Superconformal field theories and cyclic homology Richard Eager Kavli IPMU Strings and Fields Yukawa Institute for Theoretical Physics Kyoto University Kyoto, Japan Thursday, July 24th, 2014 Richard Eager Kavli IPMU Superconformal field theories and cyclic homology
Introduction to AdS/CFT Consider a stack of N D3 branes filling R 1 , 3 in R 1 , 3 × C 3 . At low energies, the open string degrees of freedom decouple from the bulk. The resulting theory on the brane world-volume is N = 4 super Yang-Mills. Richard Eager Kavli IPMU Superconformal field theories and cyclic homology
Goal: Prove AdS/CFT Less ambitious goal: Prove part of AdS/CFT for a subset of protected BPS operators and observables. This talk: Show that the BPS operators agree under the correspondence. Based on joint work with J. Schmude, Y. Tachikawa [arXiv:1207.0573, ATMP to appear] and work in progress Richard Eager Kavli IPMU Superconformal field theories and cyclic homology
AdS/CFT Cartoon Gauge Theory Gravity Theory R 3 , 1 × X 6 AdS 5 × L 5 N units of RR-flux N D 3 branes X 6 Calabi-Yau 6-manifold L 5 Sasaki-Einstein 5-manifold τ d-1 S Figure: N D3-branes Figure: AdS Space-Time Richard Eager Kavli IPMU Superconformal field theories and cyclic homology
Protected operators in N = 4 SYM N = 4 SYM has three adjoint chiral scalar superfields Φ 1 , Φ 2 , Φ 3 . Their interactions are described by the superpotential W = Tr Φ 1 � Φ 2 , Φ 3 � . Consider an operator of the form O = T z 1 z 2 ... z k = Tr Φ z 1 Φ z 2 . . . Φ z k . If T z 1 z 2 ... z k is symmetric in its indices, then the operator is in a short representation of the superconformal algebra. If T z 1 z 2 ... z k is not symmetric, then the operator is a descendant, because the commutators [Φ z i , Φ z j ] are derivatives of the superpotential W [Witten ’98]. Richard Eager Kavli IPMU Superconformal field theories and cyclic homology
Matching protected operators in N = 4 SYM Under the AdS/CFT dictionary, a scalar excitation Φ in AdS obeying ( � AdS 5 − m 2 )Φ = 0 with asymptotics ρ − ∆ near the boundary of AdS ( ρ → ∞ ) is dual to an operator of scaling dimension � d 2 m 2 = ∆(∆ − d ) → ∆ ± = d 2 ± 4 = m 2 Richard Eager Kavli IPMU Superconformal field theories and cyclic homology
Matching protected operators in N = 4 SYM H. J. KIM, L. J. ROMANS, AND P. van NIEUVPENHUIZEN )Y'( p) — — y — — ek(k+4)Y( p), k=2, 3, . . . 2e ( a 0 p h~ (2.47) 45- Iio+ Iio , b&'„'+ and bz" the two complex while fields in (2. 41) 32 4 The operator e (k+2) . The field bz'„'+ is the complex have masses conjugate of b plo — because the four-index antisymmetric tensor is real. O = Tr Φ z 1 Φ z 2 . . . Φ z k 21- A -, We now discuss the modes contained in the fields and B. These fields are purely fluctuations and contain %'e expand them into spherical no background parts. har- 12- monics as follows: has conformal dimension k and is ga p'„(x) Y '(y ), 3 „„= dual to a supergravity state of 5- a Y '(y)+a&'(x)D~ Y' '(y)], aPy8 A„~=+[a„'(x) spin zero and mass 05 (2.48) k C — A~p — +[a "(x) Y[ "p](y)+a '(x)D[~ Yp'](y)], 4 e m 2 = k ( k − 4) . B = +B '(x) Y '(y) . FIG. 2. Mass spectrum of scalars. We choose the Lorentz-type gauges — 0, DA~p — DA 0 p —— (2. 49) Figure: From Kim-Romans-van at k =0. We summarize branch of (2.34), namely the re- by first fixing the transversal which can be implemented sults of a11 scalar modes in Fig. 2. Nieuwenhuizen [Phys.Rev. D32 D Ap — I5 =0, and part of A fields, b„„ in a„ in 6A p —— DpA to gauge a Diagonal equations. The remaining then fixing the D A„part of 6A „=D Az — to set (1985) 389] D&A and P '4 in as H(z ) in h&, have diagonal h(~p) as well a& — Il — 0. The on1y gauge transformations which respect fie1d equations which read Richard Eager Kavli IPMU Superconformal field theories and cyclic homology these have A„(x), which are the gauges y-independent x6 — =0 [from (2. 19)], for az„= (x). Thus (M ' Y[''p]+ (2. 41) usual gauge parameters we may use a+ )«b z in (2.48) with a„'=a '=0. I) )(t "Y(~p) — the expansion Substituting x+CI« — — 0 [from (E3. 1)], (2. 42) 2e ( these expansions into the field equations yields I I) I) ~ [(Max+ «)a„z+2iee z "(3 a, „']Y '=0, l [T(+x++y)H(pv)+e p D(pD Hv)k H(pv) (2. 50) ' =0 [from (El. 1)] . ++« — + —, 'D(„D )HI~]Y Y[ p] +2iea e pr Dr Y 6e )a (2. 43) for k ) 1 by 2(D~aq')(D[ — Yp])=0, (2. 