Properties of Supergroup Sigma-Models and String Theory Jan Troost Work Done in Collaboration with Sujay Ashok and Raphael Beni

IHP 2011

Properties of supergroup sigma-models and string theory

work done in collaboration with Sujay Ashok and Raphael Benichou

Jan Troost

ABT : 0903.4277 ABT : 0907.1242 BT : 1002.3712 T : 1102.0153 based on work by

Babichenko, Berkovits, Bernard, Bershadsky, Candu, Creutzig, Drouot, Gaberdiel, Gerigk, Germoni, Goetz, Konechny, LeClair, Quella, Read, Roehne, Saleur, Schomerus, Vafa, Vaintrob, Witten, Zhukov, ..

and all other members of the audience Plan

0. Motivation

1. Properties of supergroup sigma-models

Current algebra, Integrability, Spectrum

2. Properties when embedded in string theory

Asymptotic Symmetry Algebra, Physical States 0. Motivation

Physics

Quantum Gravity Strongly Coupled Field Theories

Holography

String Theory in Anti-deSitter Strings

correspond to Maps from Worldsheet into Space-time 1. Properties of Supergroup Sigma-Models

1a. The Model

1b. The Current Algebra

1c. The Energy-momentum tensor and the Spectrum Saclay seminar 26-03-2010

Based on work by Sujay Ashok, Raphael Benichou and Jan Troost

March 17, 2010

Abstract Saclay seminar 26-03-2010 We discuss the conformal current algebra and its applications Based on work by Sujay Ashok, Raphael Benichou and Jan Troost

March 17, 2010 1 Formulas Ia : The Model Abstract We discuss the conformal currentg : Σ algebra andG its applications (1.1) → 1 Formulas Supergroup

g : Σ G (1.1) 1The action : 1 1 → 1 1 2 S = 2 dgg− , dgg− + k dgg− , (dgg− ) f Σ" ∗ $ B" $ ! 1 1 1 ! 1 1 2 S = dgg− , dgg− + k dgg− , (dgg− ) PCM+WZ f 2 " ∗ $ " $ !Σ !B a b ab c1 ab c2 c z¯ c a b ab c1 ab c2 c z¯ c jL,z(z)jL,z(0) κjL,z(z)jL,z+(0)fPrincipalκc +chiraljfL,zc model(0)jL,z +(0) ( +c (2c2 gwithg) ) j L,Wess-Zuminoz¯j(0)L,z¯(0) term ∼ z2 ∼ z2z z −− z2z2 g z¯ " c c #g z¯2 + f ab "(∂ jc(0) ∂ jc(0)) + 2 ∂ jc (0) + 2 − ∂ j#c (0)2 ab g z¯ c c −4 z z z¯ c − z¯ z c2 2 z L,zc 2 cz22 z¯ L,gz¯ z¯ c + f (∂zj (0)$ ∂z¯j (0)) + ∂zj (0) + − % ∂z¯j (0) c z¯ z1 z¯2 L,zz¯ 2 L,z¯ −4 z +:jajb : (0)− + Aab : jcjd :2 (0) + Bab : jcjd : (0) 2Cab zlog z 2 : jcje : (0) $ z z cd 2 z2 z¯ z¯ cd z z z¯ − cd | | z z % 2 a b + ab... 1 z¯ c d ab z¯ c d ab 2 c e +:j j : (0) + A cd : j j : (0) + B cd : j j : (0) C cd log z : j j : (0) z z a b ab2 1 z¯ abz¯ c4 c (c4 g)z c z z¯ z z j (z)j (0) 2κzc3 + f j (0) + − zj (0) − | | L,z¯ L,z¯ ∼ z¯2 c z¯ L,z¯ z¯2 L,z $ % + ... g z c c g z2 + f ab (∂ jc(0) ∂ jc(0)) + 4 ∂ jc (0) + 4 − ∂ jc (0) c 4 z¯ z z¯ − z¯ z 2 z¯ L,z¯ 2 z¯2 z L,z $ % 1 c4 (c4 g)z 2 a b ab a b ab ab c 2 c d ab z c dc ab 1 z c d jL,z¯(z)jL,z¯(0) κ c3+:jz¯+jz¯ :f (0) cA cd logjL,z z¯(0): jz¯jz¯ +: (0) + B−cd : jzjjz¯L,z: (0)(0) + C cd : jzjz : (0) ∼ z¯2 − z¯ | | z¯2 z¯ 2 z¯2 + ... $ % 2 g z c4 (c4 g) c4(c2 gg)z ab ja (z)jbc (0) c˜κab2πδc(2)(z w)+f ab c − jc (0) + − jc (0) c + f c L,z(∂zjzL,¯(0)z¯ ∼ ∂z¯jz(0))− + ∂cz¯jL,z¯z¯(0)L,z + −z L,z¯∂zjL,z(0) 4 z¯ − 2 $ 2 z¯2 % g (c g)z $ + f ab log z 2(∂ jc(0) ∂ jc(0)) + 4 − jc (0) % c −4 | | z z¯ − z¯ z z¯ L,z 2 a b ab $ 2 c d ab z %c d ab 1 z c d +:j j : (0) A log z :z¯ j j : (0) + B : j j : (0)z + C : j j : (0) z¯ z¯ +:jacdjb : (0) + Aab : jz¯cjdz¯: (0) Bab log z cd2 : jcjd : (0)z +z¯Cab : jcjd : (0) cd 2 z z − z z¯ | | cd z z¯ z¯ − cd | | z¯z z¯ cd z¯ z z 2 z¯ + ... + ... a b ab (2) ab (c4 g) c (c2 g) c jL,z(z)jL,z¯(0) c˜κ 2πδ (z w)+f c − jL,z(0) + − jL,z¯(0) ∼ ab − 2 2 1 b agz¯ cd b ag 4z A cd = f (1 + kf ) ($f cgf d( 1) + f dgf c)+ (f ) % g − 2 (−c g)z O + f ab log z 2(∂ jc(0) ∂ jc(0)) + 4 − jc (0) c −4 | | z z¯ − z¯ z 1 z¯ L,z $ % z¯ z +:jajb : (0) + Aab : jcjd : (0) Bab log z 2 : jcjd : (0) + Cab : jcjd : (0) z z¯ cd z z¯ z¯ − cd | | z z¯ cd z¯ z z + ...

1 Aab = f 2(1 + kf2) (f b f ag ( 1)cd + f b f ag )+ (f 4) cd − 2 cg d − dg c O

1 a jµ = jµta Saclay seminar 26-03-2010

t Lie G Based on work by Sujay Ashok,a ∈ Raphael Benichou and Jan Troost March 17, 2010 µ ∂ jµ =0 Abstract (0.1) We discuss the conformal current algebra and its applications

1 Formulas di(zz¯) (0.2)

g : Σ G (1.1) → lim jµ(z)jν(w) = lim jν(w)jµ(z) (0.3) z w z w → →

1 1 1 1 1 2 SPCM+WZ = 2 dgg− , dgg− + k dgg− , (dgg− ) Assume thef superΣ" Lie algebra∗ has$a zero KillingB" form $ ! jµ = jµta !

a cb f bcf d = 0 (1.2) a jµ = jµta c c z¯ ja (z)jb (0) κaband1 +a non-degeneratef ab 2 jc (0) invariant + (c gmetric) jc (0) L,z L,z ∼ z2 c z L,z 2 − z2 L,z¯ " # 2 g z¯ cκ2 c2 g z¯ + f ab (∂ jc(0) ∂ jc(0)) + ∂ab jc (0) + − ∂ jc (0) c −4 z z z¯ − z¯ z 2 z L,z 2 z2 z¯ L,z¯ $ then we have % 1 z¯2 z¯ +:jajb : (0) + Aab : jcjd : (0) + Bab : jcjd : (0) Cab log z 2 : jcje : (0) z z cd 2 z2 z¯ z¯ cd z z z¯ − cd | | z z a b ab 2 ab 2 c c d3 c +jz...(z)jz(0) κ (¯z) d1 + f c(¯z) d2zjz(0) + d3zj¯ (0) + zz¯∂zjz¯(0) ∼ − 2 ! a b ab 1 ab c4 c (c4 g)z c j (z)j (0) κ c +df2 j (0) +d2 − j (0)d3 L,z¯ L,z¯ ∼ 3 z¯2 + zcz¯∂z¯jcL,(0)z¯ + (z)z2¯2∂ jc(0)L,z + (¯z)2∂ jc(0) + ... $ z¯ z z z % z¯ z¯ 2 2 22 ab g z c c c4 c c4 g z c " + f (∂zj (0) ∂z¯j (0)) + ∂z¯j (0) + − ∂zj (0)d a c b4 z¯ z¯ ab− 2z ab2 2L,z¯ c 2 z¯2 c L,z 5 c jz¯(z)j$z¯(0) κ (z) d4 + f c(z) d5zj¯ z¯(0) + d6zjz(0) + % zz¯∂zjz¯(0) ∼ 2 2 a b ab 2 c d ! ab z c d ab 1 z c d +:jz¯ jz¯ : (0) A cddlog z : jz¯jz¯ : (0)d + B cd : jzjz¯d: (0) + C cd : jzjz : (0) − + 5 (¯z| )|2∂ jc(0) + 6 zz¯∂ jcz¯(0) + 6 (z)2∂ jc(0)2 z¯+2 ... 2 z¯ z¯ 2 z¯ z 2 z z + ... " a b ab ab c c c j (z)j (0) κ d7 + f [d8zj(c(0) +g)d9zj¯ (0) +(cd12zgz¯)∂z¯j (0) ja (z)jb z (0) z¯ c˜κab∼2πδ(2)(z w)+cf ab z 4 − jc (0)z¯ + 2 − jc z(0) L,z L,z¯ ∼ − c z¯ 2 L,zc z 2 L,z¯c +$d10(z) ∂zjz(0) + d11(¯z) ∂z¯jz¯(0)% + ... (0.4) ab g 2 c c (c4 g)z c + f log z (∂zj (0) ∂z¯j (0)) + − j (0) # c −4 | | z¯ − z z¯ L,z $ % z¯ z +:jajb : (0) + Aab : jcjd : (0) Bab log z 2 : jcjd : (0) + Cab : jcjd : (0) z z¯ cd z z¯ z¯ − cd | | z z¯ cd z¯ z z + ...

