Superstring theory and integration over the moduli space Kantaro Ohmori Hongo,The University of Tokyo Oct, 23rd, 2013 arxiv:1303.7299 [KO,Yuji Tachikawa] Todai/Riken joint workshop on Super Yang-Mills, solvable systems and related subjects
Section 1 introduction
Simplified history of the superstring theory
Simplified history of the superstring theory 70 s:bosonic string theory! supersymmetrize
Simplified history of the superstring theory 70 s:bosonic string theory! supersymmetrize 84 :1st superstring revolution
Simplified history of the superstring theory 70 s:bosonic string theory! supersymmetrize 84 :1st superstring revolution 86: Conformal Invariance, Supersymmetry, and String Theory [D.Friedan, E.Martinec, S.Shenker]
Simplified history of the superstring theory 70 s:bosonic string theory! supersymmetrize 84 :1st superstring revolution 86: Conformal Invariance, Supersymmetry, and String Theory [D.Friedan, E.Martinec, S.Shenker] 94 :2nd superstring revolution ) nonperturbative perspective
Simplified history of the superstring theory 70 s:bosonic string theory! supersymmetrize 84 :1st superstring revolution 86: Conformal Invariance, Supersymmetry, and String Theory [D.Friedan, E.Martinec, S.Shenker] 94 :2nd superstring revolution ) nonperturbative perspective 12: Superstring Perturbation Theory Revisited [E.Witten]: Superstring perturbation theory and supergeometry
Bosonic string theory
Bosonic string theory Action:S = R d2 zg ij @X i @X j
Bosonic string theory Action:S = R d2 zg ij @X i @X j ) Riemann Surface = 2d surface + metric
Bosonic string theory Action:S = R d2 zg ij @X i @X j ) Riemann Surface = 2d surface + metric Conformal symmetry ) bc ghost
Bosonic string theory Action:S = R d2 zg ij @X i @X j ) Riemann Surface = 2d surface + metric Conformal symmetry ) bc ghost )A= R M bos,g dmd mf (m, m)
Superstring theory
Superstring theory Action:S = R d2 zd 2 G ij D X i D X j
Superstring theory Action:S = R d2 zd 2 G ij D X i D X j ) Super Riemann Surface =2dsurface+supermetric =2dsurface+metric + gravitino θ θ θ θ θ θ θ θ θ
Superstring theory Action:S = R d2 zd 2 G ij D X i D X j ) Super Riemann Surface =2dsurface+supermetric =2dsurface+metric + gravitino θ θ θ θ θ θ θ θ θ Superconformal symmetry ) BC(bc ) superghost
Superstring theory Action:S = R d2 zd 2 G ij D X i D X j ) Super Riemann Surface =2dsurface+supermetric =2dsurface+metric + gravitino θ θ θ θ θ θ θ θ θ Superconformal symmetry ) BC(bc ) superghost )A= R M super,g dmd d md F (m, m,, )
Superstring theory Action:S = R d2 zd 2 G ij D X i D X j ) Super Riemann Surface =2dsurface+supermetric =2dsurface+metric + gravitino θ θ θ θ θ θ θ θ θ Superconformal symmetry ) BC(bc ) superghost )A= R M super,g dmd d md F (m, m,, )? )A= R M spin,g dmd mf 0 (m, m)
The bosonic and super moduli space M super 6=M spin odd direction M super M spin
The bosonic and super moduli space M super 6=M spin odd direction M super M spin Global integration on M super 6) Global integration on M spin (preserving holomorphic factorization)
The work of [KO,Tachikawa]
The work of [KO,Tachikawa] Global integration on M super 6) Global integration on M spin
The work of [KO,Tachikawa] Global integration on M super 6) Global integration on M spin We found special cases where Global integration on M super ) Global integration on M spin
The work of [KO,Tachikawa] (2)
The work of [KO,Tachikawa] (2) 2concreteexamples
The work of [KO,Tachikawa] (2) 2concreteexamples N =0 N =1embedding [N. Berkovits,C. Vafa 94]
The work of [KO,Tachikawa] (2) 2concreteexamples N =0 N =1embedding [N. Berkovits,C. Vafa 94] Topological amplitudes [I. Antoniadis, E. Gava, K.S. Narain, T.R. Taylor 94] [M. Bershadsky, S. Cecotti, H. Ooguri, C. Vafa 94]
