Remarks on the Green–Schwarz Terms of Six-Dimensional Supergravity Theories

Monnier, Samuel; Moore, Gregory W.

doi:10.1007/s00220-019-03341-7

Cited by 46 publications

(95 citation statements)

References 59 publications

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“…Crucially, G has to be extended to W subject to a fractional flux quantization law: the periods of 2 G modulo 2 coincide with the periods of a certain

Z_{2}

‐valued characteristic class of W , the degree

2 ℓ + 2

Wu class of W . The net anomaly due to the coupling the self‐dual field to the background gauge field is therefore

\frac{1}{8} σ_{W} - \frac{1}{2} \int_{W} G^{2} .

At the level of anomaly field theories, the quarter Dai–Freed theory whose partition function is gets tensored by a certain Wu Chern–Simons theory, whose partition function on a

4 ℓ + 3

manifold U such that

U = \partial W

is . Wu Chern–Simons theories can be seen as quadratic Chern–Simons theories of degree

2 ℓ + 1

Abelian gauge field at ‘half‐integer level’, thereby generalizing spin Chern–Simons theories to higher dimension.…”

Section: Examplesmentioning

confidence: 99%

“…Wu Chern–Simons theories can be seen as quadratic Chern–Simons theories of degree

2 ℓ + 1

Abelian gauge field at ‘half‐integer level’, thereby generalizing spin Chern–Simons theories to higher dimension. The definition of the anomaly field theory of charged self‐dual fields is implicit in [] and will be discussed in more detail elsewhere.…”

Section: Examplesmentioning

confidence: 99%

“… i)The anomaly coefficients

a, b_{i}, \frac{1}{2} b_{I I}, b_{I J}

must all be element of the lattice Λ, as opposed to arbitrary elements of the vector space

Λ \otimes R

. In fact, a stronger condition depending on the global form of the gauge group G holds, see [] for details. ii) a cannot be an arbitrary element of Λ: it has to be a characteristic element, i.e. an element such that

(x, x) = (x, a)

modulo 2 for all

x \in Λ

. Moreover, the cancellation of anomalies is possible in the present context only if the Wu Chern–Simons theory is invertible, which is the case if and only if the lattice Λ is unimodular.…”

Section: A Concrete Use Case: Anomaly Cancellation In 6d Supergravitymentioning

confidence: 99%

“…In [], we show that for gauge groups G given by arbitrary products of

U (n)

S U (n)

and

S p (n)

factors, as well as for

G = E_{8}

, all the 7‐manifolds are bordant to zero, ensuring that

A = {scriptA}_{CT}

and therefore that the Green–Schwarz term cancels all anomalies. In other cases, it turns out to be very hard to check the equality of

scriptA

and

A_{CT}

on manifolds not bordant to zero.…”

Section: A Concrete Use Case: Anomaly Cancellation In 6d Supergravitymentioning

confidence: 99%

“…In other cases, it turns out to be very hard to check the equality of

scriptA

and

A_{CT}

on manifolds not bordant to zero. Partial results were obtained in [] for supergravities with discrete gauge group, in the form of constraints on the matter representations. It was checked that these constraints are satisfied in the class of F‐theory models constructed in [], yielding yet another consistency check on string theory.…”

Section: A Concrete Use Case: Anomaly Cancellation In 6d Supergravitymentioning

confidence: 99%

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A Modern Point of View on Anomalies

Monnier

2019

Fortschritte der Physik

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We review the concept of anomaly field theory, namely the fact that the anomalies of a d‐dimensional field theory can be encoded in a d+1‐dimensional field theory functor. We give numerous examples of anomaly field theories, explain how classical facts about anomalies are recovered from the anomaly field theory, and review recent work on global anomaly cancellation in 6d supergravity where this concept was instrumental. We also sketch the status of global anomaly cancellation checks in string theory. This paper is based on a talk given at the Durham Symposium ‘Higher Structures in M‐theory’ in August 2018.

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“…Crucially, G has to be extended to W subject to a fractional flux quantization law: the periods of 2 G modulo 2 coincide with the periods of a certain

Z_{2}

‐valued characteristic class of W , the degree

2 ℓ + 2

Wu class of W . The net anomaly due to the coupling the self‐dual field to the background gauge field is therefore

\frac{1}{8} σ_{W} - \frac{1}{2} \int_{W} G^{2} .