1 1 a two-dimensional group invariant moduli space of conformal ﬁeld theories

2 1 PCM f -2

WZW

a goal : compute the spectrum (=open problem) Ib. Two-dimensional current algebras

a jµ = jµta Lie algebra valued a jµ = j ta Conserved current µ t Lie G a ∈ in Lorentz invariant t Lie G local theory a ∈ di(zz) (0.1) Hypothesis : the operators appearing as coefﬁcients of singularities are the identity, the current,di (andzz) its (0.1) lim jµ(z)jν(w) = lim jν(w)jµ(z) (0.2) derivatives. z w z w → →

lim jµ(z)jν(w) = lim jν(w)jµ(z) (0.2) z w z a w → jµ = j→µta

a jµ = j taa jµ = µjµta

a b ab 2 j ab = 2 jat c c d3 c jz (z)jz(0) κ (z) d1 + fµ c(z) dµ2zja z(0) + d3zj (0) + zz∂zjz(0) ∼ − 2 ! d2 c d2 2 c d3 2 c + zz∂zjz(0) + (z) ∂zjz(0) + (z) ∂zjz(0) + ... a b ab 22 ab 22 c 2 c d3 c jz (z)jz(0) κ (z) d1 + f c(z) d2zjz(0) + d3zj (0) + "zz∂zjz(0) a b ∼ ab 2 ab 2 c −c d52 c j (z)j (0) κ (z) d4 + f (z)! d5zj (0) + d6zj (0) + zz∂zj (0) z z ∼ c z z 2 z d2 c d2 !2 c d3 2 c + zdz∂zjz(0) + (zd) ∂zjz(0) + d (z) ∂zjz(0) + ... 2+ 5 (z)2∂ jc(0)2 + 6 zz∂ jc(0) + 26 (z)2∂ jc(0) + ... 2 z z 2 z z 2 z z " d " a ba b ab ab2 abab 2c c c c c 5 c jz (z)jjz(0)(z)j (0) κ (κz) dd4++ff [cd(zzj) (0)d5 +zjdz(0)zj (0) + d +6zjd z(0)zz∂ +j (0)zz∂zjz(0) z z ∼ ∼ 7 c 8 z 9 z 12 z z 2 ! 2 c 2 c +d10(z) ∂zjz(0) + d11(z) ∂zjz(0) + ... (0.3) d5 2 c d6 c d6 2 c + (z) ∂zjz(0) + zz∂zjz(0) + (z) ∂zjz(0) + ... 2 2 2 2 2 # x 1 x 1 d8 " a b d1 + d1# +ab d7# =0ab,d2c+ d2# + dc 8# + 2 =0, c jz (z)jz(0) 2 κ d27 + f c [d8zjz(0)2 + d9zj2z(0) +2xd12zz∂zjz(0) ∼x2 1 3 x22 c 1 2 c d + d# + d# =0, +dd10+(z) d∂#zj+z(0)d# +=0d11, (z) ∂zjz(0) + ... (0.3) 4 2 4 2 7 2 6 2 6 2 8 2 2 32 x 1 2 x # xd + 1d# + d# =0, 2d x+ d# +(1 d# dd# ) = 0 2 3 2 3 2 9 3 2 3 9 − 118 d1 + d1# + d7# =0,d2 + d2# + d8# + 2 =0, 2 x22 2 x2 2 22x 2d2 + d# + d# =0,d+ 2 d# + d# + d =0, x 6 2 16 10 3 5 x2 5 111 x2 11 d4 + d4# + d7# =0, d6 + d6# + d8# =0, 2 x2 2 1 1 2 2 2 d + d# + d# + d =0, 3 5x2 2 5 1 2 9 2x2 9 x2 d + d# + d# =0, 2d + d# +(d# d# ) = 0 2 3 2 x32 2 9 32 2 3 9 − 11 d2 + d# +(d# d# )+ (d8 d10)=0 x2 2 2 8 − 10 x2 x2− 2 2 2d6 +3 d6# + d10# x=0,d5 + d5#1+ d11# + 2 d11 =0, (2d6 d5)+ (d6# d5# )+d12# 2+ d12 =0, x 22 − 2 − x2 x 1 2 1 d5 + 3 d5# + d9# +x 2 d9 =0, 1 2(d2 d32)+ 2(dx# d# )+(d# d# d# )+ (d8 d12 d9)=0. (0.4) 2 − 2 2 − 3 8 − 12 − 9 x2 − − x2 2 d2 + d2# +(d8# d10# )+ 2 (d8 d10)=0 2 − x − 1 3 x2 1 (d d )+ (d# d# )+d# + d =0, 2 6 − 5 2 6 − 5 12 x2 12 3 x2 1 (d d )+ (d# d# )+(d# d# d# )+ (d d d )=0. (0.4) 2 2 − 3 2 2 − 3 8 − 12 − 9 x2 8 − 12 − 9

1 The current algebra contains functions of the Lorentz invariant distance between the operator insertions.

Current conservation leads to ﬁrst order differential equations that can be solved under the assumption of leading powerlike singularities.

Five free parameters at ﬁrst order in the currents (and zeroth order in the distance).

Eight free parameters at second order in the currents. 2 2 x 1 x 1 d8 d + d! + d! =0,d+ d! + d! + =0, 1 2 1 2 7 2 2 2 2 8 2x2 x2 1 3 x2 1 d + d! + d! =0, d + d! + d! =0, 4 2 4 2 7 2 6 2 6 2 8 3 x2 1 x2 d + d! + d! =0, 2d + d! +(d! d! ) = 0 2 3 2 3 2 9 3 2 3 9 − 11 x2 x2 2 2d + d! + d! =0,d+ d! + d! + d =0, 6 2 6 10 5 2 5 11 x2 11 x2 1 1 d + d! + d! + d =0, 5 2 5 2 9 2x2 9 x2 2 d + d! +(d! d! )+ (d d )=0 2 2 2 8 − 10 x2 8 − 10 3 x2 1 (d d )+ (d! d! )+d! + d =0, 2 6 − 5 2 6 − 5 12 x2 12 3 x2 1 (d d )+ (d! d! )+(d! d! d! )+ (d d d )=0. (0.5) 2 2 − 3 2 2 − 3 8 − 12 − 9 x2 8 − 12 − 9

c c z¯ ja(z)jb(0) κab 1 + f ab 2 jc(0) + (c g) jc(0) z z ∼ z2 c z z 2 − z2 z¯ g z¯ ! c c g z¯2 (∂ jc(0) ∂ jc(0)) + 2 ∂ jc(0) + 2 − ∂ jc(0) + ... −4 z z z¯ − z¯ z 2 z z 2 z2 z¯ z¯ " 1 c (c g)z ja(x)jb(0) κabc + f ab 4 jc(0) + 4 − jc(0) z¯ z¯ ∼ 3 z¯2 c z¯ z¯ z¯2 z # g z c (c g) z2 + (∂ jc(0) ∂ jc(0)) + 4 ∂ jc(0) + 4 − ∂ jc(0) + ... 4 z¯ z z¯ − z¯ z 2 z¯ z¯ 2 z¯2 z z " (c g) (c g) (c g)z ja(x)jb(0) f ab 4 − jc(0) + 2 − jc(0) + 4 − ∂ jc(0) z z¯ ∼ c z¯ z z z¯ z¯ z z # g (c + log µ2x2)(∂ jc(0) ∂ jc(0)) + ... (0.6) − 5 4 z z¯ − z¯ z $ ab a b κ c1 ab c2 c (c2 g)x− c g x− c c j+(x)j+(0) + f c j+(0) + − j (0) ∂+j (0) ∂ j+(0) Non-chiral∼ (x+)2 currentx+ algebra(x+)2 with− − 4 x+ − − − # 2 % & c2 c c2 g (x−) c + ∂+j+(0) + − ∂ j (0) + ... 2 2 (x+)2 − Logarithmic term − " ab + + a b κ c3 ab c4 c (c4 g) x c g x c c j (x)j (0) + f c j (0) + − j+(0) + ∂+j (0) ∂ j+(0) − − ∼ (x )2 x − (x )2 4 x − − − − # − − − + 2 % & as in Luscher+c 4allowc parityc4 g ( xviolation) c + ∂ j (0) + − ∂+j+(0) + ... 2 − 2 (x )2 − − " + a b ab c4 g c (c2 g) c (c4 g)x c j+(x)j (0) f c − j+(0) + − j (0) + − ∂+j+(0) − ∼ x x+ − x # − − g 2 2 c c c5 + log µ x ∂+j (0) ∂ j+(0) + ... − 4 − − − ' ( % &$ (0.7) It would be interesting to search for more general solutions to the set of diﬀerential equations.

