Schreiber notes

Contents

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Some notes.

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We want to check that the the L L_\infty-isomorphism sends IIA cocycles to IIB cocycles.

Contents

Conventions

The following computation uses the spinor conventions of CDF, II.8.1.

In particular we take Γ 0 =Γ 0\Gamma_0^\dagger = \Gamma_0 and (Γ spatial) =Γ spatial(\Gamma_{spatial})^\dagger = -\Gamma_{spatial}.

Using ψ¯=ψ Γ 0\overline{\psi} = \psi^\dagger \Gamma_0 we get

(ψ¯Γ a 1a pψ) * =ψ¯Γ a pa 1ψ =(1) p(p1)/2ψ¯Γ a 1a pψ \begin{aligned} \left( \overline{\psi} \Gamma_{a_1 \cdots a_p} \psi \right)^\ast & = \overline{\psi} \Gamma_{a_p \cdots a_1} \psi \\ &= (-1)^{p(p-1)/2} \overline{\psi} \Gamma_{a_1 \cdots a_p} \psi \end{aligned}

As a consequence, the following expressions are real

ψ¯Γ aψ i ψ¯Γ a 1a 2ψ i ψ¯Γ a 1a 2a 3ψ ψ¯Γ a 1a 4ψ ψ¯Γ a 1a 5ψ i ψ¯Γ a 1a 6ψ i ψ¯Γ a 1a 7ψ \begin{aligned} & \overline{\psi} \Gamma_a \psi \\ i & \overline{\psi}\Gamma_{a_1 a_2} \psi \\ i & \overline{\psi} \Gamma_{a_1 a_2 a_3} \psi \\ & \overline{\psi} \Gamma_{a_1 \cdots a_4} \psi \\ & \overline{\psi} \Gamma_{a_1 \cdots a_5} \psi \\ i & \overline{\psi} \Gamma_{a_1 \cdots a_6} \psi \\ i & \overline{\psi} \Gamma_{a_1 \cdots a_7} \psi \end{aligned}

Given a choice of Clifford matrices in d=9d = 9 we choose Clifford matrices in dimension 10 and 11 as discussed at Majorana spinors – In dimension 11, 10 and 9:

Γ a8(0 γ a γ a 0),Γ 9(0 id id 0),Γ 10(iid 0 0 iid). \Gamma_{a \leq 8} \coloneqq \left( \array{ 0 & \gamma_a \\ \gamma_a & 0 } \right) \;\,,\;\; \Gamma_{9} \coloneqq \left( \array{ 0 & id \\ -id & 0 } \right) \;\,,\;\; \Gamma_{10} \coloneqq \left( \array{ i id & 0 \\ 0 & -i id } \right) \,.

Notice that Γ ab\Gamma_{a b } for a<b9a \lt b \leq 9 is the same for both chiralities, except if b=9b = 9. Therefore if we write

Γ a B={Γ a |a8 (0 id id 0) |a=9 \Gamma_a^B = \left\{ \array{ \Gamma_a & \vert \; a \leq 8 \\ \left( \array{ 0 & id \\ id & 0 } \right) & \vert \; a = 9 } \right.

then the IIB spin representation is given by elements exp(ω abΓ a BΓ b B)\exp(\omega^{a b} \Gamma_a^B \Gamma_b^B). Hence we set

Definition
Γ a8 BΓ a \Gamma_{a \leq 8}^B \coloneqq \Gamma_a
Γ 9 BiΓ 9Γ 10 \Gamma_9^B \coloneqq i \Gamma_9 \Gamma_{10}

hence

Γ 10=iΓ 9 BΓ 9 \Gamma_{10} = - i \Gamma_9^B \Gamma_9
Remark

Beware that the {Γ a B}\{\Gamma_a^B\} are not a Clifford algebra, but the even part exp(ω abΓ a BΓ b B)\exp(\omega^{a b} \Gamma_a^B \Gamma_b^B) is a spin representation (namely the direct sum of two copies of 16\mathbf{16}), and for odd pp then

ψ¯Γ a 1a p Bψ =ψ Γ 0Γ a 1a p Bψ =ψ Γ 0 BΓ a 1a p Bψ \begin{aligned} \overline{\psi} \Gamma^B_{a_1\cdots a_{p}} \psi & = \psi^\dagger \Gamma_0 \Gamma^B_{a_1 \cdots a_p} \psi \\ & = \psi^\dagger \Gamma^B_0 \Gamma^B_{a_1 \cdots a_p} \psi \\ \end{aligned}

is the sum of the corresponding spin pairings in two copies of 16\mathbf{16}.

