F-theory with Quivers

Transcription

1 University of Trieste String Phenomenology 2017 (Virginia Tech) Based on work in progress with A. Collinucci, M. Fazzi and D. Morrison

2 Introduction In the M/F-theory geometric engineering, Abelian gauge symmetries emanate from reduction of C 3 along harmonic, normalizable 2-forms. C 3 A µdx µ ω The 2-form ω can be described via its Poincaré dual cycle (divisor). In F-theory, the elliptically fibered CY has extra sections, which are identified as new divisor classes giving rise to U(1)s. [Morrison,Vafa] Techniques expanded and refined over the past few years. [Grimm,Weigand; Morrison,Park; (Borchmann),Mayrhofer,Palti,Weigand; Cvetic,(Grassi),Klevers,Piragua,(Song),(Taylor); V.Braun,Grimm,Keitel; A.Braun,Collinucci,RV] In this talk, new way of detecting such divisors in varieties that admit small resolutions. We will focus on CY three-fold.

3 Introduction Simplest way to get a massless U(1) in F-theory is U(1) restriction : [Grimm,Weigand] (y a 3 )(y + a 3 ) = s(s 2 + b 2 s + 2b 4 ) where s x b 2 3 elliptic fibration has one conifold singularity at (s, y, a 3, b 4 ); after small resolution one sees the extra divisor D ω that intersects the exceptional P 1 at one point. P 1 D ω M2 couples to the 3-form: C 3 = M2 A µdx P µ ω = (P 1 D ω) 1 A µdx µ

4 Introduction U(1) restriction is so far also the only case where a matrix factorization (MF) has been worked out in F-theory. [Collinucci,Savelli] This formalism allows to deal with singular manifolds without resolution. In particular, a line bundle M on CY arises naturally. c 1 (M) ω related to the U(1) divisor. Identify massless matter charged under this U(1). Moreover, MF comes naturally with (NC) resolution and associated quiver. [Aspinwall,Morrison] Apply this formalism to more generic case with abelian gauge symmetries. This approach can give new insights.

5 Conifold In U(1) restriction, Weierstrass model factorizes as (y a 3 )(y + a 3 ) = s(s 2 + b 2 s + 2b 4 ) where s x b 2 3 y +y = s w with y ± = y ± a 3, w = s 2 + b 2 s + 2b 4 Non-Cartier divisors (y ±, s) extra section of elliptic fibration. After small resolution: (y ±, s) become Cartier divisors; one sees exceptional P 1 wrapped by M2 (charged states) Weak coupling limit: one U(1) brane and its orientifold image, intersecting away from O7 (where massless matter live).

6 Conifold - Matrix Factorization (MF) Eq y +y = s w admits a (pair of) MF, i.e. a pair of matrices (φ, ψ) s.t. φ ψ = ψ φ = (y +y s w)1 2 For the conifold: ( y s φ = w y + ) ( y+ s ψ = w y ) From φ, ψ one can define (MCM) modules over R [Eisenbud], e.g. M = coker(r 2 ψ R 2 ) R 2 / Imψ (where R = C[y +, y, s, w]/(y +y s w) is the coordinate ring. ) Line bundle over conifold ( but rank two on sing locus ) defined on sing space c 1 extra div massless U(1).

7 Conifold - Matrix Factorization (MF) Eq y +y = s w admits a (pair of) MF, i.e. a pair of matrices (φ, ψ) s.t. φ ψ = ψ φ = (y +y s w)1 2 For the conifold: ( y s φ = w y + ) ( y+ s ψ = w y ) From φ, ψ one can define (MCM) modules over R [Eisenbud], e.g. M = coker(r 2 ψ R 2 ) R 2 / Imψ (where R = C[y +, y, s, w]/(y +y s w) is the coordinate ring. ) Line bundle over conifold ( but rank two on sing locus ) defined on sing space c 1 extra div massless U(1).

8 Conifold - Matrix Factorization (MF) For the conifold: ( ) ( y s y+ s φ = ψ = w y + w y ) M = coker(r 2 ψ R 2 ) Line bundle over conifold ( except on sing locus ) c 1 (M) locus where a generic section vanishes. ( coker ψ y s = Im φ c 1 : w y + ) ( σ1 σ 2 ) = 0, i.e. σ 1 y + σ 2 s = 0, σ 1 w + σ 2 y + = 0 Family of non-cartier divisors, among which extra-section of elliptic fibration (σ 1 = 0, σ 2 = 1)..