2 The Toolbox to Fix Coefﬁcients

a) Semi-classical perturbation theory b) Exactness of low order calculation (superalgebra)

c) Recursion via Maurer-Cartan (integrability) a) Semi-classical (= large volume) perturbation theory

Develop the action near the constant map into the identity element, in an expansion in Lie algebra valued ﬁelds.

The action and the currents are an expansion in the structure constants of the Lie algebra.

Semi-classical perturbation theory is an expansion in the number of structure constants. a jµ = jµta

ta Lie G a ∈ jµ = jµta

µ t Lie G ∂ jµ a=0∈ (0.1)

µ ∂ jµ =0 (0.1) di(zz¯) (0.2)

di(zz¯) (0.2) lim jµ(z)jν(w) = lim jν(w)jµ(z) (0.3) z w z w → →

lim jµ(z)jν(w) = lim jν(w)jµ(z) (0.3) z w z w → → a jµ = jµta

a jµ = jµta

a jµ = jµta a jµ = jµta

κab κab

ab κabf c =0ab κabf c =0

κab κab

a iAat So : g = geiA= tae a

1 2 a ¯ 1 a b c i ¯ j 4 S1 = d z(∂A ∂A1 a f faijA ∂A A ∂A + O(f ) S = 4dπ2zf(2∂Aa∂¯A f−a 12f bcAb∂AcAi∂¯Aj + O(f 4) 4πf 2 ! a − 12 bc aij ! k d2zfabcA ∂A ∂¯A + O(f 3) k −12π a b c d2zf!abcA ∂A ∂¯A + O(f 3) −12π a b c ! 1 The action and the 1currents are given in terms of an expansion in the structure constants. b) Superalgebra

The two- and three-point functions of the currents are exact at zeroth and ﬁrst order in the structure constants respectively.

Bershadsky, Zhukov, Vaintrob Using complex coordinates, and after taking the trace, the kinetic term becomes:

1 2 1 1 c S = d z(∂gg− ) (∂¯gg− ) . (0.14) kin −4πf 2 c ! The ﬁeld g takes values in a supergroup.

Two-point functions ja(z)jb(w) (0.15) " µ ν # From the action we can calculate the classical currents associated to the invariance of the theory underPull left out multiplication a structure constant of from the an field arbitraryg by diagram: a group element in G and right multiplica- a L f cd (0.4) tion by a group element in GR. Thec classical∝ equations of motion for the model read: a b 2 ¯ a 2 ¯ 1 a a kf +1 ∂J + kf 1 ∂(gJg− ) =0, (0.16) f cd d fcdb −(0.4) (0.5) ∝ ∝ QED where we have used the" standard# expressions" ∅ for# the left- and right-current at the Wess- 2 Zumino-Witten point :fcdb (0.5) a b ∝ ab 2 ab 2 c c d3 c jz (z)jz(0) κ (¯z) d1 + f c(¯z) d2zjz(0) + d3zj¯ (0) + zz¯∂zjz¯(0) ∼ − 2 The two-pointsJ( functionsz,z¯)= takek ∂theirgg free1! values.and J¯(z,z¯)=kg 1∂¯g. (0.17) a b ab 2 ab 2 cd2 cc dd3−2 2c c d3 2 c − jz (z)jz(0) κ (¯z) d1 + f c(¯z) d2zj+z(0)z +z¯∂d3z¯−zj¯jz(0)(0) + + zz(¯z∂z)jz¯∂(0)zjz(0) + (¯z) ∂z¯jz¯(0) + ... ∼ 2 − 22 2 ! " d2 c d2 2 c d3 2 c The classical+ zz¯G∂z¯Lj (0)currents + (z) ∂zj are(0) + given(¯z) ∂z¯j by:(0) + ... d5 2 ja(z)zjb(0) 2 κabz (z)2d2 + f ab z¯(z)2 d zj¯ c(0) + d zjc(0) + zz¯∂ jc(0) z¯ z¯ ∼ 4 c " 5 z¯ 6 z 2 z z¯ a b ab 2 ab 2 c c d5 ! c jz¯(z)jz¯(0) κ (z) d4 + f c(z) d5zj¯ z¯(0) + d6zjz(0) + zz¯∂zjz¯(0) 2 ∼ d5 1 2 1c 2 d6 c 1 d6(1 +2 kfc ) j!z =+ (¯z) ∂z¯jz¯(0)+ +k zz¯∂∂ggz¯jz−(0) += (z) ∂zjz(0) J+ ... d5 d6 2 d6 2 2 2 2 + (¯z)2∂ jc(0) + zz¯∂ jc(0)−2 + (zf)2∂ jc(0) + ... 2kf 2 z¯ z¯ 2 z¯ z 2 z z " a b ab $ab c "% c c a b ab jz (zab)jz¯(0) c κ dc7 + f c [d8zjc z(0) + d9zj¯ z¯(0) + d12zz¯∂z¯jz(0)2 jz (z)jz¯(0) κ d7 + f c [d8zjz(0)∼ + d9zj¯ z¯(0)1 + d121zz¯∂z¯jz(0) (1 kf ) ∼ ¯2 c 1 2 c 1 jz¯ 2= c 2 +cdk10(z)∂∂ggzjz−(0)= + d11(¯z) −∂z¯jz¯(0) +(g...Jg¯ − ).(0.6) (0.18) +d10(z) ∂zjz−(0)2 + d11(¯fz)2∂z¯−jz¯(0) + ... (0.6)− 2kf2 $ #% # 2 2 2 2 x 1 x x 1 1 d8 2x 1 d8 d1 + d1" + d7" =0,d2 + d2" + d8" + =0, At the2 Wess-Zumino-Witten2 d1 + d1"2+ d27" =02,d pointx2 2 +f =1d2" + /kd8" +the2 z ¯=0-component, of the left-moving current 2 2 2 2 2 2 2x x 1 3 2 x 1 2 becomesd4 + d" + zero.d" =0, Asxd6 a+ consequence,d1" + d" =0, 3 thex z-component1 J becomes holomorphic. A similar 2 4 2 7 2 2 6 2 8 d4 + d4" + d7" =0, d6 + d6" + d8" =0, 2 phenomenon3 x2 1 happens2 x2 at2 the other2 Wess-Zumino-Witten2 2 point (f = 1/k) for the anti- d + d" + d" =0, 2d + d" +(d" d" ) = 0 2 3 2 3 2 9 3 3x2 2 3 1 ¯9 − 111 x2 − holomorphicx2 componentd3 + xd23" +gJgd9" =0−2 ., From2d3 + nowd3" +( on,d9" wed11" will) = 0 concentrate on the current j associated 2d6 + d6" + d10" =02,d5 +2 d5" +2d11" + d11 =0, 2 − to the2 left action of2 the2 group.x2 For future2 reference we note that the coefficients that relate 2 x x 2 x 1 12d6 + d" + d" =0,d5 + d" + d" + d11 =0, d + d" + d" + d =0, 6 10 5 11 2 the5 left2 5 current2 9 2x2 components9 2 to the derivative2 ofx the group element are (see section ??): x2 2x2 1 1 d2 + d2" +(d8" d10" d)++ (d8d" +d10)=0d" + d =0, 2 − 5 x22 −5 2 9 2(1x2 +9 kf2) (1 kf2) 3 x2 2 1 (d d )+ (d d )+xd + c d ==0, 2 and c = − . (0.19) 6 5 6" 5" 12" 2+12 2 2 2 − 2 −d2 + d2" +(x d8" −d10" )+2f2 (d8 d10)=0 − − 2f 3 x2 2 − 1 x − (d d )+ (d" d" )+(d" d" 2d" )+ (d d d )=0. (0.7) 2 2 − 3 2 2 −3 3 8 − 12 −x 9 x2 8 − 12 − 9 1 (d d )+ (d" d" )+d" + d =0, 0.2 Exact2 perturbation6 − 5 2 6 − theory5 12 x2 12 3 x2 1 (d d )+ (d" d" )+(d" d" d" )+ (d d d )=0. (0.7) Elegant arguments2 2 − were3 2 given2 − [6]3 for8 − the12 exactness− 9 x2 8 of− low-order12 − 9 perturbation theory for the calculation of various quantities in the supergroup model on PSL(n n). In particular, we will | use these arguments to compute the (left) current-current two-point function exactly to all orders in perturbation theory using the free theory. Similarly we also compute the current three-point functions to all orders using perturbation theory up to first order in the structure constants. Below, we2 summarize some important facts that lead to these results. For the detailed proofs of these facts, we refer to [6]. The argument is essentially based on the special feature of the Lagrangian that all in- teraction vertices are proportional to (powers of) the structure constants as well as certain properties of the Lie superalgebra, which2 we now list:

2At the Wess-Zumino-Witten point, the parameters satisfy the equation: 1/f 2 = k.

