Insights into Calabi-Yau Threefolds and Black Hole Microstates

Mock modularity of Calabi-Yau threefolds Khalil Bendriss Laboratoire Charles Coulomb, Montpellier October 2024 Based on joint work with Sergey Alexandrov to appear soon

Black holes Thermodynamical objects! [Bekenstein 72, Hawking 74] Entropy ? log ? : Black hole microstates. How to compute ? 1

String theory String theory lives in 10 dimensions Compactify on a 6d manifold ?. Often ? is a Calabi-Yau threefold. A theory of extended objects D?-brane: ? + 1 dimensional object 2

Type IIA compactified on a Calabi-Yau X Branes wrapping cycles on X Donaldson-Thomas (DT) invariants. 4d N=2 SUGRA Modularity properties BPS black holes Very rigid constraints Generating function Count # of Black Hole microstates (entropy). ? = ?? ?2?? ?? Allow to (almost) fix ? 3

Outline I. Modularity II. DT invariants III. Constraining the generating function

Outline I. Modularity II. DT invariants III. Constraining the generating function

Tori, lattices and the modular group ? is a modulus of the torus (with ? > 0). Keeps the torus invariant Preserves the orientation ? ? ? ? ??(2, ) The modular group ? ?? + ? ?? + ? Modularity ? 4

Modular forms Properties Characterestics ? ?(?) holomorphic ? ?? + ? ?? + ? = ?? + ?? ?(?) ? ? is the weight. Modular forms have a Fourier expansion: ???2?? ? ? ? ? = ?=?0 1 0 1 1 ? ? + 1 = ? ? Modular forms of fixed weight ? form a finite dimensional vector space. 5

Mock modular Modular forms Properties Characterestics ? ?(?) holomorphic ? ? ? : Modular form of weight 2-k ? ? is the weight. ? ? is the shadow. ?? + ? ?? + ? ?( ?) ? ?? ?? = ?? + ??? ? ? ?/? ? ?( ?) ? ?? ?? ? ?, ? = ? ? They have a completion: ? Two mock modular forms of fixed weight ? and shadow ? are related by a modular form. 6

Depth 1 Mock modular Modular forms Properties Characterestics ? ?(?) holomorphic ? ? ? : Modular form of weight 2-k ? ? is the weight. ? ? is the shadow. ?? + ? ?? + ? ?( ?) ? ?? ?? = ?? + ??? ? ? ?/? ? ?( ?) ? ?? ?? ? ?, ? = ? ? They have a completion: ? Two mock modular forms of fixed weight ? and shadow ? are related by a modular form. 7

Depth ? Mock modular Modular forms Properties Characterestics ? ?(?) holomorphic ? ? ? : Depth (? 1) mock modular form of weight 2-k ? ? is the weight. ? ? is the shadow. ?? + ? ?? + ? ?( ?) ? ?? ?? = ?? + ??? ? ? ?/? ? ?( ?) ? ?? ?? ? ?, ? = ? ? They have a completion: ? Two (higher depth) mock modular forms of fixed weight ? and shadow ? are related by a modular form. 8

Outline I. Modularity II. DT invariants III. Constraining the generating function

The Donaldson-Thomas (DT) invariants DT invariants Count # of Black Hole microstates (entropy). ? Type IIA compactified on a Calabi-Yau X We restrict to ?2= 1. Count D6-D4-D2-D0 brane bound states ? = ?0,?,?,?0 9

Defining the generating functions Rank 0 DT invariants Spectral flow symmetry. ?0= 0 Modular properties!! D6-brane charge ? = 0, ,??-1 ? ? = (0,?,?,?0) (?, ?0) Generating function ?0 ?,? ?0? 2?? ?0? ?,?? = Define rational invariants 1 ??? ?0 ?0 ?,?( ?0) ? = ?2 ( ? ?) ? ? ? labels different functions ? is a vector index. 10

A torus in type IIA ? M-theory /?1 Type IIA ?2 Type IIA M-theory /? ?1 /? ?1 ?1 Symmetry under action of the modular group Moduli space (vector multipliet) Instanton corrections D4-branes/ 4 ?1 Pointlike (instanton) Donaldson-Thomas invariants. Imposes modular constraints on ? 11

Outline I. Modularity II. DT invariants III. Constraining the generating function

Modularity of ? 1,?(?) is a VV modular form For p=1 For p=1 ?,?(?) is a depth (? 1) VV mock modular form Higher p [S.Alexandrov, B.Pioline 18] Completion equation ? ? (?1, ,??)(?2) ?,??, ? = ??,?1, ,?? ??,??(?) Let s look at an example ?=1 ?1+ +??=? ?=1 ?2= ??(?) 12

The modular ambiguity ? = 2 Example: (?1,?2)(?2) 1,?1 1,?2 2,??, ? = 2,?? + ??,?1,?2 Problem: the equation doesn t fix 2 completely. Solution: compute a few DT invariants (specifically the polar terms) and fix the modular ambiguity 13

