Open-Closed Super String Field Theory Mritunjay Verma INFN, Napoli (Work with Faroogh Moosavian and Ashoke Sen) (ArXiv: 1907.10632 + Ongoing work) Workshop on Fundamental Aspects of String Theory, ICTP-SAIFR/IFT-UNESP, Sao Paolo, June 2020 University of California, Davis, December 2017 Brief Outline 1. Introduction 2. Construction of open-closed string field theory 3. Generalization to unoriented strings Introduction String Field theory is the field theoretic formulation of string • theory. (Kaku, Kikkawa; Witten; Zwiebach; Hata, Zwiebach; Sen; Berkovits; ··· ) The usual string theory based on Polyakov prescription re- • lies very heavily on world sheet conformal field theory. The world sheet CFT gives powerful tools allowing us to • explore the theory deeply. However, it also puts some re- strictions on how far one can go. The world sheet calculations typically give IR divergences. • One can trace these divergences to be coming from bound- ary regions of the moduli space. Introduction Some of these divergences can be cured within the frame- • work of world-sheet theory by making an analytic continua- tion in the external momenta. (Berera; Witten) However, for some of the problems such as • Mass renormalization of generic string state • Tadpole divergences due to incorrect vacuum • Contribution of D instantons to perturbation theory, • the analytic continuation is not enough. During the last few years, string field theory has given sys- • tematic tools to deal with these IR issues. ( Rudra, Pius, Sen; Sen) Introduction String field theory has also given a deeper understanding • of the issues of unitarity and analyticity in string theory. (Pius, Sen; Sen; Lacroix, Erbin, Sen) The string field theory has also been used to study the su- • perstrings in RR background. (Cho, Collier, Yin) Apart from perturbative questions, the string field theory • has also given insights into the non perturbative aspects of string theory. Following Ashoke Sen’s work on Tachyon condensation, a • lot of results have been obtained both anlytically and numer- ically regarding the open strings in D-brane backgrounds. (Sen; Sen, Zwiebach; Gaiotto, Rastelli; Schnabl; Erler, Maccaferri;··· ) Introduction The bosonic closed string field theory was developed in • early 90s and provides the language we shall be using. (Zwiebach; Sen, Zweibach) The closed superstring field theory has been constructed • recently. (Sen) The open super string field theory was first constructed by • Edward Witten. However, it was soon found that it contained divergences which correspond to propagation of closed string states. (Freedman, Giddings, Shapiro, Thorn) Introduction There have been various attempts to reformulate the open • string field theory so that these divergences disappear. (Berkovits; Gaberdiel, Zwiebach; Kroyter, Okawa, Schnabl, Torii, Zwiebach; Earler, Konopka, Sachs; Kunitomo, Okawa; ··· ) One way to deal with these divergences is to explicitly add • the closed string sector to the open string field theory which leads to combined open closed string field theory. Such systems natuarally arise in the presence of D branes. For the bosonic case, the combined open closed string field • theory was constructed by Zwiebach. Our goal will be to describe the super symmetric version of • the combined open closed theory. (Moosavian, Sen, MV) How to Construct String Field Action ? Before proceeding further, we describe the schematic way • in which the string field theories are typically constructed. A string field is a general linear combination of all the string • states which appear in first quantized world-sheet theory X Ψ = r φr r j i where φr denote the basis states which are in one to one correspondencej i with the states in the world-sheet theory and r turn out to be the space-time fields. How to Construct String Field Action ? The usual string theory, based on Polyakov prescription, • gives a formal expression for the on-shell amplitudes given as integrals over the moduli spaces. Z g;n = [dmi] 1 n g;n A Mg;n hV ···V i where mi denote the coordinates on the moduli space and f g 1 n g;n denotes a correlation function on the genus g RiemannhV ···V i surface with n external string states. One generalizes this to construct the off shell amplitudes. • This involves making the amplitudes depend upon some off shell data (such as local coordinates around the punctures). How to Construct String Field Action ? One now divides the integration regions in ‘’1PI and 1PR • regions” Z Z g;n = + [dmi] 1 n g;n A 1PI 1PR hV ···V i The 1PR part includes those regions in moduli space which • correspond to separating type degeneration of Riemann sur- faces. On the other hand, the 1PI part excludes the separating • type degeneration. Identifying the 1PI and 1PR parts of amplitudes, we can • construct the vertices of string field theory to be used in the action. How to Construct String Field Action ? S2 S1 Separating type degeneration non separating type degeneration − Construction of Open Closed String Field Theory - Ingredients for constructing open closed SFT - 1PI effective action - Section Independence Ingredients for Constructing Open Closed SFT We start by introducing the vertices which shall be used in • the construction of the 1PI action. For this, we introduce some specific vector spaces in which • string fields live c −1;−1 ; H ≡ H ⊕ H−1=2;−1 ⊕ H−1;−1=2 ⊕ H−1=2;−1=2 c e −1;−1 −3=2;−1 −1;−3=2 −3=2;−3=2; Ho ≡ H ⊕ H ⊕ H ⊕ H −1 ; H ≡ H ⊕ H−1=2 o e −1 − = H ≡ H ⊕ H 3 2 m;n denotes the vector space of closed string states with H picture number (m; n) and m denotes the vector space of open string states with pictureH number m. Ingredients for Constructing Open Closed SFT For fields Ψc c and Ψo o, we denote the off shell 1PI • i i amplitude/vertex2 H by 2 H c c o o Ψ1 ΨN;Ψ1 ΨM f ··· ··· g Z X 2g−2+b c c o o (gs) Ωp(Ψ1; ; ΨN;Ψ1; ; ΨM) ≡ g;b≥0 Rg;b;N;M ··· ··· (g;b;N;M)6=(0;1;1;0) is the part of moduli space of punctured Riemann • g;b;N;M surfacesR which includes information about the local coordi- nates around the punctures and PCO locations. Ω is a correlation function on the Riemann surface and p = • p 6g 6 + 3b + 2N + M denotes its dimension. − Ingredients for Constructing Open Closed SFT The disc one point function is not included in the definition • of Ψc Ψc ;Ψo Ψo . We need to treat this separately. f 1 ··· N 1 ··· Mg The reason for this is that in the RR sector with picture num- • 1 1 c ber ( 2 ; 2 ), the quantity Ψ ; vanishes since it can not satisfy− the− picture number conservationf g (we need the total picture number to be -2 on the disc). However, for elements in ec, there is no problem. In the RR • H 3 3 sector, they carry the picture number ( 2 ; 2 ) and we can construct non zero disc one point function− − using them (by inserting a picture changing operator). Ingredients for Constructing Open Closed SFT Hence, for Ψ~ c c, we define Ψ~ c to be the contribution • e D from one point2 functionH on the disc.f g For the NS-NS sector, we can include the disc one point • c c contribution either in Ψ ; or Ψ~ D. For uniformity, we f g c f g include it in the definition of Ψ~ D. f g The basic reason that the disc one point function needs the • states from ec is that the D branes carry the RR charge and H couple to the RR potential which is contained in Ψ~ c (the Ψc contain the RR field strengths). Ingredients for Constructing Open Closed SFT Before proceeding further, we note some features of the • section segments g;b;N;M. R The section segments must not contain any separating type • degeneration. Given two section segments, we can generate new section • segments by repeated applications of following two opera- tions 1. Plumbing fixture (sewing) 2. Hole creation Ingredients for Constructing Open Closed SFT In the plumbing fixture, we take two section segments g ;b ;N ;M R 1 1 1 1 and g2;b2;N2;M2 . We now pick a Riemann surface from one fam- ily andR sew one of its puncture to a puncture on the Riemann surface from the other family by identifying the local coordinates around the punctures. −s−iθ Closed String : w1w2 = e ; 0 s −s ≤ ≤ 1 Open String : w1w2 = e − Ingredients for Constructing Open Closed SFT In the hole creation operation, the Riemann surfaces from the section segment g;b;N;M are glued to a disc with one closed string puncture R −s w1w2 = e This needs to be specified separately since the disc with one bulk puncture is special in the combined open closed super string field theory. Ingredients for Constructing Open Closed SFT c A very useful identity satisfied by and Ψ~ D is the so called ’main identity’ {· · · g f g c (QBAe ) D = 0 f g and, N X c c c c c o o A A (QBA )A A ; A A f 1 ··· i−1 i i+1 ··· N 1 ··· Mg i=1 M X c c o o o o o j−1 + A A ; A A (QBA )A A ( 1) f 1 ··· N 1 ··· j−1 j j+1 ··· Mg − j=1 N M 1 X X X X = g2 Ac Ac c; Ao Ao −2 s f i1 ··· ik B j1 ··· j` g k=0 fi1··· ;ik}⊂{1;··· ;Ng `=0 fj1;··· ;j`}⊂{1;··· ;Mg c c o o o +gs A A ; A A f i1 ··· ik B j1 ··· j` g 2 1 c o o c g [A A ; A A ] D ; − s f c ··· N 1 ··· M g Ingredients for Constructing Open Closed SFT where c c c o o c o c c o o o [A¯ A¯ ; A¯ A¯ ] ; [A¯ A¯ ; A¯ A¯ ] B ≡ G i1 ··· iN−k j1 ··· jM−` B ≡ G i1 ··· iN−k j1 ··· jM−` with the conditions i1; ; ik ¯i1; ;¯iN−k = 1; ; N f ··· g [ f ··· g f ··· g j1; ; j` ¯j1; ;¯jM−` = 1; ; M f ··· g [ f ··· g f ··· g The states [; ]c and [; ]o are defined via the inner products Ψc c− [Ψc Ψc ;Ψo Ψo ]c = ΨcΨc Ψc ;Ψo Ψo ; h 0j 0 j 1 ··· N 1 ··· M i f 0 1 ··· N 1 ··· Mg Ψo [Ψc Ψc ;Ψo Ψo ]o = Ψc Ψc ;ΨoΨo Ψo h 0j 1 ··· N 1 ··· M i f 1 ··· N 0 1 ··· Mg where Ψ denotes the BPZ conjugate of Ψ h j j i