9 Conifold - Non Commutative Resolution (NCCR) NCCR: enlarge coordinate ring R = C[y +, y, s, w]/(y +y s w) by replacing it with A End R (R M), i.e. Hom(R, R) Hom(M, M) Hom(R, M) Hom(M, R) = R = R = M = M e 0 e 1 α i β i This is a (in principle) non-coomutative ring. Associated quiver: α i e 0 R M e 1 β i

10 Conifold - Quiver rep and NCCR Given CY X, equivalence of Categories { coherent sheaves } { A-modules } { representations of quiver } Physically, objects are D-branes on X. ( E.g. take IIA string on X. ) Moduli space of D0-branes is the full space X. Its quiver rep d = (1, 1) α i α i or in physics C C U(1) U(1) language β i β i Moduli space: all possible α i, β i modulo relative U(1) and D-term α 1 α 2 β 1 β with α α 2 2 β 1 2 β 2 2 = θ Exceptional P 1 is β 1 = β 2 = 0 (for θ > 0). In resolved space, extra divisor is zero locus of a section of M, i.e. σ 1 α 1 + σ 2 α 2 = 0 intersects exceptional P 1 in one point.

11 Conifold - Quiver rep and NCCR Given CY X, equivalence of Categories { coherent sheaves } { A-modules } { representations of quiver } Physically, objects are D-branes on X. ( E.g. take IIA string on X. ) Moduli space of D0-branes is the full space X. Its quiver rep d = (1, 1) α i α i or in physics C C U(1) U(1) language β i β i Moduli space: all possible α i, β i modulo relative U(1) and D-term α 1 α 2 β 1 β with α α 2 2 β 1 2 β 2 2 = θ Exceptional P 1 is β 1 = β 2 = 0 (for θ > 0). In resolved space, extra divisor is zero locus of a section of M, i.e. σ 1 α 1 + σ 2 α 2 = 0 intersects exceptional P 1 in one point.

12 Conifold - Quiver rep D0-brane splits into fractional branes (D2 wrapping vanishing P 1 ): d = (1, 1) rep splits into simple reps, i.e. d = (0, 1) and d = (1, 0): C 1 1 C No moduli space: D2s wrap rigid curve. BPS particles in space-time. In the singular limit, they become massless for given choice of B-field (not both) and are charged under A µ P 1 C 3. massless M2 charged states in M-theory language.

13 Morrison-Park U(1) restriction is subcase of class of ellitpic fibrations with one extra section. Full class is described by Morrison-Park ) y 2 = s 3 + c 2 s 2 + (c 1 c 3 b 2 c 0 s + c 0 c3 2 b 2 c 0 c 2 + b2 c1 2 4 A rational section massless U(1); two curves of conifold-like sing charged matter ( At weak coupling: pair of brane-imagebrane and invariant brane. ) Is there a 2 2 MF?

14 Morrison-Park U(1) restriction is subcase of class of ellitpic fibrations with one extra section. Full class is described by Morrison-Park ) y 2 = s 3 + c 2 s 2 + (c 1 c 3 b 2 c 0 s + c 0 c3 2 b 2 c 0 c 2 + b2 c1 2 4 A rational section massless U(1); two curves of conifold-like sing charged matter ( At weak coupling: pair of brane-imagebrane and invariant brane. ) Is there a 2 2 MF?

15 Morrison-Park The answer is NO! But 4 4 MF: y + c 1b s c 2 3 b Ψ MP = c 1 c 3 + s(s + c 2 ) y c 1b b(s + c 2 2 ) c 3 c 0 c 3 c 0 b y + c 1b s 2 c 0 b(s + c 2 ) c 0 c 3 c 1 c 3 s(s + c 2 ) y c 1b 2 ) y 2 = s 3 + c 2 s 2 + (c 1 c 3 b 2 c 0 s + c 0 c3 2 b 2 c 0 c 2 + b2 c1 2 4 How can we extract massless U(1) and charged matter? Specialize to a simpler example: c 0 w, c 1 0, c 2 0, c 3 z, b w get Laufer s threefold ( also s s and y y ) y 2 + s 3 + w 3 s + z 2 w = 0 singular at (y, s, z, w).

16 Morrison-Park The answer is NO! But 4 4 MF: y + c 1b s c 2 3 b Ψ MP = c 1 c 3 + s(s + c 2 ) y c 1b b(s + c 2 2 ) c 3 c 0 c 3 c 0 b y + c 1b s 2 c 0 b(s + c 2 ) c 0 c 3 c 1 c 3 s(s + c 2 ) y c 1b 2 ) y 2 = s 3 + c 2 s 2 + (c 1 c 3 b 2 c 0 s + c 0 c3 2 b 2 c 0 c 2 + b2 c1 2 4 How can we extract massless U(1) and charged matter? Specialize to a simpler example: c 0 w, c 1 0, c 2 0, c 3 z, b w get Laufer s threefold ( also s s and y y ) y 2 + s 3 + w 3 s + z 2 w = 0 singular at (y, s, z, w).

