nLab C-field tadpole cancellation

Idea

In M-theory compactified on 8-dimensional compact fibers $X^{(8)}$ (see M-theory on 8-manifolds) tadpole cancellation for the supergravity C-field has been argued (Sethi-Vafa-Witten 96, Becker-Becker 96, Dasgupta-Mukhi 97) to be the condition

$$N_{M2} \;+\; \tfrac{1}{2} \big( G_4[X^{(8)}]\big)^2 \;=\; \underset{ I_8(X^8) }{ \underbrace{ \tfrac{1}{48}\big( p_2 - (\tfrac{1}{2}p_1)^2 \big)[X^{8}] } } \;\;\;\; \overset{!}{\in} \mathbb{Z} \,,$$

where

1. $N_{M2}$ is the net number of M2-branes in the spacetime (whose worldvolume appears as points in $X^{(8)}$);

2. $G_4$ is the field strength/flux of the supergravity C-field

3. $p_1$ is the first Pontryagin class and $p_2$ the second Pontryagin class combining to I8, all regarded here in rational homotopy theory.

If $X^{8}$ has

• Spin(7)-structure (hence in particular if it is a Calabi-Yau manifold, which has $SU(4) =$ Spin(6)-structure)

or

• Sp(2).Sp(1)-structure

then

$$\tfrac{1}{2}\big( p_2 - \tfrac{1}{4}(p_1)^2 \big) \;=\; \chi$$

is the Euler class (see this Prop. and this Prop., respectively), hence in these cases the condition is equivalently

(1)$$N_{M2} \;=\; - \tfrac{1}{2} \big( G_4[X^{(8)}]\big)^2 \;+\; \tfrac{1}{24}\chi[X^8] \;\;\;\; \in \mathbb{Z} \,,$$

where $\chi[X]$ is the Euler characteristic of $X$.

Properties

Integrality on $K3 \times K3$

That (1) should be an integer is a highly non-trivial condition on the manifold $X^8$.

One case where this is satisfied is for $X^8$ being the product space of K3 with itself.

To see this, one needs the shifted C-field flux quantization-condition

(2)$$[\tilde G_4] \;\in\; H^4(X^8, \mathbb{Z}) \to H^4(X^8, \mathbb{R})$$

for

(3)$$\tilde G_4 \;\coloneqq\; G_4 + \tfrac{1}{4}p_1(\nabla_{T X^8})$$

Proposition

With the shifted C-field flux quantization (3), the Dasgupta-Mukhi-expression (1) for the number of M2-branes is indeed an integer on the product space $X^8 = K3 \times K3$ of K3 with itself:

$$\tfrac{1}{24}\chi[K3 \times K3] - \tfrac{1}{2} \big( G_4[K3 \times K3]\big)^2 \;\in\; \mathbb{Z} \,,$$

Proof

After replacing $G_4$ by $\widetilde G_4$ (3) the expression becomes

(4)$$\begin{aligned} & \tfrac{1}{24}\chi - \tfrac{1}{2} \big( G_4\big)^2 \\ & = \tfrac{1}{24}\chi - \tfrac{1}{2}\big( \tfrac{1}{4} p_1 \big)^2 - \underset{ \in \mathbb{Z} }{ \underbrace{ \tfrac{1}{2} \widetilde G_4 \cdot \big( \widetilde G_4 - \tfrac{1}{2}p_1\big) } } \,, \end{aligned}$$

where the summand over the brace is an integral class (by this Corollary), because $K3$ is a spin manifold so that $\tfrac{1}{2}p_1$ is the Wu class $\nu_4$ (by this Prop.).

Hence it is now sufficient to show that the first two summands on the right of (4) are both integers, when evaluated on $K3 \times K3$.

But we have

1. $\chi[K3\times K3] \;=\; 24^2$ (by this Prop.)

2. $p_1[K3 \times K3] \;=\; - 2 \times 48$ (by this Prop.)