1 introduction 2 Supergeometry and Superstring 3 Reduction to integration over bosonic moduli
Supermanifold
Supermanifold supermanifold = patched superspaces
Supermanifold supermanifold = patched superspaces M :supermanifoldofdimensionp q M 3 (x 1,, x p 1,, q ) i j = j i
Supermanifold supermanifold = patched superspaces M :supermanifoldofdimensionp q M 3 (x 1,, x p 1,, q ) i j = j i x 0 i = f i (x ) 0 µ = µ(x ) U α
Supermanifold supermanifold = patched superspaces M :supermanifoldofdimensionp q M 3 (x 1,, x p 1,, q ) i j = j i x 0 i = f i (x ) 0 µ = µ(x ) M red = {(x i,, x p )},! M U α
Supermoduli space Msuper
Supermoduli space M super Space of deformations of SRS (Super Riemann Surface)
Supermoduli space M super Space of deformations of SRS (Super Riemann Surface) even deformation: µ z z 2 H 1 ( red, T red )
Supermoduli space M super Space of deformations of SRS (Super Riemann Surface) even deformation: µ z z 2 H 1 ( red, T red ) odd deformation: z 2 H 1 ( red, T 1/2 red )
Supermoduli space M super Space of deformations of SRS (Super Riemann Surface) even deformation: µ z z 2 H 1 ( red, T red ) odd deformation: z 2 H 1 ( red, T 1/2 red ) { }: basisofgravitinob.g) =
Supermoduli space M super Space of deformations of SRS (Super Riemann Surface) even deformation: µ z z 2 H 1 ( red, T red ) odd deformation: z 2 H 1 ( red, T 1/2 red ) { }: basisofgravitinob.g) = dim M super = e o =3g 3 2g 2 (g 2)
Supermoduli space M super Space of deformations of SRS (Super Riemann Surface) even deformation: µ z z 2 H 1 ( red, T red ) odd deformation: z 2 H 1 ( red, T 1/2 red ) { }: basisofgravitinob.g) = dim M super = e o =3g 3 2g 2 (g 2) M super,red 'M spin
Supermoduli space of SRS with punctures
Supermoduli space of SRS with punctures n NS NS punctures and n R Rpunctures
Supermoduli space of SRS with punctures n NS NS punctures and n R Rpunctures dim M super = e o =3g 3+n NS +n R 2g 2+n NS +n R /2 (g 2)
Coupling between gravitino and SCFT
Coupling between gravitino and SCFT S = 1 R 2 d 2 z T F
Coupling between gravitino and SCFT S = 1 R 2 d 2 z T F S e = 1 R 2 d 2 z e TF e
Coupling between gravitino and SCFT S = 1 R 2 d 2 z T F S e = 1 R 2 d 2 z e TF e S e = R d 2 z ea A / µ e µ (Flat background)
Coupling between gravitino and SCFT S = 1 R 2 d 2 z T F S e = 1 R 2 d 2 z e TF e S e = R d 2 z ea A / µ e µ (Flat background) c.f. D µ D µ 3 A µ A µ
Scattering amplitudes of superstring
Scattering amplitudes of superstring V i :Vertexop.pic.number 1 (NS) or 1/2 (R)
Scattering amplitudes of superstring V i :Vertexop.pic.number 1 (NS) or 1/2 (R) A V,g = R M super dmd d md F V,g (m,, m, )
Scattering amplitudes of superstring V i :Vertexop.pic.number 1 (NS) or 1/2 (R) A V,g = R M super dmd d md F V,g (m,, m, ) FD V,g (m,, m, ) = Q i V Q e i a=1 (R red d 2 zbµ a ) Q e b=1 (R red d 2 zb e eµ b ) Q o=1 ( R d2 red z ) Q e o =1 ( R red d 2 z e e ) exp( S S e S e )i m
Scattering amplitudes of superstring V i :Vertexop.pic.number 1 (NS) or 1/2 (R) A V,g = R M super dmd d md F V,g (m,, m, ) FD V,g (m,, m, ) = Q i V Q e i a=1 (R red d 2 zbµ a ) Q e b=1 (R red d 2 zb e eµ b ) Q o=1 ( R d2 red z ) Q e o =1 ( R red d 2 z e e ) exp( S S e S e )i m S = P R o =1 2 red d 2 z T F
Scattering amplitudes of superstring V i :Vertexop.pic.number 1 (NS) or 1/2 (R) A V,g = R M super dmd d md F V,g (m,, m, ) FD V,g (m,, m, ) = Q i V Q e i a=1 (R red d 2 zbµ a ) Q e b=1 (R red d 2 zb e eµ b ) Q o=1 ( R d2 red z ) Q e o =1 ( R red d 2 z e e ) exp( S S e S e )i m S = P R o =1 2 red d 2 z T F We get result of FMS when = (z p ) and int. out. s.
