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March 10, 2008

Exceptional F-Theory.

I’ve been reading Beasley, Heckman and Vafa’s recent 125 page opus, hoping to get through it before the promised Part II comes out.

F-theory is the fancy name for Type IIB string theory with 7-branes. If we compactify on BB (for compactifications down to 4 dimensions, we’re interested in BB a complex 3-fold), the 7-branes are wrapped on divisors in BB. The complex IIB coupling, τ\tau, has monodromies as we circle those divisors and, viewing it as the modulus of an elliptic curve, we get the total space of an elliptically-fibered Calabi-Yau 4-fold, XBX\to B.

Except for the case where one has only D7-branes (and orientifold O7 planes), Im(τ)Im(\tau) cannot be taken to be uniformly large. So perturbative string theory techniques are not applicable. General configurations of 7-branes are hard to study, except in some special cases.

The interest, here, is to study a local model for a wrapped 7-brane, or perhaps a pair of 7-branes intersecting transversally, and study the local physics from the point of view of the twisted SYM theory living on the brane.

In the local model, BB is noncompact, and is the total space of the line bundle

(1)B=Tot(N S|BS)B = Tot(N_{S|B}\to S)

where the normal bundle N S|B=K S pN_{S|B}=K_S^p, a power of the canonical bundle of the surface, SS, on which the 7-brane is wrapped. Away from the zero section, we have an elliptic curve (with affine coordinates x,yx,y) fibered over the base. Over the zero section, the curve degenerates, and the total space of XSX\to S looks like an isolated ADE surface singularity fibered over SS. Denoting the fiber coordinate of BSB\to S as zz, the total space of XSX\to S is given as the locus {f(x,y,z)=0}V\{f(x,y,z)=0\}\subset V, with

ADE 7-branes as the locus {f=0}Tot(VS)\{f=0\}\subset Tot(V\to S)
f(x,y,z)f(x,y,z) VV N S|BN_{S|B} K BK_{B} ΔΓ(K B 12)\Delta\in\Gamma\left(K_{B}^{-12}\right)
A nA_n y 2x 2z n+1y^2-x^2-z^{n+1} K S (n+1)/2K S (n+1)/2K SK_S^{(n+1)/2}\oplus K_S^{(n+1)/2} \oplus K_S K SK_S 𝒪\mathcal{O} ?
D nD_n y 2x 2zz n1y^2-x^2z -z^{n-1} K S n2K S n1K S 2K_S^{n-2}\oplus K_S^{n-1} \oplus K_S^2 K S 2K_S^2 π *(K S 1)\pi^*\left(K_S^{-1}\right) ?
E 6E_6 y 2x 3z 4y^2-x^3-z^{4} K S 4K S 6K S 3K_S^{4}\oplus K_S^{6} \oplus K_S^3 K S 3K_S^3 π *(K S 2)\pi^*\left(K_S^{-2}\right) 2 43 3z 8-2^4 3^3 z^8
E 7E_7 y 2x 3xz 3y^2-x^3 - x z^{3} K S 6K S 9K S 4K_S^{6}\oplus K_S^{9} \oplus K_S^4 K S 4K_S^4 π *(K S 3)\pi^*\left(K_S^{-3}\right) 2 6z 9-2^6 z^9
E 8E_8 y 2x 3z 5y^2-x^3-z^{5} K S 10K S 15K S 6K_S^{10}\oplus K_S^{15} \oplus K_S^6 K S 6K_S^6 π *(K S 5)\pi^*\left(K_S^{-5}\right) 2 43 3z 10-2^4 3^3 z^{10}

Now, I’m a little confused by the A nA_n and D nD_n cases. As written, these are not elliptically-fibered; away from z=0z=0, the fiber of XBX\to B looks to me like a smooth quadric. But the main focus of attention is on the exceptional cases where, indeed, we have a Weierstrass form for the equation of the elliptic fiber.

