2019/03/08

Talk@the workshop “String: T-duality, Integrability and Geometry”, Sendai, Japan

Weyl anomaly cancellation in string theories
on generalized supergravity backgrounds

Kentaroh Yoshida (Dept. of Phys., Kyoto U.)

1

0. Introduction

2String Theory
A promising candidate of the unified theory of 4 forces in nature.
However, String Theory has not been completed yet!
In particular, String Theory is defined only perturbatively, and there arevarious approaches towards the non-perturbative formulation of String Theory.
EXString Field Theory, Matrix Model, Tensor Model etc.

QuestionIs there anything to consider for perturbative string theory?

YES!

3 well-known formulations of perturbative string theory
1.

NS-R formulation (world-sheet fermions)

2.

Green-Schwarz formulation (space-time fermions + kappa symmetry)

3.

Pure Spinor formulation (space-time fermion + pure spinor condition)
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The Green-Schwarz (GS) formulation of type IIB superstring
Space-time fermions contain 32 components (= 2 x 16 comps. of Majorana-Weyl spinor ).
The on-shell condition reduces # of d.o.f. to 16, but # of physical comps. should be 8.
So it is necessary to impose an additional condition.
Kappa-symmetry (a fermionic gauge symmetry)

FACT:

the on-shell condition of the standard type IIB SUGRA
kappa-invariant GS string theory
The inverse was conjectured, but not true.

New result:[Grisaru-Howe-Mezincescu-Nilsson-Townsend, 1985]

kappa-invariant GS string theory on an arbitrary backgroundthe generalized type IIB SUGRA[Tseytlin-Wulf,
f 1605.04884]

This issue has been resolved after more than 30 years from the old work.
This is the recent fundamental progress in String Theory!
4What does this result indicate?
Low Energy Effective Theory emerging from String Theory
may be more general than the well-known SUGRA!
Generalized SUGRA = SUGRA + an extra vector field(The detail of generalized SUGRA will be explained soon)

We may have missed an important ingredient in String Phenomenology
for more than 30 years.
It is really significant to study the generalized SUGRA in more detail.
It may be possible to get a nice idea to solve the long-standing problems
such as cosmological constant problem and stabilizing de Sitter vacuum.
5The plan of my talk1.

What is the generalized type IIB SUGRA?

2.

Weyl anomaly cancelation in string theories
on generalized SUGRA backgrounds

3.

Summary and Discussion

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1. What is the generalized type IIB SUGRA?

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The generalized eqns of type IIB SUGRA[Arutyunov-Frolov-Hoare-Roiban-Tseytlin,1511.05795]

Modified Bianchi identities

8Modified parts:

3 vector fields
ButThen, so two of them are independent.

satisfy the relations given by

Supposing thatis chosen such that the Lie derivative

vanishes, the 2nd equation above can be solved by.

Thus, after all, onlyNoteWhenis independent.

, the usual type IIB SUGRA is reproduced.
9Characteristics of the generalized SUGRA
1.

Talks by Yuho, Junichi, Eoin, Suguru, Falk

Yang-Baxter deformation is a solution generation technique[Klimcik]Classical r-matrices lead to sols. of the generalized SUGRA.
[Delduc-Magro-Vicedo, 1309.5850] [Kawaguchi-Matsumoto-KY, 1401.4855]

2.

The extra vector field I can be seen as a non-geometric Q-flux
For example, T-folds can be realized as solutions of the generalized SUGRA.
[Fernandez Melgarejo-Sakamoto-Sakatani-KY, 1710.06849 ]3.

Generalized SUGRA can be obtained from DFT or EFT
A slightly modified section condition leads to generalized type IIB and IIA SUGRAs.
[Sakatani-Uehara-KY, 1611.05856] [Baguet-Magro-Samtleben, 1612.07210] [Sakamoto-Sakatani-KY, 1703.09213]

4.

Generalized T-duality rule for the I direction
A modification is to ``add a linear-dependent part to dilaton depending on I‘’[Arutyunov-Frolov-Hoare-Roiban-Tseytlin, 1511.05795]

Remarkably, the dual background is a solution of the usual SUGRA!
10Influence of the appearance of the generalized SUGRA?
Recall that there are a lot of the so-called pathological backgrounds,
which do not satisfy the on-shell condition of the standard SUGRA.
EXnon-abelian T-dualities lead to such backgrounds

However, these backgrounds may be solutions of the generalized SUGRA.
In fact, there is an intimate relation between non-abelian T-duality
and Yang-Baxter deformation(A class of) non-abelian T-duality

Yang-Baxter defs. (with CYBE)

conjecture䠖[Hoare-Tseytlin, 1609.02550]

proof䠖[Borsato-Wulff, 1609.09834]

Solutions of the generalized SUGRA

How about non-abelian T-duality which cannot be described as YB deformations?
11An example of such non-abelian T-dualized backgrounds
The Gasperini-Ricci-Veneziano background[hep-th/9308112]

This is not a solution of the usual SUGRA.
However, by adding the following extra vector field[Fernandez Melgarejo-Sakamoto-Sakatani-KY, 1710.06849 ]this background becomes a solution of the generalized SUGRA.
Further confirmation䠖

[M. Hong-Y. Kim-O Colgain, 1801.09567 ]More ``pathological’’ backgrounds may be sols. of the generalized SUGRA.
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So what kinds of pathologies have been resolved?

Classical level䠖 kappa-symmetry is ensured, so acceptable as GS string theory.
Quantum level䠖 Weyl invariance on the world-sheet?
Recall that the problem in the non-abelian T-duality argument is that the beta function
does not vanish because the background is not a solution of the usual SUGRA.

