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Gauge Fixing Problem in Cubic Superstring Field Theory Masaki Murata YITP based on work in progress with Taichiro Kugo, Maiko Kohriki, Hiroshi Kunitomo and Isao Kishimoto 1. Introduction 2. Gauge Fixing of Ramond Field 3. Gauge Fixing of Neveu-Schwarz Field (incomplete) 4. Other topic 5. Future directions Oct. 22, 2010 at YITP, Kyoto

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1. Introduction (SSFT) Open Superstring Field Theories (SSFT)

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1. Introduction (Motivation) Our goal is write down Siegel gauge action with kinetic operator L 0 (F 0 ) take into account of interaction terms Batalin-Vilkovisky (BV) formalism

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1. Introduction (Batalin-Vilkovisky ) BV formalism with "Gauge-Fixed basis" fieldanti-field standard BV formalism BV master equation Boundary conditiongauge invariant action BV master equation Boundary conditiongauge fixed action Definition of and are different

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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 fieldanti-field

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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)]

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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

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2. Gauge Fixing of Ramond (3) Siegel gauge : we don't have Y 0 !! fieldanti-field Variation of action

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3. Gauge fixing of NS (PTY) gauge PTY projection operator Propagator [Preitschopf-Thorn-Yost(1990)], [Arefeva-Medvedev-Zubarev(1990)] -2, 0-picture

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We would like to write down Siegel gauge action with kinetic operator L 0 (F 0 ) is problematic : doesn’t have Klein-Gordon operator (second derivative) retracts (part of) world sheet t extend 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.

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3. Gauge Fixing of NS 1. Constructed another projection operator (1) Ramond NS naive extension We can show

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3. Gauge Fixing of NS Projected field with The computation is much complicated. Ramond We couldn't find counterpart NS

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3. Gauge Fixing of NS 2.Construct another projection operator (2) seems to be important : difficult to find We investigated another projection operator so that we can find We found one example by slightly modifying PTY's

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3. Gauge Fixing of NS We found However, kinetic operator does not contain

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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

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5.Future directions 1. 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

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5. Future directions 2. “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

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5. Future directions Minimal BRST-closed space for Ramond Start line Second step Close projected space can be expressed as minimal space!!

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5. Future directions We want to construct ``minimal space’’ for NS RamondNS ``zero modes'' First step ?

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These two formalisms can be related through anti-canonical transformation

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5. Future directions 2. “Minimal BRST-closed space” V Iterative construction : we can express any state in terms of minimal : the number of independent is minimum BRST-closed :

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