Hiroaki Matsunaga YITP Workshop Strings & Fields 2018 BV master action for superstring field theory in the large Hilbert space ( based on JHEP 05 (2018) 020 )
Today’s topic : Gauge fixing of “large” SFT 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.)
WZW-like SFT Large space Large and small string field theories Berkovits’ WZW-like theory • S = 1 e − Φ Qe Φ , e − Φ η e Φ � � 2 � 1 − 1 e − t Φ d � �� dte t Φ , e − t Φ Qe t Φ , e − t Φ η e t Φ � � � � dt 2 0 • String field based on ηξφ -system ( not δ(β)δ(γ) -system ) η φ ≠ 0 where η ≡ η 0 • Large gauge symmetries and Various gauge conditions δφ = Q Λ + η Ω + ⋯ • Useful for constructing classical solutions • Kunitomo-Okawa’s action ( + R sector )
gauge invariance String field of β γ WZW-like SFT EKS’ SFT Sen’s formulation Classical solution is unknown Small space Large space other SFTs Easily gauge-fixable Large and small string field theories Berkovits’ WZW-like theory • S = 1 e − Φ Qe Φ , e − Φ η e Φ � � 2 � 1 − 1 e − t Φ d � �� dte t Φ , e − t Φ Qe t Φ , e − t Φ η e t Φ � � � � dt 2 0 SFTs in small space • 2 ⟨ Ψ , Q Ψ ⟩ + ∑ S = 1 1 n ⟨ Ψ , M n ( Ψ n ) ⟩ • A ∞ / L ∞ A ∞ / L ∞ • • •
gauge invariance String field of β γ WZW-like SFT EKS’ SFT Sen’s formulation Classical solution is unknown Small space Large space other SFTs field re-def. Easily gauge-fixable Large and small string field theories Berkovits’ WZW-like theory • S = 1 e − Φ Qe Φ , e − Φ η e Φ � � 2 � 1 − 1 e − t Φ d � �� dte t Φ , e − t Φ Qe t Φ , e − t Φ η e t Φ � � � � dt 2 0 SFTs in small space • 2 ⟨ Ψ , Q Ψ ⟩ + ∑ S = 1 1 n ⟨ Ψ , M n ( Ψ n ) ⟩ • A ∞ / L ∞ A ∞ / L ∞ • • •
EKS’ SFT Partial gauge field re-def. other SFTs Large space Small space WZW-like SFT fixing (non-linear) Sen’s formulation Motivation of the “large” space • Partial gauge-fixing - reduces to other SFTs Large SFT is interesting • - unusual (non-geo.) propagators A ∞ / L ∞ - to understand SFT itself
Large theory Sen’s formulation field re-def. other SFTs Large space Small space embedding (linear) Natural EKS’ SFT WZW-like SFT Motivation of the “large” space • Natural embedding - every SFT can be large One can apply Large-space technique to SFT defined in small space !! A ∞ / L ∞ ( BV string fields-antifields don’t have to be small. )
field re-def. Natural Large theory field re-def. other SFTs Large space Small space embedding (linear) Sen’s formulation EKS’ SFT WZW-like SFT Motivation of the “large” space • Natural embedding - every SFT can be large One can apply Large-space technique to SFT defined in small space !! A ∞ / L ∞ ( BV string fields-antifields don’t have to be small. )
EKS’ SFT Sen’s formulation Natural embedding (linear) Small space Large space other SFTs field re-def. field re-def. WZW-like SFT Large theory Today’s topic : “large theory” We focus on “ large theory ” A ∞ • A ∞ / L ∞
EKS’ SFT WZW-like SFT if we get a BV master action BV canonical transformations the gauge-fixing problem via field-redefinitions. off-shell equivalent to it under this embedded theory. space is unsolved, even for Sen’s formulation Large theory field re-def. field re-def. other SFTs Large space Small space Natural embedding (linear) for this “large theory”. Motivation of the “large theory” We focus on “ large theory ” A ∞ • • Gauge-fixing of SFT in large • Other SFTs in large space are A ∞ / L ∞ • Thus, in principle, we can solve
for this “large theory”. space is unsolved, even for Natural embedding SFT in small space Large theory if we get a BV master action BV canonical transformations the gauge-fixing problem via field-redefinitions. off-shell equivalent to it under this embedded theory. Motivation of the “large theory” We focus on “ large theory ” A ∞ • S [ Φ ] = 1 2 ⟨ Φ , Q η Φ ⟩ + ⋯ • Gauge-fixing of SFT in large Ψ ≡ η Φ A ∞ S [ Ψ ] = 1 2 ⟨ Ψ , Q Ψ ⟩ Ker[ η ] + ⋯ • Other SFTs in large space are • Thus, in principle, we can solve
