PowerPoint PresentationKen Intriligator (UCSD)
Based on works with Eddy Barnes, Elie Gorbatov, Brian Wecht, and Jason Wright
Plan:
* Situations where finding it is non-trivial.
* Intuitions about RG flows and Cardy's conjecture.
* a-maximixation for 4d N=1 SCFTs and extensions.
* AdS/CFT & Z-minimization of Martelli, Sparks, Yau
* A new way to determine the susy R-symm., for any d.
* Computing current 2-point fns using AdS/CFT.
Use supersymmetry to obtain some exact results. Can test and develop intuitions about quantum field theories.
Find e.g. that there is a vast landscape of 4d interacting conformal field theories. Can study them, and RG flows between them.
UV CFT
A
B
Recipe to find new CFTS: start with one, find the relevant operators, perturb by them, flow to new CFTs in IR (might be non-interacting).
Variety of exact results about 4d N=1 theories can be made, thanks to a key property: supersymmetry relates the stress tensor to an R-symmetry.
Both in supermultiplet
Anomalous dims of chiral fields related to their running R-charges.
Exact beta functions ~ linear combinations of R-charges of fields:
U(1)R ABJ anomaly.
super-R current = (conserved part) + (violated part)
Picks up anom. dim. irrelevant in IR.
s1
Superconformal R symmetry of SCFT:
.
violated JR
Had been a stumbling block for general SCFTs. We found a simple, exact sol'n. Ties in with intuitions/conjectures about RG flows.
"Course graining is a one-way process: # of massless d.o.f. ~ height of flowing water, always decreasing in RG flows to IR."
Detour on intuition/conjectures about RG flow:
Conjectures: "Weak" version: Can find some quantity "c" that counts the "# massless d.o.f." of a CFT, with c > 0 for unitary thys and cUV > cIR for endpoints of all RG flows. (Already a strong statement!)
Stronger: Can extend to a "c-function", that's monotonically decreasing along entire RG flow to IR.
Strongest: RG flow = gradient flow of c-function, with positive definite metric on space of couplings:
Zamolodchikov proved medium version for all unitary 2d theories.
No proof yet of any of these conjectures for d>2.
What about in four dimensions? Cardy conjecture ('88) for quantity that counts # massless d.o.f. of 4d CFTs: coefficient a:
no! yes?
4d "a-theorem" conjecture: Endpoints of all 4d RG flows satisfy
and also
True in every known
Stronger versions investigated perturbatively by Osborn and co. Conjecture for an a-function a(g) along RG flows. In all perturb. examples, finds indeed RG flow turns out to be gradient flow of this fn, and the associated "metric" turns out to be pos definite. Again, no general proof of any of this.
Also, what about 3d? No conformal anomaly there!
For 4d, N=1 SCFTs, conformal anomalies a and c related by supersymmetry to 't Hooft anomalies of the superconformal R-symmetry R* of the SCFT:
Anselmi, Freedman, Grisaru, Johansen.
Can use the power of 't Hooft anomaly matching to get exact a and c at strongly interacting RG fixed points, if exact R* is known. Can then check Cardy's conjecture for a.
Anselmi, Erlich, Freedman, Johansen
"a-maximization" KI, B.Wecht '03
Exact solution to the problem of finding the exact superconformal U(1)R symmetry R* of general IR SCFTs: we proved that R* is the unique, local maximum of
't Hooft anomalies, exactly computable!
over the space of all possible conserved R-symmetries:
atrial(R)
R
R*
Value of function at unique local maximum is the central charge aSCFT of Cardy's conjecture.
Here's why:
Consider a free chiral superfieldQuick example:
*
*Local max at r = 2/3, correct value for free field.
And it’s basically just as easy** for interacting theories!
** Caution: Need to maximize a(R) over the correct, full set of possibilities. Overlooking any symmetries, including any accidental ones, leads to wrong results.
Input: free UV data and anomaly and W constraints on R.
a-maximization 't Hooft anomaly matching
Exact, non-perturbative anomalous dimensions and central charges.
A general consequence of a-maximization:
R*(Qi) are solutions of quadratic equations, with integer coefficients.
They are thus always algebraic numbers.
