String Field Theories Oct. 18, 2010 at ： YITP, Kyoto University Taichiro Kugo

History of SFT – personal view 1974 Kaku and Kikkawa （加来 - 吉川） First SFT Light-cone gauge SFT String length Gave much impact to physics society Gauge symmetry is not manifest Success of QCD (local field theory) → has soon faded away, unfortunately

BRS （ T) symmetry 1978 Operator Formalism of YM Kugo and Ojima Gupta-Bleuler-Nakanishi-Lautrup 1982 Fujikawa: BRS quantization of String Anomaly of conformal and ghost number 1983 Kato and Ogawa

Kato-Ogawa, “Covariant Quantization of String based on BRS Invariance” Nucl.Phys.B212(1983)443 Infinite number of conditions A Single condition

Toward String Field Theories Single Condition Q B Φ ＝０ → kinetic term should be Interaction term should be midpoint type: 1983 – 1984 Kato, Ogawa, Uehara and TK We gave seminar talks in a few University seminars

Tetsuo Goto, “Picture of Extended Particles” (Iwanami, 1978) Chapter5. String Model §5.5 String-string Interaction Midpoint Interaction

Unfortunately,we could have confidence on the correctness neither of the Kinetic term nor of the Interaction term. Huge gauge invariance → how to gauge-fix? Bizarre midpoint interaction → reproduce Koba-Nielsen? Soon Siegel’s paper appeared 1984 July Gauge fixed and Free theory We could not convince ourselves to the last and gave up publication of our work Important Lesson: If you find a natural theory, then publish it even you cannot prove its validity.

Interacting gauge SFTs : 1986 Witten: Cubic SFT Goto-Witten-type vertex open bosonic open NSR superstring HIKKO (Hata-Itoh-Kugo-Kunitomo-Ogawa) bosonic open LC-type vertex bosonic closed : cubic

Problems: GW-type vertex NSR superstring Colliding X 2 problem (C.Wendt, 1989) ・ 1990 Modified Cubic SFT (0-picture NS) Preitschopf-Thorn-Yost Aref’eva-Medvedev-Zubarev gauge-fixing problem (Kernel of PC ops X,Y) ・ 1995 WZW-like superSFT Berkovitz problem in R-sector Closed string Non-polynomial SFT Saadi-Zwiebach Kugo-Kunitomo-Suehiro ’90 Tree level action Polyhedral vertices B.Zwiebach ’93 Quantum Action

Problems: LC-type vertex HIKKO ： unphysical α １） (identify) HIKKO Consistent gauge theory, but non-covariant 2)Covariantized LC SFT Unphysical quartet variables: α + 1(bosonic) + 2(fermionic) Siegel-Zwiebach, Neveu-West, Uehara, Kugo Cubic SFT for closed string! But, it is difficult to discuss vacuum energy Vertex depends on the string length ratio Cf: a single field condensation can be discussed in the cubic vertex Dilaton condensation: Yoneya(787), Kawano(’92)

Nongeometrical vertex? BSFT (Boundary SFT) Witten There are no propagator and no vertices explicitly at least It is not a SFT in the usual sense.

Insight on Non-perturbative Aspects from SFT At eraly stage ・ SFT is a gauge theory with Huge gauge symmetry ・ Pregeometrical SFT (Purely cubic action) Q B : only object specifying space-time metric Spontaneous generation of space-time and motion Yoneya,Friedan(‘85), HIKKO(‘86), Strominger et al. T-duality and gauge symmetry 1992 Kugo-Zwiebach based on HIKKO

Remarkable Success of SFT Exact solution by Schnabl (Nov. 2005) classical solutions and technique explosive progress made by Okawa, Ellwood, Fuchs, Kroyter, Rastelli, Zwiebach, Aref’eva, …. Stagnancy after that, at the moment ?!

We should study by SFT D-branes matrix models : SFT D-instanton complementary D-particle D-string S-duality U-duality M-theory AdS/CFT May give hints On a new formulation of SFT I hope that this workshop, this series of workshops, give a breakthrough for the SFT approach to string theory