BV master action for superstring field theory in the large Hilbert - - PowerPoint PPT Presentation

BV master action for superstring field theory

in the large Hilbert space ( based on JHEP 05 (2018) 020 )

Hiroaki Matsunaga

YITP Workshop

Strings & Fields 2018

• WZW-like string field theory — the large Hilbert space
• Its gauge-fixing problem remains unsolved since 1995

… how to gauge-fix it precisely?

• SFT’s gauge algebra
• Infinitely reducible
• open (closed up to e.o.m.)

Today’s topic : Gauge fixing of “large” SFT

• Berkovits’ WZW-like theory

WZW-like SFT

Large space

Large and small string field theories

S = 1 2

• e−ΦQeΦ, e−ΦηeΦ
• − 1

2 1 dt

• e−tΦ d

dtetΦ,

• e−tΦQetΦ, e−tΦηetΦ
• String field based on ηξφ-system ( not δ(β)δ(γ)-system )
• Large gauge symmetries and Various gauge conditions
• Useful for constructing classical solutions
• Kunitomo-Okawa’s action ( + R sector )

δφ = QΛ + ηΩ + ⋯

η φ ≠ 0 where η ≡ η0

• SFTs in small space

WZW-like SFT EKS’ SFT

Sen’s formulation

A∞/L∞ Small space Large space

• ther SFTs

Large and small string field theories

• Berkovits’ WZW-like theory

S = 1 2

• e−ΦQeΦ, e−ΦηeΦ
• − 1

2 1 dt

• e−tΦ d

dtetΦ,

• e−tΦQetΦ, e−tΦηetΦ
• gauge invariance
• String field of βγ
• Easily gauge-fixable
• Classical solution is unknown

A∞/L∞

S = 1 2⟨Ψ, QΨ⟩ + ∑ 1 n⟨Ψ, Mn(Ψn)⟩

• SFTs in small space

WZW-like SFT EKS’ SFT

Sen’s formulation

A∞/L∞ Small space Large space

• ther SFTs

field re-def.

Large and small string field theories

• Berkovits’ WZW-like theory

S = 1 2

• e−ΦQeΦ, e−ΦηeΦ
• − 1

2 1 dt

• e−tΦ d

dtetΦ,

• e−tΦQetΦ, e−tΦηetΦ
• gauge invariance
• String field of βγ
• Easily gauge-fixable
• Classical solution is unknown

A∞/L∞

S = 1 2⟨Ψ, QΨ⟩ + ∑ 1 n⟨Ψ, Mn(Ψn)⟩

• Partial gauge-fixing
• reduces to other SFTs
• Large SFT is interesting
• unusual (non-geo.) propagators
• to understand SFT itself

Motivation of the “large” space

WZW-like SFT EKS’ SFT

Sen’s formulation

Partial gauge fixing (non-linear)

Small space Large space

• ther SFTs

field re-def. A∞/L∞

• Natural embedding
• every SFT can be large

One can apply Large-space technique to SFT defined in small space !!

( BV string fields-antifields don’t have to be small. )

Motivation of the “large” space

WZW-like SFT EKS’ SFT

Sen’s formulation

A∞/L∞

Natural embedding (linear)

Small space Large space

• ther SFTs

field re-def.

Large theory

• Natural embedding
• every SFT can be large

One can apply Large-space technique to SFT defined in small space !!

( BV string fields-antifields don’t have to be small. )

Motivation of the “large” space

WZW-like SFT EKS’ SFT

Sen’s formulation

A∞/L∞

Natural embedding (linear)

Small space Large space

• ther SFTs

field re-def.

Large theory

field re-def.

EKS’ SFT

Sen’s formulation

• We focus on “large theory”

Today’s topic : “large theory”

A∞/L∞

Natural embedding (linear)

Small space Large space

• ther SFTs

field re-def. field re-def.

WZW-like SFT

Large theory

A∞

EKS’ SFT

Sen’s formulation

• We focus on “large theory”

Motivation of the “large theory”

A∞/L∞

Natural embedding (linear)

Small space Large space

• ther SFTs

field re-def. field re-def.

WZW-like SFT

Large theory

A∞

• Gauge-fixing of SFT in large

space is unsolved, even for this embedded theory.

