Eric Sharpe - Examples of homological projective duality in physics

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Since ~ 2006, many of us have come to believe that all geometries realized by GLSM’s are related by hpd. This now seems to be (close to) proven: Ballard, Favero, Katzarkov, 1203.6643 & to appear Today, I’ll discuss some (old) examples of such exotic GLSM’s, and some (newer) analyses. Matrix factorizations will play an important role.
Outline: * physical realization of hpd as phases of abelian GLSM, i.e. V // C * * detour through physics of Z 2 gerbes * some phases will be CFT’s for nc resolutions * D-brane probes of nc res’ns (w/ Nick, Ed) * analogues of GW invt’s for nc res’ns
Page 1 and 2: Examples of homological projective
Page 3: What did hpd teach us? Prior to ~ 2
Page 7 and 8: Prototype for the exotic GLSM’s I
Page 9 and 10: GLSM for P 3 [2,2] (=T 2 ): Looks l
Page 11 and 12: All phases of GLSM’s can be descr
Page 13 and 14: The other phase is more exciting: V
Page 15 and 16: A more important subtlety is the fa
Page 17 and 18: General decomposition conjecture Co
Page 19 and 20: Quick consistency check: A sheaf on
Page 21 and 22: GLSM’s Let’s now return to our
Page 23 and 24: V// − C × : Double cover P 1 Ber
Page 25 and 26: Next simplest example: // + : LG on
Page 27 and 28: The next example in the pattern is
Page 29 and 30: Check that we are seeing K’s nonc
Page 31 and 32: Summary so far: CP 7 [2,2,2,2] This
Page 33 and 34: More examples: CI of 2 quadrics in
Page 35 and 36: D-brane probes of nc resolutions Le
Page 37 and 38: Our D-brane probes of this Landau-G
Page 39 and 40: When is a matrix factorization `poi
Page 41 and 42: We’re interested in Landau-Ginzbu
Page 43 and 44: Example: Family [C 2 /Z 2 ] x,y ×
Page 45 and 46: So far: When the LG model flows in
Page 47 and 48: Case of an nc resolution, cont’d:
Page 49 and 50: So far: * examples of hpd realizing
Page 51 and 52: Let’s work through this in more d
Page 53 and 54: As a warm-up, let’s outline the G
Page 55 and 56:
P 7 [2,2,2,2], cont’d In principl
Page 57 and 58:
P 7 [2,2,2,2], cont’d After norma
Page 59 and 60:
Result for P 7 [2,2,2,2]: n N 1 512
Page 61 and 62:
Applying the same method, Compare G
Page 63:
Summary: * physical realization of
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Eric Sharpe - Examples of homological projective duality in physics

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