Gauge Fixing Problem in Cubic Superstring Field Theory

Masaki MurataYITP

based on work in progress withTaichiro Kugo, Maiko Kohriki,

Hiroshi Kunitomo and Isao Kishimoto

1. Introduction2. Gauge Fixing of Ramond Field3. Gauge Fixing of Neveu-Schwarz Field (incomplete)4. Other topic5. Future directions

Oct. 22, 2010 at YITP, Kyoto

1. Introduction(SSFT)

Open Superstring Field Theories (SSFT)

1. Introduction(Motivation)

Our goal is

write down Siegel gauge action with kinetic operator L0 (F0)

take into account of interaction terms

Batalin-Vilkovisky (BV) formalism

1. Introduction (Batalin-Vilkovisky )

BV formalism with "Gauge-Fixed basis"

field anti-field

standard BV formalism

BV master equation

Boundary condition gauge invariant action

BV master equation

Boundary condition gauge fixed action

Definition of and are different

1. Introduction (BV for bosonic SFT)

:Ghost number constraint is relaxed.

BRST transformation:

Benefit of gauge-fixed basis : action satisfying master equation is same form as original

Siegel gauge

field anti-field

2. Gauge Fixing of Ramond (1)Kinetic term

Kernel of Y : Additional gauge symmetry[Kugo,Terao (1988)]

Projection operator removing kernel of Y :

picture changing operator

,

[Arefeva-Medvedev (1988)]

2. Gauge Fixing of Ramond (2)Projected field [Kazama-Neveu-Nicolai-West(1986)]

We can rewrite the action as

Relax ghost number constraint of

BV master equation

BRST transformation

[Sazdovic(1987)]

constrained field

2. Gauge Fixing of Ramond (3)

Siegel gauge :

we don't have Y0 !!

field anti-field

Variation of action

3. Gauge fixing of NS (PTY)

gauge

PTY projection operator

Propagator

[Preitschopf-Thorn-Yost(1990)],[Arefeva-Medvedev-Zubarev(1990)]

-2, 0-picture

We would like to

write down Siegel gauge action with kinetic operator L0 (F0)

is problematic : doesn’t have Klein-Gordon operator (second derivative)retracts (part of) world sheet

textend the worldsheet retract the worldsheet

3. Gauge fixing of NS (PTY)

We haven't succeeded yet.I will show our trials to explain where difficulties come form.

3. Gauge Fixing of NS1. Constructed another projection operator (1)

Ramond

NS

naive extension

We can show

3. Gauge Fixing of NSProjected field with

The computation is much complicated.

Ramond

Ramond

We couldn't find counterpartNS

3. Gauge Fixing of NS2.Construct another projection operator (2)

seems to be important

: difficult to find

We investigated another projection operatorso that we can find

We found one example by slightly modifying PTY's

3. Gauge Fixing of NS

We found

However,

kinetic operator does not contain

4. Another topic

We searched another candidate of picture changing operator with no divergence

modified cubic SSFT has divergence

Result : we proved uniqueness of X and Y

1. 2. (Virasoro) primary,3. commutes with BRST charge,

Strategy : construct operators satisfying following conditions

study operators with different picture number

5.Future directions1. Investigate gauge transformation at linearized level

By straightforwardly calculating , we can specify what gauge is possible.The components eliminated by might be identified as anti-fields. This will make the calculation of simpler.

The computation is much complicated.

original gauge transformation

5. Future directions2. “Minimal BRST-closed space”

minimal : the number of independent component is minimum

BRST-closed :

We want to construct the simpler projected state.

We want to construct simpler projected space as minimal BRST-closed space.

“minimal BRST-closed space”

messy

for example let us consider Ramond sector

5. Future directionsMinimal BRST-closed space for Ramond

Start line

Second step Close

projected space can be expressed as minimal space!!

5. Future directionsWe want to construct ``minimal space’’ for NS

Ramond NS``zero modes''

First step

?

These two formalisms can be related through anti-canonical transformation

5. Future directions2. “Minimal BRST-closed space”

V

Iterative construction :

we can express any state in terms of

minimal : the number of independent is minimum BRST-closed :