【正文】 QUIVER GAUGE THEORY: MASSES Fundamentals Antifundamentals i a = 1, …. , Li QUIVER GAUGE THEORY: MASSES Bifundamentals i j QUIVER GAUGE THEORY What is so special about these masses? QUIVER GAUGE THEORY From the gauge theory point of view nothing special….. QUIVER GAUGE THEORY The mass puzzle! The mass puzzle The Bethe ansatz like equations Can be written for an arbitrary matrix The mass puzzle However the Yangian symmetry Y(H) would get replaced by some ugly infinitedimensional 171。 , NN, ~19971998。 , NN。 The moduli space MG of flat Gconnections = minima of the action。 explained 187。 fake 187。 ~1993。 , 。 The theory is exactly soluble () with the help of the Polyakov lattice YM action YangMills theories in 2d Physical YM Field content: Q u ic k T im e ? a n d aT I F F (U n c o m p re s s e d ) d e c o m p re s s o ra re n e e d e d t o s e e t h is p i c t u re .YangMills theories in 2d Witten found a way to map the BPS sector of the N=2 theory to the N=0 theory. The result is: Q u i ckT i me ? a n d aT I F F (U n co m p re ss e d ) d e co mp re sso ra re n e e d e d t o se e t h i s p i ct u re .YangMills theories in 2d Two dimensional YangMills partition function is given by the explicit sum Q u ickT im e ? a n d aT I F F (U n co m p re ss e d ) d e co m p re sso ra re n e e d e d t o se e t h is p i ct u re .YangMills theories in 2d In the limit the partition function putes the volume of MG Q u ick T im e ? a n d aT IF F ( U n c o m p r e ss e d ) d e co m p r e ss o ra re n e e d e d to se e th is p ict u r e .Q u ic k T im e ? a n d aT I F F (U n c o m p re s s e d ) d e c o m p re s s o ra re n e e d e d t o s e e t h is p i c t u re .YangMills theories in 2d Witten’s approach: add twisted superpotential and its conjugate Q u ickT ime ? a n d aT I F F (U n co m p re s se d ) d e co m p re sso ra re n e e d e d t o se e t h is p i ct u re .YangMills theories in 2d Take a limit In the limit the fields are infinitely massive and can be integrated out: one is left with the field content of the physical YM theory Q u ic k T im e ? a n d aT IF F ( U n c o m p re s s e d ) d e c o m p re s s o ra r e n e e d e d to s e e th i s p i c tu r e .Q u ic k T im e ? a n d aT IF F ( U n c o m p re s s e d ) d e c o m p r e s s o ra r e n e e d e d to s e e th i s p i c tu r e .Q ui c k Ti m e ? an d aT IF F (U n c o m p res s e d ) d ec o m p re s s ora re ne ed e d to s ee t h is p ic t ure .YangMills theories in 2d Both physical and cohomological YangMills theories define topological field theories (TFT) YangMills theories in 2d Both physical and cohomological YangMills theories define topological field theories (TFT) Vacuum states + deformations = quantum mechanics YM in 2d and particles on a circle Physical YM is explicitly equivalent to a quantum mechanical model: free fermions on a circle Can be checked by a partition function on a twotorus Gross Douglas YM in 2d and particles on a circle Physical YM is explicitly equivalent to a quantum mechanical model: free fermions on a circle States are labelled by the partitions, for G=U(N) Q u i ck T im e ? a n d aT I F F (U n co m p re sse d ) d e co m p r e sso ra re n e e d e d t o se e t h is p i ct u re .YM in 2d and particles on a circle For N=2 YM these free fermions on a circle Label the vacua of the theory deformed by twisted superpotential W YM in 2d and particles on a circle The fermions can be made interacting by adding a localized matter: for example a timelike Wilson loop in some representat