(the same kinetic term and gauge reducibility) It gives another representation of Berkovits’ WZW-like theory. Hilbert space to your SFTs in small space. If how to gauge-fix it is clarified, you can apply techniques of the large Natural embedding SFT in small space Large theory Motivation of the “large theory” We focus on “ large theory ” A ∞ • S [ Φ ] = 1 2 ⟨ Φ , Q η Φ ⟩ + ⋯ - Motivation 1 - Ψ ≡ η Φ It is the simplest WZW-like theory. A ∞ S [ Ψ ] = 1 2 ⟨ Ψ , Q Ψ ⟩ Ker[ η ] + ⋯ - Motivation 2 - - Motivation 3 -
Propagator Large theory has larger gauge inv. and various gauge conditions - unusual propagators available (don’t have to be geometrical unlike βγ) Kroyter-Okawa-Schnabl-Torii-Zwiebach 2012 Propagator Siegel gauge Gauge condition Techunical Motivation ( Motivation 2 ) String field theory in small space has (geometrical) propagator • P n = b 0 b 0 Ψ n = 0 L 0 • ⎛ d 0 ⎞ P b 0 · · · 0 0 0 of d = [ Q, b ξ ], ( α +1) L 0 ⎜ ⎟ . . . ... b 0 Φ ( − n, 0) = 0 ( n ≥ 0) , . . . ⎜ ⎟ α b 0 d 0 0 . . . ⎜ ⎟ ( α +1) L 0 ( α +1) L 0 ⎜ ⎟ d 0 Φ ( − n,m ) + α b 0 Φ ( − n,m +1) = 0 (0 ≤ m ≤ n − 1) , ⎜ ⎟ . . . ... ⎜ . . . ⎟ α b 0 . 0 0 . . � � ⎜ ⎟ α + η 0 d 0 b 0 d 0 Φ ( − n,n ) = 0 ( n ≥ 0) , ( α +1) L 0 ⎜ ⎟ P b = , P P n +1 ,n +2 = ⎜ ⎟ ( α + 1) L 0 L 0 . . . ... ⎜ . . . ⎟ d 0 . . 0 0 . ⎜ ⎟ b 0 d 0 Φ ( n +1 , − 1) = 0 ( n ≥ 1) , ( α +1) L 0 ⎜ ⎟ ⎜ ⎟ . . . ... ⎜ . . . ⎟ α b 0 Φ ( n +1 , − m ) + d 0 Φ ( n +1 , − ( m +1)) = 0 (1 ≤ m ≤ n − 1) . α b 0 d 0 . . . 0 ⎜ ⎟ � 1 + α Qb 0 � d 0 ( α +1) L 0 ( α +1) L 0 ⎜ ⎟ P d = , . ⎝ ⎠ L 0 ( α + 1) L 0 α b 0 0 0 0 · · · 0 ( α +1) L 0 P d
(the same kinetic term and gauge reducibility) It gives another representation of Berkovits’ WZW-like theory. Hilbert space to your SFTs in small space. If how to gauge-fix it is clarified, you can apply techniques of the large Natural embedding SFT in small space Large theory Motivation of the “large theory” We focus on “ large theory ” A ∞ • S [ Φ ] = 1 2 ⟨ Φ , Q η Φ ⟩ + ⋯ - Motivation 1 - Ψ ≡ η Φ It is the simplest WZW-like theory. A ∞ S [ Ψ ] = 1 2 ⟨ Ψ , Q Ψ ⟩ Ker[ η ] + ⋯ - Motivation 2 - - Motivation 3 -
: functional of string field s.t. Berkovits’SFT gives the following solution It gives another representation of Berkovits’ WZW-like theory. (Off course, the same kinetic term and gauge reducibility) WZW-like SFT Motivation 3 Berkovits’ WZW-like theory S = ∫ 1 dt ⟨ A t [ φ ] , Q A η [ φ ] ⟩ S = 1 e − Φ Qe Φ , e − Φ η e Φ � � 0 2 � 1 − 1 e − t Φ d A η [ φ ] � �� dte t Φ , e − t Φ Qe t Φ , e − t Φ η e t Φ � � � � dt 2 0 η A η [ φ ] − A η [ φ ] * A η [ φ ] = 0 A η [ φ ] = ( η e t φ ) e − t φ A t [ φ ] = ( ∂ t e t φ ) e − t φ
Large A ∞ theory gives another solution : functional of string field s.t. ̂ ̂ WZW-like SFT (Off course, the same kinetic term and gauge reducibility) It gives another representation of Berkovits’ WZW-like theory. Berkovits’SFT gives the following solution Motivation 3 Berkovits’ WZW-like theory S = ∫ 1 dt ⟨ A t [ φ ] , Q A η [ φ ] ⟩ S = 1 e − Φ Qe Φ , e − Φ η e Φ � � 0 2 � 1 − 1 e − t Φ d A η [ φ ] � �� dte t Φ , e − t Φ Qe t Φ , e − t Φ η e t Φ � � � � dt 2 0 η A η [ φ ] − A η [ φ ] * A η [ φ ] = 0 A η [ φ ] = ( η e t φ ) e − t φ A t [ φ ] = ( ∂ t e t φ ) e − t φ 1 1 1 A η [ Φ ] = π 1 G A t [ Φ ] = π 1 G 1 − t η Φ ⊗ Φ ⊗ 1 − t η Φ 1 − t η Φ
Natural embedding SFT in small space Large theory Today’s topic S [ Φ ] = 1 2 ⟨ Φ , Q η Φ ⟩ + ⋯ We consider a BV master action • for this “large” theory. Ψ ≡ η Φ A ∞ S [ Ψ ] = 1 2 ⟨ Ψ , Q Ψ ⟩ Ker[ η ] + ⋯
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
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