Now, the idea is that the local physics is captured by the 8D twisted SYM theory on R 3,1×SR^{3,1}\times S. The super Yang-Mills multiplet, in 8 dimensions, consists of a G SG_S-connection, a complex scalar in the adjoint representation, and some fermions, also in the adjoint. After twisting, the scalar becomes a 2-form φΩ (2,0)(S,ad(P)),φ¯Ω (0,2)(S,ad(P)) \varphi \in \Omega^{(2,0)}(S,ad(P)),\quad \overline{\varphi} \in \Omega^{(0,2)}(S,ad(P)) where PSP\to S is a principal G SG_S-bundle. The left-handed fermions are η α Γ(ad(P)) ψ α Ω (0,1)(S,ad(P)) χ α Ω (2,0)(S,ad(P)) \begin{aligned} \eta_\alpha &\in \Gamma(ad(P))\\ \psi_\alpha &\in \Omega^{(0,1)}(S,ad(P))\\ \chi_\alpha &\in \Omega^{(2,0)}(S,ad(P)) \end{aligned} and their right-handed conjugates are η¯ α˙ Γ(ad(P)) ψ¯ α˙ Ω (1,0)(S,ad(P)) χ¯ α˙ Ω (0,2)(S,ad(P)) \begin{aligned} \overline{\eta}_{\dot{\alpha}} &\in \Gamma(ad(P))\\ \overline{\psi}_{\dot{\alpha}} &\in \Omega^{(1,0)}(S,ad(P))\\ \overline{\chi}_{\dot{\alpha}} &\in \Omega^{(0,2)}(S,ad(P)) \end{aligned} The conditions for a supersymmetric solution are

(2)F S (2,0)=F S (0,2)=0 D S (1,0)φ¯=D S (0,1)φ=0 ωF S+i2[φ,φ¯]=0\begin{gathered} F_S^{(2,0)} = F_S^{(0,2)} = 0\\ D^{(1,0)}_S \overline{\varphi} = D^{(0,1)}_S \varphi=0\\ \omega\wedge F_S + \frac{i}{2} [\varphi, \overline{\varphi}] = 0 \end{gathered}

Here, F S (2,0)=(D S (1,0)) 2,F S (0,2)=(D S (0,1)) 2,F S={D S (1,0),D S (0,1)} F_S^{(2,0)} = {\left(D^{(1,0)}_S\right)}^2,\quad F_S^{(0,2)} = {\left(D^{(0,1)}_S\right)}^2,\quad F_S= \{D^{(1,0)}_S,D^{(0,1)}_S\} are, respectively, the (2,0)(2,0), (0,2)(0,2) and (1,1)(1,1) parts of the field strength on SS, and ω\omega is the Kähler form.

In the particular case1 of φ=0\varphi=0 (or, more generally, [φ,φ¯]=0[\varphi, \overline{\varphi}]=0), (2), these equations imply the Donaldson-Uhlenbeck-Yau equation, whose solution is an anti-self-dual connection, with field strength F SF_S, for some subgroup H SG SH_S\subset G_S. Correspondingly, there’s a reduction of the structure group of PP from G SG_S to H SH_S, Denoting by Γ S\Gamma_S, the commutant of H SH_S in G SG_S, we decompose ad(P) i(R i𝒯 i) ad(P) \simeq \oplus_i (R_i \otimes \mathcal{T}_i) where R iR_i are irreps of Γ i\Gamma_i and 𝒯 i\mathcal{T}_i are vector bundles associated to our H SH_S principal bundle.

One obtains massless 4D chiral multiplets, in representation R iR_i from the Dolbeault cohomology groups

(3)(δφ,χ α)H 0(S,K S𝒯 i) (δa¯,ψ α)H 1(S,𝒯 i)\begin{gathered} (\delta\varphi,\chi_\alpha) \in H^0(S,K_S\otimes \mathcal{T}_i)\\ (\delta \overline{a},\psi_\alpha)\in H^1(S,\mathcal{T}_i) \end{gathered}

As a toy example, take S=𝔽 nS= \mathbb{F}_n, the nnth Hirzebruch surface2, G S=E 6G_S= E_6, H S=U(1)H_S = U(1) and the unbroken gauge group, Γ S=SO(10)\Gamma_S= SO(10). That is, we have a nontrivial connection, satisfying the Donaldson-Uhlenbeck-Yau equation, for some line bundle S\mathcal{L}\to S. Decomposing 78=45 0+1 0+16 3+16¯ 3 78 = 45_0 + 1_0 + 16_{-3} +\overline{16}_3 We get chiral multiplets in the 1616 from H 1(S, 3)H^1(S,\mathcal{L}^{-3}), and 16¯\overline{16}s from H 1(S, 3)H^1(S,\mathcal{L}^{3}). Their number is given by the index theorem. In the notation of the footnote, n 16 =(3b1)[(3b+1)32(2a+(2n)b)] n 16¯ =(3b+1)[(3b1)32(2a+(2n)b)] \begin{aligned} n_{16} &= (3b-1)\left[(3b+1) -\tfrac{3}{2}(2a+(2-n) b)\right]\\ n_{\overline{16}} &= (3b+1)\left[(3b-1) -\tfrac{3}{2}(2a+(2-n) b)\right] \end{aligned}

