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RESOLVING SINGULARITIES IN STRING THEORYPowerPoint Presentation

RESOLVING SINGULARITIES IN STRING THEORY

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### RESOLVING SINGULARITIES IN STRING THEORY

CERN, Oct. 3, 2007.

Finn Larsen

U. of Michigan and CERN

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.

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.

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.

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.

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

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.

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.

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.

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

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

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:

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

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.

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

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)

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!

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

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

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:

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.

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.

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

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

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.

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

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

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).

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.

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

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.

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