Double dimensional reduction

For π d1: d1,1|... d2,1|...\pi_{d-1} \colon \mathbb{R}^{d-1,1\vert ...} \longrightarrow \mathbb{R}^{d-2,1\vert ...} an extension we decompose any element in

μCE( d1,1|...) \mu \in CE(\mathbb{R}^{d-1,1\vert ...})

into a piece of the same degree and a piece of one degree lower, by the identification

μ=μ| d1e 9(π d1) *μ. \mu = \mu|_{d-1} - e^9 \wedge (\pi_{d-1})_\ast \mu \,.

Here μ| d1\mu|_{d-1} is the “dimensional reduction”, while (π d1) *μ(\pi_{d-1})_\ast \mu is the “double dimensional reduction”.

Remark

If

μ= a i=0 d1ψ¯Γ a 1a pψe a 1e a p \mu = \sum_{a_i = 0}^{d-1} \overline{\psi} \Gamma_{a_1 \cdots a_p}\psi \wedge e^{a_1} \wedge \cdots \wedge e^{a_p}

is a cochain in dimension dd, then its double dimensional reduction is

(π d1) *(μ)=(1) pp a i=0 d2ψ¯Γ a 1a p1Γ d1ψe a 1e a p1 (\pi_{d-1})_\ast(\mu) = (-1)^{p} p \, \sum_{a_i = 0}^{d-2} \overline{\psi} \Gamma_{a_1 \cdots a_{p-1}} \Gamma_{d-1} \psi \wedge e^{a_1} \wedge \cdots \wedge e^{a_{p-1}}

Notice the prefactor of (1) pp(-1)^p p here, which is crucial in the following.

The M-brane cocycles

Let’s be careful and go back to the source: from DAuria-Fre 82, equation (3.27) we have (with our convention that de 10=ψ¯Γ 10ψd e^{10} = \overline{\psi}\Gamma^{10}\psi) that

d(ψ¯Γ a 1a 5e a 1e a 5ψ)=15(ψ¯Γ a 1a 2ψe a 1e a 2)(ψ¯Γ a 1a 2ψe a 1e a 2). d \left(\overline{\psi}\Gamma_{a_1 \cdots a_5} \wedge e^{a_1} \wedge \cdots \wedge e^{a_5}\psi\right) = 15 \left(\overline{\psi}\Gamma_{a_1 a_2}\psi \wedge e^{a_1} \wedge e^{a_2} \right) \wedge \left(\overline{\psi}\Gamma_{a_1 a_2}\psi \wedge e^{a_1} \wedge e^{a_2} \right) \,.

Hence if we set

μ M2icψ¯Γ a 1a 2ψe a 1e a 2 \mu_{M2} \coloneqq i c \, \overline{\psi}\Gamma_{a_1 a_2}\psi \wedge e^{a_1} \wedge e^{a_2}

for any c{0}c \in \mathbb{R}-\{0\} then

μ M5=c 215(ψ¯Γ a 1a 5ψe a 1e a 5). \mu_{M5} = -\frac{c^2}{15} \left(\overline{\psi}\Gamma_{a_1 \cdots a_5}\psi \wedge e^{a_1} \wedge \cdots \wedge e^{a_5}\right) \,.

Now for T-duality to come out right, below it turns out that we need

μ M2 =ic2 a i=0 10ψ¯Γ a 1Γ a 2ψe a 1e a 2 μ M5 =c5432 a i=0 10ψ¯Γ a 1a 5ψe a 1e a 5 \begin{aligned} \mu_{M2} &= i \tfrac{c'}{2} \sum_{a_i = 0}^{10} \overline{\psi} \Gamma_{a_1} \Gamma_{a_2} \psi \wedge e^{a_1} \wedge e^{a_2} \\ \mu_{M5} &= - \tfrac{c'}{5 \cdot 4 \cdot 3 \cdot 2} \sum_{a_i = 0}^{10} \overline{\psi} \Gamma_{a_1 \cdots a_5} \psi \wedge e^{a_1} \wedge \cdots \wedge e^{a_5} \end{aligned}

for some global prefactor cc'

So we should solve for cc and cc'.

Matching them on μ M2\mu_{M2} gives that

c=2c. c ' = 2 c \,.

