RESOLVING SINGULARITIES IN STRING THEORY

1. Theory Seminar CERN, Oct. 3, 2007. RESOLVING SINGULARITIES IN STRING THEORY Finn Larsen U. of Michigan and CERN

2. INTRODUCTION • Consider a warped geometry such as the KK-compactification • Superficially similar metrics describe black holes, RS-throats, FRW-cosmology, .….. • In each setting we often consider situations where the conformal factors U1,2 become large somewhere on the base space. • Then we should ask if the geometry is described accurately by conventional (super)gravity.

3. EXAMPLE: 5D BLACK HOLE • Geometry of 5D black hole • The scale factor U diverges at the horizon. • For conventional black holes the curvature is finite at the horizon so the geometry is smooth. • Then corrections to the solutions are small. • For “small black holes” the horizon size vanishes in the classical approximation, curvature diverges, and “corrections” are important.

4. TOY MODELS? • Much exploratory work has focused on toy models of the higher derivative interactions. • A popular toy model: • General limitation: there could be other terms at the same order so results cannot be trusted. • The challenge: compute all terms at four derivative order, then solve the corresponding equations of motion.

5. A CONCEPTUAL DIFFICULTY • Suppose we actually determined all the higher derivative corrections up to some order: • Then: if the leading order solution is regular, the corrections can be computed systematically. • The problem: if the corrections change the solution qualitatively (like resolving a singularity) then the “corrections” are as important as the “leading order” terms. • In other words: there is no systematic expansion parameter and so we must generally keep all orders in the Lagrangian, an impossible task.

6. OUR APPROACH • This talk: 5D black strings with AdS3xS2 near string geometry. • Anomalies determine the sizes of the AdS3 and S2 geometries which are then one loop exact. • Explicit computation of all terms to a given order: using a supersymmetric action. • Find explicit solution: exploit off-shell SUSY. • Discussion and applications. REFS: A. Castro, J. Davis, P. Kraus, and FL, hep-th/0702072, 0703087, 0705.1847 P. Kraus and FL: hep-th/0506176, hep-th/0508218 P. Kraus, FL, and A. Shah: hep-th/0708.1001

7. THE EXAMPLE: COSMIC STRINGS • The ansatz for a string solution is • The scale factors U1,2 diverge at the horizon. • Generically, the geometry is nevertheless regular. • An important case the matter supporting the string solution are the two-form B and the dilaton Ф, as in perturbative string theory. • This case is singular at the string source.

8. THE SETTING • Consider M-theory on CY3 x R4,1. For a small CY3 the theory is effectively D=5. • In the supergravity approximation M-theory reduces to N=2 SUGRA in D=5. • Solutions that are magnetically charged with respect to the vector fields AI are black strings. • Such solutions generically have AdS3 x S2 near horizon geometry.

9. THE SETTING: MORE DETAILS • The higher dimensional interpretation of these string solutions: they are M5-branes wrapped on 4-cycles P in CY3 . • The cIJK are the intersection numbers of the basis cycles PI. • The magnetic strings have AdS3 x S2 near horizon geometry with scale set by the self-intersection number of the M5-brane cIJKpIpJpK≠0. • Special case: the CY is K3 x T2 and the M5-brane wraps the 4-cycle P=K3. This solitonic string is the type IIA dual of the heterotic string. • The dual heterotic strings have singular near horizon geometry in the supergravity approximation since cIJKpIpJpK=0.

10. THE SIGNIFICANCE OF ADS3 • The global symmetry group of AdS3 is • Diffeormorphism symmetry enhances each of the SL(2)’s acting on the boundary at infinity to a Virasoro algebra. • Explicit computation from the standard (two-derivative) Einstein action determines the spacetime central charges • The central charge measures the number of degrees of freedom in the boundary theory but in the bulk it is essentially the size of AdS3. Brown-Henneaux

11. ANOMALY INFLOW • N=2 supergravity has a gravitational Chern-Simons term : • The interaction violates gauge symmetry and/or diffeomorphism invariance, but only by a total derivative. • Anomaly inflow: symmetries are preserved in full theory so boundary CFT anomalies must agree precisely with spacetime noninvariance. • This condition determines the boundary central charges • These expressions are exact because the underlying symmetries must be exact. Maldacena, Strominger, Witten Harvey, Minasian, Moore Kraus, FL

