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

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 U 1,2 become large somewhere on the base space.  Then we should ask if the geometry is described accurately by conventional (super)gravity.  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 U 1,2 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.  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.  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.  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 AdS 3 xS 2 near string geometry.  Anomalies determine the sizes of the AdS 3 and S 2 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.  This talk: 5D black strings with AdS 3 xS 2 near string geometry.  Anomalies determine the sizes of the AdS 3 and S 2 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/ , , P. Kraus and FL: hep-th/ , hep-th/ P. Kraus, FL, and A. Shah: hep-th/

THE EXAMPLE: COSMIC STRINGS  The ansatz for a string solution is  The scale factors U 1,2 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 ansatz for a string solution is  The scale factors U 1,2 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 CY 3 x R 4,1. For a small CY 3 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 A I are black strings.  Such solutions generically have AdS 3 x S 2 near horizon geometry.  Consider M-theory on CY 3 x R 4,1. For a small CY 3 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 A I are black strings.  Such solutions generically have AdS 3 x S 2 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 CY 3.  The c IJK are the intersection numbers of the basis cycles P I.  The magnetic strings have AdS 3 x S 2 near horizon geometry with scale set by the self-intersection number of the M5-brane c IJK p I p J p K ≠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 c IJK p I p J p K =0.  The higher dimensional interpretation of these string solutions: they are M5-branes wrapped on 4-cycles P in CY 3.  The c IJK are the intersection numbers of the basis cycles P I.  The magnetic strings have AdS 3 x S 2 near horizon geometry with scale set by the self-intersection number of the M5-brane c IJK p I p J p K ≠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 c IJK p I p J p K =0.

THE SIGNIFICANCE OF ADS 3  The global symmetry group of AdS 3 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 AdS 3.  The global symmetry group of AdS 3 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 AdS 3. 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.  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:  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 T 2, P=K3 (so M5 wraps the K3).  The intersection number C IJK p I p J p K =0 so the central charges are linear in the magnetic charges Since c 2 (K3)=24 we have c L = 12p and c R =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 AdS 3 and S 2. Its only contribution is from the higher derivative terms; so a singularity has been resolved.  The dual heterotic string: CY=K3 x T 2, P=K3 (so M5 wraps the K3).  The intersection number C IJK p I p J p K =0 so the central charges are linear in the magnetic charges Since c 2 (K3)=24 we have c L = 12p and c R =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 AdS 3 and S 2. 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 AdS 3 xS 2 near-string geometry and then consistency demands nonvanishing geometric sizes.  Motivation for assumption: fundamental string should have world-sheet CFT and so an AdS 3 dual, and SU(2) R-symmetry motivates S 2.  Superior to the indirect story: construct asymptotically flat solutions directly.  This is what we turn to next.  So far: the resolution of a singularity was inferred from an indirect argument.  A weakness: we assume AdS 3 xS 2 near-string geometry and then consistency demands nonvanishing geometric sizes.  Motivation for assumption: fundamental string should have world-sheet CFT and so an AdS 3 dual, and SU(2) R-symmetry motivates S 2.  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 v ab and scalar D. 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 v ab 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. 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) (auxiliaryWeyl) (gaugino)

THE BPS SOLUTION Assume that the metric takes the string form: The BPS conditions impose U 1 =U 2 and determine the auxiliary fields: Also, the magnetic fields are determined by the scalars (the attractor flow) The scalar fields M I and the metric function U are not determined by SUSY alone - they depend on the action! Assume that the metric takes the string form: The BPS conditions impose U 1 =U 2 and determine the auxiliary fields: Also, the magnetic fields are determined by the scalars (the attractor flow) The scalar fields M I and the metric function U are not determined by SUSY alone - they depend on the action!

CHARGE CONSERVATION The scalar fields M I 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 The scalar fields M I 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 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 SUSY completion of the 5D Chern-Simons term Definition of Weyl tensor: Covariant derivatives include additional curvature terms such as: SUSY completion of the 5D Chern-Simons term Definition of Weyl tensor: Covariant derivatives include additional curvature terms such as: Hanaki, Ohashi, Tachikawa

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 H I =1 + p I /2r is a given function. Interpretation: the special geometry constraint has been deformed. 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 H I =1 + p I /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 S 2 is The relation U 1 = U 2 determines the AdS 3 radius Note: the near horizon geometry remains smooth in the singular case c IJK p I p J p K =0 as long as c 2I p I is nonvanishing. The constraint can be solved analytically near the string where Result: the size of the S 2 is The relation U 1 = U 2 determines the AdS 3 radius Note: the near horizon geometry remains smooth in the singular case c IJK p I p J p K =0 as long as c 2I p I 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, l A, l S,m) unspecified (m defined by M I =mp I and 6p 3 =c IJK p I p J p K ) Consistency: extremizing c relates (V, D, l A, l S,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. 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, l A, l S,m) unspecified (m defined by M I =mp I and 6p 3 =c IJK p I p J p K ) Consistency: extremizing c relates (V, D, l A, l S,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~p 1/3 but the entire region r<<p is described by a p- independent equation Now: analyse the differential equation for U in the singular case. The attractor has r~p 1/3 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. 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. Blue: numerical solution extended to larger distances. Green: analytical expansion around flat space. Sen Hubeny, Maloney, Rangamani

THE DUAL OF THE HETEROTIC STRING  Dualizing our solution to the heterotic frame we find that p heterotic strings have near string AdS 3 x S 2 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.  Dualizing our solution to the heterotic frame we find that p heterotic strings have near string AdS 3 x S 2 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?  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.  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. Henneaux, Maoz, Schwimmer Lapan, Simons,Strominger Kraus, FL 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/g 2 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).  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/g 2 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 AdS 3 x S 2 near string geometry. But techniques apply in many other examples.  Black holes in 5D with AdS 2 x S 3 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.  This talk: just 5D string solutions with AdS 3 x S 2 near string geometry. But techniques apply in many other examples.  Black holes in 5D with AdS 2 x S 3 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 R 2 -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.  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 R 2 -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.  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.