• Other SFTs in large space are
• ff-shell equivalent to it under

field-redefinitions.

• Thus, in principle, we can solve

the gauge-fixing problem via BV canonical transformations if we get a BV master action for this “large theory”.

• We focus on “large theory”

Motivation of the “large theory”

Natural embedding SFT in small space

Large theory

A∞

S[Φ] = 1 2 ⟨Φ , Q η Φ⟩ + ⋯

Ψ ≡ ηΦ

S[Ψ] = 1 2⟨Ψ , QΨ⟩Ker[η] + ⋯

A∞

• Gauge-fixing of SFT in large

space is unsolved, even for this embedded theory.

• Other SFTs in large space are
• ff-shell equivalent to it under

field-redefinitions.

• Thus, in principle, we can solve

the gauge-fixing problem via BV canonical transformations if we get a BV master action for this “large theory”.

• We focus on “large theory”
• Motivation 1 -

It is the simplest WZW-like theory.

• Motivation 2 -
• Motivation 3 -

Motivation of the “large theory”

Natural embedding SFT in small space

Large theory

A∞

S[Φ] = 1 2 ⟨Φ , Q η Φ⟩ + ⋯

Ψ ≡ ηΦ

S[Ψ] = 1 2⟨Ψ , QΨ⟩Ker[η] + ⋯

A∞

If how to gauge-fix it is clarified, you can apply techniques of the large Hilbert space to your SFTs in small space. It gives another representation of Berkovits’ WZW-like theory. (the same kinetic term and gauge reducibility)

• String field theory in small space has (geometrical) propagator

Techunical Motivation ( Motivation 2 )

• Large theory has larger gauge inv. and various gauge conditions

Pn+1,n+2 = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ Pb

d0 (α+1)L0

· · ·

αb0 (α+1)L0 d0 (α+1)L0

... . . . . . . . . . . . .

αb0 (α+1)L0

... . . . . . . . . . . . . ...

d0 (α+1)L0

. . . . . . . . . . . . ...

αb0 (α+1)L0 d0 (α+1)L0

· · ·

αb0 (α+1)L0 Pd

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

• f d = [Q, bξ],

b0Φ(−n,0) = 0 (n ≥ 0) , d0Φ(−n,m) + αb0Φ(−n,m+1) = 0 (0 ≤ m ≤ n − 1) , d0Φ(−n,n) = 0 (n ≥ 0) , b0d0Φ(n+1,−1) = 0 (n ≥ 1) , αb0Φ(n+1,−m) + d0Φ(n+1,−(m+1)) = 0 (1 ≤ m ≤ n − 1) .

Propagator

Gauge condition

b0Ψn = 0

Siegel gauge Propagator

Pn = b0 L0

Kroyter-Okawa-Schnabl-Torii-Zwiebach 2012

• unusual propagators available (don’t have to be geometrical unlike βγ)

Pb =

• α + η0d0

L0

• b0

(α + 1)L0 , P , Pd =

• 1 + αQb0

L0

• d0

(α + 1)L0 .

• We focus on “large theory”
• Motivation 1 -

It is the simplest WZW-like theory.

• Motivation 2 -
• Motivation 3 -

Motivation of the “large theory”

Natural embedding SFT in small space

Large theory

A∞

S[Φ] = 1 2 ⟨Φ , Q η Φ⟩ + ⋯

Ψ ≡ ηΦ

S[Ψ] = 1 2⟨Ψ , QΨ⟩Ker[η] + ⋯

A∞

If how to gauge-fix it is clarified, you can apply techniques of the large Hilbert space to your SFTs in small space. It gives another representation of Berkovits’ WZW-like theory. (the same kinetic term and gauge reducibility)

S = ∫

1

dt ⟨At[φ] , Q Aη[φ]⟩

Aη[φ] = (η etφ)e−tφ

η Aη[φ] − Aη[φ] * Aη[φ] = 0

At[φ] = (∂t etφ)e−tφ

: functional of string field s.t. Aη[φ]

Berkovits’SFT gives the following solution

Motivation 3

It gives another representation of Berkovits’ WZW-like theory. (Off course, the same kinetic term and gauge reducibility)

