Instantons in string theory via F-theory

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1 Instantons in string theory via F-theory Andrés Collinucci ASC, LMU, Munich Padova, May 12, 2010 arxiv: in collaboration with R. Blumenhagen and B. Jurke

2 Outline 1. Intro: From string theory to M-theory 2. IIB strings and F-theory 3. Instantons

3 From particles to strings Main postulate: Particles are actually tiny strings Different species of particles are strings in different states: Spin 0 (scalar) Spin 1 (photon) Spin 2 (graviton)

4 Interactions The Feynman diagram expansion in loops of QFT is replaced by an expansion in surfaces with holes: Number of loops number of holes. What is the coupling constant g s?

5 Fields from strings Coherent states of infinitely many low energy strings can be treated as quantum fields. Exponentiate coherent state field configuration The low energy effective actions: 10-d Supergravity theories. Field content: scalars, vectors, gravitons, p-forms and fermions. Coherent state of scalar particles represented by field φ (dilaton). It turns out that g s = e φ = string coupling is dynamically determined string coupling can vary over space

6 Please add a dimension Bosonic content IIA supergravity (in 10-d): g µν ; φ ; C µ... These fields can be encoded into 11-d supergravity compactified on a circle with the following content g (11) mn Kaluza-Klein reduction: Split up dimensions m (z, µ): ( ) g mn (11) g zz (11) g zµ (11) ( ) e φ C = g zµ (11) g µν (11) = µ, C µ g µν Crucial fact: Radius of circle is measured by e φ = g s

7 Solitons: D6-branes String theory can have defects/solitions magnetic monopoles. E. g. the D6-brane in d = 10 is like a magnetic particle in d = 4. ) 6 ds 2 = (1 + M 1 /r) ( dt 1/2 2 + i=1 + (1 + M 1 /r) 1/2 ( dr 2 + r 2 dω 2 ) 2 dx 2 i e φ = e φ 0 (1 + M 1 /r) 3/4 C µ = (0, A). String coupling varies in space. It decreases as r 0. Main feature: The D6-brane backreacts on (g µν, φ, C µ ).

8 Strong coupling 11d Uplift to 11 dimensions: (g µν (10), φ, C µ (10) ) g mn (11) 6 ds11 2 = dt 2 + dxi 2 + (1 + M 1 /r) (dr 2 + r 2 d Ω 2 2) i=1 + (1 + M 1 /r) 1 (dx 10 + A φ d φ) 2. Circle vanishes as r 0.

9 Taub-NUT The D6-brane lifts to a pure geometry: The Taub-NUT space. 1 S D6 brane R 3

10 Circle fibration Sphere can be seen as a circle-fibration over the interval.

11 D7-branes as cosmic strings IIA s sister theory: IIB string theory. Has odd-dimensional defects. D7-brane in d = 10 is like a magnetic cosmic string in d = 4. B = B 3d 2d cylindrical + time translation symmetry can work in 2 dimensions.

12 The axion D6 in d = 10: monopole in d = 3 backreacts on (g µ,ν, φ, C µ ) D7 in d = 10: cosmic string in d = 3 monopole in d = 2 backreacts on (g µ,ν, φ, C (0) ) C (0) is vector potential. Q mag = = C (0) C (0) + 1 as θ θ + 2 π. B ˆr dθ = 2 π S 1 θ C (0). S 1

13 The axio-dilaton Define the complex scalar τ C (0) + i e φ. Lagrangian of IIB SUGRA: L R zτ z τ 2 (Iτ) 2. Conjectured exact symmetry of IIB: τ a τ + b c τ + d, for a, b, c, d Z and a d c d = 1. SL(2, Z), S-duality group. Same as modular group of the torus.

14 Torus-fibration!(z) D7 z compact space F-theory encodes compactification + axio-dilaton data into one dimensional geometric object.

15 Orientifolds from Sen s limit Around 7-branes, g s is typically running wild non-perturbative. Contact with IIB strings? Yes, via Sen s limit: By taking CY 4 to specific point in cplx. str. moduli space, can make contact with IIB with O7-planes and D7-branes only. T 2 ı CY 4 π B 3 CY 3 /Z 2 Base of fibration becomes quotient of IIB CY threefold.

16 D3-instantons as M5-branes Goal of the project: To understand the zero-modes of E3-instantons in IIB. Motivation: Moduli stabilization Yukawa couplings Instantons are not a choice! Witten s take on it: Lift E3 in IIB to M5 in M-theory via M/F-duality. Zero-modes are counted by Hodge numbers h 0,i (M5)

17 D3-instantons as M5-branes T 2 ı M5 CY 4 π E3/Z 2 CY 3 /Z 2 E3 CY 3 /Z 2 E3-brane wraps an internal 4-cycle in CY 3. M5-brane wraps an internal 6-cycle in CY 4. M5 brane is elliptically fibered over Z 2 -quotient of E3 brane drawing

18 Counting zero-modes: Witten s criterion Witten s approach: Count fermions on M5-brane in absence of fluxes Necessary condition: χ(m5) = 3 i=0 ( 1)i h 0,i (M5) = 1 Sufficient condition: all h 0,i (M5) = 0, i 0 Difficulties: Would still like to work in IIB: α and pert. g s corrections, DBI fluxes,... not available in M/F-theory. Duality from IIB orientifolds to F-theory incomplete until recently. Only very few dual pairs known. h 0,3 (M5) = geometric moduli of E3. No interpretation for h 0,1, h 0,2.