12. ASIDE: BLACK HOLE ENTROPY • A major string theory triumph: the black holes entropy is accounted for by string theory microstates. • Why does this work? • Central charges must agree on two sides because of anomaly inflow upholding diffeomorphism invariance and supersymmetry! • Black holes arise as excitations of the magnetic string considered here so the agreement of entropies follows from Cardy’s formula:

13. RESOLUTION OF SINGULARITIES • The dual heterotic string: CY=K3 x T2, P=K3 (so M5 wraps the K3). • The intersection number CIJKpIpJpK=0 so the central charges are linear in the magnetic charges • Since c2(K3)=24 we have cL = 12p and cR=24p. • These are the correct values for p heterotic strings in a physical gauge (Left movers=8B+8F, Right movers=24B). • The central charge measures the scale of AdS3 and S2. Its only contribution is from the higher derivative terms; so a singularity has been resolved. Dabholkar Kraus, FL

14. EXPLICIT SINGULARITY RESOLUTION • So far: the resolution of a singularity was inferred from an indirect argument. • A weakness: we assume AdS3xS2 near-string geometry and then consistency demands nonvanishing geometric sizes. • Motivation for assumption: fundamental string should have world-sheet CFT and so an AdS3dual, and SU(2) R-symmetry motivates S2. • Superior to the indirect story: construct asymptotically flat solutions directly. • This is what we turn to next.

15. THE NEED FOR OFF-SHELL SUSY • The essential interaction is the anomalous Chern-Simons term • SUSY then determines all other four-derivative terms uniquely. • Complication: on-shell SUSY closes on terms of ever higher order. • Resolution: use the off-shell (superconformal) formalism. • Unfamiliar feature: the Weyl multiplet (gravity) has auxiliary two-tensor vab and scalar D. Hanaki, Ohashi, Tachikawa

16. SUSY VARIATIONS • The off-shell action is invariant under the SUSY transformations • Simplification: these variations are symmetries of each order in the action by itself. • BPS conditions: these variations must vanish when evaluated on the solution. (gravitino) (gaugino) (auxiliaryWeyl)

17. THE BPS SOLUTION • Assume that the metric takes the string form: • The BPS conditions impose U1=U2 and determine the auxiliary fields: • Also, the magnetic fields are determined by the scalars (the attractor flow) • The scalar fields MI and the metric function U are not determined by SUSY alone - they depend on the action!

18. CHARGE CONSERVATION • The scalar fields MI are generally determined by the solving the Maxwell equations. • However, the magnetic field strength is exact because it is topological • Imposing the Bianchi identity (which is not automatic for the solution to the BPS condition) gives a harmonic equation • With the standard solution

19. OFF-SHELL SUGRA: THE LEADING ORDER • Leading order supergravity, in off-shell formalism: • The equation of motion for the auxiliary D-field gives the familiar special geometry constraint: • Eliminating also the auxiliary v-field gives the standard on-shell action • where

20. OFF-SHELL SUGRA: FOUR DERIVATIVES Hanaki, Ohashi, Tachikawa • SUSY completion of the 5D Chern-Simons term • Definition of Weyl tensor: • Covariant derivatives include additional curvature terms such as:

21. DEFORMED SPECIAL GEOMETRY • Status: the solution has been specified in terms of the metric factor U which is still unknown. • The equation of motion for the the D-field: • Evaluated on the solution • This is an ordinary differential equation for the metric factor U since HI=1 + pI/2r is a given function. • Interpretation: the special geometry constraint has been deformed.

22. NEAR STRING ATTRACTOR • The constraint can be solved analytically near the string where • Result: the size of the S2 is • The relation U1 = U2 determines the AdS3 radius • Note: the near horizon geometry remains smooth in the singular case cIJKpIpJpK=0 as long as c2IpI is nonvanishing.