WZW-like SFT

Berkovits’ WZW-like theory

S = 1 2

• e−ΦQeΦ, e−ΦηeΦ
• − 1

2 1 dt

• e−tΦ d

dtetΦ,

• e−tΦQetΦ, e−tΦηetΦ

Large A ∞ theory gives another solution

S = ∫

1

dt ⟨At[φ] , Q Aη[φ]⟩

Aη[φ] = (η etφ)e−tφ

η Aη[φ] − Aη[φ] * Aη[φ] = 0

At[φ] = (∂t etφ)e−tφ

: functional of string field s.t. Aη[φ]

Berkovits’SFT gives the following solution

Motivation 3

It gives another representation of Berkovits’ WZW-like theory. (Off course, the same kinetic term and gauge reducibility)

WZW-like SFT

Aη[Φ] = π1 ̂ G 1 1 − tηΦ

At[Φ] = π1 ̂ G 1 1 − tηΦ ⊗ Φ ⊗ 1 1 − tηΦ

Berkovits’ WZW-like theory

S = 1 2

• e−ΦQeΦ, e−ΦηeΦ
• − 1

2 1 dt

• e−tΦ d

dtetΦ,

• e−tΦQetΦ, e−tΦηetΦ

Today’s topic

Natural embedding SFT in small space

Large theory

A∞

S[Φ] = 1 2 ⟨Φ , Q η Φ⟩ + ⋯

Ψ ≡ ηΦ

S[Ψ] = 1 2⟨Ψ , QΨ⟩Ker[η] + ⋯

• We consider a BV master action

for this “large” theory.

Plan

• 1. Conventional BV approach

1.1) Minimal set, usual string field-antifield breakdown 1.2) + some remediations (+ trivial gauge transformations)

• 2. Gauge fixing fermions

Constrained BV is more powerful & elegant ( See [ JHEP 05 (2018) 020 ] )

Berkovits’ constraint BV almost (but not precisely) correct + Improved constraints precisely correct for large theory

• 1. Conventional BV approach

Is Gauge-fixing of “large” theory complicated ?

Natural embedding SFT in small space

Large theory

A∞

S[Φ] = 1 2 ⟨Φ , Q η Φ⟩ + ⋯

Ψ ≡ ηΦ

S[Ψ] = 1 2⟨Ψ , QΨ⟩Ker[η] + ⋯

• We consider “large” theory

Even for very trivial embeddings, “WZW-like structure” arises.

Why is gauge-fixing of “large” SFT so complicated ??

— It was easily gauge-fixable before embedding!! Even if it may look trivial, embeddings give NON-trivial gauge algebras.

Is Gauge-fixing of “large” theory complicated ?

Natural embedding SFT in small space

Large theory

A∞

S[Φ] = 1 2 ⟨Φ , Q η Φ⟩ + ⋯

Ψ ≡ ηΦ

S[Ψ] = 1 2⟨Ψ , QΨ⟩Ker[η] + ⋯

• We consider “large” theory

Even for very trivial embeddings, “WZW-like structure” arises.

Why is gauge-fixing of “large” SFT so complicated ??

— It was easily gauge-fixable before embedding!! Even if it may look trivial, embeddings give NON-trivial gauge algebras.

Ex.) Before embedding

• Gauge parameters can be large.
• e.o.m. of SFT in small space : where .
• gauge variation must be small :

QΨ = 0

δΨ = Qλ1

η(δΨ) = 0

ηΨ = 0

η λ1 = 0

with small gauge parameter

Ex.) Before embedding

• Gauge parameters can be large.
• e.o.m. of SFT in small space : where .
• gauge variation must be small :

QΨ = 0

δλ1 = Qλ2

δΨ = Qλ1

η(δΨ) = 0

ηΨ = 0

δλn = Qλn+1

{Ψ ; λ1 , λ2 , … , λn , …}

η λ1 = 0

with small gauge parameter

Field & Gauge parameters

: :

Ex.) Before embedding

• Gauge parameters can be large.
• e.o.m. of SFT in small space : where .
• gauge variation must be small :

Ready-made procedure : where .