Then matching that on μ M5\mu_{M5} gives

2c5432=c 253 -\frac{2 c}{5 \cdot 4 \cdot 3 \cdot 2} = -\frac{c^2}{5 \cdot 3}

hence

c4=c 2 \frac{c}{4 } = c^2

hence there is a unique solution:

c=14c=12 c = \tfrac{1}{4} \;\;\;\; \Leftrightarrow \;\;\; c' = \tfrac{1}{2}

The IIA D-brane cocycles

We get the IIA cocycle for F1, D0, D2 and D4 from double dimensional reduction of the above M-brane cocycles.

μ D0 =cψ¯Γ 10ψ μ F1 IIA =(π 10) *(μ M2) =ic a=0 9ψ¯Γ aΓ 10e a μ D2 =μ M2| 8+1 =ic2 a i=0 9ψ¯Γ a 1Γ a 2ψe a 1e a 2 μ D4 =(π 10) *(μ M5) =+c432 a i=0 9ψ¯Γ a 1a 4Γ 10ψe a 1e a 4. \begin{aligned} \mu_{D0} &= c'\overline{\psi}\Gamma_{10}\psi \\ \mu_{F1}^{IIA} & = (\pi_{10})_\ast(\mu_{M2}) \\ & = i c' \sum_{a = 0}^{9} \overline{\psi} \Gamma_a \Gamma_{10} \wedge e^a \\ \mu_{D2} & = \mu_{M2}|_{8+1} \\ & = \tfrac{i c'}{2} \sum_{a_i = 0}^{9} \overline{\psi} \Gamma_{a_1} \Gamma_{a_2} \psi \wedge e^{a_1} \wedge e^{a_2} \\ \mu_{D4} & = (\pi_{10})_\ast(\mu_{M5}) \\ & = + \tfrac{c'}{4 \cdot 3 \cdot 2} \sum_{a_i = 0}^9 \overline{\psi} \Gamma_{a_1 \cdots a_4} \Gamma_{10}\psi \wedge e^{a_1} \wedge \cdots \wedge e^{a_4} \end{aligned} \,.

In all of the following the global prefactor cc' is being suppressed.

Problem: For the (unique!) solution c=1/2c' = 1/2 from above, then μ D0\mu_{D0} is (1/2)ψ¯Γ 10ψ(1/2) \overline{\psi}\Gamma_{10}\psi, hence half of what it needs to be to give the IIA-extension. This breaks the dd-reduction. Needs a fix. But what?

dμ D4 =h 3μ D2 c32(ψ¯Γ a 1a 2a 3aΓ 10ψ)(ψ¯Γ aψ) =(c) 22(ψ¯Γ a 1a 2ψ)(ψ¯Γ aΓ 10) \begin{aligned} d \mu_{D4} & = h_3 \wedge \mu_{D2} \\ \tfrac{c'}{3 \cdot 2} \left(\overline{\psi} \Gamma_{a_1 a_2 a_3 a \Gamma_{10}} \psi \right) \wedge \left( \overline{\psi} \Gamma^a \psi \right) & = - \tfrac{(c')^2}{2} \left( \overline{\psi} \Gamma_{a_1 a_2} \psi \right) \wedge \left( \overline{\psi} \Gamma_a \Gamma_{10} \right) \end{aligned}

Deriving the IIB D-brane cocycles

Now let’s check that a IIB cocycle like μ D1\mu_{D1} will come out as it should under sending the above data through the L L_\infty-T-duality construction.

The D1-brane

So we write

μ D1 =μ D1| 8+1e 9(π 9) *μ D1 \begin{aligned} \mu_{D1} & = \mu_{D1}|_{8+1} - e^9 (\pi_9)_\ast \mu_{D1} \end{aligned}

and need to match this to the corresponding 1-forms that we get from the IIA data above. This gives

μ D1 =(π 9) *μ D2e 9(μ D0| 8+1) =i a=0 8ψ¯Γ aΓ 9ψe ae 9ψ¯Γ 10ψ =i a=0 8ψ¯Γ a BΓ 9ψe a+ie 9ψ¯Γ 9 BΓ 9ψ =i a=0 9ψ¯Γ a BΓ 9ψe a \begin{aligned} \mu_{D1} & = (\pi_9)_\ast \mu_{D2} - e^9 \wedge (\mu_{D0}|_{8+1}) \\ & = i \sum_{a = 0}^8 \overline{\psi}\Gamma_a \Gamma_9 \psi e^{a} - e^9 \wedge \overline{\psi} \Gamma_{10} \psi \\ & = i \sum_{a = 0}^8 \overline{\psi}\Gamma^B_a \Gamma_9 \psi e^{a} + i e^9 \wedge \overline{\psi} \Gamma_9^B \Gamma_9 \psi \\ & = i \sum_{a = 0}^9 \overline{\psi} \Gamma_a^B \Gamma_9 \psi \wedge e^a \end{aligned}

where in the third line we inserted Γ 10=iΓ 9 BΓ 9\Gamma_{10} = -i \Gamma_9^B \Gamma_9 from def. .

