Eric Sharpe - Examples of homological projective duality in physics

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P 7 [2,2,2,2], cont’d Details: also Z = 8 [ 3 (zz)q − ln(zz) 3 f(0) − 8π 2 f ′ (0) + 3 ln(zz) 2 f ′ (0) + ln(zz) ( 8π 2 f(0) − 3f ′′ (0) ) ] + f (3) (0) Expect ∝ − i 6 κ(t − t)3 + ζ(3) 4π 3 χ(X) + − t − t = ln(zz) 2πi i ∑ (2πi) 2 + n 2i ∑ (2πi) 3 n N n (Li 3 (q n ) + Li 3 (q n )) N n (Li 2 (q n ) + Li 2 (q n )) n(t − t) ∆(z) + ∆(z) 2πi for some ∆(z) so we use the ln(z z*) 3 term to normalize.
P 7 [2,2,2,2], cont’d After normalization, e −K = −i 16 6 also [ ln(zz) 3 (2πi) 3 + 8π2 f ′ (0) (2πi) 3 f(0) − 3 − ln(zz) 2πi ( 8π 2 (2πi) 2 − 3 f ′′ (0) (2πi) 2 f(0) = − i 6 κ(t − t)3 + ζ(3) 4π 3 χ(X) + − i ∑ (2πi) 2 n 2i ∑ (2πi) 3 ln(zz) 2 f ′ (0) 2πi (2πi) 2 f(0) ) − n 1 f (3) (0) (2πi) 3 f(0) N n (Li 3 (q n ) + Li 3 (q n )) N n (Li 2 (q n ) + Li 2 (q n )) n(t − t) ] Expect t − t = ln(zz) 2πi + ∆(z) + ∆(z) 2πi for some ∆(z) so from ln(z z*) 2 term, ∆ + ∆ = − ∂ ∂ɛ ln f(ɛ) ∣ ∣∣∣ɛ=0
Page 1 and 2:
Examples of homological projective
Page 3 and 4:
What did hpd teach us? Prior to ~ 2
Page 5 and 6: Outline: * physical realization of
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: P 7 [2,2,2,2], cont’d In principl
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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