E.g. when R = R0 + s F, a one parameter family of R-symmetries, get
quadratic irrationals:
So the dimensions of chiral primary operators and the central charges are always algebraic numbers.
Because these numbers are not continuous, they can not depend on any continuous moduli of the SCFT. E.g. for beta deformation of N=4 SYM to N=1, this shows that the chiral primary operator dimensions, and the conformal anomaly must be independent of the modulus beta.
a-maximization "almost proves" the a-theorem! Since relevant deformations generally break the flavor symmetries,
Maximizing over a subset then implies that
1) Accidental symmetries.Loopholes: 2) Only a local max.
An extension of a-maximization, away from RG fixed points, helps close the 2nd loophole (Kutasov). Implement via Lagrange multipliers. Maximize w.r.t. R,
Conjecture Lagrange multiplier = running coupling. Then gives monotonically decreasing a-function along the entire RG flow. Check: Gives anomalous dimensions that agree with the explicit calulations to 3-loops(!!!) of Jack, Jones, and North, up to an identified scheme change. Barnes, KI, Wecht, Wright; Kutasov and Schwimmer.
Extension of this approach to multiple couplings: conjecture implies that RG flow = gradient flow (BIWW). Metric on coupling space is positive definite if jacobian from Lagrange multipliers to couplings is positive definite. We also extended approach to close the 1st loophole, for certain accidental symmetries: those visible from operators hitting a unitarity bound. It's an open problem to close the 1st loophole in our a-maximization "almost proof" of the a-theorem for more general accidental symmetries.
AdS/CFT: N=1 4d SCFTs IIB on AdS5 x Y5 . Y = Sasaki-Einstein, i.e. metric ds2(X=C(Y))=dr2+r2ds2(Y) of cone is a local (non-compact) Calabi-Yau.
Henningson, Skenderis; Gubser
AdS/CFT:
(These Vols have scale L, set by Einstein cond, factored out)
Berenstein, Herzog, Klebanov.
Di-baryon, from D3 wrapped on a susy 3-cycle of Y.
is then a bizarre rel'n in mathematics.
SCFT rel'n
Our a-maximization result implies that Vol(Y) and the susy 3-cycle volumes must always be algebraic numbers, up to same factors of pi as for spheres. A prediction for mathematics.
Some more recent developments:
Gauntlett, Martelli, Sparks, Waldram: found explicit Sasaki-Einstein metrics, e.g. ds2(Yp,q). Remarkable! They indeed have quadratic irrational volumes!
Martelli, Sparks: showed Yp,q = toric, with e.g. C(Y2,1) = complex cone over F1. Computed volumes and susy 3-cycles of Yp,q . Algebraic numbers, in agreement with our general predictions about superconformal R-charges from a-maximiz'n.
Bertolini, Bigazzi, Cotrone: verified AdS/CFT predictions for F1 : a-maximization in quiver gauge theory gives same R-charges as from the volumes of susy cycles.
Benvenuti, Franco, Hanany, Martelli, Sparks: Constructed quiver gauge thys for all Yp,q. Verify for all (p,q) that R-charges computed in the gauge theory via a-maximization agree with volumes of susy cycles, using the GMSW metric.
Martelli, Sparks, Yau: Metrics not even needed! For any toric C(Y), the volume of Y, and its susy cycles can be determined entirely from the weaker info of the Reeb vector. Reeb vector determined by Z-minimization, needing only toric data. Verified in all known examples that Z-minimization leads to R-charges (highly non-trivial expressions!) that always precisely agree with results obtained by a-maximization in the dual 4d gauge theory.
Review of Martelli, Sparks, and Yau's Z-minimization:
take toric:
Reeb vector= partner (via complex structure) of radial dilatations =U(1)R isometry. bi picks it out as a particular combination of the U(1)n isometries.
Constants!
Volume of Y and its calibrated cycles can be determined entirely from the toric data va and constants bi . Don't need to determine the actual metric!
with toric polytope. They
show that the correct S.E. values of bi determined by minimizing Z[bi]=Einstein-Hilbert action on Y. Gives b1=n, and others via:
Z-minimization solves the same problem in the geometry as a-maximization solves in the gauge theory: finding the Reeb vector in geometry amounts to finding the R-symm in the gauge theory. But how precisely are the two methods related?