QΨ = 0

δλ1 = Qλ2

δΨ = Qλ1

η(δΨ) = 0

ηΨ = 0

δλn = Qλn+1

{Ψ ; λ1 , λ2 , … , λn , …}

{Ψ ; Ψ1 , Ψ2 , … , Ψn , …}

ψ = Ψ + ∑ Ψn + ∑ Ψ*

n

SBV ∼ ⟨ ψ, Qψ ⟩

η λ1 = 0

{Ψ*

0 ; Ψ* 1 , Ψ* 2 , … Ψ* n , …}

with small gauge parameter

Field & Gauge parameters

Fields :

Anti-fields :

: :

Ex.) Natural embedding of “λ”

We don’t have to use small λ itself.

• Gauge parameters can be large.
• e.o.m. of SFT in small space : where .
• gauge variation must be small :

QΨ = 0

δΨ = Qλ1

η(δΨ) = 0

ηΨ = 0

η λ1 = 0

with small gauge parameter

Ex.) Natural embedding of “λ”

We don’t have to use small λ itself.

• Gauge parameters can be large.
• e.o.m. of SFT in small space : where .
• gauge variation must be small :

QΨ = 0

δΨ = Qλ1

η(δΨ) = 0

ηΨ = 0

η λ1 = 0

with small gauge parameter

with large gauge parameter

δΨ = Q(ηΛ1)

λ1 ≡ ηΛ1 ≠ 0

Ex.) After embedding

• Gauge parameters can be large.
• e.o.m. of SFT in small space : where .
• gauge variation must be small :

δΨ = Q(ηΛ1)

λ1 ≡ ηΛ1 ≠ 0

δΛ1 = QΛ2,0 + ηΛ2,1 δΛg,p = QΛg+1,p + ηΛg+1,p+1

QΨ = 0

ηΨ = 0

η(δΨ) = 0

with large gauge parameter

: : Very different gauge reducibility

Ex.) After embedding

• Gauge parameters can be large.
• e.o.m. of SFT in small space : where .
• gauge variation must be small :

δΛ1 = QΛ2,0 + ηΛ2,1 δΛg,p = QΛg+1,p + ηΛg+1,p+1

Λ1 , Λ2,0 , Λ3,0 , … Λ2,1 , Λ3,1 , … Λ3,2 , …

with large gauge parameter

Gauge parameters

: :

Although string field is small, BV fields-antifields can be large !!

QΨ = 0

ηΨ = 0

η(δΨ) = 0

δΨ = Q(ηΛ1)

λ1 ≡ ηΛ1 ≠ 0

Ex.) After embedding

• Gauge parameters can be large.
• e.o.m. of SFT in small space : where .
• gauge variation must be small :

δΛ1 = QΛ2,0 + ηΛ2,1 δΛg,p = QΛg+1,p + ηΛg+1,p+1

Λ1 , Λ2,0 , Λ3,0 , … Λ2,1 , Λ3,1 , … Λ3,2 , …

with large gauge parameter

Gauge parameters

: :

Although string field is small, BV fields-antifields can be large !!

QΨ = 0

ηΨ = 0

η(δΨ) = 0

δΨ = Q(ηΛ1)

λ1 ≡ ηΛ1 ≠ 0

In large SFT, a string field itself is embedded : !!

Ψ ≡ ηΦ

What happens at the level of action ?

It yields an infinite tower of gauge transformations. We find the spectrum of “string fields-antifields” as

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• δΨ = Q λ0

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• h δλ−g = Q λ−1−g

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• δ(δλ−g) = 0

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g-label

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η Ψ

Free SFT in small space

Fields

Anti-fields

(linear)

A∞

Q2 = 0

It yields an infinite tower of gauge transformations. We find the spectrum of “string fields-antifields” as Free action : where . Then, the master action is given by just replacing with :

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S[Ψ] = 1 2

• Ψ, Q Ψ
• Ker[η]

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• η Ψ = 0

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• δΨ = Q λ0

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• h δλ−g = Q λ−1−g

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• δ(δλ−g) = 0

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g-label

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η Ψ

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Sbv = 1 2

• ψ , Q ψ
• where ψ ≡ Ψ +
• ghost

Ψ−g +

• antifield

(Ψ−g)∗

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where ψ

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η Ψ

Free SFT in small space

Fields

Anti-fields

(linear)

A∞

Q2 = 0

Likewise, one can find interacting BV master action.