23. C-EXTREMIZATION • The central charge is the trace anomali which is the bulk on-shell action, up to known constants of proportionality. • So: compute on-shell action for our ansatz with (V, D, lA, lS,m) unspecified (m defined by MI=mpI and 6p3=cIJKpIpJpK) • Consistency: extremizing c relates (V, D, lA, lS,m) as found previously. The value of c at the extremum gives • This agrees with the anomaly inflow. The agreement relies on most terms in the four-derivative action. Kraus, FL

24. THE RESOLVED SINGULARITY • Now: analyse the differential equation for U in the singular case. • The attractor has r~p1/3 but the entire region r<<p is described by a p-independent equation Red: analytical expansion around near string attractor. Blue: numerical solution. • Upshot: extends smoothly away from the near string attractor

25. THE SPURIOUS MODES • The numerical solution also attaches smoothly to the analytical expansion around flat space. • The quasiperiodic behavior is due to spurious modes, a characteristic of solutions to higher derivative theories. • This unphysical artifact is generally present even in flat space but can be removed by a field redefinition. Sen Hubeny, Maloney, Rangamani Blue: numerical solution extended to larger distances. Green: analytical expansion around flat space.

26. THE DUAL OF THE HETEROTIC STRING • Dualizing our solution to the heterotic frame we find that p heterotic strings have near string AdS3 x S2 with • The space is of string scale but we can still ask: what is the AdS/CFT dual to this space? • It must be a D=1+1 CFT with (0,8) SUSY and R-symmetry at least SU(2), presumably based on supergroup OSp(4*|4). • Puzzle: no SCFT with these symmetries exists! They are not consistent with the Jacobi identities. Lapan, Simons,Strominger

27. NONLINEAR ALGEBRAS? Henneaux, Maoz, Schwimmer Lapan, Simons,Strominger Kraus, FL • Suggested resolution: there exists nonlinear superconformal algebras with the correct symmetries! • Nonlinearity: the OPEs include current bilinears • Notation: • The nonlinear superconformal algebras are powerful but unfamiliar relatives to W-algebras. • We consider multistring states so the suggestion is that NSCAs are important in string field theory. Bershadsky Knizhnik

28. QUANTUM CORRECTIONS TO AdS/CFT • Intriguing fact: nonlinearities determine the central charge. For example, for OSp(4*|4) • Classical limit (large k) gives the Brown-Henneaux formula. Since k~N~1/g2 the nonlinear algebra determines the quantum corrections to all orders! • Warning: there are presently a number of loose ends in this story. • The biggest problem: it seems that non-unitary representations play a central role (the central charge is negative).

29. MANY MORE EXAMPLES • This talk: just 5D string solutions with AdS3 x S2 near string geometry. But techniques apply in many other examples. • Black holes in 5D with AdS2 x S3 near horizon geometry. There are electric charges and so the Maxwell equations are non-trivial. • Rotating 5D black holes. The solution is much more complicated (all terms in the four-derivative action contribute) but still explicit. • Black holes on Taub-NUT base space. A smooth interpolation between asymptotically flat 4D and 5D spacetimes. • Upshot: we check various indirect arguments explicitly, sort out discrepancies in those arguments, and find new results.

30. EXAMPLE: 5D CALABI-YAU BLACK HOLES • Based on 4D one loop corrections, the quantum corrections to 5D Calabi-Yau black holes were conjectured as: • Our explicit solution: • Understanding of dicrepancy: 4D charges are 5D charges as well as a R2-contribution from the interpolating Taub-NUT geometry • Topological strings is a powerful technique for computing quantum corrections to holomorphic quantities in 4D. The strong coupling limit is effectively 5D. It appears to confirm the 1/8 shift. Guica, Huang, Li, Strominger Huang, Klemm, Marino, Taranfar

31. SUMMARY • Challenge for higher derivative corrections: keep all terms at a given order. • Additional challenge for singularity resolution: understand why there are no further corrections. • Our example: controlled by anomalies (so no further corrections) and we employ the complete supersymmetric action (so all important terms are kept). • Main example: explicit construction of dual fundamental string with asymptotically flat boundary conditions.