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Theory Seminar CERN, Oct. 3, 2007. RESOLVING SINGULARITIES IN STRING THEORY. 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, .…..

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Resolving singularities in string theory

Theory Seminar

CERN, Oct. 3, 2007.

RESOLVING SINGULARITIES IN STRING THEORY

Finn Larsen

U. of Michigan and CERN


Introduction
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
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
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
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
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 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
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 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 ads 3
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
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
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
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
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 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
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
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
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
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
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
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
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
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
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 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
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
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
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
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
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
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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