Free SFT in large space

Large gauge invariances : We find the spectrum of “string fields-antifields” as

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• δΦ = η Λ−1,1 + Q Λ−1,0

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− −

δ(δΛ−g,p) = 0 with δΛ−g,p = η Λ−1−g,p+1 + Q Λ−1−g,p

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g-label 平成 年 月 日

el p-label

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Φ

Fields

Anti-fields

(linear) pair

A∞

Q2 = 0 η2 = 0 [Q, η] = 0

Kroyter-Okawa-Schnabl-Torii-Zwiebach 2012 JHEP 03 (2012) 030

Free SFT in large space

Large gauge invariances : We find the spectrum of “string fields-antifields” as Embedded free action : Then, the master action is NOT given by just replacing Φ :

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S[Φ] = 1 2

• Φ, Q η Φ
• 平成

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• δΦ = η Λ−1,1 + Q Λ−1,0

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− −

δ(δΛ−g,p) = 0 with δΛ−g,p = η Λ−1−g,p+1 + Q Λ−1−g,p

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g-label 平成 年 月 日

el p-label

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Φ

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Sbv = 1 2

• Φ, Q η Φ
• +
• g≥0

g

• p=0

(Φ−g,p)∗, Q Φ−1−g,p + η Φ−1−g,p+1

• Fields

Anti-fields

(linear) pair

A∞

Q2 = 0 η2 = 0 [Q, η] = 0

Kroyter-Okawa-Schnabl-Torii-Zwiebach 2012 JHEP 03 (2012) 030

Interacting SFT in large space

We try to construct a master action for the large SFT : Large gauge symmetry :

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S[Φ] = 1 2

• Φ, Q η Φ
• + 1

3

• Φ, M2
• η Φ

2 + 1 4

• Φ, M3
• η Φ

3 + · · · . 平成 年 月 日

− − − − −

δΦ = η Λ′ + Q Λ + M2(η Φ, Λ) + M2(Λ, η Φ) + · · ·

The same BV fields-antifields as the free theory Mutually commutative pair :

A∞

[M, η] = 0 M2 = 0 η2 = 0

(1)usual string fields-antifields (2)usual gauge generators M and η (3)no ξ , no other products, no other (non-minimal) fields

As usual, we try to construct a BV master action under

Conventional BV approach breaks down

One can perturbatively construct its BV master action : However, there is no solution for and higher parts !!

S(1)[Φ, Φ∗] =

• Φ∗

2,−1, η Φ−1,1 + π1MΦ−1,0

1 1 − η Φ0,0

• 平成

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el Sbv = S + S(1) + S(2) + · · ·

S(2)[Φ, Φ∗] =

• Φ∗

3,−1, η Φ−2,1 + π1 M

Φ−1,0 2 (ηΦ−1,0) + Φ−2,0

• 1

1 − ηΦ0,0

• +
• Φ∗

3,−2, η Φ−2,2 + π1 M

Φ−1,0 2 MΦ−1,0

• 1

1 − ηΦ0,0

• −
• − 1

2

• Φ∗

2,−1, π1 M

• Φ∗

2,−1

Φ−1,0 2 (ηΦ−1,0) + Φ∗

2,−1Φ−2,0

• 1

1 − η Φ0,0

• S(3)

Conventional BV approach breaks down

One can perturbatively construct its BV master action : However, there is no solution for and higher parts !!

S(1)[Φ, Φ∗] =

• Φ∗

2,−1, η Φ−1,1 + π1MΦ−1,0

1 1 − η Φ0,0

• 平成

年 月 日

el Sbv = S + S(1) + S(2) + · · ·

S(2)[Φ, Φ∗] =

• Φ∗

3,−1, η Φ−2,1 + π1 M

Φ−1,0 2 (ηΦ−1,0) + Φ−2,0

• 1

1 − ηΦ0,0

• +
• Φ∗

3,−2, η Φ−2,2 + π1 M

Φ−1,0 2 MΦ−1,0

• 1

1 − ηΦ0,0

• −
• − 1

2

• Φ∗

2,−1, π1 M

• Φ∗

2,−1

Φ−1,0 2 (ηΦ−1,0) + Φ∗

2,−1Φ−2,0

• 1

1 − η Φ0,0

• S(3)

But, why?? ̶ If gauge algebra is generated by M and η only, we could construct it without using ξ .