4 Using complex coordinates, and after taking the trace, the kinetic term becomes:

1 2 1 1 c S = d z(∂gg− ) (∂¯gg− ) . (0.14) kin −4πf 2 c ! The ﬁeld g takes values in a supergroup.

ja(z)jb(w) (0.15) " µ ν #

Three-point functions ja(z)jb(w)jc(y) (0.16) " µ ν ρ # From the action we can calculate the classical currents associated to the invariance of the a d theory under left multiplication of the fieldc g by a group element in GL and right multiplication by a group element inb GR. Thee classical equations of motion for the model read: ∅ 2 a 2 1 a kf +1 ∂¯J + kf 1 ∂(gJg¯ − ) =0, (0.17) invariant four-tensor by superalgebra− where we have used the" standard# expressions" for# the left- and right-current at the Wess- Zumino-Witten pointThere2: are only contributions at first order in the structure constants. 1 1 J(z,z¯)= k∂gg− and J¯(z,z¯)=kg− ∂¯g. (0.18) − The classical GL currents are given by: 2 1 1 1 (1 + kf ) j = + k ∂gg− = J z −2 f 2 2kf2 $ % 2 1 1 1 (1 kf ) 1 j = k ∂¯gg− = − (gJg¯ − ). (0.19) z¯ −2 f 2 − − 2kf2 $ % At the Wess-Zumino-Witten point f 2 =1/k thez ¯-component of the left-moving current becomes zero. As a consequence, the z-component J becomes holomorphic. A similar phenomenon happens at the other Wess-Zumino-Witten point (f 2 = 1/k) for the anti- 1 − holomorphic component gJg¯ − . From now on, we will concentrate on the current j associated to the left action of the group. For future reference we note that the coefficients that relate the left current components to the derivative of the group element are (see section ??): (1 + kf2) (1 kf2) c+ = and c = − . (0.20) − 2f 2 − − 2f 2

0.2 Exact perturbation theory Elegant arguments were given [6] for the exactness of low-order perturbation theory for the calculation of various quantities in the supergroup model on PSL(n n). In particular, we will | use these arguments to compute the (left) current-current two-point function exactly to all orders in perturbation theory using the free theory. Similarly we also compute the current three-point functions to all orders using perturbation theory up to ﬁrst order in the structure constants. Below, we summarize some important facts that lead to these results. For the detailed proofs of these facts, we refer to [6]. The argument is essentially based on the special feature of the Lagrangian that all in- teraction vertices are proportional to (powers of) the structure constants as well as certain properties of the Lie superalgebra, which we now list: 2At the Wess-Zumino-Witten point, the parameters satisfy the equation: 1/f 2 = k.

4 5 coefﬁcients ﬁxed exactly by low order perturbation theory c) Recursion via Maurer-Cartan equation

In the classical theory, the Maurer-Cartan equation reads:

1 1 1 d(dgg− )=dgg− dgg− ∧

1 1 jz = c+∂+gg− ,jz¯ = c ∂ gg− The Maurer-Cartan equation contains− − a composite Composite operatoroperator ! Normal which ordering. potentially needs UV regularization.

(c+ + c )(c2 g)+ic1 =0. (0.8) Assume MC validity− in −the quantum theory (a strong assumption, implying integrability). We similarly ﬁnd a constraint linking c3, c4 and g to c+ and c : −

(c+ + c )(c4 g)+ic3 =0. (0.9) − − (NOT ﬁve to three but ﬁve to three minus one, minus one, is two.)

1 b b Tzz(z)= : jzjz :(z), (0.10) 2c1 as we will demonstrate. ja(z) ∂ja(z) T (w)ja(z)= z + z , (0.11) z (w z)2 w z − − which shows that the current jz is a primary ﬁeld of conformal dimension one. dim G 2T (w) ∂T (w) T (z)T (w)= + + , (0.12) 2(z w)4 (z w)2 z w − − − We now switch gears and consider a concrete model in which the generic analysis of two- dimensional current algebras of section ?? can be applied. We consider a conformal supergroup sigma-model from the list given in [7]. Though we believe our analysis applies to the whole list, some facts that we use below have been proven explicitly only for the PSL(n n) | models. We will calculate two-, three- and four-point functions of currents. Later we will infer the operator algebra of the currents from those correlation functions.

0.1 The model We consider a supergroup non-linear sigma-model with standard kinetic term based on a bi-invariant metric on the supergroup. It is the principal chiral model on the supergroup. In addition we allow for a Wess-Zumino term. Therefore, we have two coupling constants, namely the coeﬃcient of the kinetic term 1/f 2 and the coupling constant k preceding the Wess-Zumino term. The action is 1:

S = Skin + SWZ

1 2 µ 1 S = d xT r"[ ∂ g− ∂ g] kin 16πf 2 − µ ! ik 3 αβγ 1 1 1 S = d y# Tr"(g− ∂ gg− ∂ gg− ∂ g) (0.13) WZ −24π α β γ !B 1Our normalizations and conventions are mostly as in [23]. In particular we deﬁne the primed trace as Tr!(tatb)=2κab where κab is normalized to be the Kronecker delta-function for a compact subgroup. The action is written in terms of real euclidean coordinates. We soon switch to complex coordinates via z = x1+ix2. a b ab c Also, we take the convention that the generators of the Lie group ta are hermitian and satisfy [t ,t ]=if ct . See [23] for further details. Again we will not be careful in always keeping track of the signs due to the bosonic or fermionic nature of the super Lie algebra generators.

3 Saclay seminar 26-03-2010

Based on work by Sujay Ashok, Raphael Benichou and Jan Troost

March 17, 2010

Abstract We discuss the conformal current algebra and its applications

1 Formulas

g : Σ G (1.1) →

1 1 1 1 1 2 S = dgg− , dgg− + k dgg− , (dgg− ) PCM+WZ f 2 " ∗ $ " $ By exploiting!Σ the MC assumption: !B

∂ja ∂¯ja + if 2f a : jbjc :=0 z¯ − z bc z z¯

We can recursivelyf a computef cb =higher 0 order terms (1.2) in the current-currentbc operatord product expansion. c c z¯ ja(z)jb(0) κab 1 + f ab 2 jc(0) + (c g) jc(0) z z ∼ z2 c z z 2 − z2 z¯ g z¯ " c c# g z¯2 + f ab (∂ jc(0) ∂ jc(0)) + 2 ∂ jc(0) + 2 − ∂ jc(0) c −4 z z z¯ − z¯ z 2 z z 2 z2 z¯ z¯ $ % a b +:jz jz : (0) 1 z¯2 z¯ + Aab : jcjd : (0) + Bab : jcjd : (0) Cab log z 2 : jcje : (0) cd 2 z2 z¯ z¯ cd z z z¯ − cd | | z z + ... 1 c (c g)z ja (z)jb (0) κabc + f ab 4 jc (0) + 4 − jc (0) L,z¯ L,z¯ ∼ 3 z¯2 c z¯ L,z¯ z¯2 L,z $ % g z c c g z2 + f ab (∂ jc(0) ∂ jc(0)) + 4 ∂ jc (0) + 4 − ∂ jc (0) c 4 z¯ z z¯ − z¯ z 2 z¯ L,z¯ 2 z¯2 z L,z $ % z 1 z2 +:jajb : (0) Aab log z 2 : jcjd : (0) + Bab : jcjd : (0) + Cab : jcjd : (0) z¯ z¯ − cd | | z¯ z¯ cd z¯ z z¯ cd 2 z¯2 z z + ... (c g) (c g) ja (z)jb (0) c˜κab2πδ(2)(z w)+f ab 4 − jc (0) + 2 − jc (0) L,z L,z¯ ∼ − c z¯ L,z z L,z¯ $ % g (c g)z + f ab log z 2(∂ jc(0) ∂ jc(0)) + 4 − jc (0) c −4 | | z z¯ − z¯ z z¯ L,z $ %