Two-step approach (??)+ ?,? (0) ?,?= ?,? Any solution The modular ambiguity Strategy: (??) Can we perform step 1 for all ? ? 1. Find a solution ? 2. Compute a few DT invariants and fix ? 0 Challenge: the completion equation for ? depends on ?? 0 for lower charges. (?1,?2)?2( 1,?1 (??)+ 1,?2 (0))( 1,?2 (??)+ 1,?1 (0)) 2,??, ? = 2,?? + ??,?1,?2 0 for all ?? ?. Result: Recipe to compute ? up to ?? (Using indefinite theta series) 14

Conclusions DT invariants of the Calabi-Yau count the number of BPS black hole microstates. Generating functions of these invariants at rank 0, posess remarkable modular properties. Mock modular We fix these functions, by solving their modular anomaly, up to computing a finite number of DT invariants. Further directions: Compute polar terms to fix ? [S. Alexandrov, S.Feyzbakhsh, A.Klemm, B.Pioline, T.Schimannek 23]) Generalize the construction for ?2> 1. 0(done for ? = 1 for eleven CYs

Type IIA compactified on a Calabi-Yau X Branes wrapping cycles on X Donaldson-Thomas invariants. 4d N=2 SUGRA 4d N=2 SUGRA Modularity properties BPS black holes Very rigid constraints Generating function Count # of Black Hole microstates (entropy). ? = ?? ?2?? ?? Allow to (almost) fix ?

Appendix

Disentangling the ambiguity Ansatz to extract the dependance of the generating functions on lower rank ambiguities ? ? (?1, ,??)(?) 0 ?,??, ? = ??,?1, ,?? ??,?? (?) ?=1 ?1+ +??=? ?=1 ?= ??? Anomalous coefficients ??? We trade the conditions on ? for conditions on ??,?? ??

The new completion equation ??1, ,??is a VV depth (? 1) mock modular form For? charges ?1, ,?? Completion equation ? ????+1, ,???+1 ??1, ,??= ??? ??1, ,?? ??=? ?=1 (??, ,??)(?) Goal: find the anomalous coefficients ??,??, ,??

Studying ? = 2 ? = 2 Example: (?1,?2)(?2) ?1,?2 = ??,?1,?2 ?1,?2+ ??,?1,?2 ??,?1,?2 Indefinite theta series

Definite theta series ? ?? ? ? ? ??? = ? +? is a ? dimensional lattice. It has quadratic form ? ? 2 ? is negative definite ??? is a Vector valued modular form of weight ?/2

Indefinite theta series 2?2 ? ? ?? ? ? ? ??? = ? +? is a ? dimensional lattice. It has quadratic form ? ? 2 ? is indefinite !!!! Kernel: ensures convergence. Holomorphic Modular When a vector ?? is null then we can have both! (Product of) difference of sign functions. (Product of) difference of error functions. ?1 ? ?1 ??? ???? ?1 ? ?1 ? ?1 ?2 ? ?2 (x) (???? ?1 ? ????(?2 ?)) ??? ???

Studying ? = 2 ? = 2 Example: (?1,?2)(?2) ?1,?2 = ??,?1,?2 ?1,?2+ ??,?1,?2 ??,?1,?2 (?1,?2): Positive definite theta series ??,?1,?2 on a 1 dimensional lattice with kernel ??? ?1 ? ???? ?1 ? Our lattice doesn t have null vectors. Extend the lattice. ?1,?2 to be an indefinite Choose ??,?1,?2 theta series with kernel ???? ?1 ? ???? ?1 ? where Q ?1 = 0.

Recipe for solution: Extend to solve. Black hole microstates ??? Ansatz Compute few DT invariants ??1, ,??(?) Refinement Limit ? 0 ??1, ,??(?,?) Lattice extension Modular derivatives ??1, ,??(?,?,{??}) Solve the equation with indefinite theta series

Vector-Valued(VV) Modular forms Properties Characterestics ? ?(?) holomorphic ? ? ? ? ? ? = ?? + ? ?? + ? ? is the weight. ??? is the multiplier system. = ?? + ?? ?? ???? ??(?) ? Modular forms have a Fourier expansion: ??,???, ? = ?2?? ? ??? = ?=?0 VV Modular forms of fixed weight ? and multiplier system ??? form a finite dimensional vector space.

Jacobi-like Modular forms Properties Characterestics Automorphy factor ? ?(?) holomorphic ? 2?? ? ? ?2 ??+? ?? + ? ?? + ?, ? ? is the weight. ? is the index. = ?? + ?? ? ? ?(?,?) ?? + ? Jacobi-like forms have a series expansion in ? : ??(?) ??, ? ?,? = ?>?0 The function ??0(?) is a weight (? + ?0) modular form.

Modularity recap Term Math. Object Charact. ?(?) Weight ? Modular form ??(?) Multiplier system ??? VV modular form ? ?,? ;?(?,?1,?2) Index m; indices ?1,?2 Shadow g(?) Jacobi-like form ?(?, ?) Mock modular form ? ?

Modularity recap Term Math. Object Charact. ?(?) Weight ? Modular form ??(?) Multiplier system ??? VV modular form ?(?, ?) Shadow g(?) Mock modular form ? ? ? 2 ?0> 0 ?? ? ?0= 0 ?? ?? 1 ?0< 0 ?? ?? Modular forms offer control on the growth of their Fourier coefficients ?