Integration over a supermanifold
Integration over a supermanifold E = {(m 1, 2 )}/ m m + + 1 2 m +1
Integration over a supermanifold E = {(m 1, 2 )}/ m m + + 1 2 m +1 R E (a + b 1 2 )dmdmd 1 d 2 = b= + p 1a/2
Integration over a supermanifold E = {(m 1, 2 )}/ m m + + 1 2 m +1 R E (a + b 1 2 )dmdmd 1 d 2 = b= + p 1a/2 1 2 mixed with m
Projectedness of supermanifolds
Projectedness of supermanifolds M:projected, an atlas in which x 0 = f (x) for all gluing ) naive integrations with s are allowed
Projectedness of supermanifolds M:projected, an atlas in which x 0 = f (x) for all gluing ) naive integrations with s are allowed, p : M! M red exists
Projectedness of supermanifolds M:projected, an atlas in which x 0 = f (x) for all gluing ) naive integrations with s are allowed, p : M! M red exists R M! = R M red p (!)
Projectedness of supermanifolds M:projected, an atlas in which x 0 = f (x) for all gluing ) naive integrations with s are allowed, p : M! M red exists R M! = R M red p (!) All supermanifolds are projected in smooth sense.
Projectedness of supermanifolds M:projected, an atlas in which x 0 = f (x) for all gluing ) naive integrations with s are allowed, p : M! M red exists R M! = R M red p (!) All supermanifolds are projected in smooth sense. Complex supermanifolds are not projected in holomorphic sense in general.
Supermoduli Space is not projected [R.Donagi, E.Witten, arxiv:1304.7798]
Supermoduli Space is not projected [R.Donagi, E.Witten, arxiv:1304.7798] M super,red 'M spin i : M spin,! M super
Supermoduli Space is not projected [R.Donagi, E.Witten, arxiv:1304.7798] M super,red 'M spin i : M spin,! M super p : M super!m spin does not exists. (g 5, non-holomorphiconeexists)
Supermoduli Space is not projected [R.Donagi, E.Witten, arxiv:1304.7798] M super,red 'M spin i : M spin,! M super p : M super!m spin does not exists. (g 5, non-holomorphiconeexists) Global integration on M super 6) Global integration on M spin (preserving holomorphic factorization)
Section 3 Reduction to integration over bosonic moduli
Reduction condition
Reduction condition Global integration on M super ) Global integration on M spin in special cases F (m, m,, ) =G(m, m) Q all i i (saturations of s)
Reduction condition Global integration on M super ) Global integration on M spin in special cases F (m, m,, ) =G(m, m) Q all i i (saturations of s) mixing of m and (m 0 = m + 1 2 )doesnot changes F
Reduction condition Global integration on M super ) Global integration on M spin in special cases F (m, m,, ) =G(m, m) Q all i i (saturations of s) mixing of m and (m 0 = m + 1 2 )doesnot changes F F does not depend on the locations of picture changing operators
Topological amplitudes in string theory [I. Antoniadis, E. Gava, K.S. Narain, T.R. Taylor 94] [M. Bershadsky, S. Cecotti, H. Ooguri, C. Vafa 94]
Topological amplitudes in string theory [I. Antoniadis, E. Gava, K.S. Narain, T.R. Taylor 94] [M. Bershadsky, S. Cecotti, H. Ooguri, C. Vafa 94] Type II string on CY (N =(2, 2)SCFT) R 1,3 ) 4d N =2supergravity model
Topological amplitudes in string theory [I. Antoniadis, E. Gava, K.S. Narain, T.R. Taylor 94] [M. Bershadsky, S. Cecotti, H. Ooguri, C. Vafa 94] Type II string on CY (N =(2, 2)SCFT) R 1,3 ) 4d N =2supergravity model F-term of 4d e ective field theory is related to topological string.