51) The last equation can be diagonalized « — 4e )a 'Y '+(D"a&'„)(D Y ')=0, (Max+ (2. 52) — — 12eb) j[(k—+1— )(k+3)] . (E3 +xCly)B 'Y' '=0. )( , n. H(~„) P(q„)+D(„D, (2. 53) (2. 44) I&o We recall that the spherical harmonies p~ are not Y~ The traceless field P(&v) is then transversal on-shell from of 6, but also of the operator only eigenfunctions (2. 30) and satisfies the Einstein equation [Ein — — — = cap k(k+4)e ]P(„„) 0, ( D ) Y[ap] — (2. 45) Dr Y[sp] where Ein stands for the Einstein operator y — Since (*D)(*D) =4( 8e P(„„) (O +2e — 6e ), we can divide the Y[ p] into 2R~„'"(P(p )) — — )P(„„) . YI~p~ and Yl~p~, where ' is the Ricci tensor of five-dimensional Here R„', (*D)Y['p] =+2i( — «+6e space- Y['p] (2. 54) ) with Rz ' and the orgi- time. One should not be confused nial R„. Recall that R& is the pv component of the full Since For k =0, the (El) equa- Ricci tensor in ten dimensions. ( — Cl +6 ')Y" — = — ", —, = '(k+ )'Y '— b. tion, together with (2. 21) and (2. 40) yields ' p] = + 2l e( k + 2 ) Y[" p]'' 4e (g„„+h„' )=0— we thus have Rp, '(g„„+h„', ) . (2. 46) (*D ) Y[ (2. 55) This clearly demonstrates that is the massless gravi- h& ton, as expected. Collecting all terms with a given spherical harmonic, I[4 one gets the d = 5 field equations The real scalars P ' in (2. 42) have masses ~
Goal: Test AdS/CFT by small deformations N = 4 SYM has superpotential W = Tr ( XYZ − XZY ) . What happens when we deform it by giving a mass to one of the scalars XYZ − XZY + mZ 2 � � W = Tr or deform the coupling constants? qXYZ − q − 1 XZY � � W = Tr Can we still match the spectrum of protected operators? Richard Eager Kavli IPMU Superconformal field theories and cyclic homology
Goal: Match Closed String States in the large-N limit Gauge Theory Gravity Theory R 3 , 1 × X 6 AdS 5 × L 5 Closed strings: Closed strings: HP • ( X , π = 0) HC • ( C Q /∂ W ) τ d-1 S Richard Eager Kavli IPMU Superconformal field theories and cyclic homology
The Superconformal Algebra The 4D superconformal algebra combines both the conformal algebra and N = 1 supersymmetry algebra. The conformal algebra consists of Lorentz generators M µν , momenta P µ , special conformal generators K µ and a dilatation D . Figure: Generators of the Superconformal Algebra Richard Eager Kavli IPMU Superconformal field theories and cyclic homology
The Superconformal Index The SCI is a 4D analog of the Witten index in quantum mechanics Defined as I ( µ i ) = Tr( − 1) F e − βδ e − µ i M i The trace is over the Hilbert space of states on S 3 Q is one of the Poincare supercharges Q † is the conjugate conformal supercharge δ ≡ 1 Q , Q † � � 2 M i are Q− closed conserved charges Richard Eager Kavli IPMU Superconformal field theories and cyclic homology
Operators contributing to the index Key commutation relations: α + 3 { Q α , Q † β } = E + 2 M β 2 r α − 3 † ˙ ˙ β } = E + 2 M β { Q ˙ α , Q 2 r ˙ α O = 0 are called chiral primaries. ˙ Operators for which Q Operators contributing to the (right-handed) index have δ = {Q , Q † } = 0. Choosing Q = Q ˙ − , operators contributing to the index satisfy E − 2 j 2 − 3 2 r = 0 . (0.1) Richard Eager Kavli IPMU Superconformal field theories and cyclic homology
The 4D Letter Index Letter ( j 1 , j 2 ) I I Letter ( j 1 , j 2 ) t 3 r − t 3 y φ (0 , 0) λ 1 (1 / 2 , 0) − t 3(2 − r ) − t 3 y − 1 λ 2 ( − 1 / 2 , 0) ψ 2 (0 , 1 / 2) t 6 f 22 (0,1) t 3 y ± 1 t 3 y ± 1 ( ± 1 / 2 , 1 / 2) ( ± 1 / 2 , 1 / 2) ∂ ±− ∂ ±− Fields contributing to the index, from a chiral multiplet (left) and from a vector multiplet (right) 1 1 [F. Dolan, H. Osborn],[A. Gadde, L. Rastelli, S. S. Razamat, W. Yan] Richard Eager Kavli IPMU Superconformal field theories and cyclic homology
Ginzburg’s DG Algebra Letter ( j 1 , j 2 ) I I Letter ( j 1 , j 2 ) t 3 r t 6 φ (0 , 0) f 22 (0,1) − t 3(2 − r ) ψ 2 (0 , 1 / 2) Table: Fields contributing to the index, from a chiral multiplet (left) and from a vector multiplet (right), after the cancellation of W α and the spacetime derivatives ∂ µ are taken into account. Ginzburg’s DG algebra is a free differential-graded algebra D = C � x 1 , . . . , x n , θ 1 , . . . , θ n , t 1 , . . . , t m � where φ, ψ 2 , f 22 correspond to x , θ, t respectively. Richard Eager Kavli IPMU Superconformal field theories and cyclic homology
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