Note that Z-minimization determines the Reeb vector for any Sasaki-Einstein space. So it applies just as well to M theory on AdS4 x Y7 , which is dual to 3d, N=2 SCFTs. On the other hand, a-maximization only applies in 4d (there are no 't Hooft anomalies in 3d). Up to now, there was no known field theory criterion to determine the superconformal R-charges for 3d SCFTs.
Something new is needed (at least in 3d)!
A new way to determine the superconformal U(1)R : (Barnes, Gorbatov, KI, Wright)
minimization. Applies for all spacetime dims d.
For CFTd
We proved that for the exact superconformal U(1)R and any non-R flavor symmetry Fi. Follows from superspace, and their being in different kinds of supermultiplets. Write general trial U(1)R
(so invertable)
Uniquely determine correct sj * by
Consider the coefficient of Rt two-point function with itself, . Quadratic function of the sj. The exact, superconformal U(1)R gives the unique global minimum of this function.
A simple example: a free chiral superfield for SCFTd:
typo!
Different fn.'s but same ans.
Recall in 4d:
It is not so easy to implement -minimization in interacting theories. But AdS/CFT is well-suited for this!
Current correlators and AdS/CFT Geometry
In AdSd+1: (Freedman and co.)
Coefficients of AdSd+1 bulk gauge field kinetic terms. Global symm of bndy CFT = gauge in AdS bulk.
Can compute in AdSd+1 by reducing SUGRA on S.E. geom.Y2n-1 .
i) U(1)R graviphoton from Reeb isometry of Y.
ii) mesonic, non-R symms from other isometries of Y.
iii) baryonic, non R symms from C reduced on Y homology cycles.
Kaluza Klein
All alter Ramond-Ramond bkgd, and thus C. At linearized level:
Each AdS gauge field has an associated 2n-3 form, that we determine.
Local metric on S.E. Y2n-1 :
Unit 1-form dual to Reeb:
Find:
(Barnes, Gorbatov, KI, Wright)
Charges of wrapped branes:
Reduce on Y. All AI gauge fields get kinetic term contributions from the C-field kinetic terms, since
For the Kaluza-Klein AI there are also the usual KK gauge kinetic term contributions from the Einstein action:
(For 11d SUGRA on S7 by Duff, Pope Warner. Generalize.)
Argue that:
Upshot: compute precise gauge kinetic terms in terms of volume integrals on Y.
(No kinetic mixing between KK and baryonic gauge fields: .)
So our minimization conditions, , are automatically satisfied for all baryonic symmetries, given that R is a KK symmetry.
Our minimization condition uniquely determines the U(1)R and hence the Reeb vector isometry, from among all the other isometries, via:
any other Killing vector
Reeb vector
For C(Y) toric, the non-R Killing vector isometries are given by
which gives and thus
minimization = Z[b] minimization!
Using the forms that we determined, we can also make new, quantitative checks of AdS/CFT duality. Compute flavor charges of wrapped branes, and compare with field theory values. Compute current-current correlation function coefficients and . Verify relation to field theory, and field theory identities.
For 4d SCFTs, current correlators and central charges all given in terms of 't Hooft anomalies = Chern-Simons terms in AdS5 . We did not directly compute these CS terms from reducing IIB on Y (would require going beyond our linearlized analysis). But did verify that these identities are ensured to hold in the effective AdS5 bulk theory, thanks to the real special geometry of 5d, N=2 SUGRA (partially overlaps with recent paper of Tachikawa).
implies the field theory identities, e.g. (expected: same SU(2,2|1) symmetry)
Conclusions and open questions:
a-maximization and extensions suggest strongest forms of 4d a-theorem, that RG flow = gradient flow of an a-function, with positive definite metric. Can the remaining loophole in our "almost proof", related to accidental symmetries, be closed?
A vast landscape of 4d SCFTs and flows; explore using a-max'n.
A new method to determine U(1)R: minimize . Why same as a-maximization in 4d? Also Rel'n to Z-minimization in geometry. Recent Butti & Zaffaroni work relates Z(b)= 1/atrial (R), even before extremizing!
Candidate decreasing c-function for 3d SCFTs? Unfortunately, minimization does not seem to help.