1 Saclay seminar 26-03-2010

Based on work by Sujay Ashok, Raphael Benichou and Jan Troost

March 17, 2010

Abstract We discuss the conformal current algebra and its applications

1 Formulas

g : Σ G (1.1) →

1 1 1 1 1 2 S = dgg− , dgg− + k dgg− , (dgg− ) PCM+WZ f 2 " ∗ $ " $ !Σ !B

∂ja ∂¯ja + if 2f a jbjc =0 Result for the current-currentz¯ − z bc OPEz z¯ up to

zeroth order in the distance,a cb and second order f bcf d = 0 (1.2) in the current operators:

c c z¯ ja(z)jb(0) κab 1 + f ab 2 jc(0) + (c g) jc(0) z z ∼ z2 c z z 2 − z2 z¯ g z¯ " c c# g z¯2 + f ab z¯(∂ jc(0) ∂ jc(0)) + 2 ∂ jc(0) + 2 − ∂ jc(0)z +:jajb : (0) +cA−ab4 z :zjcz¯jd :− (0)z¯ z Bab log2 zzz2 : jcjd :2 (0)z +2 Cz¯ abz¯ : jcjd : (0) z z¯ $ cd z z¯ z¯ cd z z¯ cd %z¯ z z a b − | | + ... +:jz jz : (0) 1 z¯2 z¯ + Aab : jcjd : (0) + Bab : jcjd : (0) Cab log z 2 : jcje : (0) cd 2 z2 z¯ z¯ cd z z z¯ − cd | | z z + ... 1 c (c g)z ja (z)jb (0) κabc + f ab1 4 jc (0) + 4 − jc (0) L,Az¯ ab L,z¯= ∼ f 2(13 +z¯2kf2) c(f bz¯ fL,agz¯ ( 1)cd z+¯2 f b L,zf ag )+ (f 4) cd $ cg d dg c % − 2 − 2 O ab ab g2z c 2 4 b c ag c4 cdc b cag4 g z 4c B cd+ f= f (1(∂ jk(0)f )(∂f jcg(0))f d +( 1)∂ j +(0)f dg +f −c)+ ∂(fj ) (0) c 4 z¯ −z z¯ − z¯ z −2 z¯ L,z¯ 2 z¯O2 z L,z ab $ 2 2 1 b ag cd b ag 4 % C cd = f (1 kf ) (f cgf d( 1) + f zdgf c)+ (f ). 1 z2 +:jajb −: (0) A−ab log2 z 2 : jcjd : (0)− + Bab : jcjd : (0)O + Cab : jcjd : (0) z¯ z¯ − cd | | z¯ z¯ cd z¯ z z¯ cd 2 z¯2 z z + ... (1 kf2) c = ± . (1.3) 2 (c g) (c g) ja (z)jb (0) c˜κab2πδ(2)(z± w−)+f2abf 4 − jc (0) + 2 − jc (0) L,z L,z¯ ∼ − c z¯ L,z z L,z¯ $ % ab g 2 2 c c (c4 g2)z c + f c logc+z (∂zjz¯(0) ∂z¯jz(0)) + −c jL,z(0) c1 =−4 | | − c3 = z¯ − $ −c+ + c −c+ + c % − − c+(c+ +2c ) c (2c+ + c ) − − − c2 = i 2 c41 = i 2 (c+ + c ) (c+ + c ) − − 2c+c c+c − − g = i 2 c˜ = (c+ + c ) c+ + c − −

a c+ a φ (w, w¯) jL,z(z,z¯)φ (w, w¯)= tR R + order zero R −c+ + c z w − − a c a φ (w, w¯) jL,z¯(z,z¯)φ (w, w¯)= − tR R + order zero R −c+ + c z¯ w¯ − − a c+ a φ(w, w¯) a jL,z(z,z¯)φ(w, w¯)= t +:jL,zφ :(w, w¯) − c+ + c z w − − z¯ w¯ + Aa log z w 2 : jc φ :(w, w¯)+Ba − : jc φ :(w, w¯)+... c | − | L,z c z w L,z¯ − a c a φ(w, w¯) a jL,z¯(z,z¯)φ(w, w¯)= − t +:jL,z¯φ :(w, w¯) − c+ + c z¯ w¯ − − z w Aa − : jc φ : Ba log z w 2 : jc φ :(w, w¯)+... − c z¯ w¯ L,z − c | − | L,z¯ −

a c a b 4 a c+ a b 4 − A c = 2 if cbt + (f ); B c = 2 if cbt + (f ). (c+ + c ) O (c+ + c ) O − −

2 2 a f (2) f 2 (2) (2) 4 h : jL,zφ : ˜ = c +1+ (1 kf )(c ˜ c )+ (f ). R 2 R 2 − R − R O !" # $

Qa = N dσja + dσ dσ #(σ σ )f a jc(τ, σ )jb (τ, σ ), (1) σ 1 2 1 − 2 bc τ 1 τ 2 % % 1 J a(x)= d2z(ja ∂¯Λ(x, x¯; z,z¯)+ja ∂Λ(x, x¯; z,z¯)) π L,z L,z¯ %

2 The fact that a solution exists depends on tensor identities, valid in these supergroups.

Open problem : complete the algebraic bootstrap. Ic : Applications i) The Sugawara energy-momentum tensor. ii) Current algebra primaries and their conformal dimension. a jµ = jµta

t Lie G a ∈

µ ∂ jµ =0 (0.1)

di(zz¯) (0.2)

lim jµ(z)jν(w) = lim jν(w)jµ(z) (0.3) z w z w → →

a jµ = jµta i) The Sugawara energy-momentum tensor

a for non-chiral currents jµ = jµta

Use κab • the current algebra the vanishing of self-contracted • κ f ab =0 structure constants ab c • vanishing of the Killing metric the Maurer-Cartan equation • κab

a b ab 2 ab 2 c c d3 c jz (z)jz(0) κ (¯z) d1 + f c(¯z) d2zjz(0) + d3zj¯ (0) + zz¯∂zjz¯(0) ∼ − 2 ! d d d + 2 zz¯∂ jc(0) + 2 (z)2∂ jc(0) + 3 (¯z)2∂ jc(0) + ... 2 z¯ z 2 z z 2 z¯ z¯ " d ja(z)jb(0) κab(z)2d + f ab (z)2 d zj¯ c(0) + d zjc(0) + 5 zz¯∂ jc(0) z¯ z¯ ∼ 4 c 5 z¯ 6 z 2 z z¯ ! d d d + 5 (¯z)2∂ jc(0) + 6 zz¯∂ jc(0) + 6 (z)2∂ jc(0) + ... 2 z¯ z¯ 2 z¯ z 2 z z " a b ab ab c c c jz (z)jz¯(0) κ d7 + f c [d8zjz(0) + d9zj¯ z¯(0) + d12zz¯∂z¯jz(0) ∼ 2 c 2 c +d10(z) ∂zjz(0) + d11(¯z) ∂z¯jz¯(0) + ... (0.4) # 1 In detail : • current / current bi-linear OPE :

• most of the terms vanish by the above rules :

∅ Use the Maurer-Cartan • The current component has dimension one equation to simplify for an appropriate choice of energy momentum tensor : this term. ⇒

• One computes the energy-momentum tensor OPE similarly : a zero Killing metric leads to a (non-chiral) conformal current algebra. z¯ z +:jajb : (0) + Aab : jcjd : (0) Bab log z 2 : jcjd : (0) + Cab : jcjd : (0) z z¯ cd z z¯ z¯ − cd | | z z¯ cd z¯ z z + ...

1 Aab = f 2(1 + kf2) (f b f ag ( 1)cd + f b f ag )+ (f 4) cd − 2 cg d − dg c O Bab = f 2(1 k2f 4)(f b f ag ( 1)cd + f b f ag )+ (f 4) cd − cg d − dg c O ab 2 2 1 b ag cd b ag 4 aCb cd = ab fz¯ (1c dkf ) (f abcgf d( 21) c +d f dgf abc)+z (cf d). +:j j : (0) + A −cd : j −j : (0) 2 B cd log z− : j j : (0) + C cd O: j j : (0) z z¯ z z¯ z¯ − | | z z¯ z¯ z z + ... (1 kf2) c = ± . (1.3) ii) Current algebra± − 2primariesf 2 and their conformal dimensions 2 2 ab 2 c 2 1 b ag cd b agc 4 A cd = f (1 + kf+ ) (f f ( 1) + f f )+ (f ) c1−= 2 cg d − c3 = dg c− O Bab = f 2(1−c+k2+f 4c)(f b f ag ( 1)cd + f b −f agc+)++ c (f 4) cd − − cg d − dg c O− c+(c+ +21 c ) c (2c+ + c ) abWe deﬁne2 current2 −b algebraag cd primariesb − ag −: 4 C cd =c2 =f i(1 kf ) (f2 cgf d( 1) c4+=fidgf c)+ (2f ). − (−c+ + c2 ) − (c+ + c O) − − 2c+c c+c − 2 − g = i 2 (1 kf ) c˜ = (c+ + c )= ± . c+ + c (1.3) ±− − 2f 2 −