At the level of anomaly field theories, the quarter Dai–Freed theory whose partition function is gets tensored by a certain Wu Chern–Simons theory, whose partition function on a

4 ℓ + 3

manifold U such that

U = \partial W

is . Wu Chern–Simons theories can be seen as quadratic Chern–Simons theories of degree

2 ℓ + 1

Abelian gauge field at ‘half‐integer level’, thereby generalizing spin Chern–Simons theories to higher dimension.…”

Section: Examplesmentioning

confidence: 99%

“…Wu Chern–Simons theories can be seen as quadratic Chern–Simons theories of degree

2 ℓ + 1

Section: Examplesmentioning

confidence: 99%

“… i)The anomaly coefficients

a, b_{i}, \frac{1}{2} b_{I I}, b_{I J}

must all be element of the lattice Λ, as opposed to arbitrary elements of the vector space

Λ \otimes R

(x, x) = (x, a)

modulo 2 for all

x \in Λ

. Moreover, the cancellation of anomalies is possible in the present context only if the Wu Chern–Simons theory is invertible, which is the case if and only if the lattice Λ is unimodular.…”

Section: A Concrete Use Case: Anomaly Cancellation In 6d Supergravitymentioning

confidence: 99%

“…In [], we show that for gauge groups G given by arbitrary products of

U (n)

S U (n)

and

S p (n)

factors, as well as for

G = E_{8}

, all the 7‐manifolds are bordant to zero, ensuring that

A = {scriptA}_{CT}

and therefore that the Green–Schwarz term cancels all anomalies. In other cases, it turns out to be very hard to check the equality of

scriptA

and

A_{CT}

on manifolds not bordant to zero.…”

Section: A Concrete Use Case: Anomaly Cancellation In 6d Supergravitymentioning

confidence: 99%

“…In other cases, it turns out to be very hard to check the equality of

scriptA

and

A_{CT}

Section: A Concrete Use Case: Anomaly Cancellation In 6d Supergravitymentioning

confidence: 99%

See 3 more Smart Citations

A Modern Point of View on Anomalies

Monnier

2019

Fortschritte der Physik

Self Cite

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Cobordism conjecture, anomalies, and the String Lamppost Principle

Montero

Vafa

2021

J. High Energ. Phys.

108

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We consider consequences of triviality of cobordism classes and anomaly cancellation in supergravity theories in d > 6. We argue that this leads to the existence of certain defects which we call “I-folds” (a generalization of orientifolds). The requirement that compactifications to lower dimensions involving these defects be anomaly free leads to conditions on the higher dimensional theory. We show that in theories with 16 supercharges in d > 6 this leads to restrictions on the rank of the allowed gauge groups and thus provides an explanation for the observed restrictions in known string theory constructions. In particular, in eight and nine dimensions the only solutions to our constraints are precisely the ones realized in string theory compactifications. We also use these techniques to place constraints on the global structure of the gauge group in eight and nine dimensions.

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Anomalies as obstructions: from dimensional lifts to swampland

Peng

Minasian

Theisen

2022

J. High Energ. Phys.

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We revisit the relation between the anomalies in four and six dimensions and the Chern-Simons couplings one dimension below. While the dimensional reduction of chiral theories is well-understood, the question which three and five-dimensional theories can come from a general circle reduction, and are hence liftable, is more subtle. We argue that existence of an anomaly cancellation mechanism is a necessary condition for liftability. In addition, the anomaly cancellation and the CS couplings in six and five dimensions respectively determine the central charges of string-like BPS objects that cannot be consistently decoupled from gravity, a.k.a. supergravity strings. Following the completeness conjecture and requiring that their worldsheet theory is unitary imposes bounds on the admissible theories. We argue that for the anomaly-free six-dimensional theories it is more advantageous to study the unitarity constraints obtained after reduction to five dimensions. In general these are slightly more stringent and can be cast in a more geometric form, highly reminiscent of the Kodaira positivity condition (KPC). Indeed, for the F-theoretic models which have an underlying Calabi-Yau threefold these can be directly compared. The unitarity constraints (UC) are in general weaker than KPC, and maybe useful in understanding the consistent models without F-theoretic realisation. We catalogue the cases when UC is more restrictive than KPC, hinting at more refined hidden structure in elliptic Calabi-Yau threefolds with certain singularity structure.

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Remarks on the Green–Schwarz Terms of Six-Dimensional Supergravity Theories

Cited by 46 publications

References 59 publications

A Modern Point of View on Anomalies

A Modern Point of View on Anomalies

Cobordism conjecture, anomalies, and the String Lamppost Principle

Anomalies as obstructions: from dimensional lifts to swampland

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