Hence

$$\begin{aligned} \tfrac{1}{24}\chi[K3 \times K3] & = \tfrac{1}{24} 24^2 \\ & = 24 \;\in\; \mathbb{Z} \end{aligned}$$

and

$$\begin{aligned} \tfrac{1}{32} (p_1[K3 \times K3])^2 & = \tfrac{1}{32} \times ( \underset{ \mathclap{ = 3 \times 32 } } { \underbrace{ 2 \times 48} } )^2 \\ & = 9 \times 32 \;\in\; \mathbb{Z} \end{aligned}$$

Related concepts

• RR-field tadpole cancellation in type II string theory

• shifted C-field flux quantization in M-theory

• 24 branes transverse to K3 in F-theory and HET-theory

References

The C-field tadpole cancellation condition is claimed in

• Savdeep Sethi, Cumrun Vafa, Edward Witten, p. 2 of: Constraints on Low-Dimensional String Compactifications, Nucl. Phys. B 480 (1996) 213-224 [arXiv:hep-th/9606122, doi:10.1016/S0550-3213(96)00483-X]

referring for proof to the computation in

• Cumrun Vafa, Edward Witten, p. 11 (12 of 14) of: A One-Loop Test Of String Duality, Nucl. Phys. B 480 (1996) 213-224 [arxiv:hep-th/9505053, doi:10.1016/0550-3213(95)00280-6]

A comment is also in

• Edward Witten, p. 3 of On Flux Quantization In M-Theory And The Effective Action, J. Geom. Phys. 22 (1997) 1-13 [arXiv:hep-th/9609122, doi:10.1016/S0393-0440(96)00042-3]

Another condition appears in

• Katrin Becker, Melanie Becker, around (2.58) in: M-Theory on Eight-Manifolds, Nucl. Phys. B 477 (1996) 155-167 [arXiv:hep-th/9605053, doi:10.1016/0550-3213(96)00367-7]

The formulas of Sethi-Vafa-Witten 96 and Becker-Becker 96 have been plugged together in

• Keshav Dasgupta, Sunil Mukhi, equation (1) in: A Note on Low-Dimensional String Compactifications, Phys. Lett. B 398 (1997) 285-290 [arXiv:hep-th/9612188, doi:10.1016/S0370-2693(97)00216-5]

Further discussion:

• Sergei Gukov, Cumrun Vafa, Edward Witten: CFT’s From Calabi-Yau Four-folds, Nucl. Phys. B 584 (2000) 69-108, Erratum-ibid. 608 (2001) 477-478 [arXiv:hep-th/9906070]

• Keshav Dasgupta, Govindan Rajesh, Savdeep Sethi, M Theory, Orientifolds and G-Flux, JHEP 9908 (1999) 023 [arXiv:hep-th/9908088, doi:10.1088/1126-6708/1999/08/023]

Lecture notes:

• Timo Weigand, p. 86 of TASI Lectures on F-theory (arXiv:1806.01854)

Application in dualities in string theory:

• Savdeep Sethi, around (2) in A Note on Heterotic Dualities via M-theory, Phys. Lett. B659:385-387, 2008 (arXiv:0707.0295)

Application in string phenomenology

• Mirjam Cvetic, James Halverson, Ling Lin, Muyang Liu, Jiahua Tian, equation (9) in A Quadrillion Standard Models from F-theory (arXiv:1903.00009)

Discussion via Hypothesis H:

• Domenico Fiorenza, Hisham Sati, Urs Schreiber, Twisted Cohomotopy implies M-theory anomaly cancellation,

  Comm. Math. Phys. 377 (2020) 1961-2025 (doi:10.1007/s00220-020-03707-2, arXiv:1904.10207)

surveyed in:

• Domenico Fiorenza, Hisham Sati, Urs Schreiber, The Character Map in Nonabelian Cohomology — Twisted, Differential, Generalizaed, World Scientific (2023) [arXiv:2009.11909, doi:10.1142/13422]