19 Alternative criteria in IIB Conditions for IIB and IIA instatons via CFT and quiver methods [Blumenhagen, Cvetic, Richter, Weigand], [Argurio, Bertollini, Franco, Kachru] Neutral zero-modes Instanton must be rigid: h 0,2 (E3) = 0. No Wilson line zero-modes: h 0,1 (E3) = 0. No τ zero-mode only Z 2 -invariant E3 O(1) gauge gp. D7/E3 intersections charged zero-modes W Φ matter e A Questions Can interpret h 0,i (M5) in IIB language? Charged zero-modes from M5 perspective?

20 IIB to F-theory lift: Example CY 4 = u 1 u 2 u 3 u 4 x y z deg π CY 3 /Z 2 = P 3 CY 3 = P [8] /Z 2 CY 3 : Deg 8 hypersurface in P with (u 1, u 2, u 3, u 4, ξ) (λ u 1, λ u 2, λ u 3, λ u 4, λ 4 ξ) λ C Z 2 involution: ξ ξ

21 E3 to M5 lift M5 on degree n divisor CY 4 π E3/Z 2 CY 3 /Z 2 /Z 2 E3 on degree n divisor CY 3 E3 on a divisor of degree n in CY 3 divisor of degree n in P 3 divisor of degree (n, 0) in CY 4

22 Naïve counting Geometric moduli: h 0,3 (M5)? = h 0,2 (E3). what did we miss? Orientifold Z 2 action 1 6 (n n) + n (n n) 1 H 0,i (E3) = H 0,i (E3) + H 0,i (E3) h 0,2 h 0,1 (E3) = allowed geom moduli h0,2(e3) : projected out (E3) = allowed Wilson lines h0,1(e3) : projected out + +

23 Z 2 -graded cohomology Can compute Z 2 -graded cohomology with equivariant index theorems. Holomorphic Lefschetz fixed point theorem: ( ) i (h 0,i + h0,i ) = 1 4 D O7 DE3 2 i h 0,2 (E3) = h0,3 (M5) h 0,2 + (E3) = h0,2 (M5) bosons + fermions fermions Lift more complicated examples [A.C.], [Blumenhagen, Grimm, Jurke, Weigand] : Resolved P [8] h 0,1 (E3) h0,2 (M5) h 0,1 + (E3) h0,1 (M5) Quintic blown-up at two points τ zero mode corresponds to h 0,0, contributes to h0,1 (M5)

24 Z 2 -graded cohomology h 0,3 (M5) h 0,2 (E3) h 0,2 + (E3) h 0,1 (E3) h 0,1 + (E3) h 0,0 (E3) h 0,0 + (E3) h 0,2 (M5) h 0,1 (M5) h 0,0 (M5) All h 0,i (M5) are accounted for. What about charged zero-modes?

25 Charged zero-modes Charged zero-modes If E3 intersects D7 charged zero-modes [B, C, R, W] charged superpotential W Φ matter e A. Can spoil mod stab, can create Yukawa s. Counting In principle, counted by h 0,i (E3 D7, F E3 F D7 ). More generally, charged zero-modes inevitable if genus(e3 D7) 0. Singularity caveat: Generic Z 2 -invariant D7 has singular shape. Whitney s umbrella [A.C., Denef, Esole] D7 : η 2 ξ 2 χ = 0 singular at η = ξ = 0. Must first resolve singularities before counting via blow-up.

26 3-cycles Empirical result from example: h 0,2 Terms on l.h.s. : (E3) + h1,1 1. allowed geometric moduli 2. allowed B-field moduli (E3) + h0,1 (E3 D7) = h1,2 (M5) 3. Z 2 -odd one-cycles from blown-up E3 D7 intersection all contribute to 3-cycles of M5. Charged modes don t contribute to Witten s h 0,i (M5)!

27 3-cycles S 1 -fiber 2-chains E3 D7 E3 D7 1-cycles trivial outside the intersection 2-chains end on them. One S 1 from T 2 -fibration collapses over the D7-brane 3-cycle.

28 Partition function Physical significance of 3-cycles? M5-brane has a self-dual 3-form T 3 on it s worldvolume. It satisfies G 4 M5 = dt 3, where G 4 = dc 3 from bulk SUGRA. Witten showed partition function Z(T 3 ) is a section of a line bundle over moduli space of bulk SUGRA C 3. L π Z(T 3 ) section of L H 3 (M5, R)/H 3 (M5, Z) Presence of 3-cycles effective superpotential can vanish along points in C 3 moduli space.

29 Conclusions Introduced F-theory geometrizes 7-branes of IIB string theory. E3-instantons important for understanding vacuum structure of the theory. Witten: Zero-modes of E3 via zero-modes of M5. Results h 0,i + (E3) h0,i (M5), h 0,i (E3) h0,i+1 (M5). τ zero-mode h 0,1 (M5). Charged zero-modes 3-cycles of M5.

30 Appendix: M/F-theory duality Unlike M-theory, extra two dimensions are not physical. However, by T-duality, one can make sense of it through M-theory. F theory 12d M theory 11d T 2 T 2 IIB 10d IIA 9d T duality

31 Appendix: Whitney s umbrella

32 Appendix: Complete picture

33 Appendix: dp 2 7 vs. dp2 6 dp 7 dp 7 dp 6 dp 6 P 1 O7 O7 Blow-up quintic CY at two points: x 1 x 2 x 3 x 4 x 5 v 1 v 2 deg FI r r r 3 r 1 < 0: 2 non-intersecting dp 6 F-theory lift: One M5 with all h 0,1 = 1 τ zero-mode. r 1 > 0: 2 intersecting dp 7 s F-theory lift: One M5 with all h 0,i = 0. τ zero-mode somehow lifted!