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I was working through the Mirror Symmetry book by Clay Math Institute. It deals with supersymmetric sigma model in 10.4 section. It doesn't derive how the action is invariant under the variation. I am trying hard, but stuck at few places. The Lagrangian and variation from the book is:

$$ L=\frac{1}{2} g_{I J} \dot{\phi}^I \dot{\phi}^J+\frac{i}{2} g_{I J}\left(\bar{\psi}^I D_t \psi^J-D_t \bar{\psi}^I \psi^J\right)-\frac{1}{2} R_{I J K L} \psi^I \bar{\psi}^J \psi^K \bar{\psi}^L, $$ where $$ D_t \psi^I=\partial_t \psi^I+\Gamma_{J K}^I \partial_t \phi^J \psi^K, $$ $$ \begin{aligned} \delta \phi^I & =\epsilon \bar{\psi}^I-\bar{\epsilon} \psi^I, \\ \delta \psi^I & =\epsilon\left(i \dot{\phi}^I-\Gamma_{J K}^I \bar{\psi}^J \psi^K\right), \\ \delta \bar{\psi}^I & =\bar{\epsilon}\left(-i \dot{\phi}^I-\Gamma_{J K}^I \bar{\psi}^J \psi^K\right), \end{aligned} $$

ludoft
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Jaswin
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  • Related question: http://physics.stackexchange.com/q/52462/2451 – Qmechanic Feb 01 '13 at 17:29
  • Please type equations instead of pasting images. Images make editing impossible, suffer the possibility of link rot, are harder to read, and cannot be searched. – DanielSank Nov 14 '15 at 06:47
  • @DanielSank : My apologies, I shall enter the equations from now on. – Jaswin Nov 18 '15 at 06:54
  • Hint: Start with flat space, and confirm the invariance of this noninteracting limit. Now promote to a hypersphere, the primitive hyperspherical σ model. The metric, Christoffels, and curvature are very simple function you may manipulate directly. Only then, bother with the full Riemannian manifold, and recall the standard identities applicable. – Cosmas Zachos Jan 28 '17 at 23:17

2 Answers2

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Showing the supersymmetry invariance of SUSY nonlinear sigma models requires the use of various identities from differential geometry; in particular, the (target space) derivative of the metric is related to the Christoffel symbol, and the the derivative of the Christoffel symbol is related to the Riemann curvature tensor. Besides these, there are various other useful geometrical identities, refer to Chapter 7 and Chapter 8 of Nakahara's Geometry, Topology and Physics for these. For example, it is quite common for the supersymmetry variation of the action to give quantities which are symmetric under the exchange of two covariant indices (due to some differential geometric identitiy), and when these indices are contracted with the contravariant indices of two identical fermionic fields, the resulting quantity is zero, due to the antisymmetry of the fermionic fields.

The OP has mentioned the case of a 1d SUSY sigma model, also called supersymmetric quantum mechanics, but points similar to those in the previous paragraph should hold for all SUSY sigma models.

Meer Ashwinkumar
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I worked on this for my thesis. The negative sign in front of the Riemann curvature tensor should be a positive sign, if you're using the positive sign convention for the Riemann curvature tensor, i.e. the definition used in MTW. You can check this using a Superfield formalism. To show that the Lagrangian L is invariant, take the variation of L. You can assume a symmetric metric tensor and normal coordinates, i.e. the first derivative of the metric tensor is always zero but higher derivatives may not be zero, to simplify the calculations. Next use the SUSY relations to simplify. Afterward, separate the terms by the number of fermionic variables, since they can only cancel with other terms with the same amount. Then use definitions, integration by parts, symmetry of the metric tensor, and the antisymmetry of the fermionic variables to show everything goes to zero. Hope this helps.

SUSYGradStudent
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