Actually, one cannot construct the BV master action without ξ !!

Conventional BV revisited

• Revisit the gauge inv. of the free action

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S[Φ] = 1 2

• Φ, Q η Φ
• δΦ = ηΛ1,1 + QΛ1,0

δΦ = ηΛ1,1 + ηξ QΛ1,0 + ξη QΛ1,0 = η (Λ1,1 + ξ QΛ1,0) + ξ Q (−ηΛ1,0)

δΦ = η Λnew

1,1 + ξ Q (ηΛnew 1,0 )

δΛnew

g,0 = η Λnew g+1,1 + ξ Q (ηΛnew g+1,0)

δΛnew

g,p = η Λnew g+1,p+1 We should have used…

If we define “new gauge parameters” as ,

we find the following (factorised) gauge transformations :

Actually, vanishing “ξ-parts of p>0” generate “trivial transformations” !!

(p>0)

(p=0)

Trivial transformations appear !!

To see it explicitly, let us consider Re-definition :

Λnew

−1,0 ≡ −Λold −1,0 ,

Λnew

−1,1 ≡ π1 ξ

• M, Λold

−1,0

• 1

1 − η Φ + Λold

−1,1

δgΛ−g,p = π1

• M, Λ−g−1,p
• 1

1 − η Φ + η Λ−g−1,p+1 δΛnew

−g,0 = π1 ξ [

[M, η Λnew

−g−1,0]

] 1 1 − η Φ + η Λnew

−g−1,1 ,

δΛnew

−g,p = η Λnew −g−1,p+1 .

δΛnew

−1,1 = π1 ξ

• M, δΛold

−1,0

• 1

1 − η Φ + δΛold

−1,1

= π1 ξ

• M, π1
• M, Λold

−2,1

• 1

1 − ηΦ

• 1

1 − η Φ + η

• π1 ξ
• M, Λold

−2,1

• 1

1 − ηΦ + Λold

−2,2

• = ξ T(Λnew

−2,1) + η Λnew −2,2 ,

• Then, ξ-part of the gauge variation generates a trivial transformation !!

As a result,

(p>0) (p=0)

BV master action in large space

• We consider the sum of fields carrying fixed picture number p :
• Decompose it into η- and ξ-exacts :
• Introduce their antifields separately :

ϕp ≡

∞

• g=p

Φ−g,p

as ϕp = ϕξ

p + ϕη p

• nent of ϕ resp

(ϕξ

p)∗ = ∞

• g=p

(Φ ξ

−g,p)∗ ,

(ϕη

p)∗ = ∞

• g=p

(Φ η

−g,p)∗ .

• Sbv =

1 dt

• ϕ0 + ξ (ϕξ

0)∗, M

1 1 − t η (ϕ0 + ξ (ϕξ

0)∗)

• +
• p>0
• (ϕη

p−1)∗, η ϕp

• .
• n Sbv = Sbv[ ϕ, (ϕξ)∗(ϕη)∗] i

BV master action is a functional of these string fields-antifields

BV master action in large space

• We consider the sum of fields carrying fixed picture number p :
• Decompose it into η- and ξ-exacts :
• Introduce their antifields separately :

ϕp ≡

∞

• g=p

Φ−g,p

as ϕp = ϕξ

p + ϕη p

• nent of ϕ resp

(ϕξ

p)∗ = ∞

• g=p

(Φ ξ

−g,p)∗ ,

(ϕη

p)∗ = ∞

• g=p

(Φ η

−g,p)∗ .

• Sbv =

1 dt

• ϕ0 + ξ (ϕξ

0)∗, M

1 1 − t η (ϕ0 + ξ (ϕξ

0)∗)

• +
• p>0
• (ϕη

p−1)∗, η ϕp

• .
• n Sbv = Sbv[ ϕ, (ϕξ)∗(ϕη)∗] i

1 2

• Sbv, Sbv
• min =

←

∂ Sbv ∂ϕξ ·

→

∂ Sbv ∂(ϕξ)∗ +

←

∂ Sbv ∂ϕη ·

→

∂ Sbv ∂(ϕη)∗ = 0 .

master eq.