The Yukawa couplings of the chiral multiplets (3) are obtained from the trilinear form c ijk:H 0(S,K S𝒯 i)H 1(S,𝒯 j)H 1(S,𝒯 i) c_{i j k}:\, H^0(S,K_S\otimes \mathcal{T}_i)\otimes H^1(S,\mathcal{T}_j) \otimes H^1(S,\mathcal{T}_i) \to \mathbb{C}

Unfortunately, for del Pezzo and Hirzebruch surfaces, H 0(S,K SE)H^0(S,K_S\otimes E) vanishes, for any vector bundle, EE, admitting a connection satisfying Donaldson-Uhlenbeck-Yau. So there are no nontrivial Yukawa couplings among these “bulk modes.”

Intersecting Branes

To get nontrivial Yukawa couplings, we need to consider intersecting branes. Let ΣS\Sigma\subset S be a smooth, irreducible curve. We can modify our setup to include 7-branes wrapping R 3,1×SR^{3,1}\times S', where S=Tot(K S pΣ)S'=Tot(K_S^p\to \Sigma).

Since SS' is noncompact, the G SG_{S'} gauge theory on it is decoupled. But there are new degrees of freedom localized on Σ=SS\Sigma= S\cap S', given essentially by 6 dimensional twisted gauge theory, for the group G ΣG S×G SG_\Sigma \supset G_S\times G_{S'}. Let αH 0(S,𝒪 S(Σ))\alpha \in H^0(S,\mathcal{O}_S(\Sigma)). As before, we want the fiber over a generic point in SS to be an isolated ADEADE surface singularity, corresponding to G SG_S. But, over SS' (roughly, taking z0z\neq 0, and allowing α\alpha to vary), we want a G SG_{S'} surface singularity.

Local model for intersecting ADE 7-branes
G SG_S G SG_{S'} G ΣG_\Sigma f(x,y,z)f(x,y,z) VV LL N S|BN_{S|B} K BK_{B}
A nA_n A mA_m A n+mA_{n+m} y 2x 2α m+1z n+1y^2-x^2-\alpha^{m+1}z^{n+1} LLK SL\oplus L \oplus K_S 𝒪 S(m+12Σ)K S (n+1)/2\mathcal{O}_S\left(\frac{m+1}{2}\Sigma\right)\otimes K_S^{(n+1)/2} K SK_S 𝒪\mathcal{O}
D nD_n U(1)U(1) D n+1D_{n+1} y 2+x 2zα 2z n1y^2+x^2 z - \alpha^2 z^{n-1} (LK S 1)LK S 2\left(L\otimes K_S^{-1}\right)\oplus L \oplus K_S^2 𝒪 S(Σ)K S n1\mathcal{O}_S(\Sigma)\otimes K_S^{n-1} K S 2K_S^2 π *(K S 1)\pi^*\left(K_S^{-1}\right)
E 6E_6 A 2A_2 E 8E_8 y 2x 3α 2z 4y^2-x^3-\alpha^2 z^{4} L 2L 3(LK S)L^2\oplus L^3 \oplus (L\otimes K_S) 𝒪 S(Σ)K S 2\mathcal{O}_S(\Sigma)\otimes K_S^{2} LK SL\otimes K_S π *(L 1)\pi^*\left(L^{-1}\right)
E 7E_7 A 1A_1 E 8E_8 y 2+x 316αxz 3y^2+x^3 - 16\alpha x z^{3} L 2L 3(LK S)L^2\oplus L^3 \oplus (L\otimes K_S) 𝒪 S(Σ)K S 3\mathcal{O}_S(\Sigma)\otimes K_S^{3} LK SL\otimes K_S π *(L 1)\pi^*\left(L^{-1}\right)
E 8E_8 E 8E_8 y 2x 3αz 5y^2-x^3-\alpha z^{5} L 2L 3(LK S)L^2\oplus L^3 \oplus (L\otimes K_S) 𝒪 S(Σ)K S 5\mathcal{O}_S(\Sigma)\otimes K_S^{5} LK SL\otimes K_S π *(L 1)\pi^*\left(L^{-1}\right)