2 2 a c+ c+ a φ (cw, w¯) jL,zc1(z,=z¯)φ (w, w¯)= c3 =tR R − + order zero −c+R+ c −c+ + c −cz+ + wc − − − − c+(c+ +2c ) c (2c+ + c ) a − c a φ− (w, w¯) − j c2(z,=z¯i)φ (w, w¯)=2 −c4 =t i R 2+ order zero L,z¯ (c+ + c ) R (c+ + c ) R − −c+ + c z¯ w¯ − 2c+c − c+−c − − g = i 2 c˜ = (c+ + c ) c+ + c − − a Their OPEc+ witha φ(w, w¯the) currenta algebra can jL,z(z,z¯)φ(w, w¯)= t +:jL,zφ :(w, w¯) − c+ + c z w abe completed− −recursively.c+ a φ (w, w¯ ) jL,z(z,z¯)φ a(w, w¯)= 2 c tR R + ordera z¯ zerow¯ c + AR c log z −wc+ +: jc φ :(z w,ww¯)+B c − : j φ :(w, w¯)+... | − | L,z− − z w L,z¯ a c a φ (w, w¯) j (z,z¯)φ (w, w¯)= − t R + order zero− a L,z¯ c a φ(w, w¯) R a R − −c+ + c z¯ w¯ jL,z¯(z,z¯)φ(w, w¯)= t −+:jL,z¯φ :(w, w¯) − c+ + c z¯ w¯ − − − a z w c a 2 c a Ac+c −a φ(:w,jL,zw¯) φ : aB c log z w : jL,z¯φ :(w, w¯)+... jL,z(z,z¯)φ(w, w¯)= − z¯ t w¯ +:−jL,zφ :(w,|w¯)− | − c+ + c − z w − − z¯ w¯ + Aa log z w 2 : jc φ :(w, w¯)+Ba − : jc φ :(w, w¯)+... c | − | L,z c z w L,z¯ a c a b 4 a c+ a b 4 A = − if t + (f ); B = − if t + (f ). a c 2 c cb a φ(w, w¯) a c 2 cb jL,z¯(z,z¯)φ((w,c+w¯+)=c ) − t O +:jL,z¯φ :(w, w¯) (c+ + c ) O −− c+ + c z¯ w¯ − − − a z w c a 2 c A c − : jL,zφ : B c log z w : jL,z¯φ :(w, w¯)+... − z¯ w¯ 2 − 2| − | a − f (2) f 2 (2) (2) 4 h : jL,zφ : ˜ = c +1+ (1 kf )(c ˜ c )+ (f ). R 2 R 2 − R − R O a ! c a$ b 4 a c+ a b 4 " − # A c = 2 if cbt + (f ); B c = 2 if cbt + (f ). (c+ + c ) O (c+ + c ) O − − a a a c b Q(1) = N dσjσ + dσ1dσ2#(σ1 σ2)f bcjτ (τ, σ1)jτ (τ, σ2), 2 2 − a % f (2)% f 2 (2) (2) 4 h : jL,zφ : ˜ = c +1+ (1 kf )(c ˜ c )+ (f ). R 1 2 R 2 − R − R O !" J a(#x)=$ d2z(ja ∂¯Λ(x, x¯; z,z¯)+ja ∂Λ(x, x¯; z,z¯)) π L,z L,z¯ % Qa = N dσja + dσ dσ #(σ σ )f a jc(τ, σ )jb (τ, σ ), (1) σ 1 2 1 − 2 bc τ 1 τ 2 % % 2 1 J a(x)= d2z(ja ∂¯Λ(x, x¯; z,z¯)+ja ∂Λ(x, x¯; z,z¯)) π L,z L,z¯ %

2 a b ab z¯ c d ab 2 c d ab z c d +:jz jz¯ : (0) + A cd : jz¯jz¯ : (0) B cd log z : jzjz¯ : (0) + C cd : jzjz : (0) z¯z − | | z z¯ +:jajb : (0) + Aab : jcjd : (0) Bab log z 2 : jcjd : (0) + Cab : jcjd : (0) + ...z z¯ cd z z¯ z¯ − cd | | z z¯ cd z¯ z z + ...

ab 2 2 1 b ag cd b ag 4 A cd = f (1 + kf ) (f cgf d( 1) + f dgf c)+ (f ) ab −2 2 1 2 b ag −cd b ag O4 A abcd = f2 (1 + kf2 4) (fb fag ( 1)cd + f b f ag )+ (f 4) B cd = −f (1 k f )(2 f cgcgf dd(−1) + f dgdgf cc)+O (f ) ab 2 − 2 4 b1 ag −cd b ag O4 B abcd = f (12 k f )(2f cgf b d( ag1) + fcd dgf b c)+ag (f ) 4 C cd = f −(1 kf ) (f cgf− d( 1) + f dgf Oc)+ (f ). − − 1 2 − O Cab = f 2(1 kf2) (f b f ag ( 1)cd + f b f ag )+ (f 4). cd − − 2 cg d − dg c O (1 kf2) c = . (1.3) (1 ±kf22) c ±= − ± 2f . (1.3) ± − 2f 2

2 2 c+ c c = 2 c = 2 − 1 c+ 3 c −c+ + c −c+− + c c1 = − c3 = − −c+ + c −c+ + c c+(c+−+2c ) c (2c−+ + c ) − − − c2 = ci+(c+ +2c )2 c4 =c i (2c+ + c ) 2 (c+ + c −) − (c+ + c− ) c2 = i 2 c4 = i 2 (c+ + c )− (c+ + c ) − 2c+c− c+c − g = i 2c+c − c˜ =c+c − − 2 − g = i (c+ + c 2) c˜ = c+ + c (c+ + c )− c+ + c − − −

a c+ a φ (w, w¯) jaL,z(z,z¯)φ (w, w¯)= c+ a tφR (Rw, w¯) + order zero jL,z(z,z¯)φ R(w, w¯)= −c+ + ctR R z w+ order zero R −c+ + c − z w− − − a c a φ (w, w¯) ja (z,z¯)φ (w, w¯)= c − a tφ (Rw, w¯) + order zero j L,z(¯z,z¯)φ (w, w¯)= − t RR + order zero We ﬁnd: L,z¯ R −c+ + c R z¯ w¯ R −c+ + c − z¯ w¯ − − −

c+ φ(w, w¯) jaa (z,z¯)φ(w, w¯)= c+ atφa(w, w¯) +:aja φ :(w, w¯) jL,zL,z(z,z¯)φ(w, w¯)= t +:jL,zL,zφ :(w, w¯) −−c+c+++c c z z ww −− −− a a 2 2 c c a z¯a z¯w¯ w¯ c c ++AA cloglogz z ww : j: jL,zφ :(φ :(w,w,w¯)+w¯)+B B c− −: j : jφL,:(z¯φw,:(w¯w,)+w¯)+...... c | |−− | | L,z c z zw w L,z¯ − − aa c c a φa(φw,(w,w¯)w¯) a a jjL,L,z¯z¯((z,z,z¯z¯))φφ((w,w,w¯w¯)=)= −− t t +:+:jL,jz¯L,φz¯:(φ w,:(w,w¯)w¯) −−c+c+++c c z¯ z¯ w¯w¯ −− −− a azz ww c c a a 2 2 c c AAc −− : j: j φ φ: : B Bc loglogz z w w: j : jφ :(φw,:(ww,¯)+w¯)+...... −− zc¯z¯ w¯w¯ L,zL,z −− c | −| −| | L,z¯L,z¯ −−

c c aa c a a b b 4 4 a a + c+ a ab b 4 4 AA c == −− ififcbt t++ (f(f););B Bc = = if ifcbt +t +(f )(.f ). c (c + c )2 2 cb c (c + c )2 2 cb (c++ + c ) OO +(c+ + c ) O O −− − −

2 2 f 2(2) f 2 (2) (2) a f (2) f 2 (2) (2) 4 h : jL,za φ : ˜ = c +1+ (1 kf )(2c ˜ c )+ (f ). 4 h : jL,zφ : ˜ = 2 Rc +1+2 (1− kf )(c−˜ R c )+O (f ). R 2 R 2 − R − R O !" # R$ R !" # $

Qa = N dσja + dσ dσ #(σ σ )f a jc(τ, σ )jb (τ, σ ), Q(1)a = N dσσja + d1σ d2σ #(1σ− 2σ )bcf aτ jc(τ1, στ )jb (τ2, σ ), (1) % σ % 1 2 1 − 2 bc τ 1 τ 2 % % 1 J a(x)= 1 d2z(ja ∂¯Λ(x, x¯; z,z¯)+ja ∂Λ(x, x¯; z,z¯)) J a(x)=π d2z(L,zja ∂¯Λ(x, x¯; z,z¯)+L,jz¯a ∂Λ(x, x¯; z,z¯)) π% L,z L,z¯ %

2 2 Current primaries (large radius) conformal dimension: An overcomplete basis of operators is:

:j:j:j:j ... phi ::::

Compute their conformal dimension perturbatively. z¯ z +:jajb : (0) + Aab : jcjd : (0) Bab log z 2 : jcjd : (0) + Cab : jcjd : (0) z z¯ cd z z¯ z¯ − cd | | z z¯ cd z¯ z z + ...