BV master action is a functional of these string fields-antifields

BV master action in large space

• We consider the sum of fields carrying fixed picture number p :
• Decompose it into η- and ξ-exacts :
• Introduce their antifields separately :

ϕp ≡

∞

• g=p

Φ−g,p

as ϕp = ϕξ

p + ϕη p

• nent of ϕ resp

(ϕξ

p)∗ = ∞

• g=p

(Φ ξ

−g,p)∗ ,

(ϕη

p)∗ = ∞

• g=p

(Φ η

−g,p)∗ .

• Sbv =

1 dt

• ϕ0 + ξ (ϕξ

0)∗, M

1 1 − t η (ϕ0 + ξ (ϕξ

0)∗)

• +
• p>0
• (ϕη

p−1)∗, η ϕp

• .
• n Sbv = Sbv[ ϕ, (ϕξ)∗(ϕη)∗] i

δϕ0 =

• ϕξ

0 + ϕη 0 , Sbv

• min = π1 ξ M

1 1 − η (ϕ0 + ξ (ϕ0)∗) + η ϕ1 , δϕp =

• ϕξ

p + ϕη p , Sbv

• min = η ϕp+1 ,

δ(ϕ0)∗ =

• (ϕξ

0)∗ + (ϕη 0)∗ , Sbv

• min = π1 M

1 1 − η (ϕ0 + ξ (ϕ0)∗) , δ(ϕp)∗ =

• (ϕξ

p)∗ + (ϕη p)∗ , Sbv

• min = η (ϕ η

p−1)∗ .

It generates appropriate BV-BRST transformations !!

BV master action is a functional of these string fields-antifields

Summary

• Naive conventional approach works up to antifield number 2.
• Gauge algebra is generated by M and η, but ξ-parts generate

“trivial transformations”.

• Therefore, “ξ” must appear in the BV master action.

Comments

Since string-field redefinitions connect different SFTs, in principle, BV master actions for other SFTs are obtained via BV canonical transformations. Even for Berkovits’ theory, “ξ” generates “trivial transformations”.

(Thus, we need “ξ” even for the BV master action for Berkovits’ theory)

EKS’ SFT

Sen’s formulation

A∞/L∞ Small space Large space

• ther SFTs

field re-def. field re-def.

WZW-like SFT

Large theory

• 2. Gauge-fixing

fermions (results only)

a) Reduction to small BV b) Direct gauge fixing

a) Partially gauge-fixing fermion

• Our “large” BV master action reduces to known “small” BV master action.
• Consider the following trivial pairs and (partially) gauge-fixing fermion :

Strivial =

• g,p
• N η

1−g,p−1, ξ0 (Ψ2+g,−1−p)∗

+

• N ξ

−1−g,1+p, (C−1−g,1+p)∗

. F =

• g,p
• Φ−g,p, Ψ2+g,−1−p
• +
• C−1−g,1+p, η Ψ2+g,−1−p
• Sbv + Strivial
• F = S′

bv[ψ] ≡

1 dt

• ψ , ξ M

1 1 − t ψ

• ≡

ψ ≡

∞

• g=0

g

• p=0

δp,0

• Ψ1−g,p−1 + Ψ2+g,−1−p
• .

After some computations, we find

where ψ is given by

b) (Direct) gauge-fixing fermions

• Gauge conditions studied by S.Torii

is given by the following trivial pairs and gauge-fixing fermion :

Strivial =

∞

• n=0
• B−(n+2) Nn+3 , (Ψn+2)∗

+

• (C2)∗ , N2
• +

∞

• n=0
• (C−n)∗ , N−n
• F =

∞

• n=0
• Φ−n , Ψn+2
• +
• C2, b0 ξ0 Ψ2
• +

∞

• n=0
• C−n, B−(n+2) Ψn+3
• B−(n+2) =

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ b · · · yn ζn xn b0 . . . yn ζn xn b0 ... yn ζn xn b0 · · · ζn ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

B−(n+2) ⎡ ⎢ ⎣ Φ−n,0 . . . Φ−n,n ⎤ ⎥ ⎦ = 0

You can apply technique of large-space to your SFT in small-space !!