The original claim [Arutyunov-Frolov-Hoare-Roiban-Tseytlin, 1511.05795]
When the background is given by a solution of the generalized SUGRA,
scale invariance is ensured, but Weyl invariance would not be.
Actually, an appropriate counter-term has not been discussed so much.
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2. Weyl anomaly cancelation in string theories
on generalized SUGRA backgrounds[Sakamoto-Sakatani-KY, 1703.09213][Fernandez Melgarejo-Sakamoto-Sakatani-KY, 1811.10600]
accepted in PRL.

For simplicity, we will focus upon the bosonic string case hereafter.
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Weyl invariance of the bosonic string theory (D=26)
The classical action

At classical level,
Weyl invariant
But at quantum level, the trace anomaly appears[Callan-Friedan-Martinec-Perry, ’85]

where,15

Quantum scale invariance[Hull-Townsend, ’86]

Suppose that the beta functions take the following forms:

Then scale invariance is preserved at quantum level.
In fact, the trace anomaly can be rewritten into a total derivative form:

where the eom of X has been utilized.
NOTE
This supposition is satisfied for the solutions of the generalized SUGRA.
Here Z and I are arbitrary vector fields, and of course these are nothing but those in the generalized SUGRA!

The origin of the generalized SUGRA
16Quantum Weyl invariance
As a special case of Hull and Townsend, one may take.
Then the trace anomaly is given by.
This anomaly can be cancelled out by adding the Fradkin-Tseytlin (FT) term:alpha’ is not contained!

becauseNote: the FT term itself should be regarded as quantum contribution.
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A generalization of the FT term
Question:Can one generalize the FT term for the case with?

Generalized FT term: [Sakamoto-Sakatani-KY, 1703.09213]
Dual coordinates,

Exactly cancels out Hull-Townsend’s trace anomaly!
Here we have used the eom of Hull’s double sigma model,

Note that 2D Ricci scalar is locally a total derivative, so this cancellation is local.
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A two-dim. Ricci scalar can be written locally as

Then by integrating by part, the counter-term can be rewritten as

where J is a Noether current associated with the Killing vector I,

The integrand of the 2nd term is local, but the integrated one would not be…
(Be careful for the global topology)

Also, be careful for diffeomorphism invariance of the 2nd term.
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What is an appropriate ߙ ?
 An obstacle
If[Fernandez Melgarejo-Sakamoto-Sakatani-KY, 1811.10600][Deser-Jackiw, hep-th/9510145]

is constructed only from the metric, it is not covariant.

 A possible resolution: a normalized vector field[Yale-Padmanabhan, 1008.5154],A remarkable point:
By introducing, one can avoid Deser-Jackiw’s argument.
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NOTE:

does not appear in.

ሺ‫)׶‬

where we have used the following identity intrinsic to 2D:

A key observation:[Fernandez Melgarejo-Sakamoto-Sakatani-KY, 1811.10600]

may depend onwhileanddoes not containthrough,.

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A naive choice ofis the Noether current:.

By adopting this choice, we can expressas.

This is manifestly local and covariant.

A reasoning of this choice:
The Killing direction I is specialized in target space-time and hence
the world-sheet counter-part J should be taken into account.

22Another derivation of the modified counter-term
Let us start from the classical action of a gauged NLSM:

whereThis action is invariant under the gauge transformation:

Then the e.o.m. for an auxiliary fieldis given by23

By using the gauge transformation,

vanishes to the leading order of.

So one can take the following gauge:Hence
By using the e.o.m. forone can eliminate,from the action.

The resulting action is given by

The previous action has been reproduced from a gauged NLSM!
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3. Summary and Discussion

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Summary & discussion
Generalized SUGRA and Weyl anomaly cancellation
Kappa-symmetry of GS superstring

generalized SUGRA

A modified Fradkin-Tseytlin term

Weyl anomaly cancellation[Sakamoto-Sakatani-KY, 1703.09213]

However, for Weyl invariance, need to check a lot of things, e.g. higher genus cases.
The integrand of the counter-term is local and covariant, but non-polynomial
The integration would be non-local.

[Fernandez Melgarejo-Sakamoto-Sakatani-KY, 1811.10600]

The non-polynomial and non-local property might be compatible
with non-geometric b.g. like T-folds. (where the order of alpha’ expansion is quite unclear)
Of course, it is useful to check the counter-term for simple exactly solvable backgrounds[Sakatani-Tseytlin-KY, in progress]
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Thank you!

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Back up28

An example of solutions of the generalized SUGRA¥

The deformed background:[Kyono-KY, 1605.02519]The Bianchi identity is broken.

The e.o.m. of B2 is not satisfied as well.
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Some comments• Special case

When a1=0, the classical r-matrix becomes unimodular.

The background is a solution of type IIB SUGRA.

• General case

[Hubeney-Rangamani-Ross, hep-th/0504034]

The resulting background satisfies the generalized equations with
[Kyono-KY, 1605.02519]

It is more interesting to perform ``generalized T-dualities’’ for this solution.
A modification is to ``add a linear-dependent part to dilaton depending on I‘’[Arutyunov-Frolov-Hoare-Roiban-Tseytlin, 1511.05795]

a solution of the usual type IIB SUGRA
Furthermore, this ``T-dualized’’ background is locally equivalent to AdS5 x S5
via a non-linear field redefinition.

[Orlando-Reffert-Sakamoto-KY, 1607.00795]
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Weyl invariance of the bosonic string theory (D=26)
The classical action

At classical level,
Weyl invariant
But at quantum level, the trace anomaly appears[Callan-Friedan-Martinec-Perry, ’85]

where,

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