1 Aab = f 2(1 + kf2) (f b f ag ( 1)cd + f b f ag )+ (f 4) cd − 2 cg d − dg c O ab 2 2 4 b ag cd b ag 4 B cd = f (1 k f )(f cgf d( 1) + f dgf c)+ (f ) − 1 − O Cab = f 2(1 kf2) (f b f ag ( 1)cd + f b f ag )+ (f 4). cd − − 2 cg d − dg c O

(1 kf2) c = ± . (1.3) ± − 2f 2

2 2 c+ c c1 = c3 = − −c+ + c −c+ + c − − c+(c+ +2c ) c (2c+ + c ) − − − c2 = i 2 c4 = i 2 (c+ + c ) (c+ + c ) − − 2c+c c+c − − g = i 2 c˜ = (c+ + c ) c+ + c − −

a c+ a φ (w, w¯) jL,z(z,z¯)φ (w, w¯)= tR R + order zero R −c+ + c z w − − a c a φ (w, w¯) jL,z¯(z,z¯)φ (w, w¯)= − tR R + order zero R −c+ + c z¯ w¯ − − a c+ a φ(w, w¯) a jL,z(z,z¯)φ(w, w¯)= t +:jL,zφ :(w, w¯) − c+ + c z w − − z¯ w¯ + Aa log z w 2 : jc φ :(w, w¯)+Ba − : jc φ :(w, w¯)+... c | − | L,z c z w L,z¯ − a c a φ(w, w¯) a jL,z¯(z,z¯)φ(w, w¯)= − t +:jL,z¯φ :(w, w¯) − c+ + c z¯ w¯ − − z w Example:Aa − : jc φ : Ba log z w 2 : jc φ :(w, w¯)+... − c z¯ w¯ L,z − c | − | L,z¯ − :j phi:

a c a b 4 a c+ a b 4 − A c = 2 if cbt + (f ); B c = 2 if cbt + (f ). (c+ + c ) O (c+ + c ) O Perturbative− conformal dimension: −

2 2 a f (2) f 2 (2) (2) 4 h : jL,zφ : ˜ = c +1+ (1 kf )(c ˜ c )+ (f ). R 2 R 2 − R − R O !" # $ Interpolation Qa = N dσja + dσ dσ #(σ σ )f a jc(τ, σ )jb (τ, σ ), (1) σ 1 2 1 − 2 bc τ 1 τ 2 % % between WZW value 1 and particle approximation J a(x)= d2z(ja ∂¯Λ(x, x¯; z,z¯)+ja ∂Λ(x, x¯; z,z¯)) π L,z L,z¯ %

2 see also

First order perturbative prove of the Hirota equation for cosets (e.g. string on AdS5 x S5).

Raphael Benichou 2. Properties in String Theory

2a : The Physical String States

2b : An Asymptotic Symmetry Group 2 2 c+ c c1 = c3 = − −c+ + c −c+ + c − − 2 2 c c+(c+ +2c ) cc (2c+ + c ) c = i+ − c = i − − c1 = 2 (c + c )2 c3 = 4 −(c + c )2 −c+ + c + −c+ + c+ − − − − 2c+c c+c cg+=(c+i +2c −) c˜c=(2c+ +− c ) − 2 − − c2 = i (c+ +2c ) c4 = i c+ + c 2 (c+ + c ) − (c+ + c −) − − 2c+c c+c − − g = i 2 c˜ = a (c+ + c ) c+ a φc+(w,+ wc¯) jL,z(z,z¯)φ −(w, w¯)= tR R − + order zero R −c+ + c z w − − a a c+ c a φ a(φw, w(¯w,) w¯) jL,z(z,jzL,¯)z¯φ(z,(z¯w,)φw¯)=(w, w¯)= −tR RtR R + order+ order zero zero R R −c+ −+cc+ + c z z¯w w¯ − − − − a c a φ (w, w¯) jL,z¯(z,z¯)φ (w, w¯)= − tR R + order zero a R c+ −acφ+(w,+wc¯) az¯ w¯ jL,z(z,z¯)φ(w, w¯)= t −+:jL,z−φ :(w, w¯) − c+ + c z w − c+ φ(w, w¯)− ja (z,z¯)φ(w, w¯)= a ta 2+:cja φ :(w, w¯) a z¯ w¯ c L,z + A c log z w : jL,zL,zφ :(w, w¯)+B c − : jL,z¯φ :(w, w¯)+... − c+ + c z| −w | z w − − − a a c 2a φ(w,c w¯) a a z¯ w¯ c jL,z¯(z,z¯)φ(w,+w¯)=A c log z − w t : jL,zφ :(+:w,jL,w¯z¯)+φ :(Bw,cw¯)− : jL,z¯φ :(w, w¯)+... − c|+ +−c | z¯ w¯ z w − − − a c a za φ(ww, w¯c) a a 2 c jL,z¯(z,z¯)φ(w, w¯)= −A ct − : jL,z+:φ :jL,Bz¯φc:(logw,zw¯)w : jL,z¯φ :(w, w¯)+... − c+−+ c z¯ z¯w¯ w¯ − | − | − − − a z w c a 2 c A c − : jL,zφ : B c log z w : jL,z¯φ :(w, w¯)+... a −c z¯ aw¯ b 4− | −a | c+ a b 4 − A c = 2 if− cbt + (f ); B c = 2 if cbt + (f ). (c+ + c ) O (c+ + c ) O − − a c a b 4 a c+ a b 4 − A c = 2 if cbt + (f ); B c = 2 if cbt + (f ). (c+ + c ) O (c+ + c ) O − 2 2 − a f (2) f 2 (2) (2) 4 h : jL,zφ : ˜ = c +1+ (1 kf )(c ˜ c )+ (f ). 2 2 R 2 2 − − R O a R f (2) f 2 (2) R (2) 4 h : jL,z!φ" : ˜ #=$ c +1+ (1 kf )(c ˜ c )+ (f ). R 2 R 2 − R − R O ! $ " a # a a c b Q(1) = N dσjσ + dσ1dσ2#(σ1 σ2)f bcjτ (τ, σ1)jτ (τ, σ2), Qa = N dσja + dσ dσ #(σ σ −)f a jc(τ, σ )jb (τ, σ ), (1) %σ %1 2 1 − 2 bc τ 1 τ 2 % % a 1 2 a ¯ a J (1x)= d z(jL,z∂Λ(x,0x¯; z,z¯)+jL,z¯∂Λ(x, x¯; z,z¯)) a π2 a ¯ a (1.4) J (x)= d z%(jL,z∂Λ(x, x¯; z,1z¯)+jL,z¯∂Λ(x, x¯; z,z¯)) π ! " % 1 (γ x)(¯γ x¯)e2φ 2φ Λ(x, x¯; γ, γ¯, φ)=1 (γ x)(¯−γ x¯−)e 2φ Λ(x, x¯; γ, γ¯, φ)= −γ x 1+(−a bγ −x)(¯γ x¯)e −2aγ :Q Thex Physical=−1+(K Stringab& Fγ StatesF x)(¯−γ x¯−)e2φ ' (1.5) − & − − ' Simplify : A Particle on a Supergroup 1 1 2 2 T (x)=T (x)= d2z (d∂ zj (∂∂xjL,z∂ Λ∂x(∂x,z¯Λx¯(;x,z,x¯z¯;)+2z,z¯)+2∂2j∂xj∂L,zΛ∂(z¯x,Λ(x¯x,; z,x¯;z¯z,))z¯)) −2π x L,z x z¯ 1 x 1L,z z¯ −2π % L = dτe ∂τ gg− , ∂τ gg− % " ! " 2 +(∂xjL,z¯∂x∂zΛ(x, x¯; z,z¯)+22 ∂xjL,z¯∂zΛ(x, x¯; z,z¯)) "+(∂xjL,z¯∂x∂#zΛ(x, x¯; z,z¯)+2∂ jL,z¯∂zΛ(x, x¯; z,z¯)) Worldsheet reparameterization invariancex # leads to physical states in the cohomology #

QQ == cCcC2 2==cLcL00 = c ∆ (1.2)(1.2)

01 C = 01 (1.3) 2 C = 00 (1.3) 2 00 ( ( ) ) 0 2 (1.4) 1 ( )

a b Q = Kab F F (1.5)