Conclusion

• BV master action in large space

We can gauge-fix SFT having “large gauge symmetries” — hidden trivial transformations!! You can apply large-space technique to SFT defined in small space via embeddings. (Constrained BV gives more elegant constructions of BV master actions.)

• Gauge-fixing fermion

Large theory indeed reduces to the original small theory by partial gauge fixing. — Namely, we obtained at the level of BV master actions. Gauge-fixing fermion imposing [KOSTZ]’s gauge-conditions was constructed.

Thank you for your attentions

Appendix : BV ??

4-slides review of BV (1/4)

Let us consider a Lagrangian Its e.o.m. is given by When there is a gauge invariance you find the Noether identities : When gives a generating set of the gauge transformations, So, for gauge-fixing, we need ghosts

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概 要

S[φi] =

• dDx L
• φi, ∂µφi, ∂µ1µ2φi, . . . , ∂µ1...µnφi

.

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概 要

δS δφi ≡ ∂S ∂φi − ∂µ ∂S ∂(∂µφi) + · · · + (−)n∂µ1...µn ∂S ∂(∂µ1...µnφi) = 0 .

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概 要

δφi(x) = Riaa ≡

• dy
• Ria(x, y)a(y) + Ri µa(x, y)∂µa + · · · + Ri µ1...µna(x, y)∂µ1...µna(y)
• 平成

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概 要

δS = δS δφi δφi = δS δφi Riaa = 0 .

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概 要

δφi(x) = Riaa

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概 要

δS δφi Ria = 0 ⇐ ⇒ Ria : null vectors exist .

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概 要

Ria a = ⇒ Ria ca : ghost fields ca appear .

4-slides review of BV (2/4)

If null vectors of are degenerate, its gauge symmetry is reducible. This “gauge symmetry of gauge symmetry” requires “higher ghosts” Likewise, higher gauge symmetries need further higher ghosts. The antifield formalism defines a BRST-like operation for these ghosts. As BRST, the physical states are given by its cohomology.

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概 要

δS δφi Ria = 0

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概 要

Ria Uaα = 0 ⇐ ⇒ Uaα : further null vectors exist .

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概 要

δλa = Uaα λα = ⇒ Uaα C2nda : ghosts for ghosts C2ndα appear .

4-slides review of BV (3/4)

In the above, we ignored a trivial but important symmetry. Not only gauge theories, every theories have this gauge invariance. It is called as “trivial gauge transformations” : However, it may not be factorised and the gauge algebra may be open : Namely, one may find the following gauge commutator On-shell vanishing terms make your BRST procedure terrible. “Antifields” resolve it:

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δωS = δS δφi δωφi = 0 with δωφi = ωij δS δφj , ωij = −ωji .

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• trivial , ∀gauge transf.
• = trivial
• 平成

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• non trivial , non trivial
• = non trivial + trivial

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[δa, δb]φi = δRib δφj Rja − δRia δφj Rjb

• λb λa = Ric Λc

ab + δS

δφj Ωji

ab .

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δωφi = ωij δS δφi = ⇒ Antifield (φi)∗ s.t. δBRST(φi)∗ = ωij δS δφi

4-slides review of BV (4/4)

a) Introduce appropriate (higher) ghosts, which we also call “fields” . b) Introduce an “antifield” for each “field”. c) Define the antibracket on the space of all fields and antifields d) Find a solution of the master equation, “the BV master action” is an intrinsic object of the gauge theory and gives the generator of BRST : . e) Fix your gauge by constructing appropriate gauge-fixing fermion F :

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d (φi)∗

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Fields φi

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re ( , ),

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• F , G
• ≡
• i

δrF δφi δlG δ(φi)∗ − δrF δ(φi)∗ δrG δφi

• .

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Sbv = Sbv[φ, φ∗]

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• Sbv , Sbv
• = 0

with the initial condition Sbv|φ∗=0 = S

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, Sbv

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δBRSTF =

• Sbv , F
• (ϕi)* ≡ ∂F[ϕ]

∂ϕi