SOURCE

var vm = new Vue({
  el: '#demo',
  data: {
    page_dir_list:[],
    fullName: 'a,b,c'
  },
  watch: {
    page_dir_list: function(val) {
    	this.fullName = val.join();
      return this.fullName.split(',');
    },
  }
})

<script src="https://cdn.jsdelivr.net/npm/vue/dist/vue.js"></script>

<div id="demo">
  {{ fullName }}
        <input type="checkbox" v-model="page_dir_list" value="a">游泳
        <input type="checkbox" v-model="page_dir_list" value="b">健身
        <input type="checkbox" v-model="page_dir_list" value="c">旅游
</div>

	body{
		color:white;
	}