Over Σ\Sigma, there’s a G S×G SG_S\times G_{S'} principal bundle, obtained by restricting the respective principal bundles over SS and SS'. Decomposing ad(G Σ)=ad(G S)ad(G S)[ aU aU a] ad(G_\Sigma) = ad(G_S) \oplus ad(G_{S'}) \oplus \left[\bigoplus_a U_a \otimes U'_a\right] The twisted theory on 3,1×Σ\mathbb{R}^{3,1}\times\Sigma has complex bosons, σ a\sigma_a, and left-handed fermions, λ aα\lambda_{a\alpha}, transforming as3 sections of K Σ 1/2𝒰 a𝒰 aK_\Sigma^{1/2}\otimes \mathcal{U}_a\otimes \mathcal{U}_a', and their Hermitian conjugates. The equations for a supersymmetric background are D (0,1)σ=D (0,1)λ α=0 D^{(0,1)} \sigma = D^{(0,1)} \lambda_{\alpha} =0 As before, we take the background gauge field on each 7-brane to be an anti-self dual connection for subgroups H SG SH_S\subset G_S and H SG SH_{S'}\subset G_{S'}. Under G S=Γ S×H SG_S = \Gamma_S\times H_S and G S=Γ S×H SG_{S'}= \Gamma_{S'}\times H_{S'}, we decompose 𝒰 a = nR n𝒱 an 𝒰 a = nR n𝒱 an \begin{aligned} \mathcal{U}_a &= \bigoplus_n R_{n} \otimes \mathcal{V}_{a n}\\ \mathcal{U}'_a &= \bigoplus_n R'_{n} \otimes \mathcal{V}'_{a n}\\ \end{aligned} So we obtain 4D massless chiral multiplets, (σ,λ α)H 0(Σ,K Σ 1/2𝒱 an𝒱 an) (\sigma,\lambda_\alpha) \in H^0(\Sigma, K_\Sigma^{1/2} \otimes \mathcal{V}_{a n}\otimes \mathcal{V}'_{a n}) in the representation (R n,R n)(R_{n}, R'_{n}) of Gamma S×Gamma SGamma_S\times Gamma_{S'}.

These have Yukawa couplings with the “bulk” fields discussed previously, given by the trilinear form

(4)H 0(Σ,K Σ 1/2𝒱 an𝒱 an)H 0(Σ,K Σ 1/2𝒱 bm𝒱 bm)H 1(Σ,𝒯 i| Σ)H^0(\Sigma, K_\Sigma^{1/2}\otimes \mathcal{V}_{a n}\otimes \mathcal{V}'_{a n})\otimes H^0(\Sigma, K_\Sigma^{1/2}\otimes \mathcal{V}_{b m}\otimes \mathcal{V}'_{b m})\otimes H^1(\Sigma, \mathcal{T}_i\vert_\Sigma) \to \mathbb{C}

As a more refined version of the previous example, consider intersecting 7-branes, with G S=SO(12),H S=U(1),G S=SU(2),H S=U(1),G Σ=E 7 G_S = SO(12),\, H_S=U(1),\quad G_{S'}= SU(2),\, H_{S'}=U(1),\quad G_\Sigma = E_7 Under E 7SO(12)×SU(2)E_7\supset SO(12)\times SU(2), the adjoint decomposes as 133=(66,1)(1,3)(32,2) 133 = (66,1)\oplus (1,3)\oplus (32,2) and decomposing further under SO(12)SO(10)×U(1)SO(12)\supset SO(10)\times U(1) 66=45 0+1 0+10 2+10 2 32 =16 1+16¯ 1 \begin{aligned} 66 = 45_0 + 1_0 + {10}_2 + {10}_{-2}\\ 32 &= {16}_1 +\overline{16}_{-1} \end{aligned} So we have 4D chiral multiplets in representations of SO(10)SO(10)

