A concrete construction of 𝛼
Is there a local expression of?

 A problem [ Deser-Jackiw, 9510145 ] [ Yale-Padmanabhan, 2010 ]If we construct
only from the metric,
it is not covariant.

 A resolution [ Yale-Padmanabhan, 2010 ]: normalized vector field

An important point
itself does not contribute to the EH action at all.
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A concrete construction of 𝛼Assumption
depends onand.

A possible candidate for n is the Noether current

By using the normalized vector, we can define:asThis is manifestly covariant and a local function ofthe fundamental fields.
We obtain a local counterterm
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Summary
 Weyl invariance of bosonic string theories ingeneralized supergravity backgrounds
 Construction of an appropriate counterterm byusing the doubled formalism The counterterm is local
In particular, the final expression can be describedwithout introducing the doubled formalism.
String theories may consistently be defined on
generalized supergravity backgrounds.
24Discussions
 Studying quantum structures of higher order in
It may be useful to construct a simple solvable model.

 What is?

At this moment, there is no good reason thatis proportional to a Noether current.

However, a Killing vector
always exists on
generalized supergravity backgrounds.
Therefore, it might be natural to expect thatis a world-sheet counterpart of the Killing vector.
25Thank you

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A comment on the divergence of alphaTaking the divergence of it, we obtain

In two dimensions, the following identity holds:

is a local function of the world-sheet metric only.
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