3 2 2 c+ c c1 = c3 = − −c+ + c −c+ + c − − c+(c+ +2c ) c (2c+ + c ) − − − c2 = i 2 c4 = i 2 (c+ + c ) (c+ + c ) − − c2 2c+c c2 c+c + − − c1 = g = i 2 c3 = − c˜ = −c+ + c (c+ + c ) −c+ + c c+ + c − − − − c+(c+ +2c ) c (2c+ + c ) − − − c2 = i 2 c4 = i 2 a (c+ + c ) c+ (c+ a+φc )(w, w¯) jL,z(z,z¯)φ− (w, w¯)= tR −R + order zero 2c+c R −c + cc+c z w g = i 2 − c˜+= − 2 c+ 2 − c − (c+ + c ) c+ + c c1 = a − c3c= a −φ− (w, w¯) jL,−z¯c(+z,+z¯)cφ (w, w¯)= − −ct+R +Rc + order zero −R −c+ + c z¯− w¯ a c+(c+ +2c ) c+ a φ (w, w−¯c) (2c+ +−c ) jL,zc(z,=z¯)iφ (w, w¯)= − tR cR = i −+ order zero− 2 R 2 4 2 (c+ + c )−c+ + c z w (c+ + c ) a −c+ a φ−(w, w¯)− a − jL,z(z,z¯)aφ(w, w¯)=2c+c ct a φ (+:w, w¯j)L,zc+cφ :(w, w¯) jL,z¯(gz,=z¯)iφ (w,−w¯c)=−+ + c − z tR wRc˜ = + order− zero R 2−c− + c z¯ w¯ (c+ + c ) + − c+ + c −a − 2 c− − a z¯ w¯ c + A c log z w : jL,zφ :(w, w¯)+B c − : jL,z¯φ :(w, w¯)+... a c+ a φ(w, w¯| ) − a| z w jL,z(z,z¯)φ(w, w¯a)= t +:c+ jL,zaφ φ:(w,(w,w¯)w¯) − j (z,−z¯)cφ+ +(w,c w¯)=zc w φ(w,tw¯)R + order zero a L,z − − a R a jL,z¯(z,z¯)φ(w, w¯)=R −−c+t + c +:z jwL,z¯z¯φ :(w¯w, w¯) a c +2 c c z¯− w¯ a c + A c log−z +w : jL,zφ :(w, w¯)+−B c − : jL,z¯φ :(w, w¯)+... a | − | − c − a φ (w, w¯z) w jL,z¯(z,z¯)φ (w, w¯a)=z w − c tR R a −+ order2 zeroc a Rc Aa φc(w,−−w¯c)+ +: jcL,za φ : z¯B wc¯log z w : jL,z¯φ :(w, w¯)+... jL,z¯(z,z¯)φ(w, w¯)= −− t z¯ w¯+:j−L,z¯φ :(−w,−w¯) | − | − c+ + c z¯ −w¯ − − a a cz+ w a φc(w, w¯) a a 2 c jL,z(z,z¯)φ(w,a w¯)= Ac c − t:aj bφ : +:B c4jlogL,zφz :(ww, w¯: )j a φ :(w, w¯c)++ ... a b 4 − L,z L,z¯ A c = −− c+ z+¯ c2wif¯ cbzt +w− (f );| − | B c = 2 if cbt + (f ). (c+ + c −) − − O (c+ + c ) O a − 2 c a z¯ w¯ c − a c + A ac logb z w4 : jL,zφ :(w,a w¯)+Bc+c − a : jbL,z¯φ :(4w, w¯)+... − A c = 2 if cbt +| −(f );| 2 B c = 2 z 2 ifw cbt + (f ). (c+ + c ) a O f (2) (fc+ + c )− 2 (2) O (2) 4 a − c a φ(w, w¯) a − h : jL,zφ−: ˜ = c +1+ (1 kf )(c ˜ c )+ (f ). jL,z¯(z,z¯)φ(w, w¯)= t 2+:RjL,z¯φ :(w,2 w¯) − − R O − c+ + cR 2 z¯ w¯ 2 R a !" #−f$ (2)− f 2 (2) (2) 4 h : jL,zφ : ˜ z= w c +1+ (1 kf )(c ˜ c )+ (f ). a 2 Rc 2 a − 2− cR O RA c − : jL,zφ : B c log z wR : jL,z¯φ :(w, w¯)+... !" Q−a# $ =z¯ Nw¯ dσja +− dσ dσ| #−(σ | σ )f a jc(τ, σ )jb (τ, σ ), (1) − σ 1 2 1 − 2 bc τ 1 τ 2 a a % % a c b Q = N dσj + dσ1dσ2#(σ1 σ2)f j (τ, σ1)j (τ, σ2), a (1)c a b σ 4 a bc τ c+ τ a b 4 A = − if t + 1(f );−B = if t + (f ). c 2 %acb % 2 a ¯ c a2 cb (c+ + c ) J (x)=O d z(jL,z∂Λ(x, x¯; (z,c+z¯)++ cjL,) z¯∂Λ(x, x¯; z,Oz¯)) − − a 1 2 πa ¯ a J (x)= d z(jL,z%∂Λ(x, x¯; z,z¯)+jL,z¯∂Λ(x, x¯; z,z¯)) π 2 2 2φ a % f (2) f 1 2 (γ(2) x)(¯γ(2) x¯)e 4 h : j φ : = c +1+ (1 kf )(c 2φ c )+ (f ). L,z ˜ Λ(x, x¯; γ, γ¯, 1φ)= (γ x)(¯γ x¯)e˜− − 2φ Λ(x,Rx¯; γ, γ¯, φ)=2 R −2γ −−x 1+(− Rγ −xR)(¯γ Ox¯)e ! $ −γ x 1+(γ− x&)(¯γ x¯)e−2φ − ' " # − & − − ' a 1 a 2 a c 2 b Q(1) T =(x1)=N 2 dσjσ +d z d(∂σx1jdL,zσ2#∂(xσ∂1z¯Λ(σx,2)fx¯2; z,bcjz¯τ)+2(τ, σ1∂)xjjτL,z(τ,∂σz¯2Λ),(x, x¯; z,z¯)) T (x)= −d 2zπ(∂xjL,z∂x∂z¯Λ(x, x¯; z,z¯−)+2∂xjL,z∂z¯Λ(x, x¯; z,z¯)) −2π % % % % " 2 1 "+(∂ j ∂+(∂∂Λxj(x,L,z¯x¯∂; xz,∂z¯z)+2Λ(x,∂x¯2j; z,z∂¯)+2Λ(x,∂x¯x; jz,L,z¯z¯))∂zΛ(x, x¯; z,z¯)) J a(x)= d2xz(L,jza¯ x∂¯Λz (x, x¯; z,z¯)+jax ∂L,Λz¯(x,z x¯; z,z¯)) π L,z L,z¯ % # # Q = 1cC = (cLγ x)(¯γ x¯)e2φ (1.2) Λ(x, x¯; γ, γ¯, φ)= Q2 = −0cC2 =−cL0 (1.2) −γ x 1+(γ x)(¯γ x¯)e2φ − & − − ' 01 1 C = 01 (1.3) T (x)= d2z (∂ j 2 ∂ ∂ Λ(Cx,00x¯; =z,z¯)+2∂2j ∂ Λ(x, x¯; z,z¯)) (1.3) −2π x L,z x z¯ ( 2 ) 00x L,z z¯ % ( ) " 2 +(∂xjL,z¯∂x∂zΛ(0x, x¯; z,z¯)+2∂xjL,z¯∂zΛ(x, x¯; z,z¯)) not closed0 (in Siegel gauge) (1.4) 1 # (1.4) ( ) 1 ( ) Q One= obtainscC2 =athatcLb 0 becomes diagonal in (1.2) Q =cohomologyKab F. F (1.5) a b Q = Kab F F (1.5) The signature01 of logarithmicity disappears on C = (1.3) 2 physical2 string00 states. ( ) 2 0 (1.4) 1 ( )

a b Q = Kab F F (1.5)

2 2 2 c+ c c1 = c3 = − −c+ + c −c+ + c − − c+(c+ +2c ) c (2c+ + c ) − − − c2 = i 2 c4 = i 2 (c+ + c ) (c+ + c ) − − 2c+c c+c − − g = i 2 c˜ = (c+ + c ) c+ + c − − a c+ a φ (w, w¯) jL,z(z,z¯)φ (w, w¯)= tR R + order zero R −c+ + c z w − − a c a φ (w, w¯) jL,z¯(z,z¯)φ (w, w¯)= − tR R + order zero R −c+ + c z¯ w¯ − − a c+ a φ(w, w¯) a jL,z(z,z¯)φ(w, w¯)= t +:jL,zφ :(w, w¯) − c+ + c z w − − z¯ w¯ + Aa log z w 2 : jc φ :(w, w¯)+Ba − : jc φ :(w, w¯)+... c | − | L,z c z w L,z¯ − a c a φ(w, w¯) a jL,z¯(z,z¯)φ(w, w¯)= − t +:jL,z¯φ :(w, w¯) − c+ + c z¯ w¯ − − z w Aa − : jc φ : Ba log z w 2 : jc φ :(w, w¯)+... − c z¯ w¯ L,z − c | − | L,z¯ − a c a b 4 a c+ a b 4 − A c = 2 if cbt + (f ); B c = 2 if cbt + (f ). (c+ + c ) O (c+ + c ) O − −

2 2 a f (2) f 2 (2) (2) 4 h : jL,zφ : ˜ = c +1+ (1 kf )(c ˜ c )+ (f ). R 2 R 2 − R − R O !" # $ Qa = N dσja + dσ dσ #(σ σ )f a jc(τ, σ )jb (τ, σ ), (1) σ 1 2 1 − 2 bc τ 1 τ 2 % % 1 J a(x)= d2z(ja ∂¯Λ(x, x¯; z,z¯)+ja ∂Λ(x, x¯; z,z¯)) π L,z L,z¯ % 1 (γ x)(¯γ x¯)e2φ Λ(x, x¯; γ, γ¯, φ)= − − −γ x 1+(γ x)(¯γ x¯)e2φ − & − − ' 1 T (x)= d2z (∂ j ∂ ∂ Λ(x, x¯; z,z¯)+2∂2j ∂ Λ(x, x¯; z,z¯)) −2π x L,z x z¯ x L,z z¯ % " 2 +(∂xjL,z¯∂x∂zΛ(x, x¯; z,z¯)+2∂xjL,z¯∂zΛ(x, x¯; z,z¯)) #

Q = cC2 = cL0 (1.2)

01 C = (1.3) 2 00 Superstring( on a) Supergroup Local Fermionic Invariance of (unknown) ancestor of Berkovits0 formalism (1.4) Leads to cohomological1 problem of the type ( )

a b Q = Kab F F in (1.5)

2 0 (1.4) 1 ! "

a b Q = Kab F F (1.5)

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