(5)10 H 1(Σ, 2)H 1(Σ, 2) 16 H 1(Σ,K Σ 1/2)H 1(Σ,K Σ 1/2() 1) 16¯ H 1(Σ,K Σ 1/2 1)H 1(Σ,K Σ 1/2 1() 1) \begin{aligned} 10 &\in H^1(\Sigma, \mathcal{L}^2) \oplus H^1(\Sigma, \mathcal{L}^{-2})\\ 16 &\in H^1(\Sigma, K_\Sigma^{1/2}\otimes \mathcal{L}\otimes\mathcal{L}')\oplus H^1(\Sigma, K_\Sigma^{1/2}\otimes \mathcal{L}\otimes{(\mathcal{L}')}^{-1})\\ \overline{16} &\in H^1(\Sigma, K_\Sigma^{1/2}\otimes \mathcal{L}^{-1}\otimes\mathcal{L}')\oplus H^1(\Sigma, K_\Sigma^{1/2}\otimes \mathcal{L}^{-1}\otimes{(\mathcal{L}')}^{-1})\\ \end{aligned}

where the 1010s are actually the restrictions to Σ\Sigma of “bulk” modes on the 7-brane wrapped on SS and ,\mathcal{L},\mathcal{L}' are appropriately-chosen line bundles. The Yukawa couplings are obtained by applying the trilinear form (4) to the modes in (5).

Yet more interesting structure arises when Σ\Sigma is reducible, Σ= iΣ i\Sigma= \bigcup_i \Sigma_i, and one gets further contributions from the intersections, Σ iΣ j\Sigma_i\cap \Sigma_j.

1 φ\varphi necessarily vanishes, when SS is a Hirzebruch or del Pezzo surface, and H 0(S,K S n)=0,n>0H^0(S,K_S^n)=0,\, \forall n\gt 0.

2 A rational surface, 𝔽 n=Tot((𝒪(1)𝒪(n)) 1)\mathbb{F}_n = Tot\left(\mathbb{P}(\mathcal{O}(1)\oplus \mathcal{O}(n))\to \mathbb{P}^1\right). H 2(𝔽 n)H_2(\mathbb{F}_n) is generated by two classes, f,σf,\sigma, with f 2=0,fσ=1,σ 2=n f^2=0,\, f\cdot\sigma=1,\, \sigma^2 = -n The Kähler class, ω=αf+βσ\omega= \alpha f +\beta \sigma, must satisfy ω 2 >0 ωf >0 ωσ >0}{β >0 αnβ >0 \left. \begin{aligned} \omega^2 &\gt 0\\ \omega\cdot f &\gt 0\\ \omega\cdot\sigma &\gt 0 \end{aligned} \right\} \implies \left\{ \begin{aligned} \beta &\gt 0\\ \alpha - n\beta &\gt 0 \end{aligned} \right. A line bundle, 𝔽 n\mathcal{L}\to \mathbb{F}_n admits a connection satisfying the Donaldson-Uhlenbeck-Yau equation provided ωc 1()=0\omega\wedge c_1(\mathcal{L})=0. Writing c 1()=af+bσc_1(\mathcal{L})= a f + b \sigma, ωc 1()=aα+b(αnβ) \omega\wedge c_1(\mathcal{L})= a \alpha + b(\alpha - n\beta) so, in particular, we require ab<0a b\lt 0.

3 At first sight, this statement seems rather puzzling. While K ΣK_\Sigma admits a square-root, if the genus is greater than zero, there is no canonical choice of square-root. In fact, Beasley et al argue that N Σ|S=𝒪 S(Σ)| ΣN_{\Sigma|S} = \mathcal{O}_S(\Sigma)|_\Sigma is isomorphic to K S| ΣK_S|_\Sigma. Therefore, the adjunction formula, K Σ=K S| ΣN Σ|S K_\Sigma = K_S|_\Sigma \otimes N_{\Sigma|S} implies a canonical choice of square-root, K Σ 1/2=K S| Σ K_\Sigma^{1/2} = K_S|_\Sigma

Posted by distler at March 10, 2008 9:34 AM

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Re: Exceptional F-Theory.

Hi Jaques,

RE: In section 4.3 Unfolding singularites via surface operators:
Does this imply the possibility that symmetries may be folded rather than broken?

Posted by: Doug on April 13, 2008 8:14 AM | Permalink | Reply to this

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