17 Laufer 4 4 MF: [Curto,Morrison; Aspinwall,Morrison] y s z w Ψ L = s 2 y s w z w z w 2 y s s w 2 w z s 2 y y 2 + s 3 + w 3 s + z 2 w = 0 M = coker(r 4 Ψ L R 4 ) is now rank 2. Claim is that also in this case c 1 (M) gives extra divisor, i.e. new massless U(1) gauge boson. Matter at sing, i.e. when Ψ L becomes zero rank.

18 Laufer Ψ L = y s z w s 2 y s w z w z w 2 y s s w 2 w z s 2 y M = coker(r 4 Ψ L R 4 ) Vector bndl over Laufer ( except on sing locus ) c 1 (M) locus where a two sections become parallel. coker Ψ L = Im ΦL with specific choice of sections, c 1 : z 2 + w 2 s = 0, s z + w y = 0, y z w s 2 = 0, intersected with y 2 + s 3 + w z 2 + s w 3 = 0. In general, family of non-cartier divisors, among which extra (rational) section of elliptic fibration, i.e. y = z3 w 3 s = z2 w 2

19 Laufer - NCCR Again NCCR: enlarge coordinate ring R = C[y, s, z, w]/(y 2 + s 3 + w 3 s + z 2 w) by replacing it with A End R (R M), Associated quiver: d R M a b c with relations (b 2 +dc)d = 0, c(b 2 +dc) = 0, ab+ba = 0, a 2 +bdc+dcb+b 3 = 0.

20 Laufer - NCCR Full resolved space given by moduli space of D0 ( d = (1, 2) rep ): d C C 2 a b with relations c (b 2 +dc)d = 0, c(b 2 +dc) = 0, ab+ba = 0, a 2 +bdc+dcb+b 3 = 0. Resolved space given by all possible values of maps a, b, c, d (except SR-ideal ), subject to realtions and modded by gauge transformations U(1) U(2) and D-terms: d d cc = 2θ dd c c + [a, a ] + [b, b ] = θ 1 2

21 Laufer - Exceptional P 1 Exceptional P 1 : locus that is pushed down to (y, s, w, z) in sing space. For both phases: and a 2 = 0, b 2 = 0, ab + ba = 0 c = 0 for θ > 0, while d = 0 for θ < 0 Interpolate between the two phases by taking ( ) ( ) 0 α 0 β a =, b =, c = ( 2γ 0 ) ( ), d = δ D-terms become δ 2 γ 2 = θ, α 2 + β 2 = δ 2 + γ 2 Now, take e.g. phase θ > 0. Then γ = 0 and P 1 spanned by α, β with residual gauge transformation (α, β) (λα, λβ). Changing continuously θ, follow the flop transition through the singularity.

22 Laufer - U(1) divisor Consider phase θ > 0. Extra divisor given by locus where two sections of M become parallel: σ 1 (ad d) + σ 2 (bd d) + σ 3 (abd d) = 0. ( Sections of M are d, ad, bd, abd. ) Again family of divisors intersecting the P 1 at one point: σ 1 α + σ 2 β = 0. It corresponds to what one finds working in the sing space: sw 2 z 2 yw + sz ws 2 xz σ 1 yw + zs w 3 s 2 zw 2 + ys σ 2 = 0 s 2 w yz zw 2 + ys sw 3 y 2 σ 3 ( Before we specialized σ = (1, 0, 0). )

23 Laufer - Fractional branes Fractional branes are D0 R with d = (0, 1) and D0 L with d = (1, 0). Mobile D0 splits into D0 L 2 D0 R. D0 R has D2-charge equal to 1. D0 R has D2-charge equal to 2. Charge-1 fractional brane gives massless charged state in the singular limit. [Aspinwall,Morrison] If it worked like in the conifold, changing the B field the charge-2 state may become massless. ( Possibility to have double charge states. )

24 Conclusions Matrix factorization and quiver for fourfolds with massless U(1). Naturally encode extra U(1) divisor (already in the singular limit): family of representatives that includes extra section of elliptic fibration. Associated quiver gives resolution. Exceptional P 1. Flop transition. Massless matter as fractional branes (quiver). Open questions: More complicated geometries. Higher charge states? Results presented here for three-fold. Four-fold at the edge of math research.

25 Conclusions Matrix factorization and quiver for fourfolds with massless U(1). Naturally encode extra U(1) divisor (already in the singular limit): family of representatives that includes extra section of elliptic fibration. Associated quiver gives resolution. Exceptional P 1. Flop transition. Massless matter as fractional branes (quiver). Open questions: More complicated geometries. Higher charge states? Results presented here for three-fold. Four-fold at the edge of math research.

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