This matches the expression in (Sakaguchi 00, equation (2.10)) if we identify our Γ 9\Gamma_9 with his σ 1\sigma_1 (which makes sense).

The D3-brane

Similarly:

μ D3 =e 9(μ D2| 8+1)+(π 9) *(μ D4) =i12 a i=0 8ψ¯Γ a 1a 2ψe a 1e a 2e 9123 a i=0 8ψ¯Γ a 1a 3Γ 9Γ 10ψe a 1e a 3 =12 a i=0 8ψ¯Γ a 1a 2(Γ 9 BΓ 9Γ 10)ψe a 1e a 2e 9+123 a i=0 8ψ¯Γ a 1a 3Γ 9Γ 10ψe a 1e a 3 =123(3 a i=0 8ψ¯Γ a 1a 2 BΓ 9 B(Γ 9Γ 10)ψe a 1e a 2e 9+ a i=0 8ψ¯Γ a 1a 3 BΓ 9Γ 10ψe a 1e a 3) =123 a i=0 9ψ¯Γ a 1a 2a 3 B(Γ 9Γ 10)ψ \begin{aligned} \mu_{D3} &= -e^9 \wedge (\mu_{D2}|_{8+1}) + (\pi_9)_\ast (\mu_{D4}) \\ & = -i \tfrac{1}{2} \sum_{a_i = 0}^8 \overline{\psi}\Gamma_{a_1 a_2} \psi \wedge e^{a_1}\wedge e^{a_2} \wedge e^9 - \tfrac{1}{2\cdot 3} \sum_{a_i = 0}^8 \overline{\psi} \Gamma_{a_1\cdots a_3} \Gamma_{9}\Gamma_{10}\psi \wedge e^{a_1}\wedge \cdots \wedge e^{a_3} \\ & = -\tfrac{1}{2} \sum_{a_i = 0}^8 \overline{\psi}\Gamma_{a_1 a_2} (\Gamma_9^B \Gamma_9 \Gamma_{10}) \psi \wedge e^{a_1}\wedge e^{a_2} \wedge e^9 + \tfrac{1}{2\cdot 3} \sum_{a_i = 0}^8 \overline{\psi} \Gamma_{a_1\cdots a_3} \Gamma_{9}\Gamma_{10}\psi \wedge e^{a_1}\wedge \cdots \wedge e^{a_3} \\ & = -\tfrac{1}{2 \cdot 3} \left( 3 \sum_{a_i = 0}^8 \overline{\psi}\Gamma^B_{a_1 a_2} \Gamma_9^B (\Gamma_9 \Gamma_{10})\psi \wedge e^{a_1}\wedge e^{a_2} \wedge e^9 + \sum_{a_i = 0}^8 \overline{\psi} \Gamma^B_{a_1\cdots a_3} \Gamma_{9}\Gamma_{10}\psi \wedge e^{a_1}\wedge \cdots \wedge e^{a_3} \right) \\ & = -\tfrac{1}{2 \cdot 3} \sum_{a_i = 0}^9 \overline{\psi} \Gamma^B_{a_1 a_2 a_3} (\Gamma_9 \Gamma_{10}) \psi \end{aligned}

where in the third line we inserted i=Γ 9 BΓ 9Γ 10-i = \Gamma_9^B \Gamma_9 \Gamma_{10} from def. .

This matches with Sakaguchi 00, equation (2.11).

References

The IIA DD-brane cocycles are in section 6.1 of

  • C. Chrysso‌malakos, José de Azcárraga, J. M. Izquierdo and C. Pérez Bueno, The geometry of branes and extended superspaces, Nuclear Physics B Volume 567, Issues 1–2, 14 February 2000, Pages 293–330 (arXiv:hep-th/9904137)

The IIB DD-brane cocycles are in section 2 of

  • Makoto Sakaguchi, section 2 of IIB-Branes and New Spacetime Superalgebras, JHEP 0004 (2000) 019 (arXiv:hep-th/9909143)

Last revised on November 1, 2016 at 07:07:21. See the history of this page for a list of all contributions to it.