we have three approach to quantizing bosonic string:
1.1 covariant canonical quantization
1.2 light cone gauge ....
2 path integral ....
for 1.1, we keep Lorentz symmetry along the way and add Virasoro constraint to pick out physical state.
question: why level matching must hold? since in 2 there still may have gravitational anomaly.
for 1.2, we first solve the constraint and then check the consistency with Lorentz algebra, or in another way let the first non-tachyon state to be massless or do some Riemann zeta function regularization.
question: why 26D-Lorentz symmetry linked with massless state (I think its within Lie group representation theory but im not familiar with that).

for 2, just like 1.1 we keep Lorentz everywhere and using ghosts to cancel the Weyl anomaly and then using BRST method to distinguish physical states.
question: in this approach what if ignoring Weyl anomaly ie just leaving conformal symmetry away, then we have no critical dim and get a theory suitable for every dim.
(i know that in Polyakov s original paper the authors just follow this idea but i cannot get the access to that paper.)
without conformal symmetry, does the D-branes case like string(though they have continuous spectrum), what other kinds of anomaly may emerge?

finally in my opinion the canonical and the path integral are really different, though they give same results. is there a theorem saying that they are totally equivalent?
i think this question makes sense, since we have mathematical rigorous theory for 2D conformal field theory.

这些问题比较专业，我建议你多看看书，基本都能有答案。1.1你得先说清Level matching具体什么意思。先说1.2。26D Lorentz symmetry和massless state没什么关系…质量谱的态是由粒子数算符，也就是N决定，对于闭弦还有NL和NR之分。显然基态是N=0的态，但它是tachyon显然不合理，所以原则上说合理的第一激发态就是N=1的那个态。在Bosonic string里面Tachyon并不能被剔除掉，这也就是string引入SUSY的原因，但这是后话。

对于2，显然忽略Wely反常会使得理论自洽性有问题。数学上讲，路径积分里面的ghost低调非物理部分其实要保证BRST仍然零幂。这意思是要在Hilbert空间里面找到物理态和非物理态的边界，也就是BRST上同调。如果忽略Wely反常，那么这说明理论中仍然会存在非物理自由度，这显然会使得string不自恰。

刘维尔场论大概是用额外的场来补偿Weyl反常，从而得到conformal不变的场论，所以困难还是在于不平坦的没法处理吗？那D-brane怎么处理这个问题？@CloudK @波子的弦

there is not supposed to be any special point on the string，which means the translations in the σ direction should leave the state alone，by the Noether procedure，we can find the the generator P_σ=N-\tilde｛N｝and it is supposed to be 0，so the level matching holds

Liouville field theory can give some no trivial central charge of Virasoro algebra.