α'阶边界上的数据
The data on the boundary at order $α'$
AI总结：
本文通过受限制的T对偶变换和斯托克斯定理，证明杂化弦理论α'阶有效作用量在边界上的数据由无质量场及其一阶导数构成，并验证了边界项在T对偶下精确抵消。
摘要：
最小作用量原理表明，对于开时空流形，边界上存在数据。最近有人提出，α'阶有效作用量的数据是无质量场及其一阶导数的值。这些数据应当受到α'阶T对偶变换的尊重。此外，T对偶变换不应改变指向边界的单位矢量，这反过来意味着基空间度规也应当保持不变。假设这种受限制的T对偶变换，我们证明，在Buscher规则下，杂化弦理论α'阶有效作用量的奇宇称部分的圆约化变换被一些全导数项和一些α'阶受限制的T对偶变换所抵消。然后利用斯托克斯定理，我们证明基空间中与全导数项对应的边界项被上述受限制的T对偶变换下Gibbons-Hawking边界项的圆约化变换精确抵消。这些计算证实了上述关于α'阶有效作用量边界数据的提议。
英文摘要：
The least action principle indicates that for the open spacetime manifolds, there are data on the boundary. Recently, it has been proposed that the data for the effective actions at order $α'$ are the values of the massless fields and their first derivatives. These data should be respected by the T-duality transformations at order $α'$. Moreover, the T-duality transformations should not change the unit vector to the boundary which in turns implies that the base space metric should be also invariant. Assuming such restricted T-duality transformations, we show that the transformation of the circular reduction of the parity-odd part of the effective action of the heterotic string theory at order $α'$ under the Buscher rules is cancelled by some total derivative terms and by some restricted T-duality transformations at order $α'$. Using the Stokes' theorem, we then show that the boundary terms in the base space corresponding to the total derivative terms are $exactly$ cancelled by transformation of the circular reduction of the Gibbons-Hawking boundary term under the above restricted T-duality transformations. These calculations confirm the above proposal for the data on the boundary for the effective actions at order $α'$.
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