Reduction to topological string Type II string on CY(6d) R 1,3 ) 4d N =2supergravity model
Reduction to topological string Type II string on CY(6d) R 1,3 ) 4d N =2supergravity model A g :zeromomentalimitofg loop amplitude of 2g 2 graviphotons (RR) and 2 gravitons (NSNS)
Reduction to topological string Type II string on CY(6d) R 1,3 ) 4d N =2supergravity model A g :zeromomentalimitofg loop amplitude of 2g 2 graviphotons (RR) and 2 gravitons (NSNS) A g = R M super
Reduction to topological string Type II string on CY(6d) R 1,3 ) 4d N =2supergravity model A g :zeromomentalimitofg loop amplitude of 2g 2 graviphotons (RR) and 2 gravitons (NSNS) A g = R M super A g =(g!) 2 F g = R M bos
Vertex operators
Vertex operators Graviphoton: V T = spacetime (ghost) p = exp(i 3 2 (H(z) H(z))) e H : U(1) R boson : U(1) R charge (3/2, 3/2)
Vertex operators Graviphoton: V T = spacetime (ghost) p = exp(i 3 2 (H(z) H(z))) e H : U(1) R boson : U(1) R charge (3/2, 3/2) Graviton: V R = R red spacetime,ghost
Saturation of s
Saturation of s A g = R D Q 2g 2 i =1 (x i ) U(1) R :(3g 3, 3g 3) (spacetime) (bc) Q o=1 ( R d2 red z ) Q e o =1 ( R red d 2 z e e ) R exp( red (T F + T e E F e + A e )) m
Saturation of s A g = R D Q 2g 2 i =1 (x i ) U(1) R :(3g 3, 3g 3) (spacetime) (bc) Q o=1 ( R d2 red z ) Q e o =1 ( R red d 2 z e e ) R exp( red (T F + T e E F e + A e )) m nonzero contribution ) 3g 3 and 3g 3e
Saturation of s A g = R D Q 2g 2 i =1 (x i ) U(1) R :(3g 3, 3g 3) (spacetime) (bc) Q o=1 ( R d2 red z ) Q e o =1 ( R red d 2 z e e ) R exp( red (T F + T e E F e + A e )) m nonzero contribution ) 3g 3 and 3g 3e There are only 3g 3 and 3g 3 e
Saturation of s A g = R D Q 2g 2 i =1 (x i ) U(1) R :(3g 3, 3g 3) (spacetime) (bc) Q o=1 ( R d2 red z ) Q e o =1 ( R red d 2 z e e ) R exp( red (T F + T e E F e + A e )) m nonzero contribution ) 3g 3 and 3g 3e There are only 3g 3 and 3g 3 e The correlation function does not depend of the choice of
Conclusion
Conclusion Superstring amplitude cannot be represent as integration over M spin in general
Conclusion Superstring amplitude cannot be represent as integration over M spin in general Special cases exist.
Conclusion Superstring amplitude cannot be represent as integration over M spin in general Special cases exist. Amplitudes which is equivalent topological amplitudes are the case.
End.